Reliability of multiphysical systems set. Volume 2, Nanometer-scale defect detection using polarized...

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Nanometer-scale Defect Detection Using Polarized Light

Reliability of Multiphysical Systems Set coordinated by

Abdelkhalak El Hami

Volume 2

Nanometer-scale Defect Detection Using

Polarized Light

Pierre Richard Dahoo Philippe Pougnet

Abdelkhalak El Hami

First published 2016 in Great Britain and the United States by ISTE Ltd and John Wiley amp Sons Inc

Apart from any fair dealing for the purposes of research or private study or criticism or review as permitted under the Copyright Designs and Patents Act 1988 this publication may only be reproduced stored or transmitted in any form or by any means with the prior permission in writing of the publishers or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address

ISTE Ltd John Wiley amp Sons Inc 27-37 St Georgersquos Road 111 River Street London SW19 4EU Hoboken NJ 07030 UK USA

wwwistecouk wwwwileycom

copy ISTE Ltd 2016 The rights of Pierre Richard Dahoo Philippe Pougnet and Abdelkhalak El Hami to be identified as the authors of this work have been asserted by them in accordance with the Copyright Designs and Patents Act 1988

Library of Congress Control Number 2016943672 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-936-6

Contents

Preface xi

Chapter 1 Uncertainties 1

11 Introduction 1 12 The reliability based design approach 2

121 The MC method 2 122 The perturbation method 3 123 The polynomial chaos method 7

13 The design of experiments method 9 131 Principle 9 132 The Taguchi method 10

14 The set approach 14 141 The method of intervals 15 142 Fuzzy logic based method 18

15 Principal component analysis 20 151 Description of the process 21 152 Mathematical roots 22 153 Interpretation of results 22

16 Conclusions 23

Chapter 2 Reliability-based Design Optimization 25

21 Introduction 25 22 Deterministic design optimization 26 23 Reliability analysis 27

231 Optimal conditions 30

vi Nanometer-scale Defect Detection Using Polarized Light

24 Reliability-based design optimization 31 241 The objective function 31 242 Total cost consideration 32 243 The design variables 33 244 Response of a system by RBDO 33 245 Limit states 33 246 Solution techniques 33

25 Application optimization of materials of an electronic circuit board 34

251 Optimization problem 36 252 Optimization and uncertainties 39 253 Results analysis 43

26 Conclusions 44

Chapter 3 The WavendashParticle Nature of Light 47

31 Introduction 48 32 The optical wave theory of light according to Huyghens and Fresnel 49

321 The three postulates of wave optics 49 322 Luminous power and energy 51 323 The monochromatic wave 51

33 The electromagnetic wave according to Maxwellrsquos theory 52

331 The Maxwell equations 52 332 The wave equation according to the Coulombrsquos gauge 56 333 The wave equation according to the Lorenzrsquos gauge 57

34 The quantum theory of light 57 341 The annihilation and creation operators of the harmonic oscillator 57 342 The quantization of the electromagnetic field and the potential vector 61 343 Field modes in the second quantization 66

Chapter 4 The Polarization States of Light 71

41 Introduction 71 42 The polarization of light by the matrix method 73

421 The Jones representation of polarization 76 422 The Stokes and Muller representation of polarization 81

Contents vii

43 Other methods to represent polarization 86 431 The Poincareacute description of polarization 86 432 The quantum description of polarization 88

44 Conclusions 93

Chapter 5 Interaction of Light and Matter 95

51 Introduction 95 52 Classical models 97

521 The Drude model 103 522 The Sellmeir and Lorentz models 105

53 Quantum models for light and matter 111 531 The quantum description of matter 111 532 JaynesndashCummings model 118

54 Semiclassical models 123 541 TaucndashLorentz model 127 542 CodyndashLorentz model 130

55 Conclusions 130

Chapter 6 Experimentation and Theoretical Models 133

61 Introduction 134 62 The laser source of polarized light 135

621 Principle of operation of a laser 136 622 The specificities of light from a laser 141

63 Laser-induced fluorescence 143 631 Principle of the method 143 632 Description of the experimental setup 145

64 The DR method 145 641 Principle of the method 146 642 Description of the experimental setup 148

65 Theoretical model for the analysis of the experimental results 149

651 Radiative relaxation 152 652 Non-radiative relaxation 153 653 The theoretical model of induced fluorescence 160 654 The theoretical model of the thermal energy transfer 163

66 Conclusions 170

viii Nanometer-scale Defect Detection Using Polarized Light

Chapter 7 Defects in a Heterogeneous Medium 173 71 Introduction 173 72 Experimental setup 175

721 Pump laser 176 722 Probe laser 176 723 Detection system 177 724 Sample preparation setup 180

73 Application to a model system 182 731 Inert noble gas matrix 182 732 Molecular system trapped in an inert matrix 184 733 Experimental results for the induced fluorescence 188 734 Experimental results for the double resonance 198

74 Analysis by means of theoretical models 203 741 Determination of experimental time constants 203 742 Theoretical model for the induced fluorescence 209 743 Theoretical model for the DR 214

75 Conclusions 216

Chapter 8 Defects at the Interfaces 219

81 Measurement techniques by ellipsometry 219 811 The extinction measurement technique 222 812 The measurement by rotating optical component technique 223 813 The PM measurement technique 224

82 Analysis of results by inverse method 225 821 The simplex method 232 822 The LM method 234 823 The quasi-Newton BFGS method 237

83 Characterization of encapsulating material interfaces of mechatronic assemblies 237

831 Coating materials studied and experimental protocol 239 832 Study of bulk coatings 241 833 Study of defects at the interfaces 244 834 Results analysis 251

84 Conclusions 253

Chapter 9 Application to Nanomaterials 255

91 Introduction 255 92 Mechanical properties of SWCNT structures by MEF 256

921 Youngs modulus of SWCNT structures 258

Contents ix

922 Shear modulus of SWCNT structures 259 923 Conclusion on the modeling results 260

93 Characterization of the elastic properties of SWCNT thin films 260

931 Preparation of SWCNT structures 261 932 Nanoindentation 262 933 Experimental results 263

94 Bilinear model of thin film SWCNT structure 265 941 SWCNT thin film structure 266 942 Numerical models of thin film SWCNT structures 268 943 Numerical results 269

95 Conclusions 274

Bibliography 275

Index 293

Preface

The various actions decided on at a global level to stimulate sustainable development and to respond to climate issues bring forth increasingly stringent regulations in terms of greenhouse gas emissions and hazardous substances In the automotive sector these regulations drive industrial companies to develop new mechatronic systems using electricity to replace the various mechanical functions of vehicles International competition and constant pressure to improve the performance of innovative products compel the companies supplying embedded mechatronic devices to innovate in increasingly shorter lead times to remain competitive

To improve the performance of embedded systems in terms of volume or mass reduction or to reduce energy losses the mechatronic industry implements new packaging methods (such as those based on multimaterials) or incorporates new materials (for instance carbon nanotubes) Modeling and simulation are used to limit cost increase durability and reduce lead time to market The Physics of failure provides the knowledge to predict and reduce potential failures in application and optimize design before activating serial production In this respect Reliability Based Design Optimization (RBDO) is a numerical tool used to optimize design and reduce industrial fabrication risks This approach can only be applied efficiently when the underlying physical phenomena are thoroughly understood and

xii Nanometer-scale Defect Detection Using Polarized Light

when the models used accurately represent the conditions under which the device operates

To model a dynamic system consisting of interacting sub-parts a simplified system behavior model based on realistic hypotheses and key parameters is first used Dynamic behavior is controlled by Partial Differential Equations (PDE) based on the characteristics of the system By incorporating elements or parameters that were initially not included and by improving the PDE (for instance by taking into account non linearities or novel coupling schemes hellip) this model is extended and improved leading to an increasingly precise simulation of the real functioning behavior as used in the process like approach

Theoretical models are usually built following an analysis of the complex system which leads to equations based on fundamental laws from the bottom-up Consequences are deduced from realistic hypotheses and known physical laws Either analytical or digital methods are applied to solve the equations Whenever possible experiments are conducted to compare expected results and real data A top-down approach can also be applied using experimental methods This approach is based on data obtained by applying specific stresses or external constraints and from the study of the system response Data from these tests are compared to simulation results from theoretical or empirical models Both bottom-up and top-down approaches can lead to some uncertainties in data analysis This can be evaluated through statistical analysis which provides predictions and margins of error The objective is to reduce the margin of error in order to obtain realistic predictions and to better understand the properties of active materials

This book describes experimental and theoretical methods which are developed in fundamental research to better understand the physical chemistry and physical processes in complex systems and which on the nanometric scale are the root cause of the outstanding properties of the materials used in innovative technological devices It presents optical techniques based on polarized light which can be applied to detect material or interface defects which have an impact on their performance It also describes how to measure the mechanical

Preface xiii

properties of nanomaterials and how to analyze experimental data taking into account the range of uncertainties using theoretical models

This book is written for students at Master and Doctoral levels teaching academics and researchers in Materials Science and Experimental Studies as well as engineers and technical staff from industrial sectors involved in systems where embedded electronics mechatronics and electronic and optical materials are employed

Chapter 1 describes various approaches which take into account uncertainties and are applied to analyze the static and dynamic behavior of systems and structures Chapter 2 presents an approach to optimizing the design of a system which matches design cost with the guarantee of functioning without failure in the planned use conditions This approach is based on taking into account uncertainties and on simultaneously solving two problems optimizing the production cost of the structures performing the expected functions and ensuring an acceptable probability to fulfill its function Chapters 3 and 4 give an overview of the classical and quantum theories of light as well as the various methods established to describe the polarization state of light

Chapter 5 reviews theories on the interaction of light and matter and various condensed phase materials used in industrial applications The notion of incomplete information about a quantum system is presented using the density matrix to take into account the problem of the interaction of the quantum system with the environment Chapter 6 describes lasers sources of polarized light and the experimental methods based on lasers to study either bulk materials using Laser Induced Fluorescence and IR-IR Double Resonance techniques or the surface of materials using techniques to analyze the reflexion of a probe over the ultrasonic waves created by a pump laser These methods make it possible to discriminate the different paths through which energy dissipates in materials when defects are present This approach is used to build theoretical models to understand and analyze the thermal effects in composite materials

xiv Nanometer-scale Defect Detection Using Polarized Light

Chapter 7 describes how to apply these methods to model systems before describing the apparatus used to prepare the systems composed of molecules which are trapped at low temperature in a solid matrix (rare gases or nitrogen) The various lasers and infrared detectors used in Laser Induced Fluorescence and Double Resonance techniques are presented The results obtained on O3-GR CO2-GR and N2O-GR systems are analyzed using theoretical models developed to determine the energy relaxation rate constants according to the various paths through which a system may transfer energy Predictions and extrapolations applying the results of the highlighted transfer mechanisms to other sytems are proposed

Chapter 8 describes the study of the interfaces of assembled materials using the IR spectroscopic ellipsometry technique This technique is summarized as well as the necessary equipment and the analysis process which is based on an inverse method applied to the models describing the interaction of light and matter through optimization algorithms The results obtained on various types of interfaces found in the assembly of mechatronic power devices are presented and discussed The ellipsometry technique is used to determine the possible modifications that occur in the properties of the materials when they come into contact as a result of physical or physical-chemical processes as well as to follow the evolution of interfaces as a function of temperature in a dry or humid atmosphere

Chapter 9 describes how to determine the properties of carbon nanotubes by applying the RBDO approach which correlates theoretical models and statistical methods to characterization and fabrication methods

Pierre Richard DAHOO Philippe POUGNET

Abdelkhalak EL HAMI June 2016

1

Uncertainties

Taking into account uncertainty in the design process is an innovative approach This includes dimensioning the structure of the systems the use of safety coefficients and the most advanced techniques to calculate reliability The aim is to design a system that statistically achieves the best performance since the system is subject to variations For a given risk probability satisfactory system performance can be targeted which has low sensitivity to uncertainties and respects a minimum performance threshold From a mathematical point of view an innovative approach to system design can be considered as an optimization problem under constraints In this chapter various methods are presented to calculate systems subject to uncertainties

11 Introduction

The methods used to take uncertainties into account are mathematical and statistical tools that make it possible to model and analyze systems whose parameters or use conditions are likely to vary These methods are used to optimize the design and to balance cost and performance

These methods are based on

ndash the development of an approximate mathematical model of the physical system under study

ndash the identification and characterization of the sources of uncertainty in the model parameters

Nanometer-scale Defect Detection Using Polarized LightPierre Richard Dahoo Philippe Pougnet and Abdelkhalak El Hami copy ISTE Ltd 2016 Published by ISTE Ltd and John Wiley amp Sons Inc

First Edition

2 Nanometer-scale Defect Detection Using Polarized Light

ndash the study of the propagation of these uncertainties and their impact on the output signal (response) of the system

Analysis and estimation of the statistics (moments distribution parameters etc) of the system response are performed in the next step The methods used to analyze the propagation of uncertainties vary according to the mathematical tools on which they are based These methods include a reliability based design approach a probabilistic approach based on design of experiments and a set based approach

12 The reliability based design approach

The reliability based design approach is based on modeling uncertainties Depending on the methods used uncertainties are modeled by random variables stochastic fields or stochastic processes These methods make it possible to study and analyze the variability of a system response and to minimize its variability

The most common methods are the Monte Carlo (MC) method perturbation method and polynomial chaos method [ELH 13]

121 The MC method

1211 Origin

The first use of this mathematical tool dates back to Fermirsquos research on the characterization of new molecules in 1930 The MC method has been applied since 1940 by Von Neumann et al to perform simulations in the field of atomic physics The MC method is a powerful and very general mathematical tool Its field of applications has widened because of the processing power of todayrsquos computers

1212 Principle

The MC method is a calculation technique which proceeds by successively solving a determinist system equation in which uncertain parameters are modeled by random variables

Uncertainties 3

The MC method is used when the problem under study is too complex to solve by using an analytical resolution method It generates random draws for all uncertain parameters in accordance with their probability distribution laws The precision of the random generators is very important because for each draw a deterministic calculation is performed using the number of parameters defined by this generator

1213 Advantages and disadvantages

The main advantage of the MC method is that it can be very easily implemented Potentially this method can be applied to any system whatever their dimensions or complexity The results obtained by this method are exact in a statistical sense that is their uncertainty decreases as the number of draws increases This uncertainty of precision for a given confidence level is defined by the BienaymeacutendashChebyshev inequality A reasonable precision requires a large number of draws This sometimes makes the MC method very costly in terms of calculation time which is the main disadvantage of this method

1214 Remark

The simplicity of the MC method has made its application popular in the field of engineering sciences This is a powerful but costly method Its results are often used to validate new methods that are developed in the framework of fundamental research It is applied in Chapter 9 in order to characterize carbon nanotubes

122 The perturbation method

1221 Principle

The perturbation method is another technique used to study the propagation of uncertainties in systems [KLE 92 ELH 13] It consists of approximating the random variable functions by their Taylor expansion around their mean value According to the order of the Taylor expansion the method is described as being the first second or


nth order The conditions of existence and validity of the Taylor expansion limits the scope of this method to cases where the random variables have a narrow dispersion around their mean value [ELH 13 GUE 15a]

With the perturbation method the random functions in the expression of the modelrsquos response to input parameters are replaced by their Taylor expansions Terms of the same order are grouped together and as a result a system of equations is generated The resolution is then carried for each order starting with the zeroth order The mathematical formalism as well as the general equations for the resolution can be found in the books by El Hami and Radi [ELH 13] and Guerine et al [GUE 15b]

1222 Applications

There are many applications of the perturbation method This method makes it possible to study the propagation of uncertainties in static and dynamic systems as well as in linear and nonlinear systems However it provides precise results only when the uncertain parameters have a low dispersion [ELH 13 GUE 15a]

Guerine et al [GUE 15b] have used the perturbation method in order to study the aerodynamic properties of elastic structures (stacked flat) subject to several uncertain parameters (structural and geometrical parameters) in the field of modeling and analysis of the vibratory and dynamic behaviors of systems This work is the first published application of the stochastic finite element method (FEM) combined with the perturbation method for the analysis of aerodynamic stability

In another study El Hami and Radi [ELH 13] combine the finite difference method and the perturbation method to model vibration problems in uncertain mechanical structures This method is used for example to determine the probabilistic moments of eigen frequencies and eigen modes of a beam in which the Young modulus varies randomly

The second order is usually sufficient to determine the first two moments with good precision In [MUS 99] Muscolino presents a

Uncertainties 5

dynamic analysis method for linear systems with uncertain parameters and deterministic excitations This method improves the first-order perturbation method which is limited when the dispersion of uncertain parameters is high The results obtained are compared to the results of the MC method and to the second-order perturbation methods The results are closely correlated

1223 Remark

The perturbation method consists of expressing all the random variables by their Taylor expansions around their mean values However the use of this method is difficult to implement particularly in the case of systems with many degrees of freedom and in cases where the uncertain parameters have a low dispersion around their mean

EXAMPLE 11ndash Application of the perturbation method

The objective of this example is to demonstrate the advantages of the Muscolino perturbation method to determine the beam response

A beam which is fixed at its extremities and free to vibrate in the (Oxy) plane is considered (Figure 11)

Figure 11 Biembedded beam

The mass and stiffness matrices are given by

[11]

x O

y F

[ ]2 2

2 2

156 22l 54 13l22l 4l 13l 3lmM54 13l 156 22l42013l 3l 22l 4l

minus minus = minus minus minus minus

[ ]2 2

3

2 2

12 6l 12 6l6l 4l 6l 2lEIK12 6l 12 6ll

6l 2l 6l 4l



The beam has a square section of side b which is modeled as a Gaussian random variable

The stiffness matrix [K] can be written as follows

[K] = b4middot[A]

where [A] is a deterministic matrix

Likewise the mass matrix [M] can be written as

[M]= b2middot[B]

where [B] is a deterministic matrix

The beamrsquos response to a force F = 600 sin (800t) applied at the beam midpoint is studied The mean value and standard deviation of the displacement of the beam midpoint are calculated with the second-order perturbation method and the proposed new method The results are compared to those obtained with the MC method as the reference using 10000 draws

The results (Figures 12 and 13) show that the two perturbation methods give the same results as the MC method

Figure 12 Mean of the displacement of the beam midpoint

helliphellip Monte Carlo Simulation____ Second order perturbation method Muscolino perturbation method

Uncertainties 7

Figure 13 Standard deviation of the displacement of the beam midpoint

123 The polynomial chaos method

1231 Origins and principle

The polynomial chaos method is a powerful mathematical tool that was developed by Wiener in the framework of his theory on homogeneous chaos [GUE 15a GUE 15b] This method formalizes a separation between the stochastic components and deterministic components of a random function The polynomial chaos leads to a functional expression of a random response by decomposing its randomness on the basis of orthogonal polynomials

In a general sense a set of second-order stochastic variables (with finite variance) can be expressed by a series expansion of Hermite polynomials orthogonal functions of some Gaussian and independent random variables modeling uncertainty The deterministic part is



modeled by coefficients jx called stochastic modes weighting the Hermite polynomial functions

[12]

The polynomial family Φi forms an optimal orthogonal basis and allows for a convergence of the expansion [ELH 13] in the sense of least squares However the convergencersquos rapidity and the expansionrsquos accuracy in Hermite polynomials are not verified anymore with non-Gaussian processes In fact in the case of Gaussian processes the Hermite basis optimality is a consequence of the Gaussian characteristic of the probability density function (PDF) The mathematical expression of this PDF is then equivalent to the weight function associated with the scalar product defined in this Hermite basis This principle can be generalized and used to establish a correspondence called the Askey scheme [ASK 85] among families of orthogonal polynomials and probability distributions The concept of expansion in a generalized chaos polynomial can then be defined An exponential convergence is thus demonstrated and generalized to arbitrary probability laws (not necessarily Gaussian) [GHA 99]

1232 Remark

Polynomial chaos is a concept that is well suited to the modeling of random functions and processes It is a tool that allows the consideration of uncertainties and nonlinearities in modeling and systems analysis The numerical schemes by which polynomial chaos is implemented differ in the way they make use of the model which is subject to uncertainty propagation The intrusive numerical scheme has the advantage of requiring only one calculation to determine the stochastic methods This calculation is tedious when the original model contains a lot of uncertain parameters The calculationrsquos complexity is greater in the case of systems with many degrees of freedom that are highly nonlinear This is due to the fact that the original model is transformed via its projection on the basis of the polynomial chaos in a system of deterministic equations

( ) ( )j jj 0

X xinfin

=

ξ = φ ξ

Uncertainties 9

whose dimension and complexity depend significantly on the original modelrsquos number of uncertain parameters and degrees of freedom

In contrast the non-intrusive scheme has a significant advantage in that it does not require modifications or transformations of the original model In [ELH 13] we find numerous applications of this method

13 The design of experiments method

131 Principle

The design of experiments (DOE) method makes it possible to implement or simplify in terms of complexity and cost an experimental protocol to determine the parameters impacting the performance of an industrial product The objective of the DOE method is to obtain a design which is almost insensitive to variation in system parameters By setting the number of experiments to be performed this method makes it possible to determine the impact of several parameters on the system responses When applied to a given system its efficiency depends on the control of the values given as input to the system parameters and on the precision of the measurements of the corresponding responses Several techniques are based on the DOE concept The various concepts which can be applied to design sets of experiments are described by Chatillon [CHA 05]

The Taguchi DOE method makes it possible to significantly reduce the number of trials [TAG 86] This method is implemented by using two matrices conjointly a control matrix representing the adjustable factors and a noise matrix representing the noise factors (uncertain parameters) The trials are performed for each combination of the factors identified in these matrices The statistical data such as the mean value and standard deviation of the response signal are measured To evaluate the quality of the results the Taguchi method


uses as a quality criterion the signal to noise ratio and a loss function The method developed by Huang-Chang [HUA 05] which is based on the concept of orthogonal columns makes it possible to simultaneously study multiple design parameters thus reducing the minimum number of trials

132 The Taguchi method

This statistical method is used to set an experimental protocol which renders the main response of a system insensitive to the different values of its parameters A set of experiments is defined along with the various sets of system parameters that will be used The number of experiments to be performed depends on the adjustable design parameters the number of random (uncertain) parameters possible interactions between these parameters and the effect of these parameters on the response (linear effect or not etc)

By taking into account the variability of multiple parameters the Taguchi method optimizes the system response Originally this method used the signal to noise ratio as a quality indicator thus combining mean and variance

The advantage provided by the Taguchi method is to simplify the implementation of the design of experiments It proposes a selection of experiment matrices tools for helping to choose the most suitable table and advice to take into account the interactions between the adjustable factors of the design

Taguchirsquos tables make it possible to

ndash choose the matrix of trials to perform according to the number of factors modalities and interactions

ndash verify using linear graphs that the selected table takes into account all the factors their interactions and is representative of the problem under study

ndash identify using the interaction table the columns corresponding to the interactions that have not been taken into account

Uncertainties 11

EXAMPLE 12ndash Application of the design of experiments in robust design

The aim of this example is to highlight the advantages of the DOE method in making the system response insensitive to variations of input parameters A microcontroller component assembled on a rectangular printed circuit board is considered This microcontroller has 256 pins that are connected to the circuit board by solder joints The printed circuit is attached to an aluminum alloy case with five screws (one in each corner of the circuit and one in its central region) Applying the Finite Element Method (FEM) a model of the printed circuit board equipped with this microcontroller component is developed The input parameters are geometric (position of the fifth screw thickness of the printed circuit) and the physical properties of materials (printed circuit board layers solder pin the composite molding compound of the microcontroller) The response of the model is the strongest stress applied during thermal loading on the solder joints among the 256 solder joints of the component

To select the only input variables that have a significant effect on the response a screening design of experiments is performed As the number of variables is 35 a PlackettndashBurman design is developed This design leads to a selection of only 15 parameters that have an effect on the response

To obtain the hypersurface of the response in the multidimensional space a design of experiments of the hypercube latin (LHS) type is carried out on the selected variables An LHS design with n trials is an experimental design for which

ndash each parameter is divided into n levels identified by a set of numbers (the higher the number the thinner the ldquomeshrdquo and the more easily the fitted model can reach the optima)

ndash each parameter can take a level once only

Each parameter is divided into n levels that are equally distributed between its minimum and maximum value A sampling that is representative of the hypervolume is thus obtained The LHS design is tailored to digital testing because of its simplicity of implementation


and to the spatial interpolation method (Kriging) which provides the hypersurface of the response

The considered response is the mechanical stress applied on the solder joint which among the 256 joints of the component has the largest median stress 70 of the trials of the LHS DOE are used to build the Kriging model These trials are randomly drawn among 200 possible trials The remaining 30 are used to validate the predictive efficiency of the model

The obtained response surface makes it possible to approximate the stress on the most exposed solder joint Fifteen variables are considered in this response model To identify the variables that have the most impact on the stress a global sensitivity analysis is performed using the Sobol indices method Several simulations (total number n) of the input parameters are carried out The response is calculated using the Kriging model Then to study the sensitivity of a given parameter another draw on all other parameters except this one is conducted This step is repeated many times by bootstrap The Sobol indices of the parameters under study are then calculated from the total variance and variances relative to the studied parameter A statistical distribution for each index is obtained This distribution is represented by a boxplot and used to estimate a confidence interval on the index value

One thousand simulations to simulate the input parameters and 100 calculations of indices are performed The parameters that appear as the most influential on the response are in the order of importance

ndash Parameter X1 (COMP_Z) the thickness of the solder joint

ndash Parameter X4 (EX_SOLDER) the Youngrsquos modulus of the solder

ndash Parameter X5 (ALP_SOLDER) the expansion coefficient of the solder

ndash Parameter X14 (ALPX COMP) the component of the expansion coefficient in the design

Uncertainties 13

Once the influential factors are identified MC simulations are performed to determine the distribution of the constraint on the solder joint as a function of the variations of the influential factors

ndash a nominal value for each non-influential factor is then assigned

ndash for each influential factor a draw of a uniform law in its field of variation is conducted

ndash finally the value of the constraint on the solder joint is calculated by the Kriging model

The procedure described above is iterated a large number of times (107) to obtain the distribution of the stress applied on the solder joint

Figure 14 Histogram of the stress on the solder joint

This distribution can be estimated by a parametric model such as a Gaussian mixture model The obtained result is shown in Figure 15

The level of control factors is finally adjusted to reduce the systemrsquos sensitivity to sources of variability (noise factors) and adjust the systemrsquos response to its target (goal)

The factor ALPX COMP (CTEX of the component) has a strong effect on the constraint (positive influence) This factor can be adjusted by the composite structure of the component coating material To minimize the stress small values of Alpx COMP must be

Constraint

Den

sity


drawn By reducing the range of variation of the Alpx COMP factor to the interval [5 7] instead of [5 23] initially the average value of the stress and its variability are reduced (Figure 16)

Figure 15 Density of the stress on the solder joint

Figure 16 Density of the initial stress (red) and optimized constraint (blue) For a color version of this figure see wwwistecoukdahoonanometerzip

14 The set approach

The methods of the reliability approach described in section 13 require as a prerequisite that the probability laws governing the

Den

sity

Constraint

Constraint

Den

sity

Uncertainties 15

uncertain parameters are known The methods of the set approach have the advantage of not requiring models of the laws of probability and uncertainties to be known beforehand Two main methods are implemented in the set approach the first is based on interval arithmetic [MOO 66] the second on the formalism of fuzzy logic [ZAD 65]

141 The method of intervals

1411 Principle

The foundations of the calculation by intervals date back to the work of Moore and Baker [MOO 66] This method is based on the modeling of uncertain parameters by intervals whose bounds are the minimum and maximum settings of the parameters The error between the modelrsquos output and the systemrsquos response is then considered to be bounded with known bounds These limits take into account the noise in measurements and modeling errors No value of parameters that minimizes a convergence criterion is necessary but only a set of acceptable values is looked for This method is easy to use Unlike probabilistic methods no information is needed on the nature of the dispersions or on how they operate However it poses difficulties of convergence

1412 Interval arithmetic and stability analysis

The interval arithmetic is applied to the analysis and stability of uncertain linear dynamic systems Jaulin et al [JAU 01] proposed a method to characterize the set of all the values of uncertain parameters linked to a stable dynamic behavior To study the stability the solution is determined using the interval analysis according to the criteria of Rooth By defining two sets A and B the stability analysis is reduced to a problem of inclusion A is the admissible set of possible values of uncertain parameters while B is the set of values for which the system is stable An algorithm based on the interval theory allows for testing the inclusion of A into B which is a necessary and sufficient condition for stability The convergence of the algorithm is tested on digital systems


EXAMPLE 13ndash Method of intervals case of suspensions in a vehicle

A massndashspringndashdamper system is considered as an application example (Figure 17) This system is defined by the following equations

( ) ( ) ( )( )( ) ( ) ( ) ( ) ( )( )

s s

u u

3

s s s u s s u s s u

s

3 3

u s s u s s u s s u t u r t u r

u

1

1

x vx v

v c x x k x x K x xm

v c x x k x x K x x k x x K x xm

= = = minus minus + minus + minus

= minus minus + minus + minus minus minus minus minus

[13]

where ms and mu represent the masses c represents the damping parameter ks and kt represent the linear stiffnesses Ks and Kt represent the cubic stiffnesses

The initial conditions are

[ ] [ ]0 0000s u s u tx x v v = =

Figure 17 Massndashspringndashdamper system

Uncertainties 17

The parameters cs ks and kt are uncertain and modeled by intervals as indicated in Table 11

Parameters ms (kg) Cs (Nsm) Ks (Nsm) Kt (Nsm) Ks (Nm3) Kt (Nm3)

Mean value 375 1000 15000 200000

15 times 106 2 times 107 Interval ndash [900 1100] [13500 16500] [18 22] times104

Table 11 Parameters of the massndashspringndashdamper system model

Figure 18 Mean value of the displacement xu(t) for the intervals method (blue) and deterministic method (red)

The average value of the displacement is calculated with the method of intervals The result (Figure 18) is compared with that obtained by the deterministic method There is an agreement between the result of the method of intervals and that given by the deterministic method

1413 Conclusion

The arithmetic of intervals allows us to model uncertainties only by their physical limits which are for the most part identifiable and

temps (s)

Deacutepla

cement

(m)

time (s)

Disp

lace

men

t(m

)D

ispla

cem

ent (

m)

Time (s)


measurable No information on the evolution of uncertainties in their intervals is required

142 Fuzzy logic based method

1421 Principle

Fuzzy logic based methods have been introduced to represent and manipulate uncertain data when we possess no probabilistic or statistical information

These methods are based on the concept of fuzzy sets An element of a fuzzy set such as a modelrsquos input value has a degree of membership in the set This notion that is formally called a membership function is different from the concept of probability It defines a quantitative measure regardless of imperfect data This definition allows us to establish a fuzzy logic associated with degrees of truths assigned to propositions ranging from zero (false) to one (true) with all possible graduations which leads to the vocabulary a little moderately etc The application of fuzzy logic is thus appropriate for approximate reasoning

EXAMPLE 14 Application of the fuzzy logic based method

A two-dimensional application is considered for this example a freely vibrating cross-beam The system shown in Figure 19 in the (OXY) plane consists of three beams of the same square cross-section The beam cross-section is considered to be the only random parameter The objective is to determine the stochastic displacement of the horizontal beam of the frame under a given sinusoidal excitation (F (t))

F(t) = 20 sin(80t)

The mean value and standard deviation of displacement are calculated using the fuzzy logic method The results (Figures 110 and 111) are compared to those of the reference method ie the MC method The stochastic response of the cross-beam (mean and standard deviation of the beam displacement) that is calculated from

Uncertainties 19

the method of fuzzy logic is consistent with the responses calculated by the MC method as shown in Figures 110 and 111

Figure 19 Two-dimensional cross-beam

1422 Conclusion

The method which accounts for uncertainties by fuzzy logic allows us to manipulate information that is vague inaccurate or described in a linguistic manner These uncertainties are modeled by shape functions called membership functions The main advantage of this method is that it does not require statistical or probabilistic information The determination of the membership functions is on the other hand difficult

Figure 110 Average value of the displacement as a function of time

stemps

Deacutep

lacem

ent (

m)

( )time

Dis

plac

emen

t(m

)

helliphellip Fuzzy logic method____ Monte Carlo Simulation

Dis

plac

emen

t (m

)

Time ( ) s


Figure 111 Standard deviation of the displacement as a function of time

15 Principal component analysis

The purpose of the principal component analysis method is to determine the most significant components of a system depending on several variables For example analyzing a sample of N individuals who are characterized by P characters (or variables) There are multiple relationships between N and P that we want to analyze

To analyze the relationship between two variables the values of variables are positioned on two orthogonal axes and the effect of these variables are analyzed using statistical tests For three variables this approach leads to a three-dimensional graph For four or more variables it is no longer possible to proceed graphically Working with pairs or with triplets of variables can mask complex interactions and hence the idea of establishing a technique to extract the most relevant information This technique uses linear combinations of the variables which is well adapted to linear relationships

stemps

Deacutep

lacem

ent (

m)

( )time

Disp

lace

men

t(m

)helliphellip Fuzzy logic method____ Monte Carlo Simulation

Disp

lace

men

t (m

)

Time ( ) s

Uncertainties 21

151 Description of the process

Let X1 X2Xp be the initial quantitative and centered variables (with a mean of zero) A new variable Y1 and a linear combination of Xi are calculated

Y1 = C1X1 + C2X2 + hellip+ CpXp [14]

where c1 c2hellipcp are constants to be determined such that Y1 has a maximum variance with the following constraint

2 2 21 2 p + + + 1c c c = [15]

Of all the possible linear combinations of Xi the one that suppresses the least possible information is that which has the greatest degree of dispersion If Y1 has zero dispersion Y1 is a constant The problem to solve is to look for constants c normalized to 1 which maximize the variance of Y1 Thus the constants c can be determined (and therefore Y1) The variable Y1 is called the first principal component and V1 its variance

In general the calculation of Y1 does not use all of the variance of the original variables A second variable Y2 of maximum variance uncorrelated to Y1 is then sought as a linear combination of Xi

2 12 1 22 2 p2 p+ + +Y c X c X c X= [16]

where c12 c22hellipcp2 are constants to be determined under the constraint of normalization

2 2 212 22 p2 + + + 1c c c = [17]

REMARKndash

ndash c11 can be replaced by c1 and c21 by c2 and so on Then it can be shown that the constants c (and therefore Y2) are uniquely determined V2 is the variance of the new variable Y2 By construction 1 2 V Vge


ndash Y2 is called the second principal component New variables Y3 Y4Yp can be constructed in the same manner These variables are uncorrelated with the preceding ones with maximum variance (with the normalization condition on the coefficients of the linear combination)

Let V3 V4hellipVp be the variances of these new variables then

3 4 5 pV V V Vge ge ge [18]

152 Mathematical roots

The determination of the constants c (or those of Y) is a problem of the determination of eigenvalues Different c are the coordinates of the eigenvectors (normalized) of the covariance matrix of the initial variables X Variances V1 V2 Vp are the associated eigenvalues The different properties cited (existence and uniqueness) are deduced If the rth variance Vr + 1 is very small the variables Yr + 1 Yr + 2 Yp are almost constant for all individuals It is thus natural to keep only the main components Y1 Y2Yp In practice it is estimated that Vr + 1 is small if the following relation holds

1 2 r

1 2 p

( +V + +V ) 90 ( +V + +V )VV

asymp [19]

In the best case three principal components are sufficient The P correlated variables are then reduced to three uncorrelated variables that can easily be represented graphically

153 Interpretation of results

Generally the initial aim of extracting the most relevant information is achieved A smaller number of variables (principal components) are necessary They are uncorrelated and can easily be represented graphically without much distortion Two approaches exist one is based on variables the other on individuals

Uncertainties 23

1531 Method based on the variables

The correlation between the main components the originate variables is determined If only the first r principal components Y1 Y2Yr are considered then rP correlation coefficients are used to calculate the correlation of Y1 with X1 X2Xp Y2 with X1 X2Xp and Yp with X1 X2Xp The main components are interpreted on the basis of the observed values of these coefficients

1532 Method based on the individuals

The principal components can be interpreted using the position of individuals with respect to the principal componentsrsquo axes Individuals whose contributions relative to the axes involved are too small are considered to be poorly represented It is possible to interpret the position of individuals in the planes formed by the components

16 Conclusions

In this chapter various methods which take into account uncertainties in systems are presented If experimental data can be described by the laws of probability then the Monte Carlo approach is recommended If these data are within a given range with no other information the algebraic interval approach is better suited However a problem of convergence sometimes occurs When no probabilistic or statistical information are available and when there are no recommended ranges the fuzzy logic approach is more appropriate Several DOE methods greatly reduce the complexity of the problem posed by rendering the system responses insensitive to uncertainties in system parameters The principal component analysis approach leads to the determination of the components which most impact the system according to given indicators

2

Reliability-based Design Optimization

The optimization of mechanical structures aims to determine the best possible design in terms of cost and quality Generally design optimization uses criteria based on constraints and design variables and deterministic procedures such as the frequently used regression or stochastic or hybrid algorithms methods However in the last two cases variables are most often considered as numbers which means they are deterministic One of the major issues of the reliability-based design optimization is to establish a rigorous monitoring that is able to predict and detect failure modes of the systems under study This chapter presents the advances in the fields of optimization and reliability by taking uncertainties in mechanics into account This coupling is the basis for the competitiveness of companies in the automobile aerospace civil engineering and defense fields

21 Introduction

Traditionally a design engineer optimizes the design of the structures of a system by successive experiments Drawing on his experience and accumulated know-how a first version is developed and is then checked by calculation to ensure meets the specification requirements If not the design is adapted until it complies with functional requirements and constraints In most cases several iterations are required which makes this method costly in realization time and prototype construction To increase its efficiency engineering firms use digital modeling and optimization software to analyze the various possibilities and thus automize the search for the


First Edition


optimal solution The optimization is based on the achievement of performance goals and minimization of the bill of materials However in this approach the design engineer does not take into account the accuracy of the mechanical properties of the materials geometry and loading nor the degradation of reliability caused by cyclical use and operating conditions

This chapter presents a reliability-based design optimization method This method balances the design cost of the system and the assurance of its performance in the intended use conditions This method takes into account uncertainties and simultaneously solves two problems optimizing cost and ensuring satisfactory operation in operating conditions (reliability)

22 Deterministic design optimization

The formulation of the problem of Deterministic Design Optimization (DDO) is obtained by applying a mathematical process as described in [ELH 13] This can be expressed by

( )( ) ( )( )( ) ( )( )

1 1 1

2 2 2

min Under 0

0

t

t

f xg x G x Gg x G x G

= minus ge= minus ge

[21]

where ( )x is the deterministic vector of the design parameters

In deterministic optimization geometric and material properties and loads are analyzed to provide a detailed behavior of the structure Figure 21 shows deterministic optimization based on the safety factor

This deterministic approach has limits Variability of the properties of the structures as well as modeling approximations may have a negative impact on the ability of the system to function correctly under operational conditions Hence the need to analyze the influence of uncertainties and their effects on the concerned productrsquos reliability is essential

Reliability-based Design Optimization 27

23 Reliability analysis

The basic principles of reliability applied to problems of mechanical structures are summarized in [ELH 13]

Figure 21 Deterministic design optimization based on the safety factor

If Y is a random vector of design variables then the realizations of Y are noted as y Reliability is expressed by the probability of success of a scenario represented by a limit state G(xy) which is a function of random variables y and deterministic variables x

( )reliability 1 1 prob 0fP G x y = minus = minus le [22]

G(xy) = 0 defines the limit state G(xy) gt 0 indicates the safe functioning state and G(xy) lt 0 the state of failure A reliability index β is a measure of the level of reliability Calculating an exact and invariant index implies its definition not in the space of

2nd Limit state

1st Limit state

Feasable Region

Increasing direction of the objective function


physical variables y but in a space of statistically independent Gaussian variables u of zero mean and unitary standard deviation (Figure 22) An isoprobabilistic transformation is defined for the transition between these two spaces by

u= T(x y)

Figure 22 Normal physical space

In this normed space the reliability index β represents the minimum distance between the origin of the space and the limit state function H(xu) = G(xy) the point closest to the origin is called the design point The calculation of the reliability index is done by an optimization procedure under the constraint of belonging to the domain of failure

( )min( ) 0

Tu uwith H x u

β =le

[23]

where u is the vector module in normal space measured from the origin

Normed space

ReliabiltyDomain

FailureDomain


The solution for optimizing under the restrictions of equation [23] is called the design point P This solution is sensitive to nonlinear programming issues such as local minima gradient approximations and run time calculation Although equation [23] can be solved by any suitable optimization method specific algorithms have been developed to take advantage of this particular form of reliability problems In [ELH 13] four criteria are used to compare these different tools generality robustness efficiency and capacity Five algorithms are recommended for assessing the reliability of the structures the sequential quadratic programming the modified RackwitzndashFiessler algorithm the projected gradient the Lagrangian augmented and the penalty methods In the analysis of nonlinear finite elements (FE) the projected gradient method is less effective

Figure 23 Evaluation process of the reliability index

NO

YES

Define random variables and their statisticalproperties

Evaluate the function of merit

Compute the optimal conception point

Compute the reliability index

Analyzis of the sensitivity in the physicalspace and the probability space

Compute mean values and standard deviation types in the normal space

Verify convergence

End


231 Optimal conditions

The optimization problem [23] is equivalent to the minimization with or without constraints via the definition of a Lagrangian

( ) ( ) TH HL u u u H x yλ λ= + [24]

where Hλ is the Lagrangian multiplier

The optimal conditions of Lagrangians are as follows

[25]

( ) 0H

L H x uλ

part = =part

[26]

This method involves the assessment of the Lagrangian derivatives in the normalized space Usually the limit state function H (x u) is unknown Its evaluation is the result of a finite element analysis which consumes considerable calculation time especially for nonlinear and transient problems

In addition the analysis of the normalized gradient j

Hu

partpart

is not

instantly accessible because the mechanical analysis is performed in the physical space and not in the standard normalized space The normalized gradient calculation is performed by applying the chain

rule to the physical gradient k

Gy

partpart

[27]

These derivatives are generally obtained by finite difference techniques which require a large computation time The integration of reliability analysis in engineering design optimization is called reliability-based design optimization (RBDO)

02 =partpart+=

partpart

jHj

j uHu

uL λ

( )jkj u

uxTyG

uH

partpart

partpart=

partpart minus

1


24 Reliability-based design optimization

The aim of design in the presence of uncertain parameters is to achieve a match between the level of reliability and the optimal design cost of the structures Figure 24 compares the optimal solutions of the Deterministic Design Optimization (DDO) and RBDO methods The solution of the deterministic method is located in the vicinity of point A which is on the border of a boundary condition that can lead to an incorrect solution The solution of the RBDO method is within the feasible region around point B

Figure 24 Comparison of the RBDO and DDO approach

241 The objective function

Several objective functions are proposed for the RBDO method These include the cost and utility functions that should be minimized and maximized respectively The optimal lifetime cost and utility can be established (see [KHA 04])


242 Total cost consideration

An optimized design that does not take into account reliability aspects may not provide economic solutions as an increasingly higher failure rate in application may induce a higher operation cost Indeed codes based on dimensioning do not ensure a homogeneous reliability and admissible regulatory solutions can have various reliability levels The expected total cost of a structure CT is expressed by the linear combination of the initial failure and maintenance costs (Figure 25) as follows

[28]

where Cc is the building cost of the structure Cf is the cost of failure due to direct and indirect damage of a structural component CIr is the inspection cost CMs is the cost of maintenance and repair Pf is the probability of failure PIr is the probability of non-failure until detection at the rth inspection and PMs is the probability of repair

Figure 25 Total failure and initial costs

+++=s

MMr

IIffcT ssrrPCPCPCCC

Cost

Pf

CT

Cf Pf

Cc


The objective of the RBDO approach is to minimize the total cost CT of the structure The difficulty of quantifying the failure cost Cf (especially in the case of immaterial damages) makes the use of equation [28] difficult For this reason the optimization problem becomes more significant when the initial cost is reduced it is represented by the objective function f(x) with the constraint to meet a target confidence level β gt βt The aim is to minimize the initial and failure costs

243 The design variables

The major design variable classes [KHA 08] are as follows scale configuration based topological and material variables They can be continuous or discrete

244 Response of a system by RBDO

Structures having a linear static behavior are usually studied by the RBDO method However few dynamic analysis studies have been done as described by Mohsin et al [MOH 10]

245 Limit states

Most design optimization approaches use sequential limit states (Figure 24) and do not take into account the interactions between limit states On the contrary the RBDO method takes all relevant limit states of a structural system into account RBDO leads to a structure design that will meet reliability requirements for a limit state

246 Solution techniques

In recent years several digital RBDO optimization techniques have been developed the various proposed algorithms are discussed in [ELH 13] To solve a RBDO problem an optimization algorithm


can be coupled with specific reliability software [KAY 94 ELH 13b] However most of the solution techniques used in the RBDO method have been used for small- or moderate-sized structural systems Hence solution techniques take large size systems into account

Among the techniques that were recently proposed we have the reliability index approach and performance measure analysis methods However these methods are based like all conventional RBDO methods on resolution in two spaces the normed space of random variables and the physical space of design variables Another method was developed by taking advantage of the combination of these two spaces in a single hybrid space The latter has shown its effectiveness compared to the traditional reliability-based approach [KHA 14] Other methods based on positive points of the hybrid space are developed [MOH 10]

25 Application optimization of materials of an electronic circuit board

In order to perform their specified tasks onboard electronic systems integrate more and more functions This gradual increase in complexity impacts their overall ability to operate flawlessly in applied conditions and for the required duration (reliability) In [ELH 13a] El Hami and Radi demonstrate that the failure rate of electronic systems increases with miniaturization and subsequent higher component density on the printed circuit board (PCB)

Reliability and sustained performance of PCBs are crucial characteristics

PCBs use a support called substrate generally consisting of a laminated composite material (FR2 FR3 FR4 etc) which is a mixture of fiber reinforcement (glass fiber aramid fiber etc) and an organic or inorganic matrix (epoxy resin glass resin etc) as well as copper layers (Figures 26 and 27)


Figure 26 Diagram of a printed circuit board with six layers of copper

Figure 27 Architecture of the PCBrsquos fiber reinforcement a) overview b) detail of fiber fabrics

The composite materials are selected because of their low cost appropriate temperature resistance excellent adhesion to copper and electromechanical behavior To improve the mechanical performance of the PCB structure the fiber architecture is adapted by adjusting the fiber volume ratio and the orientation angles of the folds Once the structural optimization process is performed the designer has the data needed to study the various materials and the feasibility of the fabrication process

Copper foilSubstrate core

Bonding sheet

C-stage (Component)

B-stage (Glue layer)

C-stage (Component)


C-stage (Component)

Layer 1

Layer 2

Layer 3

Layer 4

Layer 5

Layer 6

Fill bundle Warp bundle

Matrix layer

a) b)


Matrix layer

Warp fibers

Fill fibers

(a) (b)


The PCB is a stack of copper foils and layers combining unidirectional fabrics and resin whose direction can be varied from layer to layer (Figure 28) This stack can be modeled as any basic laminate type

Figure 28 a) Constitution of a laminate b) designation of a laminate

251 Optimization problem

The miniaturization of embedded electronic systems requires an optimal design of the printed PCB in order to increase the frequency of eigenmodes and reduce its mass Due to the diversity of materials composing the PCB the use of structural analysis of the layers of FR4 composite materials is recommended

The structure of a PCB is composed of four copper foils and three layers of FR4 composite material Each FR4 layer is a combination of a fiber-type reinforcing glass and epoxy resin The FR4 laminate is a stack of a number Np of layers of fiberglass impregnated with resin characterized by a thickness hi an angle of orientation θi and a fiber volume content Vf An optimization procedure that consists of checking frequency ranges in order to find optimal values for the PCBrsquos constituents is developed in three steps

In the first step the structural variables characterizing the FR4 composite material are entered in a finite element model In the second step the influence of fiber volume and orientation of folds on the vibratory behavior is studied In the last step the design is optimized in order to reduce mass and improve mechanical performance

a) b)

Layers

Stratified

Stratified

Designation

(a) (b)


Figure 29 Comprehensive approach of PCB optimization

The PCB under study is a rectangular board measuring 170 mm times 130 mm times 16 m (Figure 210)

Elastic moduli of FR4 composites can be expressed in terms of elastic moduli functions of the fiber reinforcement and matrix materials using the HalpinndashTsai equations

According to the rule of mixtures of composite materials the moduli (Young modulus Poisson coefficient) are given by

[29]

4 (1 )fFr f m fV Vρ ρ ρ= + minus [210]

Design Optimization

Implementingvariables in the numericalmodel

Study of the influence of fiber volume ratio and orientation of the folds

Multiconstraint optimization and choice of parameters

Structural Analyzis

Layer of FR4Fiber reinforcingResinLayers of copper

Halpin-Tsai Equations FR4

Resolving method1) Metaheuristic2) Determinist

Verifying the vibratorybehaviour

Minimizationof the structure

4

4

(1 )(1 )

x y Fr x y f f m f

xy Fr xy f f m f

E E V E VV Vν ν ν

= + minus

= + minus


Figure 210 Cross-section and EF mesh of PCB

The general expression for the other moduli is

4 1

1fFr

m f

VMM V

ξηη

+=

minus [211]

where the coefficient η is expressed by

1

f

m

f

m

MMMM

ηξ

minus

=

+

[212]

whereby

ndash f is the index relative to the fibers

ndash m is the index relative to the matrix

ndash M is the transverse Youngrsquos modulus shear or transverse Poisson coefficient

ndash Mf is the corresponding fiber modulus

ndash Mm the corresponding matrix modulus

Copper


The factor ξ is a measure of the fiber reinforcement that depends on the geometry and arrangement of the fibers and the module considered ξ = 2 to determine the Youngrsquos modulus ξ = 1 to determine the shear modulus

252 Optimization and uncertainties

The impact of the fiber volume Vf and sequences of stacked layers θ on the eigenfrequencies characterizing the vibration modes of the PCB structure are analyzed The various parameter values that characterize the copper fiber and matrix are given in Table 21 With the aim of measuring the impact of design variables such as orientation and fiber volume on the normal modes it is assumed that the orientation of the folds is identical θi =θ

Parameterunits Value

ρcu (kgm3) 8930

ρf (kgm3) 2750

ρm (kgm3) 1200

Vf () 10

Exf (Gpa) 725

Eyf (Gpa) 725

Gxyf (Gpa) 30

Em (Gpa) 26

Gm (Gpa) 0985

nf 02

nm 032

Table 21 Parameters used for the numerical simulations

Figure 211 shows the obtained results in a map form For the first mode where the resonant frequency of the printed circuit must be greater than 169 Hz the optimal choice is within the range defined by


42deg le θ le60deg and 35 le Vf le 40 For the second mode where the imposed maximum frequency is greater than 216 Hz (Figure 211(b)) the optimal choice is located in the two ranges defined by 0deg le θ le 22deg and 35 le Vf le 40 78deg le θ le 90deg and 35 le Vf le 40 This parametric study defines an optimum design range based on the frequency imposed constraint for each vibration mode

The control of the laminate architecture improves the mechanical performance of the PCB The values of the fiber volume are varied as are the orientation and thickness of the folds and the thickness of the copper layers This is a multiconstraint optimization problem whose objective function is

( )

4 4

4

4

1( )( )

objCu Cu FR f FR

FR pl pl

f FR Cu

F Xh V h

h N h

X V h h

ρ ρ

θ

= + = =

[213]

where Fobj is a nonlinear function of the design variables grouped in vector X hpl is the thickness of the fold hFR4 is the thickness of FR4 for a number of folds and Npl and hCu are the thickness of the copper layer

The choice of the effective parameters of the PCB contributes greatly to obtaining Fobj The number of design variables depends on the properties of the materials (copper fiber and resin)

The minimization of the PCBrsquos mass is equivalent to the maximization of the objective function Fobj The constraints imposed on the terminals of the resolution space are respected

4 _ 4 4 _

_ _

01 04

0 90

f

FR m FR FR M

Cu m Cu Cu M

Vh h hh h h

θ

le le le le le le le le

[214]


where hFR4_m and hFR4_M are respectively the lower and upper limits of the variation interval of hFR4 hCu_m and hCu_M are respectively the lower and upper limits of the variation range of hFR4 The frequency constraints on the vibration mode i are expressed by the following inequalities

( )i i bi IN f X fforall isin ge [215]

where fi and fib are respectively the eigenfrequency and low frequency imposed on each mode i

Figure 211 Mapping of the fibers volume and orientation effects on the eigenfrequencies f1 a) f2 b) f3 c) and f4 d) of the PCB

As it is impossible to find an optimum analytically a metaheuristic resolution approach based on the genetic algorithm inspired by the mechanisms of natural selection and genetics is adopted This algorithm which is based on the natural evolution of species uses the


principle of the survival of the fittest In this context genetic properties (selection crossover mutation etc) are added

A population P0 that contains Np chromosomes (solutions) is generated by a random process (Figure 210) In order to create successive generations the chromosomes undergo a correction process to check the constraints imposed by the optimization problem This requires calling the FE algorithm for each generation of solutions until the frequency constraints are matched This selection process identifies individuals in a population that could be crossed The principle of selection by roulette is adopted In this optimization problem (maximization) each chromosome m is attributed a selection probability Pseli which is proportional to the value of the objective function

( )

( )

1

1 p

ip sel i N

ii

f Xi N P

f X=

forall isin =

[216]

Each chromosome is reproduced with probability Pseli Then the solutions are classified in three classes the strongest (Cb) that are the most reproduced the intermediate (Ci) and the weak (Cf) which are eliminated The determination of the different classes is based on the maximization of the probability of selection

max

max max

max

1 05 0 105

i b sel i s s

p i i s s sel i s s s

i f sel i s s

Chr C P Pi N Chr C P P P

Chr C P P

λλ λ λ

λ

isin geforall isin isin le lt lt lt isin lt

[217]

This probabilistic aspect is achieved by crossing the determined classes during the selection process The earliest crossing operators used a system of one point and two point operators on two binary-coded chromosomes The mutation operator brings the random genetic algorithms necessary for efficient exploration of the space This operator ensures that the genetic algorithm is likely to reach the most attainable points The genes of the randomly selected chromosomes with a low probability of mutation Pm are mutated Crossing aims at


enriching the population diversity by manipulating the chromosome components Conventionally crossing is performed with two parents and generates two children and is applied with a probability

253 Results analysis

The frequency constraint is imposed on the first mode with a given value f1b The population contains 50 chromosomes and the total number of iterations is 125 Table 22 gives the different values used in this digital simulation

Numerical parameters Value

Np 50

λs 07

Iter 125

Pm 006

f1b 200

Table 22 Parameters used in the genetic algorithm

In Figures 212 and 213 the convergence history of the genetic algorithm in achieving the optimal variables corresponding to the fiber reinforcement and copper layer is presented The convergence is very slow for the fiber volume (107 iterations) It is faster for the fold orientation angle and the ratio between the thickness of the FR4 and copper layer Table 23 summarizes the optimal values of the optimization variables

Variables Vf () hFR4 (mm) θ (deg) hCu (mm)

Values 1038 07542 200135 003736

Table 23 The optimal values of PCB design variables


a) b)

Figure 212 Evolution of the orientation angle a) and the fiber volume b) as a function of the number of iterations

Figure 213 Evolution of the ratio between the thickness of the copper and FR4 layer as a function of the number of iterations

26 Conclusions

This chapter describes the deterministic and probabilistic optimization of structures Combining optimization and reliability

Number of iterations Number of iterations

deg V f

Number of iterations

h FR4

hcu


methods (RBDO) leads to more efficient designs than conventional deterministic methods (DDO)

The RBDO method solves the optimization problem in a single hybrid space combining two types of variables design variables and random variables In this hybrid space the parameters of the problem can be controlled Optimizing the fiber glass content of a printed circuit with respect to volume and mass reduction parameters is an example which illustrates the RBDO method

3

The WavendashParticle Nature of Light

Light behaves like a wave or a particle This dual aspect has been shown experimentally in various light matter interactions (interferences photoelectric effect Compton effect etc) The qualitative theory proposed by C Huygens concerning the wavelike nature of light is contested by I Newton who supports the corpuscular theory However consolidating the earlier studies by T Young A Fresnel has not only analyzed the diffraction patterns but also established that light is a transverse wave hereby validating the optical wave theory In this approach light is represented by a scalar function which is the solution to the wave propagation equation the square of which is linked physically to the transported energy JC Maxwell takes this approach further and establishes that light is not a scalar quantity but a vector quantity His studies show that light is an electromagnetic wave that propagates at the speed of 3 times 108 msminus1 (2 99792458 times 108 msminus1) The range of optical frequencies varies between 1012 Hz (far IR) and 1016 Hz (deep UV) The visible spectrum covers 4 times 1014ndash8 times 1014 Hz The wave power density is described by the Poynting vector the vector product of the electric and magnetic fields This chapter applies a mathematical formalism based on the Maxwell equations to describe light A gauge is used to describe simply the state of polarization of a wave and obtain an electromagnetic wave propagation equation in the transversal representation By using this gauge it is possible to express the electromagnetic field as a sum of independent harmonic oscillators (through the use of creation and annihilation operators) This approach leads to the quantum description of light in terms of photons or light particles proposed by A Einstein to explain the photoelectric effect In this way light energy is shown to be associated with a frequency and to be discontinuous To conclude the Glauber approach combines the classical and quantum approaches and confers a physical reality to coherent states using a Poisson distribution law for photons


First Edition


31 Introduction

Various terms are used to describe the luminous phenomena found in nature light electromagnetic waves or photons The drive to understand the luminous phenomena observed when the Sunrsquos rays a natural source of light interact with matter led to the theory of geometrical optics This approach has explained phenomena like the mirage effect rainbows that result from light reflection on water drops sun eclipses and light reflection on mirrors When the Sun is vertical to the Earth geometrical optics can be used to calculate the Earthrsquos circumference [BRU 65 HEC 05 MEI 15]

In geometrical optics theory a light beam consists of rays propagating in straight lines In homogeneous media rays propagate according to SnellndashDescartes laws However if light passes through a pin hole diffraction phenomena are observed These phenomena go against geometrical optical theory light rays cannot be localized The approach developed by Fresnel and Huygens explains the diffraction and fringes phenomena and is validated by Maxwell theoretical works [MAX 54] Newtonrsquos particle theory does not explain the interference and diffraction phenomena However Einsteinrsquos particle approach that considers light as having an energy grain structure which explains the photoelectric effect Light interacts with matter through energy quanta E = hν where ν is the frequency associated with light color In the case of the photoelectric effect an induced absorption process is involved Modeling a blackbody as a source of radiating energy in discrete packets Planck resolved the so-called ultraviolet catastrophe for blackbody radiation In 1915 Einstein explained blackbody radiation by adding a symmetrical emission process to a discontinuous absorption process Light is thus considered as a wave or as a particle [BRO 68] These different approaches are more detailed in more specialized papers The differences between the classical and quantum approaches of light are made more apparent by applying the statistical theories of light Quantum mechanics (QM) [MES 64 COH 73] notions are necessary to help understand the corpuscular nature of light as a photon as well as Glauberrsquos approach All these

The WavendashParticle Nature of Light 49

theories are useful to understand the techniques which are polarized light to characterize matter

32 The optical wave theory of light according to Huyghens and Fresnel

321 The three postulates of wave optics

A mathematical description of light in wave optics theory [BRU 65 SIE 86 HEC 05] requires that the associated wave function be a solution to the propagation equation The velocity at which the wave propagates in a given medium depends on its index of refraction The energy flux through a closed surface is related to the square of the wave function and is conserved during its propagation in a vacuum Numerous optical phenomena such as interferences and diffraction can be explained using such a scalar wave function These properties are expressed in the three following postulates

POSTULATE 31ndash Any optical wave can be described mathematically by a real or complex function u(rt) at position r = (xyz) and time t called wave function This function is a solution of the wave equation

22

2 21 0uuc t

partnabla minus =part

[31]

where 2 2 2

22 2 2x y z

part part partnabla = + +part part part

is the Laplacian operator expressed in the

Cartesian coordinate system

Any function respecting equation [31] represents an optical wave As the wave equation is linear the superposition principle can be applied If u1(rt) and u2(rt) represent two optical waves then u(rt) = αu1(rt) + βu2(rt) is also an optical wave α and β being two real or complex constants


POSTULATE 32ndash The velocity v of an optical wave in a medium of refractive index n is given by

cvn

= [32]

If the medium is homogenous the index n is constant and light propagates in a straight line If the medium is non-homogeneous the index of the medium depends on space variables r and light follows a curved trajectory (mirage phenomenon)

POSTULATE 33ndash The optical intensity I(rt) defined as the optical power per unit surface area (Wm2) is proportional to the average of the square of the wave function

(r) 2 (r ) (r )I u t u tlowastprop times [33]

The operator lt gt represents the average over a time interval that must be longer than the duration of a wave train of the optical wave At 600 nm wavelength this time interval is about 2 fs (2 times 10minus15 s) In wave optics theory the physical meaning of the scalar wave function is not established whereas its square is linked to a measurable physical quantity which is the luminous intensity

Luminous intensity can also be expressed in the following manner

( ) ( ) ( )I r t u r t u r tlowast= times [34]

Using these three postulates and the qualitative description of the wavelike properties of light developed by Huygens Fresnel creates a model for the propagation of light based on wave surfaces Fresnel considers that when a point source emits a wave it is spherical Each point of one of the wave surfaces behaves in turn as a secondary source sending spherical waves in all directions The secondary waves interact so that the envelope of all the secondary wave surfaces constitutes a new wave surface This approach makes it possible to understand the physical phenomena of light interferences


(Youngrsquos experiments) and diffraction (Grimaldirsquos experiments) Fresnel assumes further that light is a transverse wave relative to its propagation direction (as suggested by Ampere in 1816) He finally shows that when the incidence angle is different from zero the reflection coefficients at the interface of two media have different expressions for an S wave (when the vibration is perpendicular to the incidence plane) and for a P wave (when the luminous vibration is within the incidence plane)

322 Luminous power and energy

The luminous power P(t) (in watts) that flows through a surface area A which is normal to the propagation direction is given by

SurfaceA

( ) ( )P t I r t dA= [35]

An obliquity term is introduced when the incident light makes an angle with the surface The luminous energy (J) collected during time τ corresponds to the integral of the luminous power P(t) over the time interval τ

323 The monochromatic wave

The wave propagation equation [FEY 65 JAC 98] is usually valid for describing a monochromatic wave In the case of propagation along the z axis there are two solutions that correspond to propagation along either the positive z direction (u(z ndashvt)) or the negative z direction ( ( )u z vt+ ) These are expressed by

( v ) cos( ( ))u z t a k z ctminus equiv minus or ( v ) cos( ( ))u z t a k z ct+ equiv + [36]

where v is the wave propagation velocity in a medium of index n k = 2πnλ is the wave vector λ is the wavelength and c is the wave light speed in the vacuum


The temporal Fourier transform of the propagation equation is used to obtain the Helmholtz equation in which the spatial and temporal dependencies are separated

2

2 0E EcωΔ + =

[37]

A second spatial Fourier transform of the wave propagation makes it possible to link the wave vector k and the wave pulsation ω

22

2( ) 0k Ecωminus + =

[38]

This leads to the dispersion equation

22

2 0kcωminus + =

[39]

33 The electromagnetic wave according to Maxwellrsquos theory

331 The Maxwell equations

Maxwell has shown that in vacuum when charges and currents are present the electric and magnetic phenomena are described by four equations [MAX 54 BRU 65 FEY 65 MIZ 72 JAC 98 HEC 05]

0

E divE ρε

nabla = =

[310]

BE rot Et

rarr partnabla and = = minuspart

[311]

0B divBnabla = =

[312]

0 0EB rotB jt

μ ε partnabla and = = +part

[313]


These equations unify the electric and magnetic phenomena and describe the local properties of the electrical field E and magnetic field B in terms of their sources ρ (the volume charge density) and j (the current vector density) and where μ0 is the magnetic permeability of vacuum and ε0 is the electric permittivity of vacuum In the MKS unit system these fields and sources are defined by E (in V mminus1) B in (T mminus1) ρ (in C m ndash3) and j in (A mminus3)

When the Maxwell equations are used to describe the wave properties of light the physical nature (electric or magnetic vector fields) of the mathematical function follows automatically Likewise the S and P waves (two independent components of the electrical field which are perpendicular to the wave propagation direction) introduced by Fresnel to calculate the reflection and transmission coefficients of light at the interface of two different optical media are straightforward The formula

rot(rot ) grad(div )F F FΔrarr rarr rarr

= minus

[314]

applied to the Maxwell equations leads to a wave propagation equation for the E and B fields

2 2

0 0 2 2 21 0F FF F

t c tμ ε part partΔ minus = Δ minus =

part part

[315]

In this equation F corresponds either to the electric field E or to the magnetic field B These equations demonstrate that all radiations displayed in Table 31 travel at the same speed c in the vacuum This speed is function of ε0 and μ0 With ε0 = 886 times 10minus12 F mminus1 and μ0 = 4π 10minus7 H mminus1 c is equal to 3 times 108 m sminus1 (299792458 times 108 m sminus1)

The Maxwell equations also show that light is a transverse electromagnetic wave which can be split in two linearly independent components corresponding to light polarization Light propagates like two mutually coupled vector fields E and B

Table 31 shows the domains and the uses of the electromagnetic waves


Domain Uses Associated frequency range

Cosmic rays Physics astronomy 1014 GHz and above

Gamma rays Cancerotherapy 1010ndash1013 GHz

X-rays X-ray examinations 108ndash109 GHz

Ultraviolet radiation Sterilization 106ndash108 GHz

Visible light Human vision 5105ndash106 GHz

Infrared radiation Medical 104ndash5 times 105 GHz

Tera Hertz Photography security scanners 5 times 102ndash104 GHz

Microwaves (SHFEHFUHF)

Radar microwaves satellite communication 3ndash300 GHz

Radiowaves (UHF) UHF television 470ndash806 MHz

VHF VHF television FM waves 54ndash216 MHz

HF Short wave radio 3ndash25 MHz

MF AM waves 535ndash1605 KHz

Table 31 Electromagnetic waves and their associated uses and frequency range

In a material medium the electric and magnetic vacuum parameters (ε0 and micro0) are replaced by the corresponding parameters of the medium (ε and micro) These values are defined with respect to the vacuum such as

εr = εε0 and μr = μμ0 [316]

where εr and μr are the relative permittivity and the permeability of the medium respectively The relevant Maxwell equations are then expressed as

divE ρε

=

and ErotB jt

μ ε part= +part

[317]


and the wave equation cast as

2 2

2 2 2 0cr rF FF F

t tμ εμε part partΔ minus = Δ minus =

part part

[318]

Light speed in a material depends on the refractive index of the medium through the square root of the product of εr by μr This implies that this product must be positive and materials with simultaneous negative εr and μr are physically compatible with this definition (meta-materials and photonic crystals [VES 68 PEN 99]) In this respect a new field of research connected to emerging technologies for security issues imaging in medical field imaging in artwork wireless sensors and communications in the terahertz (THz) domain is now active [DAR 02 DAV 02 FER 02] THz waves (between 300 GHz and 3 GHz or 1 mmndash100 microm) formerly known as the submillimeter waves between microwaves and infrared regions are non-ionizing and can penetrate non-conducting materials

Electromagnetism is the first gauge theory that is recognized in physics It is based on the principle of relativity From the properties of the B and E fields in space

0 divB A B rotA= lArr exist =

[319]

and

0 rotE V E gradV= lArr exist = minus

[320]

The existence of a scalar potential V and vector potential A is based on equations [319] and [320] respectively The electric and magnetic fields can be expressed by

B rotA=

[321]

and

AE gradVt

part= minus minuspart

[322]


The E and B fields are unchanged if V and A are replaced by

V Vt

ψpart= minuspart

[323]

and by

A A divψ= +

[324]

It can be shown that the scalar and vector potentials verify the following equations

0

AV divt

ρε

partΔ = minus minuspart

[325]

and 2

0 0 02 21( )A VA j div divA

t c tΔ μ ε μpart partminus = minus minus +

part part

[326]

332 The wave equation according to the Coulombrsquos gauge

The Coulombrsquos gauge implies that

0div A =

[327]

In that case equations [325] and [326] lead to

0

V ρε

Δ = minus [328]

and

2

0 0 02 21A VA j div

t c tμ ε μpart partΔ minus = minus minus

part part

[329]

Equation [328] is the electrostatics Poissonrsquos equation from which the scalar potential V is determined Equation [329] gives in this


framework the evolution of the potential vector A for given initial conditions

333 The wave equation according to the Lorenzrsquos gauge

The Lorenzrsquos (LV Lorenz Danish physicist) or Lorentzrsquos (HA Lorentz Dutch physicist) gauge implies

21div 0VAc t

part+ =part

[330]

Equations [325] and [326] lead to

2

2 20

1 VVc t

ρε

partΔ minus = minuspart

[331]

and

2

0 0 02AA j

tμ ε μpartΔ minus = minus

part

[332]

In this framework the scalar potential V and the potential vector A verify a wave equation in the presence of charges and currents as sources The electromagnetic fields can be determined from V and A

The Lorenz or Lorentz condition is an invariant of Lorentzrsquos transformations It allows transformation from one frame to another in uniform relative translational movement according to the special relativity principle (non-existence of absolute time)

34 The quantum theory of light

341 The annihilation and creation operators of the harmonic oscillator

In the QM theory the electromagnetic field is quantized as a sum of independent harmonic oscillators [MES 64 COH 73 COH 87


LAN 89 FEY 98 MEI 15] In the case of a harmonic oscillator (a mass m at the end of a spring of stiffness k submitted to an elongation x) the Lagrangian L which is the difference of the kinetic energy T and the potential energy V of the system is expressed in the non-relativistic approximation by Landau and Lifchitz [LAN 66]

22 2 2 21 1 1

2 2 2 2pL T V mx kx m xm

ω= minus = minus = minus [333]

where ω is the pulsation of the oscillator and p its momentum

In classical mechanics position x and momentum p are conjugated variables

Lp mxx

part= =part

[334]

In QM these variables are described by Hermitian operators that satisfy the commutation rule [ ]x p i= and act over a space of quantum states that are functions (or kets ψ ) of an Hilbert space The kets ψ and bra ψ formalism which is a representation-free notation was introduced by Dirac to simplify the notation in QM The Hilbert space states are determined by the Schroumldinger eigenvalue stationary equation

k k kH Eψ ψ= [335]

where H is the Hamiltonian operator of the physical system the sum of its kinetic energy operator T and potential energy operator V The eigen functions are mutually orthogonal They are usually normalized ( i k ikψ ψ δ= ) and define a complete set Each state vector Ψ of the physical system is expressed by a linear combination of kψ such

that 0

k kk

CΨ ψinfin

=

= As the coefficients ck are determined by

k kC ψ Ψ= the state vector can be written as 0

k kk

ψ ψinfin

=

Ψ = Ψ


where k k kP ψ ψ= is the projection operator This operator fulfills the relation 2

k kP P= In QM it is admitted that a given system is described by states defined by Ψ a vector of an Hilbert space In theory if all the possible Ψ vectors are determined and known then the probabilities of all possible results of a given measurement on an observable are also known Such states are termed pure states and the probability attached to each measurement given by the principles of QM The time evolution of the system is determined by

H it

ψ ψpart=part If H is independent of time then

ˆ( ) (0)

iHtt eψ ψminus= It can be written as ˆ( ) ( ) (0)t U tψ ψ= where

ˆ ( )U t is the time evolution operator of the physical system such that ˆˆ ( )

iHtU t eminus=

When the state of the system is not completely known the description of the quantum system requires the introduction of a density operator In QM there are two postulates connected to the result of a measurement of an observable QM postulate 31 ldquoan observable is represented by a Hermitian operator A and the result of a measurement is one of the eigen values of this operator with a given probabilityrdquo QM postulate 32 ldquoif a quantum system is in state Ψ the average value of the observable is given by AΨ Ψ )rdquo If pk is the probability that state kψ is known the average which takes into account the quantum and statistical aspects is expressed by

ˆ ˆ ˆˆ( )k k kk

A p A Tr Aψ ψ ρ= = where Tr represents the Trace (sum

of the diagonal elements of the matrix ˆˆ Aρ ˆ k k kk

pρ ψ ψ= is the

density matrix of the system where k k kPψ ψ = is the projection operator) The average of an operator associated with an observable consists of a quantum average and classical statistical average These average values are not separable in the density matrix In this case the


evolution of the density matrix is given by the Von Neumann equation

ˆ ˆ ˆˆ ˆ ˆ î H H Ht

ρ ρ ρ ρpart = = minus part [336]

In the case of the harmonic oscillator H is expressed by

22 2 2 2 2ˆ 1 1 1ˆ ˆ( )

2 2 2 2pH T V m x i m xm m x

ω ωpart= + == minus = minus minuspart [337]

where the symbol ^ over p and x means that p and x are operators

The harmonic oscillator is more easily described by introducing the creation a+ and annihilation a operators (second quantization)

ˆ ˆ2x ipa ω

ω+=

and ˆ ˆ2x ipa ω

ω+ minus=

[338]

These operators are Hermitian conjugates and obey the commutation rule

1a a+ = [339]

The expressions of operators x and p are then given by

ˆ ( )2

x a aω

+= + and ˆ ( )2

p i a aω += minus [340]

The Hamiltonian operator H can be written as

1( )2

H a aω += + [341]

The eigen value equation becomes

1 1( ) ( )2 2

H n a a n nω ω+= + = + [342]


where the kets n are eigen vectors of H with eigen values equal to 1( )2

n ω+ with n = 0 1 2 etc

Furthermore by applying operators to the eigenvectors the following relations are obtained

1 1a n n n+ = + + [343]

1 1a n n n= minus minus [344]

0 0a = [345]

( ) 0

nann

+

= [346]

In terms of particles two consecutive energy levels of a harmonic oscillator are separated by a quantum of energy ω State n can be described as a system of n bosons (phonon photon etc) having all the same characteristics specifically energy ω momentum k

and in

the case of photons polarization e The operator N a a+= is per construction the number of particles while the operators a+ and a create and annihilate a particle respectively The state vector 0

represents the vacuum and its associated energy is 2ω For bosons

particles can all be in the same quantum state with an arbitrary number n

342 The quantization of the electromagnetic field and the potential vector

Quantum electrodynamics (QED) theory [MES 64 FEY 85 COH 87 LAN 89 FEY 98] is applied to describe the electromagnetic


interactions between charged particles and an electromagnetic field Using the Coulomb gauge the electromagnetic field is expressed as a sum of independent oscillators The Coulomb gauge is useful to study the interaction of light with matter at low energies since it is not necessary to account for the creation of particlendashantiparticle pairs

Using equation [322] the electric field E can be split into a longitudinal part (parallel field) which is a contribution of the scalar potential and a transverse part (perpendicular field) which is a contribution of the vector potential as follows

AE gradV E Et perp

part= minus minus = +part

[347]

In the Coulombrsquos gauge the divergence of the transverse field is null and V which fulfills the Poisson equation is the contribution of the Coulombrsquos potential of the instantaneous charge distribution

In the quantum electromagnetic theory the amplitude of the vector potential A is quantized by considering that the radiation is confined in a cubic box of dimensions L and that the fields and their derivatives fulfill periodic boundary conditions As plane waves are solutions of the propagation equation the wave vectors are quantized The components of the wave vector along the Ox Oy and Oz axes are given by

2 2 2( )x y zk n n nL L Lπ π π=

[348]

where nx ny and nz are positive or null integers

The vector potential A is expanded as a superposition of monochromatic plane waves

0

( ) ( ( ) ( ) ( ) ( ))2 n n n nn

n

A r t a t u r a t u rα α α αα ε ωlowast lowast= +

[349]


with

0( ) ni tn na t a e ωα α

minus= and

3

1( ) nik rn nu r e e

Lα α=

[350]

In this equation the unα(r) form a basis of normalized orthogonal vectors enα are the two polarization vectors (α = 1 or 2) and kn is the wave vector In the Coulomb gauge enαkn = 0 Neglecting the spin of the particles the quantization of the electromagnetic field energy is obtained from the Lagrangian of a system of non-relativistic particles interacting with a radiation field and equation [349] giving the vector potential The Lagrangian is expressed as the sum of three terms comprising the Lagrangian of the system of N isolated particles the Lagrangian of the radiation field and the Lagrangian of the interaction between the field and the particles as follows

2 3 2 2 2 30

1( ) ( ) ( )

2 2

Ni

ii

mL x V j A d r E c B d rερ=

= + minus + + minus [351]

The first term of equation [351] corresponds to the kinetic energy of the isolated system (mi being the mass of the ith particle and ix its speed) The second term corresponds to the interaction within the systemrsquos volume between the volumic charge density ρ and the scalar potential V and between the current j and the vector potential A The third term is the Lagrangian of the radiation energy

In the Coulomb gauge replacing E and B by their expression relative to the scalar potential V and the potential vector A this Lagrangian is expressed by

2 3 2 2 2 301

1( ) ( ) (( ) ( ) )

2 2

Ni

i coul Ni

mL x U x x j Ad r A c rotA d rε=

= + + + minus [352]

where the second term of equation [351] is split into two parts The former corresponds to the Coulomb interaction and the latter corresponds to the current potential vector interaction


From the expression of A in the basis of orthogonal vectors unα(r) the Lagrangian of the radiation field is written as follows

2 2 2 30 (( ) ( ) )2

L A c rotA d rε= minus [353]

The temporal derivative of the first term of this integral leads to a term 2

na α which originates from 0( ) ni tn na t a e ωα α

minus= (the index 0 is

suppressed for clarity) Terms 2nω and 2

na α which come from the

equation 2 22 2 2 2 2 2( ) ( )n n n n nc rotA ik A c k a aα αω= and = minus = minus

appears in the second term of the integral The following equations are thus obtained

22 3 30 03

0

1( ) (2 ) ( )2 2 2 n

n n

A d r L aL α

α

ε εε ω

= [354]

and

22 2 3 3 20 03

0

1( ) ) (2 ) ( ) ( )2 2 2 n n

n n

c rotA d r L aL α

α

ε ε ωε ω

= minus [355]

Finally as a function of the independent discrete variables i i n nx x a aα α (i = 1 hellipN n isin V) and (α = 1 or 2) the Lagrangian

[352] is written as

21

12 22

1

( ) ( )2

( ) ( )( )2

Ni

i coul NiN

i i i n n ni n n

mL x U x x

q x A x t a aα αα

ωω

=

=

= + +

bull + minus

[356]

The last term of [356] represents the Lagrangian of the field Lfield

2 22field

( )( )2 n n n

n n

L a aα αα

ωω

= minus [357]


If usual operations transforming a Lagrangian into a Hamiltonian are applied to this system of particles interacting with the radiation field then the Hamiltonian of the free field is written by

field

1( )2n n n

nH a aα α

αω += + [358]

where the operators αα nn aa+

represent the operator number of bosons

and na α+ na α the creation and annihilation operator Hfield is a sum of

independent harmonic oscillators

Thus in QED light is composed of photons which are bosons created by the creation operators and destroyed by the annihilation operator The radiation energy is produced by a set of oscillators

The total Hamiltonian of the system of particles interacting with the radiation field is expressed by

21

1

1 ( ( )) ( )2

1( )2

N

i i i coul Ni i

n n nn

H p q A x t U x xm

a aα αα

ω

=

+

= minus +

+ +

[359]

Equation [358] shows that in QED the free field states originate from a space which is the tensor product of the spaces of independent oscillators The creation operator na α

+ leads to the creation of a photon

of mode nα energy nω polarization nαε and momentum nk The

operator na α destroys this photon and the operator n n nN a aα α α+= is the

observable of the number of photons of the mode nα In this representation the vacuum state has an infinite energy equal to

2n

n α

ω

This result is fundamentally different from the rule that applies in the classical approach of the electromagnetic field The vacuum state


is null when no charges are present In QM because of the Heisenberg uncertainty principle the electric and magnetic fields cannot be equal to zero at the same time The electromagnetic field of vacuum fluctuates and if its average value is zero its standard deviation is not leading to vacuum fluctuations These fluctuations are for example responsible for the ldquoLamb Shiftrdquo observed in atomic spectra

343 Field modes in the second quantization

Classical theory describes the electric field as two superposed complex conjugates [GLA 63 SUD 63 GLA 67 ARE 72 DAV 96]

[360]

where

[361]

and

[362]

Ck are the coefficients of the expansion of ( ) ( )E r t+ on the basis of the functions which are the solutions to the Helmholtz equation for the mode k and the angular frequency ωk

When the Ck coefficients are known it is possible to determine the classical field

( ) ( )E r t+ If the radiation field from classical sources

are statistically random the probability density P(Ck) of the set of coefficients Ck needs to be evaluated

The field ( ) ( )E r t+ has the property of a time-dependent stochastic process In diffraction and interference experiments the intensity of a field which is the superposition of fields at different positions in time and space is measured by quadratic detectors The measured intensity

( ) ( )( ) ( ) ( )E r t E r t E r t+ minus= +

( )( ) ( )( ) ( )E r t E r tlowastminus +=

( ) ( ) ( ) ki tk k

kE r t C u r e ωminus+ =


is thus expressed in terms of a field correlation function which in the classical approach is expressed by

[363]

This average is evaluated over the random distribution of the Ck coefficients

In QM this field superposition has a fundamental importance since these fields are linked with the creation and annihilation operators presented in section 341 and which act in the Fock space A field amplitude is associated with each mode k When the modes are not coupled the amplitude of each mode k is a solution to the equation of an isolated harmonic oscillator

An arbitrary pure state is expressed as a superposition of Fockrsquos space states for each mode k so that

0k

ki C k

infin

=

= [364]

The Ck terms are the expansion coefficients in the Fockrsquos space states basis

In the classical approach the electromagnetic field is completely defined by its amplitude and phase In the quantum approach the average value of this amplitude is zero and the phase is not defined when Fock states are used to describe the field Consequently Fock states are not the most appropriate representation of the electromagnetic field Introducing the concept of coherent state α of an electromagnetic field makes it possible to define a representation of wavelike states of the electromagnetic oscillator [GLA 63 SUD 63 GLA 67 ARE 72 DAV 96] α is the eigenstate of the annihilation operator a of the photon and α is its eigenvalue Since a is a non-Hermitian operator the phase α is a complex number and it corresponds to the complex wave amplitude in classical optics

( ) ( )( ) ( ) ( )moy

G r t r t E r t E r tminus +prime prime prime prime=


Using the recurrence relation 1k kkC Cα minus= α can be written in the k kets basis of Fockrsquos space as

[365]

Equation [365] connects the wavelike nature to the particle-like nature of light and shows that in a coherent state the number of photons is indefinite while the phase is well defined (respecting Heisenbergrsquos uncertainty principle) These coherent states represent quasi-classical states They can be characterized by a phase θ and an average amplitude r Furthermore they verify minimal quantum fluctuations represented by a circle of constant area in a two-dimensional phase space The fluctuations are symmetrical relative to the quadrature Hermitian operators (linear combination of the creation and annihilation operators) which obey the commutation rule [ ]ˆx p i= The coherent states can also be obtained from application of the unitary displacement operator dagger exp( )D a aα α α= minus to the vacuum ground state 0 They are normalized but are not orthogonal

In the so-called super-complete basis of Fockrsquos space the k states and coherent states α verify the closure relations and the eigen value equations

0

1 1k

d k kα α α απ

infin

=

= = [366]

a α α α= [367]

The probability to have k photons in a coherent state α is 2

( )P k k α= It can be shown that this distribution of photons is a Poisson distribution

2 2

( )

k

P k ek

α αminus= [368]

2 2

0

ki

ke k re

kα θαα

infinminus

== equiv


where the term 2α corresponds to the average of the number of photons k associated with the operator dagger( )N k a a= This average is

given by 2( )k N kα α α= = and the variance by 2( ) ( )k N kσ α α= minus

2 2( )N kα α α=

The predictions of QM are probabilistic Two types of uncertainties must be considered in the case of a quantum system If the systemrsquos quantum state is perfectly known its probability is calculated by applying the rules of QM If the knowledge of the quantum states is incomplete the uncertainty is introduced via the density matrix As the pure states are usually not accessible then the coefficients Ck of equation [364] are known with an uncertainty This specific case is taken into account by introducing density operators for each mode expressed by

meanρ α α= [369]

Here the photon detection is proportional to the correlation function

( ) ( ) ( ) ( )mean( ) ( ) ( ) ( ) ( G r t r t E r t E r t Tr E r t E r tα α ρminus + minus + = = [370]

The field operators E+ and Eminus do not commute The order in which they are applied in equation [370] is important when dealing with the case of absorption The usual order is for the annihilation operator to precede the creation operator from right to left (Glauber normal ordering) For the emission case this order should be reversed

4

The Polarization States of Light

The polarization of light was discovered in transmission by C Huyghens in 1690 and in reflection by E Malus in 1808 In classical optics theory polarization is described by the trajectory of the tip of the electric field vector associated with light In the plane perpendicular to the wave vector giving the propagation direction of a plane wave two independent directions of polarization can be defined In quantum optics theory polarization is described by the projection of the photon spin over an axis (S = +1 or minus1) This particle-like nature of light corresponds to Newtonrsquos hypothesis Various mathematical models describing light polarization have been developed Jonesrsquo approach efficiently describes states that are completely polarized Stokes and Mullerrsquos approach describes any polarizing state and Poincareacutersquos approach [POI 92] represents polarizing states by means of a sphere As these mathematical models are based on matrix algebra numerical calculations can easily be performed to determine how a material in which a light wave propagates modifies the state of polarization of light As an inverse problem it is also possible to study a material and its properties from the modification of the state of polarization of light

41 Introduction

Using the Coulomb gauge the QED theory [MIZ 72 COH 87 LAN 89 FEY 98] shows that light can be described by photons characterized in each mode indexed by nα two polarization states

nαε in the plane perpendicular to the wave vector their energy equal to nω and their momentum equal to nk

In wave optics theory [BRU 65 LAN 66 MIZ 72 BOR 99 JAC 98 HEC 05] Fresnel showed that light is characterized by two


First Edition


transverse vibrations one of type p which is parallel to the plane of incidence and one of type s which is perpendicular to the incidence plane The Maxwell approach specifies the physical nature of these vibrations They are due to an electric field E possessing a movement periodic in time and space and characterized by its angular frequency ω and its wave vector k connected by the dispersion equation kc = ω They vibrate perpendicularly to the propagation direction defined by k The polarization of type p is a transverse magnetic wave TM and the polarization of type s is a transverse electric wave TE Using the Fresnel relations and taking into account the nature of the luminous vibrations as components of an electric field the amplitude of the electric fields of the reflected and transmitted waves relative to the incident wave can be calculated This results in the amplitude reflection coefficients which are particularly useful in ellipsometry and whose expressions are

Type p wave 1 0

1 0

cos coscos cos

rp i rp

ri i r

E n nr

E n nθ θθ θ

minus= =

+

[41]

Type s wave 0 1

0 1

cos coscos cos

rs i rs

ri i r

E n nrE n n

θ θθ θ

minus= =+

[42]

where 0n and 1n are the complex optical constants of the medium and the material respectively and θi and θr are the incidence and refraction angles In most cases a material is characterized by its complex refractive index n n ik= minus The real part n is linked to the light dispersion The imaginary part k is linked to the light absorption

In wave optics theory the light polarization states can be described by various theories Stokes [STO 52] used a four-component vector to represent polarized light and Poincare [POI 92] gave its geometrical representation in the form of a sphere described for this vector The formulation in the form of vectors with two components of Jones [JON 41] is the one most commonly used Materials are represented by 2 times 2 matrices in the representation of Jones and by 4 times 4 Muller matrices [MUL 48] in the representation of Stokes The sphere of Bloch [BLO 46 FEY 57 SIE 86] another geometrical representation is also used in quantum optics theory to represent the

The Polarization States of Light 73

spin of the photon All these representations are introduced in the following sections These models are based on matrix algebra that is convenient for numerical simulations (MATLAB MAPLE etc) of the interaction of matter and polarized light

The formulation of Stokes was used a century later by Chandrasekhar [CHA 50 CHA 56] to interpret the polarization of light through Rayleigh scattering of sunlight by particles of Earthrsquos atmosphere The book by Azzam and Bashara [AZZ 77] on ellipsometry and polarized light has long been a leading reference for studies in polarimetry and ellipsometry It gives a description of the different formalisms developed on polarized light and their applications in ellipsometry The study of the fluorescence emitted by materials developed for use as laser sources also requires the use of these formalisms as well as experiments using polarized light

There are different books dealing with polarized light either explicitly or partly in chapters For a deeper exploration of these formalisms on polarized light see [BRU 65 BOR 99 GOL 03 CET 05 HUA 97 LAN 66 YAR 84]

42 The polarization of light by the matrix method

Usually a light wave that propagates along a direction z can be described by two components in the plane perpendicular to the propagation direction

Vibration over Ox 0( ) ( )cos( )x x xE z t E z t kz tω φ= minus + [43]

Vibration over Oy 0( ) ( ) cos( )y y yE z t E z t kz tω φ= minus + [44]

Any polarization state is considered as a linear combination of these two vibrations In the complex notation the wave is described by

( ) ( )0 0ˆ ˆ( ) yx i kz ti kz t

x yE z t E e x E e yω φω φ minus +minus += +

[45]


Separating the harmonic part of the amplitude of the wave a description of the amplitude as a complex number is obtained

( )0 0ˆ ˆ( ) ( )yx i i kz ti

x yE z t E e x E e y eφ ωφ minus= +

[46]

This complex amplitude contains all the information of the wave A wave is characterized by its amplitude its wavelength λ or wave vector k = 2πλ and its polarization state

The polarization is represented by the curve described by the tip of the electric field vector (Figure 41) It can be shown that the equations [43] and [44] lead to the following expression at time t

222

0 0 0 0

2cos( ) sin ( )y yx x

x y x y

E EE EE E E E

φ φ

+ minus = [47]

where the phase difference φ = φy minus φx and the amplitudes E0x and E0y are both positive This ellipse can be traced in one direction of rotation or the other according to the value of φ This corresponds to either a right-handed (clockwise) or a left-handed rotation (anticlockwise) There are two conventions for defining this ellipse If the electromagnetic wave comes toward the observer the polarization is defined either as clockwise (right) if the tip of the electric field vector describes a clockwise ellipse or anticlockwise (left) in the other case In the case where the wave propagates away from the observer the right and left turning polarizations defined in the above sentence are inverted

As an ellipse is characterized by four parameters such as half the length of its minor axis a half the length of its major axis b the angle Ψ between the major axis and Ox axis and its direction of rotation (Figure 41) four corresponding parameters are required to characterize polarized light The parameters of an elliptic polarization are the angle α defined by tan α = E0yE0x (diagonal of the rectangle containing the ellipse in Figure 41) and the phase difference φ


Figure 41 Relations between the parameters in the frame Oxy in the plane normal to the wave vector k and the

ellipse axes Oab of the components of the electric field

Figure 42 Linear circular and elliptic polarizing states [WIK 38] For a color version of this figure see wwwistecoukdahoonanometerzip

When the x and y components of the electric field are in phase the polarization is linear When the vibrations of the x and y components differ in phase by 90deg the polarization is elliptic If moreover the

βα


amplitudes of these components are equal the polarization is circular Figure 42 illustrates these three situations

421 The Jones representation of polarization

When light is fully polarized the Jones approach is used to describe polarization In this approach the relative amplitudes (E0x E0y) and the relative phases (φ = φy minus φx) of the components of the complex wave amplitude determine the state of polarization

0 0 0ˆ ˆ( )yx iix yE E e x E e yφφ= + [48]

This complex amplitude is expressed by a 2 times 1 column matrix or Jones vector such that

00 0 0

0

ˆ ˆx

yx

y

ixii

x y iy

E eE e x E e y E

E e

φφφ

φ

+ = =

[49]

Two basis vectors J1 and J2 are used to define a complex two-dimensional vector space A wave polarization state is expressed by a linear combination of these two basis vectors Equation 48 can thus be expressed by

00 0 1 0 2

0

x

yx

y

ix ii

x yiy

E eE E e J E e J

E e

φφφ

φ

= = +

[410]

where the vectors J1 and J2 are defined by

1

10

J =

and 2

01

J =

[411]

For example a polarized wave along the Ox axis is expressed by

00 0 0 1

100

xx x

ii ix

x xE e

E E e E e Jφ

φ φ = = =

[412]


To describe a polarization state the normalized Jones vector J is used as follows

0

2 200 0

1 x

y

ixx

iy yx y

E eJJ

J E eE E

φ

φ

= = +

[413]

The norm of the Jones vector is then equal to 1 and is expressed by

1x x y yJ J J J J Jlowast lowast lowastbull = + =

and 1 2 2 1 0J J J Jlowast lowastbull = bull =

[414]

The linear polarization Jα that subtends an angle α with the Ox axis is obtained by multiplying J1 with the rotation matrix of angle α relative to the propagation direction defined by the wave vector k as follows

cos sin 1 cossin cos 0 sin

Jαα α αα α α

minus = =

[415]

The vector basis defined by the JD and JG vectors corresponding to the clockwise (right) and anticlockwise (left) circular polarizations can also be used It is defined as

112DJ

i

=

and

112GJ

i

= minus

[416]

J1 and J2 can be defined in the vector basis as formed by JD and JG and vice versa The elliptic polarization drawn by the ellipsersquos own axes is expressed by

cos( )

siniJe φ

αφ α

α

=

[417]

The vectors basis J1 and J2 can be expressed in the vector basis defined by the vectors JD and JG and vice versa


In the case of an elliptic polarization the change in axes from Ox and Oy to Oa and Ob is obtained by the use of the parameters defining the ellipse encompassed by the tip of the electric field An elegant method of linking these parameters is proposed in Landaursquos field theory [LAN 66] The complex electric field is expressed in the plane z = 0 (equation [46]) and in the Ox and Oy coordinate system by

0 0ˆ ˆ( ) ( )yx ii i tx yE r t E e x E e y eφφ ωminus= +

[418]

In this coordinate system the tip of the electric vector defines an ellipse (equation [47]) However the Ox and Oy axes are not the principal axes of this ellipse Rotating the coordinate axes by an angle θ the principal axes Oa and Ob are obtained in which the expression of the electric field is written as

2 ( )ˆˆ( ) ( )i i tE r t aa be b eπ ω θminus minus= +

[419]

where a and b are real numbers In this new coordinate system it can be established that

2 22 2cos ( ) sin ( ) 1a bE E

t ta b

ω θ ω θ + = minus + minus =

[420]

The components of the coordinates in each system obey the relation

( ) ( )1 12 22 2 2 2

0 0 0 0 0 0 0 02 sin 2 sin

2x y x y x y x yE E E E E E E E

a bφ φ+ + plusmn + minus

= [421]

The angle Ψ defined by the Oa and Ox axes fulfills the relation

0 02 20 0

2 costan(2 ) x y

x y

E E

E E

ϕ Ψ = minus

[422]


Finally using the temporal average of the Poynting vector R which makes it possible to calculate the energy carried by the wave in the two coordinate systems it can be shown that

( )2 2 22 2

0 0

0 0 0 0

1 ˆ ˆ ˆ2 2 2 2

x yE E Ea bR e E B z z zmicro micro c micro c micro c

lowast ++= real and = = =

[423]

This implies that the ellipse is contained in the rectangle with its sides defined by 2a 2b or 2Eox 2Eoy The two coordinate systems of Figure 41 (angular parameters (α ϕ) and the ellipse shape (Ψ β)) verify the following relations

0 02 20 0

0 02 20 0

2sin 2 sin 2 sin sin

2tan 2 tan 2 cos cos

tan 2 sin 2 tan

x y

x y

x y

x y

E EE E

E EE E

β α ϕ ϕ

α ϕ ϕ

β ϕ

= =+

Ψ = =minus

= Ψ

[424]

The angle Ψ (polarization angle minusπ2 le Ψ lt π2) defines the axes and the angle β (tan β = plusmnba and minusπ4 le β le π4) in the principal axes defines the ellipse shape such that the linear polarizations correspond to β = 0 values and the circular polarizations correspond to β = plusmnπ4 The sign depends on the choice of the convention as discussed above

The other two parameters 2 2 2 20 0x ya b E E+ = + which are

proportional to the wave intensity (or to the amplitude of the electric vector field) and the phase shift between the initial vector position E(t) and the major axis of the ellipse (the projection of E(t) on a circle of radius equal to half the major axis) are not required to describe the polarization state

Thus the Jones vectors 1 21 1and25 5i i minus

can be viewed as

being the left and right elliptic polarizations respectively


The modules of components 15xE = and 2

5yE i= or 25xE =

and 15yE iminus= are different and the phase difference φ is π2 or minusπ2

respectively which imply elliptic polarization states

The direction of rotation of vector E is determined by the sign of the component of the vector product 1 2 1 2 ˆb a b b b b zand =

with

1 2 0 0 sin( )x yb b E E ϕ= (equation [422]) Consequently the sign is determined by sin(φ) The polarization is left if the sign of sin(φ) is gt 0 and right if the sign of sin(φ) is lt 0 As sin (π2) = 1 gt 0 and sin (minusπ2) = minus1 lt 0 the polarization is thus left elliptic for the former case and right elliptic for the latter

In the case of the following normalized Jones vectors

10

01

1112

11

2 i

11

2 i minus

[425]

The polarization states are linear polarization over Ox axis linear polarization over Oy axis linear polarization at an angle of 45deg left circular polarization (sin(φ) = sin(π2) gt 0) and right circular polarization (sin(φ) = sin(minusπ2) lt 0)

In the Jones approach optical devices are represented by 2 times 2 matrices In the case of a birefringent plate through which polarized light propagates the neutral lines of the plate are defined by the directions where the linear polarization is constant when light waves propagate through the plate at the normal incident angle Light propagates through the plate at the phase velocity v1 = cn1 when the electric field propogates in the direction D1 and at the phase velocity v2 = cn2 when the electric field propagates along the direction D2 which is normal to D1 If v1 lt v2 (n1 gt n2) the axes corresponding to D1 and D2 are called slow and fast axes respectively The plate causes a phase difference φ between the field components


which are parallel to D1 and D2 For a blade thickness e this phase difference is expressed by

1 2 1 22 2 2 ( )n e n e e n nπ π πφλ λ λ

= minus = minus [426]

When the phase difference φ is equal to π (mod 2π) the platersquos

thickness is called a half wave (2

e λ= or integer multiple) When the

phase difference φ is equal to π2 (mod 2π) the plate is called a

quarter wave plate (4

e λ= or integer multiple)

Other devices that are widely used in experimental set-ups based on polarized light are polarizers delay lines phase retarders and rotators All these devices can be represented by a Jones matrix that makes it possible to calculate the transformation of the polarization state after propagation in the plate Usually the principal axes of these devices do not match the principal axes of the polarization state To take that into account rotation matrices are used (equation [416])

For a polarizer along Ox axis polarizer along Oy axis quarter wave plate half wave plate or dephasor the Jones matrices are respectively expressed by

4 20 0 1 0 1 0 1 0 00 1 0 0 0 0 1 0

ii i

i

ee e

i eπ π

φ

φplusmn plusmn

minus minus [427]

422 The Stokes and Muller representation of polarization

Usually natural light is not polarized This means that there is no favored direction for the electric field E Its direction fluctuates rapidly relative to the response time of the detector used for the light phenomenon under study In that case the light phenomenon can be described by a scalar field A radiation that is not polarized is described by a vector that fluctuates stochastically over a time scale


which is large relative to the radiation period but small relative to the time interval during which the fluctuation is measured

Light which is partially polarized can be considered as the superposition of completely polarized light and non-polarized light It is difficult to model this particular polarization state using an electrical field In this case the model developed by Stokes [STO 52] can be used This model is based on light intensities used to describe the polarization states by the introduction of four parameters S0 S1 S2 and S3 or I Q U and V defined as follows

0

1

2

3 ( )

x x y y

x x y y

x y y x

x y y x

S E E E EIS E E E EQS E E E EUS i E E E EV

lowast lowast

lowast lowast

lowast lowast

lowast lowast

+ minus = = + minus

[428]

In the following only the notation S0 S1 S2 and S3 will be used and the fluctuations of the electric field vector will be taken into account in the following definition

0

1

2

3 ( )

x x y y

x x y y

x y y x

x y y x

E E E ESE E E ES

S E E E ES i E E E E

lowast lowast

lowast lowast

lowast lowast

lowast lowast

+ minus = + minus

[429]

where lt gt is the temporal average of the parameters between brackets S0 represents the total intensity of the wave and describes the polarized light and non-polarized light contributions S1 and S2 represent the linear part of the polarization S3 represents the circular part of the polarization

The Stokes description of polarization is more appropriate for experimental studies than that of Jones Each of the Stokes parameters corresponds to the sum or to the difference of intensities of different polarization states Stokes representation is widely used in astronomy


for space observations To determine the Stokes vector six intensity measurements are required Ii (i = 0deg 90deg 45deg minus45deg D G) which can be achieved by using linear polarizers at different angles (0deg 90deg 45deg minus45deg) and two left and right circular polarizers

0 0 90

1 0 90

2 45 45

3 D G

S I IS I IS I IS I I

minus

+ minus = minus

minus

[430]

The Stokes parameters are related to the angles Ψ and β which define the ellipse orientation and the polarization ellipticity by the following formula

0

1

2

3

cos 2 cos 2sin 2 cos

sin 2

S IS IS IS I

ββ

β

Ψ = Ψ

[431]

Although the Jones representation can be linked to a two-dimensional complex vector space the Stokes representation cannot be easily linked to a vector space From a mathematical standpoint it is possible to show that there is a one to one correspondence between the Stokes parameters and the space of non-negative Hermitian operators (defining a closed convex type cone in the four-dimensional real space) acting on a C2 type Hilbert space S0 is the trace of the operator and the linear combinations of the other parameters are the elements of the operator matrix The eigenvalues and the eigenvectors of the operator can be calculated from the polarization ellipse parameters (S0 Ψ β and

the polarization coefficient 2 2 2

1 2 320

S S SVS

+ += )

When S0 is equal to 1 (operators of trace equal to 1) there is a one to one correspondence between the Stokes parameters and three-dimensional unit closed ball of the mixed states (or density operators) of the C2 quantum space the boundary of which is the Bloch sphere


The Jones vectors correspond to C2 space and form the pure states (non-normalized) of the system There is a simple relation between the Stokes and the Jones vector which is expressed as

k kS J Jσ+lowast= [432]

where J +lowast is the transpose matrix of the Jones vector J (line matrix 2 times 1) and kσ represents a Pauli matrix (2 times 2 null trace Hermitian matrices)

0 1 2 3

1 0 1 0 0 1 0

0 1 0 1 1 0 0i

iσ σ σ σ

minus = = = = minus

[433]

The Stokes parameters appear as the average values of the kσ matrices (quantum approach)

From these definitions completely polarized light is characterized by

2 2 2 20 1 2 3S S S S= + + [434]

Partially polarized light is characterized by

2 2 2 21 2 3 00 S S S Slt + + lt [435]

The optical systems acting on the polarization of light are modeled by Stokes-Mueller matrices which make it possible to determine how the polarization states change

0 00 01 02 03 0

1 10 11 12 13 1

2 20 21 22 23 2

3 30 31 32 33 3

s e

s es e

s e

s e

S M M M M SS M M M M S

S MS orS M M M M SS M M M M S

= =

[436]

In the case of an interaction of a wave with matter that does not depend on the light intensity (linear hypothesis nonlinear effects are excluded) the incident input Stokes vector Se and the transmitted


output Stokes vector Ss are connected by a matrix relation such that Ss = MSe M is a 4 times 4 real matrix analogue to the 2 times 2 Jones matrices called the Mueller matrix of the physical system The elements of this Mueller matrix are connected to the polarization effects of the optical device These matrix elements can be understood in the following way

ndash the first line and the first column correspond to the polarizing properties

ndash the diagonal terms (M11 M22 and M33) describe the depolarizing effect

ndash the non-diagonal terms M12 M13 M21 M23 M31 and M32 are used to study the medium birefringence

For instance the following matrices represent a linear horizontal polarizer a linear polarizer at +45deg a quarter wave plate with horizontal fast axis and a homogeneous left-handed circular polarizer

4

1 1 1 1 1 10 0 0 0 0 01 0 0 02 2 2 2 2 21 1 0 0 0 0 0 1 0 0 0 0 0 00 0 2 2 1 1 0 0 0 1 0 0 0 00 00 0 0 0 2 2 0 0 1 0 1 10 00 0 0 0 0 0 0 0 2 2

ie

π

minus minus minus

If the wave propagates through a set of optical devices M is the product of the corresponding Mueller matrices the order of the matrices being the inverse of the order of the optical devices

1

n

ii

M M=

= prod [437]

If non-depolarizing devices are considered then the following relation between the Mueller and Jones matrices can be established

1( )M P J J Plowast minus= otimes [438]


where the symbol otimes indicates the dyadic product that is used to transform a 2 times 2 matrix into a 4 times 4 matrix and where P is the matrix defined by

1 0 0 11 0 0 10 1 1 00 0

P

i i

minus =

minus

[439]

43 Other methods to represent polarization

There are other representations of the polarization of light that do not use a matrix representation but allow a geometric representation of the state of polarization (Poincareacute sphere or Bloch sphere) or fall within the quantum description of the light in the form of photon and operators associated with the polarization state

431 The Poincareacute description of polarization

Poincareacute [POI 92] proposed a geometrical representation of the light polarization state based on a sphere of radius S0 and a vector S of Cartesian coordinates S1 S2 and S3 where the Si (i = 0 1 2 3) are the Stokes parameters (Figure 44)

1 0

2 0

3 0

cos 2 cos2sin 2 cos

sin 2

S SS S S

S S

ββ

β

Ψ = = Ψ

[440]

The longitude of a point on the sphere of radius S0 is equal to twice the polarization angle and the latitude to twice the angle defining ellipticity such that

2

1

32 2 2

1 2 3

1 arctan( )2

1 arcsin( )2

SSS

S S Sβ

Ψ = + +

[441]


Figure 43 The Poincareacute sphere defined by the Stokes parameters and examples of the fundamental polarization states

The Poincareacute space that is defined by the points of a Euclidian tridimensional set built from the Stokes parameters is a clear representation of the polarization state and is not based on a reference basis The square of the sphere radius is equal to the radiation intensity and the polarization state is represented by a point of longitude 2Ψ and latitude 2β

The fundamental polarization states (Figure 44) have the following geometrical characteristics

Along a meridian line the orientation angle Ψ is constant Along a latitude line the ellipticity β angle is constant The center of the sphere corresponds to the completely depolarized state Within the sphere states are partially polarized Everywhere on the sphere surface polarization is elliptic except along the equatorial plane and at the poles

2β

2Ψ

Fixed ellipticity

Fixed

orientation


The equator is the location of the linear polarization states Along the equator linear polarization varies continuously from the horizontal position (β = 0 and Ψ = 0) to the vertical position (β = 0 and Ψ = π2) (Figure 43)

The North Pole corresponds to the anticlockwise (left-handed) circular polarization (β = π4) and the South Pole to the clockwise (right-handed) circular polarization (β = π4)

Two orthogonal polarization states E+ and E- with (E+)t(Eminus) = 0 correspond to two diametrically set points (antipode points) on the Poincareacute sphere This implies for such states that Ψminus = Ψ+ + π2 and βminus = minusβ+ With this definition the orthogonal condition does not depend on the absolute wave phase

432 The quantum description of polarization

In quantum mechanics (QM) the space of the states of a system comprising different particles is described by the tensor product of the space of each particle ( 1 2 3 kΩ = Ω otimes Ω otimes Ω otimes Ω ) In the case of identical particles this is not possible because the physical kets must be either symmetrical ( SΩ sub Ω bosons with integer spin photons mesons gluons etc) or antisymmetrical ( AΩ sub Ω fermions with half-integer spin number such as electrons positrons muons etc) This means that only certain kets of the space of identical particles can describe physical states The quantum approach of the electromagnetic wave in the Coulomb gauge as presented in Chapter 3 describes light as composed of spin 1 bosons respecting the Bose Einstein statistics This approach leads to a physical understanding of the particle nature of light

A massless particle with a spin equal to 1 can be described by a vector wave function localized at the origin of the coordinate system by the function Φ( r ) = Λδ3(r) whereby Λ is a constant vector of

components Λi (i = 123) 3

1

ˆ( )i ii

x=

Λ = Λ in the Cartesian basis (xyz

or x1 x2 x3) When a rotation about the xi axes is applied the wave


function Φ( r ) = Λδ3(r) is transformed into another wave function

Φrsquo(r) = Λrsquoδ3(r) In this expression Λrsquoi = Rik Λk 11 12 13

21 22 23

31 32 33

R R RR R R R

R R R

=

is the rotation matrix in the Cartesian axis system and δ3(r) is the Dirac distribution in the three-dimensional ordinary space As the set of rotations is a non-commutative group (SO(3)) rotation group around the origin in the three-dimensional Euclidian space) it is shown that as a function of the rotation axis the matrix R is linked to operators that respect the commutation rule [MES 64 MIZ 72 COH 73]

For a rotation angle α around an axis defined by the unit vector u(θϕ) an operator Ru(α) is defined by the expression

( )i S u

uR e αα minus=

[442]

For example the rotation matrix Rz(α) of angle α around the Oz axis is linked to the operator Sz by the following relations

cos sin 0sin cos 0

0 0 1

izSR e α

α αα α minus

minus = =

where 0 0

0 00 0 0

z

iS i

minus =

[443]

The operators ( )x y zS S S S=

verify the commutation rule of the

components of a moment such that i j ijk kS S S ie S = (where eijk is the

antisymmetric tensor of Levi-Civita e123 = e231 = e312 = 1 and e213 = e321 = e132 = minus1 and where Einstein convention is applied by summing up when indices are repeated) The square of S fulfills the relation

2 ( 1)S s s= +

where s =1

The standard basis 1 101m m = minus of the eigenvectors (1)0zS S=

and (1)1

1 ( )2 x yS S Splusmn

plusmn= plusmn obeys the relation


1 1 and 1 (1 )(1 ( 1)) 1 1zS m m m S m m m mplusmn= = plusmn + plusmn [444]

and can be expressed by the following relation

10 1 ( )2z x ye and e e= plusmn = plusmn [445]

in which the value 1 of spin is not included

In the case of a non-zero mass localized at the origin there are three states In the case of a photon of momentum nk

parallel to the

Oz axis which can be described by the vector wave function( ) ( )np t p kδΦ = Λ minus

a rotation of angle α around the Oz axis transforms Λ

according to equation [444] The corresponding

operator is linked to Sz and does not change ( )np kδ minus As it is the

case for a non-zero mass particle the spin of the photon is 1 however with Λ

normal to the Oz axis Only the sates m = +1 of the standard

basis are concerned and correspond to the clockwise and anticlockwise circular polarizations

1 ( )2n x ye e eplusmn = plusmn [446]

Unlike a non-zero mass particle of spin equal to 1 the photon spin states or its polarization states for np k=

define a two-dimensional (and not a three-dimensional) space It can also be noted that the orbital angular momentum L and spin angular momentum S are not separate physical observables as is the case for a non-zero mass particle Only the total angular moment J = L + S is an observable because there are no photons at rest That situation makes it impossible to define the three S components as observables However it is possible to define the component Sz of the spin parallel to the linear momentum of the photon as a physical observable called helicity For a massless particle having a spin equal to s helicity has an eigenvalue equal to s for a particle with a non-zero mass helicity has an eigenvalue equal to 2s + 1


In QM the algebra of the operators of a two-level quantum system can be described by using a pseudo-spin S The components of S in an arbitrary direction of the three-dimensional space can take values equal to plusmnћ2 only By analogy a geometrical representation called the Bloch sphere (unit sphere Figure 44) can thus be used to represent the Hilbert space of a two-level system This representation is similar to the one proposed by Poincare The most general observable of this system can be expressed as a linear combination with real coefficients

of the 2 times 2 Pauli matrices 2 with ( )ii

S i x y zσ = =

and the identity matrix I (equation [433]) The Pauli matrices verify the commutation rules

2i j ijk kieσ σ σ = [447]

and the anticommutation ones

2i j ij Iσ σ δ= [448]

The eigenvalues of the operators are equal to plusmn1 If 1 and 0 are the eigenstates of the operator zσ linked to the eigenvalues +1 and minus1 respectively then the eigenstates of the operators xσ and yσ

can be expressed by linear combinations such as ( )0 1 0 1 2

x= plusmn and ( )0 1 0 1 2

yi= plusmn

The most general state of the spin σ in a direction defined by the polar angles θ and ϕ (Figure 44) can be expressed in the Pauli matrices basis by

cos sinsin cos sin sin cossin cos

i

x y z i

ee

ϕ

ϕ

θ θσ θ ϕ σ θ ϕ σ θ σθ θ

minus = + + =

minus


with eigenvalues equal to plusmn1 The corresponding eigenvectors can be expressed by

0 cos( 2) 0 sin(( 2) 1

1 cos( 2) 0 sin(( 2) 1

i

i

e

e

ϕσ

ϕσ

θ θ

θ θ

= +

= minus [449]

In the case of a state defined by 0 1a bψ = + equation [450] shows that such a state is linked to a pseudo-spin having an eigenvalue equal to 1 and a direction defined by the angles θ and ϕ which fulfill the relation tan( 2) ie b aϕθ = An analogy can be made with the representation of the elliptic polarization of light

Figure 44 Bloch sphere and pure states (pseudo-spin) of a two-level system

The analogy can be taken further using a sphere of radius equal to 1 (Figure 44) to describe the states of a two-level quantum system in the same way as the representation of any given polarization on Poincareacutersquos sphere In the Bloch representation the two eigenvectors are defined by directions that are aligned symmetrically relatively to

1

ϕ

θ

x

z

y

0

σ

0x

1x

1y0y

σx

σz

σy


the center of the sphere (σ(θϕ) and minusσ(θ + πϕ + π)) For example the North and South Poles correspond to the 0 and 1 eigen vectors respectively A two-level quantum system is equivalent to a spin equal to frac12 with the corresponding relations 0 and 1e grarr rarr where

ande g are the excited and the ground states respectively

Therefore the fact that a polarization state is described by m = plusmn1 means that an analogy with the quantum description of a two-level system can be drawn (system described by a spin of value s = frac12 or a pseudo spin) The corresponding 2 times 1 column Jones vector can be written as a spinor (unlike a vector it is transformed into its inverse by a rotation of angle 2π) defined by its longitude α and its latitude ϕ

2

2

cos 2( )sin 2

i

i

eJ

e

φ

φ

αφ αα

minus

+

=

[450]

44 Conclusions

In classical optical theory the photon polarization states can be described on the basis of wave theory by using either the Jones vectors (completely polarized states) or the Stokes vectors (completely or partially polarized states) These approaches make it possible to model and easily simulate the effect of a medium or of an optical device on light polarization In the case of completely polarized light there is a relation between these vectors The Stokes vector can be directly linked to the geometrical approach developed by Poincareacute which makes it possible to represent the different states of polarization on a spherical surface if light is completely polarized or in the volume of this sphere if light is partially polarized

In the QED approach the concept of polarization is associated with the existence of a specific momentum of the spin of the photon The angular momentum of the photon is equal to 1 This leads to three possible values for the projection of this momentum in the direction of propagation of the wave (wave vector) m = +1 0 minus1 However for photons with no mass the state m = 0 cannot exist (Maxwell gauge


invariance) or in an alternative formulation because the electromagnetic wave is transverse (Ez = 0 if the wave propagates along the Oz axis) The states m = plusmn1 describe the clockwise or anticlockwise circular polarizations or helicity of the photon and can be expressed by the Jones vectors

Moreover an analogy can be drawn with the quantum description of a two-level system (a system described by a spin equal to s = frac12 or a pseudo spin) Each state of polarization can be associated with the direction of a pseudo spin Thus the polarization states can be associated with directions defined by α and ϕ or equivalently with a set of points of the Poincareacute sphere Therefore all the descriptions of the polarization state of a wave can be linked whether their origin be classical or quantum

5

Interaction of Light and Matter

Light interacts with matter that is a set of atoms or molecules through electrons As electrons are lighter than ions they move more easily in response to an electrical field The centroids of the positive and negative charges are no longer superposed and the material is polarized Using Maxwells equations this polarization can be modeled by using the constitutive equation that relates the displacement field to the electric field through the dielectric function of the material This linear response of the material is described in different forms the refractive index n the dielectric function ε the impedance Z of the medium the susceptibility χ the conductivity σ or the skin depth δ The classical Drude model shows that the dielectric function of a conductor depends on the wavelength and the frequency of the electromagnetic radiation To account for the quantum properties of matter the Lorentz model is used This model is based on optical transitions between two electronic bands either from full valence bands to states of the conduction band or from states of the conduction band to empty higher energy bands The models of Cauchy or Sellmeier are widely used for weakly absorbing insulators In the case of semiconductors the energy of the band gap is determined by using the model of TaucndashLorentz or CodyndashLorentz In all these cases the refractive index can be calculated from the relative permittivity In insulators the exponentially decreasing Urbach formula is used to model absorption in the band gap In quantum mechanics (QM) when two systems interact an interaction term is added to the sum of each systems Hamiltonian In the case of the interaction of light and matter this term includes the scalar potential and the vector potential of the light in addition to the charge of the electron The same interaction Hamiltonian is used to calculate the eigenstates of electrons in metals in which the electrons are free dielectrics in which electrons are bonded and semiconductor or semimetals

51 Introduction

In the field of optics (near UV to near IR) experimental studies or observations on the interaction between light and matter (plasma gas


First Edition


liquid or solid) give access to different physical parameters for characterizing the environment According to the technique used it is possible to determine the coefficients of reflection transmission (spectroscopy reflectometry ellipsometry transmittance reflectance etc) or the coefficients of dispersion and absorption (refractive index absorbance etc) when light propagates in a medium or the luminescence properties (fluorescence phosphorescence luminescence etc) or the properties of elastic or inelastic light scattering (Mie Rayleigh Raman Brillouin etc)

Figure 51 Interaction of light and matter

The response of the medium to excitation by light can be described by a complex refractive index n n ik= minus where the real part n is related to the scattering of light and the imaginary part k is related to its absorption This macroscopic parameter reflects the interaction of the material with the electric field of the wave at the microscopic level Maxwell equations are used to connect n to the microscopic parameters It is thus possible to characterize the optical properties by the dielectric function or dielectric permittivity ε of the medium by the relation 2nε =

This chapter focuses on matter in solid form and the materials considered (metals semiconductors insulators or dielectrics) are

ABSORPTIONDIFFUSIONLUMINESCENCEETC

GAZ LIQUIDE SOLIDE PLASMA Incident wave

Reflected wave

Transmitted wave

EnvironmentGas Liquid Solid Plasma

Absorption Diffusion Luminescenceetc

Interaction of Light and Matter 97

classified according to their electrical properties This approach is justified by the fact that an electromagnetic wave interacts with matter primarily through electrons as introduced in Chapter 3 Classically it is shown that the interaction between the electric field of the wave of angular frequency ω = ck = ck (k wave vector and c speed of light) and the electrons give them a vibrating motion at the same frequency ν = ω2π The radiation emitted due to the acceleration of the electrons will interfere constructively or destructively with the incident wave Since all dipoles radiate on the path of the wave this results in the dispersion property though the refractive index of the medium In QM light is responsible for the resonant coupling or not between quantum states of matter (discrete energy levels in the atoms or molecules or valence and conduction band in solids) through the vector potential of the wave This coupling appears as the second interaction term in the Hamiltonian described in Chapter 3 (equation [352])

The description of the interaction of light and matter can be found in different textbooks some of which are listed as references in the bibliography either from the classical approach [MAX 54 BRU 65 LAN 66 GIN 78 JAC 98 BOR 99 HEC 05] in the frame of QM (QED) [MES 64 LOU 64 MIZ 72 COH 87 LAN 89 FEY 98 SAK 11] or treating both approaches specifically [TAU 66 FLU 67WOO 72 BUB 72 ROS 98 SIM 00 DRE 03 CSE 04 MES 04 WOL 06 SHU 09 FOX 10]

52 Classical models

In 1836 Cauchy [CAU 36] interpreted the variation of the refractive index of a material with wavelength as

2 4( ) B Cn Aλλ λ

= + + [51]

where A B and C are positive constants determined experimentally and n is a decreasing function of the wavelength λ and depends only


even powers of 1λ The formula is valid for a material transparent in the visible range outside the absorption zones (normal dispersion)

In 1871 Sellmeier [SEL 71] proposed an empirical law expressed differently for modeling the refractive index of transparent materials as a function of the wavelength λ as

22 22 31 2

2 2 2 2 2 21 2 3

( ) 1 BB Bn λλ λλλ λ λ λ λ λ

= + + +minus minus minus

[52]

where Bi and λi (i = 1 2 3) are constants determined experimentally (λ expressed in micrometers) In this expression n is given as a series

of oscillators characterized by the term 2

2 2i

λλ λminus

(i = 1 2 3)

At the beginning of the 20th Century the first relevant theoretical models in their classical forms were developed on the one hand by Drude for metals and on the other hand by Lorentz for dielectrics to interpret the variations of n or ε with the angular frequency ω (or the frequency ν or the wavelength λ) of the light In both models it is assumed that the electrons either free (metal) or linked (dielectric) by means of a restoring force are subjected to a damping force of fluid type in addition to the external force due to the electric field of the electromagnetic wave The models yield expressions that allow for the interpretation of both normal dispersion and anomalous dispersion Although these theories are based on incomplete or not very realistic assumptions in the light of quantum theory they eventually lead to expressions not so different from those determined by QM

From a mathematical point of view for an electron of mass me the same equation of the movement of the electron can be used to determine the optical properties of a metal a dielectric or a plasma such as

2202

( ) ( ) ( )( ) ( ) ( )e e ee e e e

d r t dr t dr tm m r t m eE t e B tdt dt dt

ω γ= minus minus minus minus and [53]


In equation [53] re(t) is the vector position of the electron at time t

referenced from its equilibrium position 2

2e

ed rmdt

is the force of

Newton 20e em rω

- is the restoring force of the electron (zero in the case of a metal) by the positive ions of the crystal or the plasma (ions that

are much heavier and static with respect to electrons) ee

drmdt

γminus

is the

viscous friction force (electronndashelectron collisions type process or

electronndashlattice or electronndashion) and edreE e Bdt

minus minus and

is the Lorentz

force on the electron due to the electromagnetic field

In forced regime the differential equation can be solved to determine the position re(t) and the speed ve(t) of the electron as a function of the angular frequency ω In the absence of magnetic effects the magnetic component of the Lorentz force is zero and the resolution leads to the following expressions for the position re(t) and the speed ve(t)

2 20

(0)( )( )

i t

ee

eE er tm i

ω

ω ω γω

--

=- -

[54]

2 20

(0)( )( )

i t

ee

i eE ev tm i

ωωω ω γω

minus

=minus minus

[55]

For a numerical density Ne of electrons the mean value of the microscopic polarization ( ) ( )ep t er t Eα

=- = (α polarizability tensor)

leads to the macroscopic polarization P(t) of the medium expressed by

0( ) 1 ( ) 1 ( ) 1 ( ) ( ) ( )k ke k e ek k

P t V p t V n er t V n E t N er t E tα ε χ= = minus = = minus =

[56]

and as a result

220

02 2 2 20 0

(0)(0)( ) ( ) (0)( ) ( )

i ti tp i te

e ee

E eN e E eP t N er t E em i i

ωωωε ω

ε χω ω γω ω ω γω

minusminusminus= minus = = =

minus minus minus minus

[57]


where 20p e eN e mω ε= is the plasma frequency and

2 2 20( )p iχ ω ω ω γω= minus minus is the susceptibility The relation between the

microscopic polarization p(t) of the medium by the electric field on the one hand and the macroscopic polarization P(t) (mean value in terms of volume over a volume of dimensions large relative to the wavelength of the field λ) on the other hand leads to the relation between the refractive index n and the dielectric permittivity ε = εrε0 (ε0 is the dielectric constant of vacuum)

Using the expression of the speed ve(t) the current vector density j(t) is expressed by

220

2 2 2 20 0

(0)(0)( ) ( ) (0)( ) ( )

i ti tp i te

e ee

i E ei N e E ej t N ev t E e

m i i

ωωωωε ωω σ

ω ω γω ω ω γω

minusminusminusminusminus

= minus = = =minus minus minus minus

[58]

where 2 2 20 0( )pi iσ ωε ω ω ω γω= minus minus minus is the conductibility of the

medium The quantities associated with a metal and a dense dielectric medium are determined by taking respectively ω0 = 0 and

(0) (0)locE E=

in the expressions of P(t) and j(t)

The introduction of these quantities in Maxwellrsquos equations (see Chapter 3 where the magnetic effects are neglected) leads to the expression of the displacement field D which is used to determine the complex dielectric function ε of a dielectric from

0 0 (1 )D E P E Eε ε χ ε= + = + =

[59]

For a metal the complex conductibility is determined from the following equation

0 0Erot B micro Et

σ ε part= +part

[510]


These expressions can then be used in those of the optical constants defined by

2

0

n εε

= where 0

iε ε εε

= + and 2 2( )n n ik= minus [511]

From equation [511] the following relationships between the real part and imaginary part of the complex dielectric function and the complex optical constant can be established

2 2 n kε = minus and 2nkε =

or

2 2 2

n ε ε ε+ += and [512]

2 2 2

k ε ε εminus + +=

The following equivalent relations ESIc = Ecgs (E and B have the same dimensions in the Gauss-cgs units system) ε0=14πc micro0= 4πc (εε0)SI = εcgs and (micromicro0)SI = microcgs) must be applied to switch from SI units system to the Gauss-cgs system

These equations show that there is a close relationship between the real refractive index n and the extinction coefficient k or between the real part εprime of the dielectric function and its imaginary part εprimeprime These relationships are due to the linear and causal properties of the system (the linear systemrsquos response function cannot precede the cause which is the excitation of electrons by the electric field) They are general and can be put in the form of integral equations called KramersndashKronig relations for the complex dielectric function or the complex refractive index as

2 20

2 ( )( ) 1 ( )

P dω ε ωε ω ωπ ω ω

infin primeprime primeprime primeminus =prime minus and 2 2

0

2 ( ) 1( )( )


infin prime prime minusprimeprime prime= minusprime minus [513]

2 20

2 ( ) ( ) 1kn P dω ωω ω

π ω ω

infin prime primeminus =minus and 2 2

0

2 ( )( )

nk P dωω ωπ ω ω

infin

prime=minus [514]


where P denotes the Cauchyrsquos principal part to take into account the singularity when ωprime = ω The KramersndashKronigrsquos relations are valid for stationary systems (convolution products can be used) and are also called dispersion relations as they relate absorption and dispersion process The constant minus1 in expression of εprime(ω) represents the high frequency limit of εprime(ω) (dielectric constant of vacuum) The integral over εprimeprime(ω) should be calculated at all angular frequencies where the material absorbs High frequencies correspond to transitions of electrons close to the nucleus (X-rays and far-ultraviolet) and nuclear resonances The low frequencies correspond to the transitions of the valence electrons (near visible and ultraviolet) the resonances of ionic vibrations (near infrared) resonances of ionic and molecular vibrations (mid-infrared) and the frequencies associated with molecular rotations and relaxation mechanisms (far infrared microwave radio frequency) Very low frequencies are associated with the electrical charges at interfacial space fillers to the relaxation of defects impurities or the grain boundaries

In general the high frequency contribution can be separated from the low frequency contribution as expressed in the first integral of equation [515] Similarly in the case of metals because of the pole in ω = 0 which accounts for the static conductivity (dc conduction) the second integral of equation [513] is modified as expressed in the second integral of equation [515]

2 20

( )2( ) ( )( )

lowP dω ε ωε ω ε ωπ ω ω

infin primeprime primeprime primeminus infin =

prime minus and

02 2

0

4 2 ( ) 1( )( )

P dπσ ω ε ωε ω ωω π ω ω


The real index n may also be calculated by an integral equation as given in the first term of equation [516] or when measured at low frequencies by the second term

2 20

2 ( ) ( ) 1( )

kn P dω ωω ωπ ω ω

infin prime primeminus =minus or 2

0

2 2 ( ) ( )( ) ( ) low

n kn P dω ωω ε ωπ ω

infin prime primeminus infin = [516]


521 The Drude model

By considering the metal as a free electron gas (a numeric density Ne of free charge carriers (Ne cmminus3)) to which the assumptions of the kinetic theory of ideal gas according to Boltzmann statistics is applied Drude constructed a model for interpreting the optical properties of a metal under the effect of the electric field of an electromagnetic wave on the free electrons In the case of sodium for example the radius of the cation is worth 9 times 10minus11 m and the volume occupied by the cations in a cell is 15 of the total volume The peripheral layer valence electrons can move in a large volume and despite the potential well of the positive nuclei shielded by the core electrons the electron can be considered to be free

Using the Drude model the general characteristics of the dielectric function of a metal and its dependence as a function of wave angular frequency can be determined It also provides qualitatively the static (dc) and alternating (ac) conductivities the Hall effect and the magneto-resistance behavior

Despite its imperfections with respect to the improvements brought by the quantum treatment especially the fact that electrons are fermions which obey FermindashDirac statistics and can propagate without dissipation in a periodic potential of the perfect crystal this model was used to find the WiedemannndashFranz law which stipulates that the ratio of the thermal and electrical conductivities of a metal is equal to the Lorentz number

In the Drude model it is assumed that after excitation the electrons return to their original equilibrium state as a result of damping of viscous type due to collisions with the crystal lattice The frictional force is proportional to the moving velocity of the electrons and the damping parameter γD can be connected to the mean time τ between collisions τ=1γD (it is assumed that the radiation wavelength is small compared with the mean free path between collisions)


When ω0 = 0 the conductivity σ and equations [58] and [510] and the relationship

2 2

2 20

( ) ( ) ( ) ( 1) ( )irot rotE grad divE E E i rot B E Ec cω σ ωω ε ω

ωεΔ Δ

= - - = = + = [517]

lead to the complex dielectric constant of a metal ε as

2 220

2 2 20

1 1 1 1 p pe

D D

N e mii i i

ω ωεσεωε ω ωγ ω ωγ ω ω τ

= + = minus = minus = minus+ + +

[518]

where 2 20 ( )pi iσ ωε ω ω γω= + is the conductivity of the metal Ne is the

numerical density of the electrons contributing to σ ωp is the plasma frequency and ω is the angular frequency of the wave Using the notation εinfin for the dielectric constant at high frequencies this equation can be written as 2 2

p Diε ε ω ω ωγinfin= minus minus where γD the imaginary part of the dielectric constant represents the absorption that is inversely proportional to the mean optical collision rate of the electrons The term εinfin represents the lattice contribution and the second term represents the contribution of free electrons The real and imaginary parts of the dielectric function are expressed as

2

2 2 p

D

ωε ε

ω γinfin= minus+

and 2

2 2 pD

D

ωγεω ω γ

=+

[519]

Typically the frequencies of the collisions are very low in comparison to the optical frequencies such that the real and imaginary parts can be written as 2 2 1 pε ω ω= minus and 2 3 p Dε ω γ ω= and such that at the plasma frequency the real part of ωP is zero It can be shown from Maxwellrsquos equations that at this frequency the magnetic field vanishes and that no volumic propagation of the electromagnetic wave occurs For frequencies lower than pω the electrons can follow the fieldrsquos variations ε(ω) is real and negative ( 2 2 1 pε ω ω= minus ) and the electromagnetic wave is damped in the metal For much higher


frequencies the electrons cannot follow the fieldrsquos variations which can thus propagate through the metal (ε(ω) is real and positive) At the plasma frequency pω the quantum of energy is given by pω and the associated quantum quasi-particle is called a Plasmon At the surface of the metal the interaction is of the type light-Plasmon The plasma frequency ωp of metals is located in general in the UV range and is of the order of 10ndash20 eV (Al 153 eV Mg 106 eV) and can be calculated by the Drudersquos model for alkalines or trivalent metals (intraband component of the conduction band) For noble metals it is necessary to add an interband contribution (polarization of core electrons and transition from the valence band to the conduction band) which shifts the frequency in the visible range toward the blue and gives to these noble metals their color which is different from gray For transition metals two main contributions are responsible for the color of metals the large variation of the reflection coefficient at

pω or the photoexcitation of bound electrons of the d band and the absorption associated with this photoexcitation

The value of this model is that it also allows to deduce the electrical resistivity ρe (static conductivity σ0 at ω = 0) which is equal to 2

0(1 2 )e D pcρ πε γ ω= When ω rarr 0 then 2 2 1 p Dε ω γ= minus and 2 p Dε ω γ ω= which leads to the following HagenndashRubens

relationship 202 2 0003p D dc dcn k ω ωγ σ ωε λ ρasymp = = =

522 The Sellmeir and Lorentz models

With the Lorentz model it is necessary make a distinction between a dilute medium and a dense medium In a dilute medium the local electric field is slightly different from the excitation electric field of the wave The restoring force is due to ions heavier than the electrons that are assumed as elastically bound to the positive centroid when the electric field of the wave polarizes the medium (Figure 52)


Figure 52 a) Positive and negative charges centers are superposed b) polarization of matter from the displacement of centroids

In a dense medium it is necessary to take into account the effects of the depolarization of the medium The restoring force that acts on the bound electrons is based on the hypothesis that the displacement of the electrons is small (linear effect) If the concept of valence and conduction bands is introduced then the restoring force is related to optical transitions between two electronic bands fully occupied bands (valence) to states of the conduction band or from the latter to empty bands at a higher energy

Using equations [59] and [510] for a dilute medium the following expression of the complex dielectric constant can be obtained in the framework of Lorentz model

220

2 2 2 21 1 pe

i ii iL i iL

N e mi i

ωεεω ω ωγ ω ω ωγ

= + = +minus minus minus minus [520]

where Ne is the density of the electrons contributing to the transition Denoting εinfin the dielectric constant at high frequencies it is possible to put this equation as

20

2 2 2 2

epe

i ii iL i iL

N e mi i

ωεε ε εω ω ωγ ω ω ωγinfin infin= + = +

minus minus minus minus [521]

++-

_

a

b


where ωi is the central frequency of oscillator i and γiL is the broadening of oscillator i The term εinfin represents the lattice contribution and the second term represents the contribution of oscillators

Thus for a dilute medium including N oscillators having the same frequency ω0 per unit volume formula [519] leads to the refractive index given by

22

2 20

1 1 pr

L

ni

ωε

ω ω ωγminus = minus =

minus minus [522]

The shape of the real and imaginary parts of the dielectric function (Figure 53) can be discussed

2 22 0

2 2 2 20 0

22 2 2 20 0

1( ) ( )

( ) ( )

pL

Lp

L

ω ωε ωω ω γ ω

γ ωε ωω ω γ ω

minus= +minus +

=minus +

[523]

Figure 53 Variation of the real part εprime and the imaginary part εprimeof the dielectric function about the oscillation at resonance frequency ω0

ε(ω)

ω

εrsquorsquoεrsquo

ω0


When |ω minus ω0 | gtgt γ very far from resonance then for ω ltlt ω0

2

2 20 0

1 11 2( )

pωε

ω ω ω= +

minus and

2

4 20 0

11 2( )

p Lω γ ωε

ω ω ω=

minus [524]

and for ω gtgt ω0

22 2

1 1 pL

ε ωω γ

= ++

and 22 2 1 L

pL

γ ωε ωω γ

= ++

[525]

In the first zone ε εgt and 1Lγ ω ltlt the absorption is negligible ie the medium is transparent As 0d dε ω gt the transparent region is characterized by an abnormal dispersion law The expansion of ε and ε in power series of ω leads to 2ε ωprop and

3ε ωprop In the second zone ε εlt and the absorption again negligible with 0d dε ω gt thus showing that the dispersion is normal

When |ω minus ω0 | lt γ in the resonance zone then

20

220 0

12

12

p

L

ω ω ωεω ω ω

γ

minus= + minus+

and 2

220 0

22

12

p L

L

ω γεω ω ω

γ

= minus+

[526]

and the absorption becomes predominant because of the relation |ω minus ω0 | lt γ As 0d dε ω lt the dispersion law in this zone is abnormal

From the expression of the absorption coefficient 4 kα π λ= and following equation [512] it can be shown that the resonance zone between the fundamental vibration frequency of the oscillator and the wave are characterized by an absorption curve of Lorentz shape centered on ω0 and with a width at half-height equal to γL This width is associated with τ the inverse of the mean time between two inelastic collisions of the electron with the ions of the medium


From QM it is known that an electron can have several oscillation frequencies and if the different absorption zones are taken into account then the refractive index can be written as

22 0

2 2 2 21 1

1N N

k e k

k kok k ok k

N e m fni i

εω ω ωγ ω ω ωγ= =

minus = =minus minus minus minus [527]

with fk = (Nke2 ε0 me) being the strength of the oscillator k

Outside the absorption bands |ω minus ω0|gtgt γ and then

22 0

2 2 2 21 1

1N N

k e k

k kok ok

N e m fn εω ω ω ω= =

minus = =minus minus [528]

If one expresses the relationship in terms of wavelengths then outside the absorption bands |ω minus ω0|gtgt γ (transparent zone or weak absorption) the Sellmeier formula can be derived as

2 2 2 22 22

2 2 2 20 01 1

11 ( )2 2N N

ok k okk k

e ek kok ok

fN e N en c m m cλ λ λ λ

π ε ε πλ λ λ λ= =


For resonant frequencies in the UV (λ2gtgt λok2) the Cauchy

formula in the visible range is used

n2 = A + B λ2 + Cλ4 [530]

For resonant frequencies in the IR (λ2ltlt λok2) the Briotrsquos formula

in the visible range is used

n2 = Aprime λ2 + A + B λ2 + C λ4 [531]

In a dense medium the average over the volume of the microscopic polarization (equation [56]) is calculated in a spherical cavity (radius r) surrounding an atom or a molecule (radius a a ltlt r ltlt λ ) and inside which the local electric field (Eloc ) is different from the field E of the wave To determine the field the medium is supposed to be homogeneous outside the cavity (macroscopic) such that the polarization field P induces charges at the surface of the


cavity They are the sources of the electric field (Ed) which superposes to the field E inside the cavity (Eloc = E + Ed) By assuming that the average of the effects of the induced dipoles is zero inside the cavity then for a simple cubic lattice it is determined that

0 0

1 1( ) ( ) ( ) ( ) ( )3 3dipocircles locE r t P r t E r t E r t P r tε ε

lt gt= = + [532]

Such that according to equation [56] and ( ) ( )locp r t E r tα= it

can be written that

0

0

1( ) ( ) ( ( ) ( )) ( )3

( )(1 3 )

locP r t N E r t N E r t P r t P r t

N E r tN

α αε

αα ε

= = +

=minus

[533]

Equations [59] and [533] lead in that case to the Clausius Mossotti relation

0

0

1 11 )2 31 3

rr

r

N NNα ε εε α

εα ε minus= + = lt gt +minus

[534]

Finally for dense isotropic media consisting of different oscillators it can be written that (εr minus 1εr + 2 ) = sum(Nkαk

2 3 ie (n2 minus 1n2 + 2 ) = sum(Nkαk

2 3 It is shown that it is possible to write εr minus 1 = n2 minus 1= (Ne2 ε0 m) 1(ω1

2 minus ω2 minus iγ ω) where ω12 = ω0

2minus (Ne23ε0 m) Because the medium is dense there results a shift in the absorption frequency In the absorption zone anomalous dispersion occurs as n decreases with lambda and it is necessary to use QM and consider the thermodynamic equilibrium to calculate n Finally the following expression is determined

22 1 2

0 0 1

( ) 11 12 ( )r

e L

N N fenm i

εε ω ω ω γminusminus = minus =

minus minus [535]


where N1 and N2 are the populations of the energy levels involved in the absorption processes and f is a term that depends on the probability of the transition between the levels

53 Quantum models for light and matter

QM was developed in the early 20th Century after Max Planck had removed the ambiguity on the ultraviolet catastrophe (RayleighndashJeans) by introducing the quantization of energy (E = ħω or E = hν h = 663 times 10minus34 Js) in his theory developed to interpret the emission of black body and that Einstein used the same quantization (E = ħω or E = hν) for interpreting the experimental results on the photoelectric effect The notion of quantification (quantification of the action

0

( )T

S L q q t dt n= = where L is the Lagrangian of the hydrogen

system consisting of one proton and one electron) is also involved in the Bohrrsquos theory for interpreting the line spectrum of hydrogen (although not adapted for the many-electron atoms) or in the famous de Broglie relation that associates a wave to any particle

orp k p h λ= = In the first two examples demonstrating the

limits of classical mechanics which considers only continuous states of energy for matter consisting of particles it is question of the lightndashmatter interaction radiation in thermodynamic equilibrium from discontinuous exchange of energy with matter in the case of the black body and the notion of packets of energy grain (later called photons) for the processes of absorption of light by a metal in the case of the photoelectric effect It is therefore more appropriate to use the framework of QM to interpret the lightndashmatter interaction

531 The quantum description of matter

In QM the energy states of atoms or molecules in dilute or condensed phase are discrete as calculated by the Schroumldinger eigenvalue equation [MES 64 LOU 64] These states are the eigenstates of the Hamiltonian operator of the physical system the sum of its kinetic energy (dynamic) and its potential energy


(configuration) corresponding to the eigenvectors Different wave functions may be associated with the same eigenvalue the energy of each state (degeneracies) The interaction between light and matter results in a transition between discrete energy levels and occurs between an initial state (or set of initial states) and a final state (or set of final states) When two systems interact an interaction term that reflects the coupling between the two systems is added to the sum of the Hamiltonians of each system In the case of lightndashmatter interaction this term (see Chapter 3) comprises the scalar potential V and vector potential A of light in addition to the charge of the electron

In the case of hydrogen-like atoms with one electron for example the energy states are characterized by quantum numbers n (principal quantum number n ge 0) l (azimuthal quantum number 0 le l le n minus 1) m (2l + 1 magnetic quantum numbers minusl le m le +1) and ms (spin quantum number) Quantum numbers n and l refer to the radial part ( ( )n l rreal ) of the wave function and the quantum numbers l m are related to the angular part ( ( )m

ly θ ϕ ) of the wave function and ms for the projection of the spin) (Figure 54(a)) for 3d n = 3 l = 2 m = 0 plusmn1 plusmn2 4s n = 4 l = 0 m = 0 4p n = 4 l = 1 m = 0 plusmn1) To determine the energy levels of atoms with several electrons it is better to use the methods of quantum chemistry that involve the density functional theory (DFT) The DFT is a self-consistent method to calculate the energy as a functional of the electron density The one-electron KohnndashSham equation [KOH 65] is resolved to determine the orbitals driving the movement of electrons Then the electronic density is calculated from which another orbital is determined This procedure is iterated until convergence (two consecutive orbitals are the same) The method is based on the optimization of the electron density rather than multielectronic wave function of the HartreendashFock theory In this approach it is assumed that each electron is submitted to the field of the other charges (electrons and nuclei) and the Slater determinant is used to calculate the wave functions

In the case of molecules in addition to electronic states we must also consider the states of vibration and rotation of the nuclei whose overall movement is controlled by the electronic wave function of the ground electronic state [AMA 53 BAR 61 BAR 67 PAP 97] To


determine the vibrationndashrotation energy levels of a molecule the eigenvalue Schroumldinger equation of the molecular system must be solved This equation involves the degrees of freedom of nuclei and electrons constituting the molecule It is necessary to use approximations for its resolution The Born and Oppenheimer (BO) approximation allows for the decoupling of the rapid movement of electrons from that of the nuclei which are much slower For each electronic state the nuclei then move in a mean potential that depends on the nuclear coordinates The movements of the nuclei can be separated from the movements of the electrons because the electrons are lighter than the nuclei (BO memN ltlt 1) The electronic states are then determined for fixed configurations of the nuclei and then in the electronic ground state the movement of the nuclei can be calculated In the case of a diatomic molecule of type AB for example this electronic state is different from the eigenstates (ΨA(r θ ϕ) and ΨB(r θ ϕ)) of each molecule A resonance phenomenon occurs when the two atoms come closer to form the molecule This leads to the formation of a binding state the symmetrical superposition of the eigenstates Ψs(R Ω) = Ns ( ΨA(r θ ϕ) + ΨB(r θ ϕ)) and a non-binding state the unsymmetrical superposition the eigenstates Ψas(R Ω) = Nas ( ΨA(r θ ϕ) minus ΨB(r θ ϕ)) (Figure 54(b))

Figure 54 Discrete electronic energy levels a) atom and b) diatomic molecule

4s3d4p

ATOME

E

EA EB

El

Eal

ΨA ΨBΨs

Ψas

SeacuteparationAtom Separation

4s

3d 4p

E


Usually the movement of a set of N nuclei can be decomposed into movements of independent oscillators (3N-5 for a linear molecule and otherwise 3N-6) as for photons (see Chapter 3) Each oscillator is identified by a quantum number qi the normal vibration coordinate and possibly its degeneracy gi CO2 for example is linear and has three normal vibrations (Figure 55(a)) one which is doubly degenerate (Q2 g2 = 2 or Q21 and Q22) while the nonlinear triatomic molecule O3 has three non-degenerate normal vibrations (Figure 55(b)) The associated frequencies are noted νi i = 1 2 3 and lie in the mid-infrared

Figure 55 Normal vibrations of molecules of a) CO2 and b) O3 (Q1 symmetrical stretch Q2 bending mode Q3 antisymmetrical stretch)

In the case of condensed matter [KIT 96] by applying the approximation of BO we can also treat the movement of electrons (weakly or strongly bound) and the vibrating movement of the nuclei (oscillating in an electronic mean potential) separately The one electron model is used to determine the electronic energy levels and solving the Schroumldinger equation (equation [535]) leads to electronic levels in an energy band structure (Figure 56) The band gap can be interpreted as being due to a type of Bragg reflections of free electrons on the crystal lattice periodic planes Theories are based on ideal crystalline solids although a solid may be in crystalline form polycrystalline or amorphous (thin films multilayers polymers


ceramics) The periodicity of the direct lattice can be represented by a vector defined by 0 1 1 2 2 3 3r r n a n a n a= + + + where r0 is vector position of the origin placed at an occupied node ni (i = 1 2 3) are integers and the ai are three non-coplanar vectors (period of the Bravais lattice) constituting the primitive cell (the smallest) of volume

1 2 3( )a a aΩ = and

Considering a one-electron model the resolution of the Schroumldinger equation

2

( ) ( ) ( ) ( )2nk nk nk nk

pH r V r r E rm

Ψ = + Ψ = Ψ

[536]

where p is the linear momentum of the electron m is the mass of the electron and V( r ) is the periodic Coulomb potential in which the electron moves This potential has the symmetry properties of the crystal lattice and its shape depends on the interatomic bonding type Electronic levels are thus calculated as eigenstates which are functions of Bloch ( ) exp( ) ( )n k n kr ik r u rΨ =

(the product of a wave function of the free electron by a function possessing the same periodicity as the lattice according to Blochrsquos theorem) where

0 0 1 1 2 2 3 3k k G k h b h b h b= + = + + +

is a wave vector associated with the reciprocal lattice The vectors bi are defined by

3 1 2 1 2 3 2 3 1(2 )( ) (2 )( ) (2 )( )b a a b a a b a aπ π π= Ω and = Ω and = Ω and [537]

with 31 2 3( ) (8 )b b b πand = Ω This lattice has the same properties of

periodicity and symmetry of the direct lattice To represent the energy states a reduced reciprocal space (Brillouin zone) (Figure 56(a)) is used in which the variations of E with k are given according to certain symmetry directions of the first Brillouin zone

Finally it is shown that resolution of the Schroumldinger equation leads to electronic energy levels grouped in a band structure (Figure 56(b)) which are the allowed energy bands separated by band gaps When these bands are filled with the electrons of the atoms


forming the crystalline system taking into account the Pauli exclusion principle (the electrons are spin frac12 fermions obeying the FermindashDirac statistics) the different cases with the last band to be filled depending on the position of the Fermi level (Ef) (Figure 56(b)) are determined It is called conduction band (BC) if it is partially filled (for metals with free electrons) and valence band (BV) if it is completely filled (dielectric case with bound electrons)

Figure 56 Energies E of electrons as a function of the distance r between the atom in a solid material BC conduction band BV valence band EV top of the valence band EC minimum of the conduction band Eg = EC minus EV energy gap or bandgap Ef Fermi level

The conduction band is always located above the valence band being empty or partially filled separated from the valence band by the energy gap Eg (Figure 56(b)) also band gap Depending on the energy gap and temperature the insulating materials can be distinguished from the semiconductors At ambient temperature for instance Eg = 0 eV for metals Eg cong 3 meV for semiconductors and Eg gt 3 meV for dielectric or insulators

To illustrate the case of a complex magnetic compound an example is the class of compounds to which the lightndashmatter interaction results in a transition between two spin states (ldquospin transitionrdquo (ST) or ldquospin crossoverrdquo) These compounds are

E

r

Ec

Ev

BC

BV

ELECTRONSLIBRES

GAP Eg

SOLIDE ATOMEISOLE

METAL

Ef

qK

εε K

Free electrons

Solid Metal Isolated atom


complex-based organic materials in which Mn+ metallic cations of a transition group 3d4 to 3d7 are incorporated (n = 3 l = 2 and m = plusmn2 plusmn1 0) in a octahedral geometric configuration The local electronic structure of the complexes ST that is determined at the molecular level using the ligand field theory shows the lifting of the degeneracy of the d orbitals into two groups in the octahedral environment The orbitals dxy dyz dxz of the irreductible representation t2g (group symmetry) which are directed in between the ligands have a lower energy than the orbitals dx

2-y

2 and dz2 of the irreducible representation

eg which are directed toward the electrons

Figure 57 BS and HS states of the ion Fe(II) and Fe(III)

The difference between these two groups (a measure of the strength of the ligand field) which depends on the distance between the metal cation and the ligand is noted ΔO (O for octahedral) The distribution of the d electrons of the metal ion in the orbitals follows the Pauli exclusion principle and Hundrsquos rule In a strong field the electron pairing energy (energy cost when two electrons are in the same orbital) is not sufficient to comply with Hundrsquos rule and only t2g orbitals are filled (low spin state BS) In a weak field the two groups of orbitals t2g and eg can be filled (high spin state HS) (Figure 57) Physical properties (magnetic optical thermal electrical mechanical)

Fen+ Fen+

T hν P

Etat BS

eg

t2g

S=12

Δeacutel

Etat HS

S=52

eg

t2g

ΔeacutelFe3+

3d6

eg

t2g

S=0

Etat BS

Δeacuteleg

t2g

S=2

Etat HS

ΔeacutelFe2+

3d6T hν P

BS State HS State

BS State HS State


depend on the change in the spin state of the metal cation These complexes can be classified with regard to the cooperativity across the solid They are molecular materials having bistable switching properties between the BS state and the HS state of multiphysics type (thermal piezoelectric magnetic and photonic)

Another class of materials having a complex structure consists of polymers which are assemblies of monomers by covalent bonds leading to macromolecules The macromolecular skeleton is generally constituted by carbonndashcarbon bonds (polyethylene polypropylene polystyrene) or the bonding of carbon atoms with other atoms such as oxygen (polyethers polyesters) or nitrogen (polyamides) There are also polymers based on Si-O bonds (polysilanes polysiloxanes polydimethylsiloxanes) Such materials interact with light in the UV-visible and mid-infrared region through functional groups present on the backbone or in ramifications or branches of the parent structure

532 JaynesndashCummings model

A purely quantum description of lightndashmatter interaction requires that the electromagnetic field be considered as an operator The theoretical model JaynesndashCummings (JC) was proposed in 1963 [JAY 63] to study the relationship between quantum theory of radiation and the semi-classical theory when describing spontaneous emission This model is applied to a two-level atomic system interacting with light which is treated as an electromagnetic radiation bath responsible for spontaneous emission or absorption of photons In QM two interacting systems may be in an entangled state so none of the systems is in a particular state However measurements on each system show correlations that can be understood in classical terms The JC model helps to show the peculiarity of quantum systems that has been observed in cavity quantum electrodynamics (ldquocavity QEDrdquo) in the study of the resonant interaction between an atom and a field mode in a cavity leading to the Rabi oscillations Concerning the evolution of the states of a two-level system interacting with the field the JC model helps to interpret the collapse that occurs after a period of Rabirsquos oscillation and their revivals This


phenomenon is due to the entanglement of the field and the atom that is a purely quantum effect

This model is presented to show the difficulties to develop a purely quantum theoretical model for complex systems with the purpose to interpret the interaction of light and matter which explains the success of the semi-classical models However it is necessary to familiarize with purely quantum models in order to fully understand the observations that result from any experience or characterizations requiring the interpretation of the interaction between light and matter

Consider an atom with two levels noted e for excited and g for fundamental (ldquoground staterdquo) and a mode of the electromagnetic field confined in a resonator or FabryndashPerot type cavity The two-level atomic system is similar to a spin frac12 and the Hamiltonian of the free atom can be expressed as

ˆˆ ( )2 2

aatom aH e e g gω σω= minus = [538]

where ωa is the frequency of the transition between the two levels and σz is one of the Pauli matrices in the ( e g ) basis (see Chapter 4)

The Hamiltonian of the total system atom and field is written as

intˆ ˆ ˆ ˆ

field atomH H H H= + + [539]

where daggerˆ ˆ ˆfield cH a aω= is the Hamiltonian of the free field ˆˆ2atom aH σω=

is the Hamiltonian of the atom and int

ˆˆ ˆ2

H ESΩ= is

the interaction Hamiltonian of JC The interaction Hamiltonian can be expressed in terms of annihilation a and creation daggera operators of the field E such that daggerˆ ˆ ˆ( )E a a= + on the one hand and ladder operators of the pseudospin S defined by ˆ e gσ+ = and ˆ g eσminus = on the other hand


Finally in the frame of the rotating wave approximation ( c a c aω ω ω ωminus ltlt + ) and the JC Hamiltonian can be written as

dagger daggerˆˆ ˆ ˆ ˆ ˆ ˆ ˆ( 1 2) ( )2 2c aH a a a aσω ω σ σ+ minus

Ω= + + + + [540]

To simplify the calculations it is more convenient to change the Hamiltonianrsquos form and express it as a sum of two operators that commute and which may have common basis vectors In that case δ a term corresponding to the detuning in frequency between ωa the atomic transition and ωc the frequency of the field mode is introduced as expressed in equation [540] The resulting Hamiltonian is well adapted to study the coherent evolution of the atomndashfield system and is expressed as

dagger dagger0 1

ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) ( )2 2 2cH H H a a a aσ σω δ σ σ+ minus

Ω= + = + + + + [541]

where dagger

0

ˆˆ ˆ ˆ( )2cH a a σω= + with the vacuum energy term put equal to

zero to simplify and dagger1

ˆˆ ˆ ˆ ˆ ˆ( )2 2

H a aσδ σ σ+ minusΩ= + +

The eigenstates of H0 (uncoupled states basis) are of the form n g and n e where n is the number of quanta in the field mode

(one mode only per frequency ωc) On this basis the Hamiltonian H is block diagonal each block representing a subspace spanned by the states ne and n + 1 g In this subspace the matrix elements of H are expressed by

12 2ˆ1 ( 1)

2 2

ac

i ja

c

n nH

n n

ωω

ωω

Ω + + Ψ Ψ =

Ω + + minus

[542]

where i n eΨ = or 1i n gΨ = + with i =12 The diagonalization of the 2 times 2 matrix leads to the eigenstates and eigenenergies of the


atomndashfield system (dressed atom) Then for a given n it is possible to write

cos( ) sin( ) 1

sin( ) cos() 1n n

n

n n e n g

n n e n g

α αα

+ = + +

minus = minus + + [543]

for eigenstates where 12 tan ( 1 )n nα δminus= Ω +

and

2 2( ) ( 1 2) ( 1) 2cE n n nω δplusmn = + plusmn + + Ω [544]

for the energy The energy levels of the atom are split into doublets and the separation in energy between each doublet depends on the number of photons n and the detuning δ The corresponding angular frequency 2 2( 1)nδ + + Ω is termed as Rabirsquos angular frequency At resonance δ = 0 and αn = π4

The quantum specificity of the evolution of the atomndashfield system can be determined from the equation of evolution of stationary states in the Schroumldinger representation Starting from δ = 0 the atom prepared in the state e and the field in a Fock state with n photons with αn = π4 the initial state of the whole system is written as

( )1(0) 2

n e n nΨ = = + + minus (δ=0 and αn=π4) In absence of

relaxation the state is expressed at time t as

( ) cos( 1 ) sin( 1 ) 12 2

t n t n e i n t n gΩ ΩΨ = + + + + The probability

that the system is in the ground state is then given by

( )2 1( ) 1 ( ) 1 cos( 1 )2

ngP t n g t n t= + Ψ = minus Ω + The back and forth

between the two excited and fundamental levels occurs at Rabirsquos pulsation Ω

Due to the dissipative coupling of the field and atom the system with their environment evolution of the system is incoherent The


dissipation processes may be described by the master equations of the density operators of the atom and field [MES 64 LOU 64 COH 87 FEY 98] If the field is in a Glauberrsquos coherent state

0(0)champ n

nC nα

infin

=

Ψ = = (2

2

n

nC en

α αminus= equation [362]) and

the atom in an excited state then the initial state of the system is given

by ( )(0) cos( ) sin( ) n n nn

C n nα αinfin

Ψ = + minus minus and the state of the

system at time t is given by ( )tΨ =

( )( ) ( )cos( ) sin( ) iE n t iE n tn n n

nC n e n eα α+ minus

infinminus minus+ minus minus

The probability to find the atom in its ground state is then given by

( )0

1( ) ( ) 1 cos( 1 )2g

nP t p n n t

infin

=

= minus Ω + the sum of 1( )2

ngP t =

( )1 cos( 1 )n tminus Ω + weighted by the probability to find n photons in the

field mode (equation [364])

This probability is a superposition of several Fourier components each describing the Rabi oscillation of the atom in the n photon field This probability is independent of the coherence of the initial field and its Fourier spectrum has a discrete structure that is the result of quantization of the field The peaks in the spectrum are located at Ωradic(n + 1) and the width of each line is proportional to the statistical weight of each number n of photons in the initial state of the field When the number of photons in the mode is well defined the probability of detecting the atom in one of these levels oscillates sinusoidally at a frequency equal to the separation between the dressed state of the corresponding doublet However if the field is in a state in which the number of photons is not defined oscillation is no longer sinusoidal This is particularly the case when the field is in a coherent state the oscillation becomes blurred after a while then it regenerates then it blurs again and so on The revival of oscillations of the probability pf(t) is a signature of the quantification of the field since in a semiclassical description of the interaction between the atom and


the radiation the revival cannot occur This model was taken up and extended to treat for example two-level atomic systems with a coupling term depending on the intensity of the field [BUC 81] or depending on the intensity of the field and varying over time [BUZ 90]

54 Semiclassical models

The semiclassical models are based on the classical models modified by the introduction of a contribution of quantum origin Matter is treated quantum mechanically and the electromagnetic field is given by Maxwell equations The total Hamiltonian of the physical system under consideration and the field is written by replacing atom by system in equation [538]

intˆ ˆ ˆ ˆ

field systemH H H H= + + [545]

In equation [545] ˆ ˆ ˆ ˆsystem N e e NH H H H minus= + + where ˆ

NH is the

Hamiltonian operator of the nuclei êH is the Hamiltonian operator of

the electrons and ê NH minus is the Hamiltonian operator of the interaction

between the nuclei and the electrons

In dielectrics and semiconductors the free charge carriers the imperfection centers and the collective vibration of the crystal lattice interact with electromagnetic radiation in the range of long wavelengths The region of mid-infrared and far infrared concerns mainly lattice vibrations of ionic crystals Most of the methods of characterization by the electromagnetic radiation are used in the optical region extending from near UV to near infrared (200ndash 1000 nm) In this region the absorptions are due to interband transitions since the photonrsquos energy is higher than the energy gap measured from the top of the valence band to the minimum of the conduction band The study of the adsorption band edge provides useful information on the energy gap and the band structure near the extrema of these bands These characteristics are used to distinguish the structures of materials and to study the effects of stresses such as


applied pressure temperature or vibrations [MIR 13 KHE 14] These studies are complementary to that by laser-induced fluorescence on color centers and which also provide information on the piezoelastic or piezoelectric properties of materials [FOR 72 BAR 73 MUR 02 JAY 05 BAS 11 TRO 13]

In QM the optical transitions are triggered by the interaction of the electromagnetic field with the electrons and are calculated from the interaction Hamiltonian int

ˆ ( )H r t As given in equation [359] of Chapter 3 it is expressed as

22

intˆ ( ) ( ) ( )

2k k kk

e eH r t A r t p A r tm mminus= + The matrix elements of

this operator are calculated in the Hilbert space spanned by the eigen functions of the unperturbed Hamiltonian operator

0ˆ ˆ ˆ

field systemH H H= +

The physical system composed of the electromagnetic field on the one hand and charged particles on the other hand can be considered as being subjected to a time-dependent perturbation through the interaction term The optical transition rate Γif is calculated by applying the Fermi golden rule

2

int2 ˆ ( )if f if H i E Eπ δ ωΓ = minus minus

[546]

where i is the initial state and f is the final state A matrix element must be calculated for each set of initial and final states If the system is a solid the matrix elements vary little when one considers the successive band states Then the total rate of optical transitions can be expressed as

2

int

2 ˆ ( )if f ii f i f

f H i E Eπ δ ωΓ = Γ = minus minus

[547]


In the case where the initial state i and final state f are states associated with the valence and conduction bands respectively the matrix element is expressed as int int

ˆ ˆf H i BC H BV= The transition rate Γ is then given by

2

int v v2 ˆ ( ( ) ( ) )c cBC H BV E k E kπ ρ ωΓ = minus minus

[548]

where v v( ( ) ( ) )c cE k E kρ ωminus minus is the joint density of states defined by

v v32( ) ( ( ) ( ) )

8c cdk E k E kρ ω δ ωπ

= minus minus

The shape of the joint density of states is responsible for prominent features in the optical constants and gives useful information about the structures at the band edges It corresponds to specific points of high symmetry close to the band edges points in the Brillouin zone By transforming the integral over k to an integral over the energy E ( v v( ) ( )k n k c n cdE Edk E dk E E dk d E E= nabla = nabla nabla minus = minus

)

and by introducing the surface of constant energy S(E) in k space such that ndk dSdk=

the joint density of states can be expressed as

v 32( )

8cρ ωπ

=v

v ( ) ( )( )

ck cEspace k E k E k

dSE E

ωminus =nabla minus

At critical points

defined by v( ) 0k cE Enabla minus rarr (Van-Hove singularities) the contribution of v ( )cρ ω to the transition rate Γ becomes very large that has an impact on the dependency of optical constants with frequency

There are four such types of critical points If a Taylor expansion of v( ( ) ( ))cE k E kminus is performed in the vicinity of k0 the constant energy surface can be written as follows

32

v 0 01

( ( ) ( )) ( ) ( )c g i i ii

E k E k E k a k k=

minus = + minus where i = 1ndash3 represent the


three (x y and z) coordinates and where 2

v2 ( ( ) ( ))i ci

a E k E kkpart= minuspart

Depending on the sign of the airsquos there are four types of points minimum (all gt 0) the maximum (all lt 0) or saddle points (two gt 0 and one lt 0 or vice versa) These critical points (Van Hove) appear as a result of the periodicity of the crystal lattice The optical properties of solids are determined by including in the calculations the density of optical states which becomes extremely high for electronic transitions in the vicinity of the various critical points They are responsible for prominent features in the intrinsic absorption or emission edge [MIR 13]

From the BeerndashLambert law the absorption coefficient α(ω) at

angular frequency ω is given by 1( ) dII dz

α ω = where I is the time

average of the Poynting vector (equation [423] in Chapter 4) or light

intensity ( 20

2cnI E= equation [33] in Chapter 3) and dz is the length

through which the light wave propagates Its calculation from QM

leads to1( ) dII dz SV

ωα ω Γ= = where S is the Poynting vector

representing the incident electromagnetic flux V is the volume of the first Brillouin zone Γ is the transition probability per unit time and ħω is the absorbed photon energy For interband transition the angular frequency dependence of the absorption coefficient α(ω) is dominated by the joint density of states Then for insulators and semiconductors it can be shown that the absorption coefficient near the threshold can

be written as 2( ) ( )n

gC Eα ω ωω

= minus

where n = 1 for direct

transitions and n = 3 for forbidden direct transitions and Eg is the band gap between the valence band and the conduction band [MIR 13 KHE 14]

The calculations in the quantum framework are not always easy and in this case the empirical parametric forms of dielectric functions


are used They are valid in a narrow region of the electromagnetic spectrum as that of the TaucndashLorentz and CodyndashLorentz based on the Lorentz models for bound electrons

541 TaucndashLorentz model

This semiempirical model is a parametric model of the dielectric function of materials mainly in determining the optical properties of amorphous semiconductors dielectrics and polymers It was developed from the model established by Tauc et al [TAU 66a TAU 66b] and the standard Lorentz oscillator model It is called the TaucndashLorentz model because it combines in the expression of the imaginary part of the full dielectric function ε2TL the product of the imaginary part of the Tauc dielectric function εiT by that of Lorentz εiL

The imaginary part of the Tauc dielectric function εiT (E) is expressed by

2

( ) gi T g T

E EE E A

Eε

minus gt =

[549]

where AT is the Tauc coefficient related to the amplitude (strength of the transition) of the dielectric function E is the photon energy (E = hν = ħω) and Eg is the optical band gap

As a function of the energy a simple Lorentz oscillator εiL(E) is described by the following mathematical form

0 2 2 2 2

0

( )( )

Li L

A E CEEE E C E

ε =minus + [550]

where AL is the coefficient related to the amplitude (strength of the transition) of the Lorentz oscillator C is the broadening term of the extinction peak (γL) and E0 (E0 = ħω0) is the center of the extinction peak in terms of energy


This gives the expression of the imaginary part of the full dielectric function ε2TL of the model of TaucndashLorentz as

2

20

2 2 2 2 2 20

( ) ( ) ( )for( )

( ) ( )( )

for0

TL i T i L

gg L

g

E E EE EE E A E CEG E L E

E E E C EE E

ε ε ε= times

gt minus= times= minus +

le

[551]

Taucrsquos law (G(E)) in equation [539] corresponds to the relationship 2 2

2 ( ) ( )gE Eω ε ωasymp minus In this expression Tauc supposes that the energy bands are parabolic and that the matrix elements of the momentum operator are constant for interband electronic transitions The product conducted in space of the energy or frequency corresponds to a convolution in the time space Then the real part of the dielectric function is calculated from the integral KramersndashKronigrsquos relations such that

21 1 2 2

( )2( ) ( )g

TLTL

E

E P dE

ξε ξε ε ξπ ξ

infin

= infin +minus [552]

where in P stands for the Cauchyrsquos principal part of the integral This function was first used by Jellison and Modine [JEL 96a JEL 96b] to reproduce the optical constants n and k of amorphous silicon Si SiO As2S3 and Si3N4

A critical comparison of the dielectric function was carried out with the parametric form of Forouhi and Bloomer [FOR 86] used for amorphous materials and giving an extinction coefficient in the form

2

2

( )( ) g

FB

A E Ek E

E BE Cminus

=minus +

[553]

where in A B C and Eg are fitting parameters The refractive index is calculated by integration using the KramersndashKronig relationship including a term n(infin) as an additional fiiting parameter Three


non-physically acceptable situations have been corrected with the model of TaucndashLorentz namely that

ndash interband transitions cannot contribute to the optical absorption when E lt Eg (kFB(E) gt 0 for E lt Eg)

ndash k(E) 0 as 1E3 or faster than E + infin from theoretical and experimental results (when E +infin kFB(E) constant)

ndash the authors Fourouhi and Bloomer did not use the symmetry of time reversal to calculate nFB(E) in the KramersndashKronigrsquos integration of kFB(E) which imposes the relationship kFB(minusE) = kFB(E)

An extension of the TaucndashLorentz model was proposed to improve the shape of the dielectric function It includes a contribution of the absorption of defects present in the material in the form of a decreasing exponentially from the band edge in the band gap called the Urbach tail This model is expressed as follows

20

2 2 2 2 20

2

( )1( )

( )0exp

L gc

UTLu

cu

A E C E Efor E E

E E E C EE

A E for E EE E

ε

minustimes ge minus +=

lt ltminus

[554]

where the first term (when E ge Ec) is identical to the TaucndashLorentz function and the second term (when 0 lt E lt Ec) represents the Urbachrsquos exponential

Parameters Au and Eu are determined taking into account the continuity property of the optical function and its first derivative The following relations are therefore used for Au and Eu

2 2 20

2 2 2 2 20

20 0

2 2 2 2 20

2( )( ) 2 2 ( )( )

( )exp

( )

cu c g c c g

c c

gcu

u c c

C E EE E E E E EE E C E

AE C E EEAE E E C E

+ minus= minus minus minus minus +

minus = minus minus +

[555]


542 CodyndashLorentz model

The CodyndashLorentz model is a parametric model like that of TaucndashLorentz It was proposed by Ferlauto et al [FER 02] to characterize an amorphous material when the photon energies are around the energy gap between the valence band and the conduction band The model combines the expression of Cody (G(E)) with that of Lorentz (L(E)) and a contribution to the absorption band gap of the type proposed by Urbach [URB 53] leading to

20

2 2 2 2 2 2 20

2

1

( )( ) ( ) for( ) ( )

( )for 0exp

gt

g pCL

tt

u

E E AE CEG E L E E EE E E E E C EE

E E E E EE E

ε

minus= times ge

minus + minus += minus lt lt

[556]

The expression of Cody assumes parabolic bands and matrix elements of the dipole moment operator which are constant for interband electronic transitions It is a better representation of the start of absorption of some amorphous materials than the function given by Tauc [TAU 66a TAU 66b] Et is the limit between the transition of the Urbach tail which defines the top of the valence band and the interband transition It represents the transition energy between the Urbachrsquos domain and the CodyndashLorentz domain For 0 lt E lt Et the absorption coefficient is given by ( ) exp( )uE E Eα asymp Urbachrsquos energy Eu represents the width of the states at the border within the band gap It is a measure of the structural disorder in the material E1 is defined such that ε2CL is a continuous function when E = Et ie E1= EtG(Et)L(Et) Ep defines a second transition energy which separates the start of the absorption E lt Ep + Eg from that of the Lorentz oscillator E gt Ep + Eg The other parameters are defined in the [542] same way as in equation [541]

55 Conclusions

In the condensed phase the optical properties of a material are less sensitive to material structure changes than mechanical properties For example although the size of a crystallite is smaller than the


wavelength of light there are only slight differences between the dielectric functions of a crystal and those of a polycrystal which is made up of crystallites In the condensed phase the spectra resulting from the interaction of light and matter are observed as absorption bands Those observed in the far-infrared range are due to transitions between energy levels of the phonons of the lattice and the vibrations of the nuclei Those located in the near-infrared visible or ultraviolet are due to electronic transitions In comparison the interaction of light and matter in the gaseous phase leads to a line spectrum for transitions between discrete energy levels (electronic vibrational rotational) or to a continuous spectrum for electronic transitions between a discrete level and the continuum

The optical properties of solids are determined by including the density of optical states in the quantum calculations The specific features that appear in the spectra result from the very high value of the density of states for electronic transitions in the vicinity of various critical points The energies of these critical points (Van Hove) are due to the periodicity of the crystal lattice When quantum calculations are too difficult empirical parametric forms of dielectric functions are used These functions are simpler to use and are representative of the differences in properties which are due to structural differences They are not valid throughout the entire electromagnetic spectrum but on specific ranges The TaucndashLorentz and the CodyndashLorentz use oscillators developed in the Lorentz model to interpret the measurements more accurately These models are realistic and are applied in ellipsometry to characterize materials In the case of amorphous solids the atomic or molecular orientations are random on the scale of distance of a few close neighbors The overall optical properties of the amorphous materials are mainly determined by local bonds at the atomic scale

6

Experimentation and Theoretical Models

Polarized light from a laser source is applied in non-intrusive laser-induced fluorescence (LIF) and double resonance (DR) techniques in order to study the interactions of a physical system (atoms ions molecules clusters of molecules etc) with its environment (matrix nanocage thermostat etc) These time- and frequency-resolved spectroscopic methods are complementary For measurements they require only a small illumination surface (laser beam diameter) and a small volume corresponding to the product of this surface by the laser penetration depth for measurements A pulsed femtosecond (10minus15 s) picosecond (10minus12 s) or nanosecond (10minus9 s) laser creates a non-equilibrium thermodynamic state of the system by bringing it to an excited state Return to equilibrium occurs by radiative relaxation as fluorescence (duration shape intensity and emission energy) or by non-radiative relaxation that can be probed by a second laser This laser can be pulsed with an adjustable time delay relative to the laser pump or continuous It is used to track the redistribution of the energy deposited by the pump from the variations in its transmission or reflection (duration form intensity) The return to equilibrium may alter the physical properties of the system as a result of thermal expansion the presence of defects interface changes or structural rearrangements This can have an effect on the characteristics of the fluorescence or the probe signal Theoretical models make it possible to analyze these variations in terms of the systemrsquos characteristic interaction parameters investigated at the nanometer scale by an inverse method The mechanisms implemented in the theoretical models can be transposed to mechatronic systems that exchange and dissipate energy in assemblies Thus the channels through which energy can be exchanged and processed in the system and its environment can be determined in a mechatronic device


First Edition


61 Introduction

The experimental techniques of LIF and DR are based on methods proposed by Brossel and Kastler [BRO 49] and Kastler [KAS 50] These techniques are applied within the framework of optical detection methods in magnetic resonance for studying structures at atomic levels Laser spectroscopy resolved in time LIF and DR has evolved in parallel with the development of lasers Shortly after the theoretical model proposed by Schawlow and Townes [SCH 58] showing the feasibility of the infrared and optical maser the first lasers that of Maiman in 1960 [MAY 60] (solid-state laser ruby pulsed 339 microm) and that of Javan et al [JAV 61] in 1961 (He-Ne gas laser continuous 115 microm) opened the way to light sources more interesting than conventional sources for studies by LIF For more details see the initial studies performed in close collaboration with the inventors of laser sources themselves such as the measurements of the vibrational relaxation parameters of CO2 by the LIF technique [HOC 66] or the selective laser photocatalysis of bromine reactions with a laser source that excites the bromine molecules in the gas phase up to the first binding quantum states close to the dissociation continuum [TIF 67] In [TAN 68] concerning the use of the LIF technique to study the potassium dimer with a He-Ne laser designed in the laboratory the possibility of using laser sources for photochemistry molecular spectroscopy or energy transfer studies through the development of intense monochromatic lasers (high power per unit area) is suggested A review of the LIF technique is given in [ZAR 12]

The pump-probe technique or DR method is complementary to the LIF method It requires lasers with pulses of durations lower than microseconds which is the characteristic time of the flash lamps used before the invention of lasers for time resolved spectroscopy [POR 50 POR 68 ETS 78 DEM 96] The technique has evolved from the scale of nanoseconds (1 times 10minus9 s) (Q-Switch laser credited to Gould) [GOU 77 BER 04 05 HUM] to the picosecond (1 times 10minus12 s) and femtosecond (1 times 10minus15 s) (laser with passive or active modes locking)

Experimentation and Theoretical Models 135

The characteristic time of the motion of atoms in molecules corresponding to the dynamics of chemical bonding at the atomic scale is the femtosecond ultrafast pump-probe spectroscopy (femtosecond spectroscopy) which involves the use of ultrashort pulsed laser and methods of highly specialized detection results in femtochemistry [ZEW 00] In a picosecond regime a laser generates sound waves in condensed matter Using the pump-probe technique it is possible to determine the variations of the reflection of the probe at the surface of the pumped material and by an inverse method to determine the structural properties of the material To study energy transfers it is preferable to use a nanosecond laser to overcome the effects of sound waves propagating in the material

This chapter recalls the characteristics of a laser and describes the LIF and DR techniques Different theoretical models developed to interpret observations concerning LIF or DR experiments on condensed matter are then described

62 The laser source of polarized light

A classical light source is obtained through excitation of the source system for example by electron bombardment (neon lamp emitting a line spectrum) heating by current (filament lamp emitting a continuous spectrum like a black body) or by passing a current in a semiconductor (light emitting diode (LED)) emitting monochromatic or white light (blue LED combined with a yellow phosphor) In these devices the source system which after excitation is in a non-equilibrium thermodynamic state returns to its original state by spontaneously emitting photons (vacuum fluctuations)

Laser is the acronym for Light Amplification by Stimulated Emission of Radiation this acronym was created by Gould [GOU 77 BER 04 HUM 05] A laser is a device that emits an intense polarized monochromatic light beam This beam usually has a very small cross-section and a small divergence The beam is almost a


perfect plane wave Its emission properties result from photons that are generated by the induced emission mechanism and not by spontaneous emission

The first laser was a solid (Ruby) pulsed laser [MAY 60] emitting in the red region (6943 nm) The first continuous laser [JAV 61] is a He-Ne gas laser emitting in the near infrared (115 microm) Thereafter the laser worked on other wavelengths in the visible red (6328 nm) and also infrared regions (339 nm) Today there are various types of lasers gas solid (diode) liquid molecular electronic and X-ray lasers

621 Principle of operation of a laser

In its simplest setup a laser consists of three basic elements an amplifying medium that is also the light source a pumping system to excite the amplifying medium and an optical cavity of a FabryndashPerot (FP) type to select an emission mode This device is analogous to an electronic oscillator that includes a source of electrical power (pumping) a selective frequency amplifier (FP cavity) and a feedback loop (back and forth in the cavity) To operate the gains of the oscillator must be greater than the losses and if the setup of the oscillator is stable the laser emission locks in to the photon noise resulting from spontaneous emission (analogous to thermal noise in electronics) The oscillator can enter saturation mode

Two conditions are necessary to obtain a coherent light from stimulated emission the physical system needs to be excited by a pumping system in order to bring it to a higher energy state This means that a population inversion is necessary (this is a system state where there are more atoms in the upper level than in the lower level so that the photon emission dominates the absorption) The higher state needs to be a metastable state that is a state in which the atoms remain a long time enough so that the transition to the lower state is achieved by a stimulated mechanism rather than by spontaneous emission


Figure 61 Typical laser mounting back and forth standing wave

The pumping system brings the physicalndashchemical system gas (He-Ne ionized argon) solid (ruby sapphire titanium) liquid dye or semiconductor to a non-equilibrium thermodynamic state The resulting population inversion of the energy levels favors spontaneous emission of several classes of photons caused by vacuum fluctuations in the excited medium As a result of the back and forth paths in the FP cavity only one class of photons remains The final mode is characterized by the wave vector parallel to the path followed in the cavity and to the selected polarization in the cavity Generally plates positioned at Brewster angle are used to select the p polarization A typical laser assembly is illustrated in Figure 61 The FP cavity is used as a filter It generates standing waves at the wavelengths λn = 2ln where l is the cavity length It selects the wavelength λn

(frequency nn

cνλ

= ) and provides feedback for the amplification of a

single mode In the absence of the amplifying medium and if the diffraction losses are neglected the quality factor Q of the cavity shown in Figure 61 is given by

22(1 )

lQr

πλ

=minus

R=100

PUMPING

AMPLIFYING MEDIUM

R=98 T=2

2L = nλ

Photons

ν = nc2L


for a given wavelength λ such that Q = 109 for l = 1 m λ = 600 nm (frequency ν = 5 1014 Hz) and r = 098 The width of the resonance of

the cavity is equal to c QννΔ = or 05 MHz compared with the interval

between modes

150MHz2ncl

νΔ = =

In the case of a semiconductor-based laser the population inversion is achieved in a p-n junction and the light is produced by radiative recombination of an electron-hole pair Because of the small size of a diode the cavity is built in situ on the semiconductor It is obtained by polishing the front and back sides at the ends of the junction placed in a heterostructure in order to confine the electron-hole pair The low dimensions of the exit window for the output of photons leads to a divergence of the light beam by diffraction that is corrected by a suitable lens In the case of a laser diode the light characteristics (intensity wavelength) depend on two parameters the temperature and the injection current in the junction Diode lasers are thus tunable because the wavelength of the laser light can be scanned over a certain range by varying these parameters

Lasing conditions are achieved if the gain of the amplifying medium exceeds the cavity losses and if emission locks in to the noise (spontaneous emission or thermal radiation) The laser can also enter in a saturation mode

The interaction of light and matter in the amplifying medium can be modeled according to the theory described in Chapter 5 If the amplifying medium is assumed to be diluted in a matrix the total electric field tE

in the cavity is the sum of the incident field iE

and

depolarizing field dE

dE

is due to the phenomenon of polarization of the material by the incident field ( ( ) ( )ep t er t Eα= minus =

and is


determined by equations [56] and [57]) The total field is then expressed as

( )( ) ( ) ( ) (00) i nkz tt i d iE z t E z t E z t E e ωminus= + =

[61]

where0

1 1 12 2

Nn χ αχε

= + + = + The polarizability iα α αprime primeprime= +

susceptibility iχ χ χprime primeprime= + and refractive index n n inprime primeprime= + are complex The real parts αprime and χ prime lead to the real part of the refractive index that accounts for the dispersion The imaginary parts αprimeprime and χprimeprime are connected to mechanism of the energy exchange between the field and the atoms or ions of the amplifying medium In the usual case of absorption tE

lt iE

with and χprimeprime being positive When they are negative the medium is an amplifier tE

gt iE

We can define a characteristic length z0 which is the inverse of the gain (or

extinction) coefficient kχprimeprime of the medium from 0z

zk ze eχ minusprimeprimeminus =

The macroscopic formulation may be connected to the microscopic point of view at the atomic level (atoms ions etc) by applying the golden rule of Fermi to the levels g and e in resonance with the electric field of the incident wave (using the notations of Chapter 5 of the JC model)

The following expression is finally obtained for the polarizability

( ) 2

0

1 1ee gg

a

e p gi

α ρ ρω ω τ

= minusminus minus

[62]

where kkρ is the population at level k and τ is the relaxation time constant reflecting a damping factor The term e p g is the matrix element that represents the probability of the transition (transition moment) from the state g to the state e under the effect of the operator p (dipolar moment) p = minuser (here e is the electronrsquos charge and not the excited state )

α centcent

e


The imaginary part of the polarizability is written as

( )2

2

2 20

1( ) 1ee gg

a

e p g τ τα ρ ρω ω τ

primeprime = minusminus +

[63]

The power transferred to the field by the atomic system is expressed as

2 20

2 2i i

time

E cEdpP Edt

α ω ε σprimeprime

= minus sdot = =

[64]

where σ is the absorption cross-section

0cα ωσεprimeprime

=

If ( )0 0ee ggρ ρminus lt then a phenomenon of absorption occurs P and

σ gt 0 (induced absorption) and if ( )0 0ee ggρ ρminus gt a phenomenon of

amplification takes place P and σ lt 0 corresponds to the stimulated emission When a population inversion is achieved by pumping that brings the system in a non-equilibrium thermodynamic state If one expresses the gain G by

0

G kNk αχεprimeprimeprimeprime= =

where N is the number of atomic systems (atoms ions molecules etc in a solid liquid or gaseous medium) αprimeprime is the imaginary part of the polarizability (inversion rate) and k is the wave vector the dynamic variation of k is between 0 and 108 mminus1 and more

The parity of the dipole moment p is odd such that the p matrix elements are non-zero between states of different parities ( 0 0e p g e p e g p gne = = ) The maximum value of the

( )0

0ee ggρ ρ- gt


dipole moment is given by ( )max12

p e p g g p e e p g= + =

which is real This property results in a limiting value for the

amplification given by 2

se p g E e p gτ =

where Es is the

field amplitude beyond which there is saturation of the amplification (nonlinear zone of the polarization) The power at saturation is then given by

22

0 02

12 2

ss

cE cPe p g

ε ετ = =

For a transition moment value of 29max 0 10p e p g qa minus= = asymp

(a0 is the Bohr radius = 529 times 10minus11 m) 910 sτ minusasymp PS is determined to be of the order of 1 Wmminus2 (Ps significantly varies from a system to another)

622 The specificities of light from a laser

The emission properties of a laser source are closely related to the coherence of the light beam which can be defined temporally or spatially Lasers may emit beams that are characterized by maximum theoretical spatial and temporal degrees of coherence Their descriptions in terms of waves show no randomness All emitted photons are in phase in time and space

Figure 62 represents a real divergent beam of section S measured in the plane where the cross-section of the beam is least (in the FP cavity of a laser the ldquowaistrdquo is the region where the section S is least) either at the source or at its image When the electric field of the wave is in phase on a surface Σ smaller that S then the following relationship holds S ΔΩ gtgt λ2 In the case of a laser beam Σ rarr S and consequently S ΔΩ cong λ2 The notion of ldquodirectivityrdquo is therefore closely linked to the spatial coherence It is the size of the coherence area Σ that determines ΔΩ


The spectral purity is associated with the temporal coherence The inverse of the emission width Δν defines the time necessary for the adjacent areas Σ that cover the section S of the beam to be renewed The notion of temporal coherence is therefore closely linked to the property of monochromatic radiation For ordinary beams time and coherence area are mainly statistical quantities

Figure 62 Divergence of a beam from a source in terms of areas Σ of sources in phase (spatial coherence) and solid angle ΔΩ

6221 Monochromaticity (temporal coherence)

The light from a laser is concentrated in a narrow band of frequencies Typically Δν le 103 Hz with Δνν = Δλλ le 10minus12 or 01 nm per 100 m or 1 s per 105 years

6222 Directivity (spatial coherence)

Light from a laser source has a low divergence (θ) Its value is θ sim 10minus5 radians which corresponds to a spot of diameter of 1 km at 100000 km distance

6223 Power or high radiance (BndashE statistics)

Lasers can be classified into two categories depending on whether they operate in a continuous or pulsed mode the concept of power

REAL CASE Diffraction

Case A Angle α Case B Solid Angle ΔΩ

S ΔΩ=λ2 Throughput is a Constant


delivered by a laser depends on its operating conditions A continuous laser delivers only powers of the order of milliwatts (He-Ne laser semiconductor diodes) possibly 10 kilowatts for industrial CO2 lasers A pulsed laser can deliver powers higher than gigawatts The irradiation power of a laser is high generally of the order of 1012 W If the laser beam is focused on a surface S of area 10 microm2 the value of the waversquos electric field is equal to

1 213

0

10 VmPEcSε

=

This value is to be compared to the interatomic field which is in the order of 1011 Vm The laser provides short pulses of the order of 10minus12 s the wave trains being 300 microm long

6224 Frequency tunability

In the optical field the selectivity of the laser sources is possible across the entire spectral range between ultraviolet and infrared radiation because of the dye and solid lasers that are tunable in frequency

63 Laser-induced fluorescence

The width of a spectral line is directly related to the lifetime of the energy levels in resonance with the transition that gives rise to the line LIF technique is applied to study the relaxation of these levels resulting from their interactions with the surrounding environment It was widely used in the UV or visible range to study in model systems (molecules trapped in a matrix of inert gas at very low temperatures ranging from 5 to 30 K) the electronic and vibrational relaxation of trapped systems interacting with their environment

631 Principle of the method

At low temperatures (5ndash30 K) the trapped molecules are all in their ground state The method consists of upraising the molecule


from the ground state to an excited vibrational level by tuning the laser frequency on the selected transition (thick black line in Figure 63) The pump laser is then in resonance with a transition between two vibrational levels of the ground state electronic level The molecular system finds itself in a non-equilibrium thermodynamic state During the equilibrium recovery by the decay of the populated excited level a fluorescence can be emitted from all the levels by which the molecules transit below the excited level as shown in Figure 63 (lines 1 2 and 3 at 16 microm and dotted lines 10 microm) for the 13C16O2 molecule trapped in an argon matrix The spectral analysis of this fluorescence allows the identification of the emitting levels and the corresponding transitions while the time analysis allows the characterization of the dynamics of the energy relaxation

Figure 63 Vibrational levels of 13C16O2 pump (ν3) and transitions of fluorescence at 16 microm (1 2 3) and 10 microm

227951 227366

0 00 1 (1)

1 11 0 (1)

125710125841 125801

1 00 0 (1)

0 22 0 (1)

64310 64491 64451

0 11 0 (1)

0 00 0 (0)

1 00 0 (2)

0 33 0 (1)

1 11 0 (2)

ν3

(ν1+ ν23ν2)

Fermi resonance

(ν12ν2)

Fermi resonance

ν2

1 2

3

10 μm

Wave number cm-1 (Argon)

128610128841 128801

203482 203398 203322

193034 193569 193440

188210 188 441 188401

137302136954 136981


The duration of a fluorescence emission gives in principle information on the lifetime of the level that relaxes If the molecule is totally isolated as in molecular beams it represents the radiative lifetime of the level which is connected to the Einstein coefficient of spontaneous emission of the transition (equation [65]) This is the maximum duration that the emission may have and in this case the number of fluorescence photons is equal to the number of excited levels In matrices spontaneous fluorescence is the most studied purely radiative relaxation If the molecule is in a solid or gaseous environment energy losses can occur by collisions (gas phase) or by interaction with the cage (solid phase) The observed lifetime is then shortened by these non-radiative phenomena which decrease the population of the emitting level The amplitude of the start of the fluorescence signal is still the same but the duration is shorter and its amplitude is lower it lacks the photons corresponding to the molecules which are not relaxing radiatively

The spontaneous fluorescence can sometimes be drastically attenuated by a stimulated fluorescence This coherent emission requires a population inversion between two levels of the trapped molecule The stimulated emission can thus become one of the most effective relaxation channels at low temperature [APK 84] However the systems for which the stimulated fluorescence is predominant do not allow direct determination of the energy transfer constants They are indirectly accessed through modeling and simulation

632 Description of the experimental setup

Figure 64 shows the diagram of the experimental setup used to observe the fluorescence induced in a sample as a result of excitation by a pump laser

64 The DR method

In the gas phase the method of double IR-IR resonance was applied for the first time to study the relaxation of the CO2 energy


levels by collisions by Rhodes et al [RHO 68] and used for the first time in matrix isolation spectroscopy field by Abouaf et al [ABO 73] Various experimental studies were carried out to explore and discriminate the contribution of different relaxation pathways in the energy transfers processes pertaining to diatomic molecules and some small polyatomic molecules (2 le n le 7)

Figure 64 Diagram of the experimental setup for spectroscopy by laser-induced fluorescence


The pump-probe technique consists of the use of two sources of electromagnetic waves (laser maser etc) to study materials Generally the pump sources are used in pulse mode (femtosecond picosecond or nanosecond) The probe sources are used in continuous or pulsed mode The pump source disturbs the medium (creation of ultrasonic waves excited energy levels etc) It is a pulsed laser in resonance with a transition as in LIF which modifies the thermodynamic equilibrium of the population of the levels of a physical system In pulse mode the probe source analyses the recovery of the equilibrium state of the populations with an adjustable time delay with respect to the excitation by the pump The evolution of a disturbance is thus studied taking into account the structural

BEAMABSORBENT

TUNABLE LASER SOURCE

COLLECTING LENS

FLUORESCENCE PROBE VOLUME

PROBED MEDIUMFOCALIZING

LENS

COMPUTER SIGNAL PROCESSING

LIGHT DETECTION

FILTER or SPECTROMETER


characteristics (homogeneous medium medium with defects and heterogeneous medium with interfaces) of the excited medium The acoustic-optical RamanndashNath effect diffraction of light by a phase grating generated by an acoustic wave allows us for example to reveal inhomogeneities and structural defects

Figure 65 IR-IR double resonance signal of the fundamental band

Figure 66 IR-IR double resonance signal of the hot band

This method eliminates the need for a sophisticated detection system The time resolution is limited only by the pulsersquos duration The probe is detected after its interaction with the medium When the probe reaches the medium the two limiting situations encountered are as follows

1) the excited level has not relaxed yet and the probe is weakly absorbed by the medium

probepump

pumpprobe

Transmitted probe

pump

time

Transmitted probe

probeprobe

pump pumppump

time


2) the excited level has relaxed and the probe is strongly absorbed by the medium

The observation of the modulation of the probe signal in time between these two limiting situations allows us to deduce the state of the system during its way back to equilibrium after excitation by the pump The absorption can take place only during the duration of the pulse As a function of the latter the observation of the probe signal can provide information about the dynamics of the absorption by the studied system In general the probe intensity is measured as a function of the delay with respect to the pump Its modulation describes the population dynamics of the energy levels

In continuous mode the probe is tuned on a transition of the pumped system or of another system if energy transfers between subsystems are to be studied The probe laser can be tuned to a fundamental transition or on a hot band In the first case a ldquopositiverdquo signal (Figure 65) is observed because the transmission of the sample increases with the depopulation of the ground state level In the other case the signal is said to be ldquonegativerdquo (Figure 66) as the population created on intermediate levels causes a transient absorption of the probe The DR technique has some advantages over LIF A DR signal carried by the probe can be modulated in the case of a continuous laser In IR its trajectory can thus be followed and this facilitates its alignment In the case of an isotropic fluorescence it is necessary to focus the radiation onto a detector in order to increase the solid angle of detection The DR signal is both proportional to the intensity of the transmitted probe and the number of pumped molecules [ABO 73] and the method is thus more sensitive than the LIF that depends only on the pumping efficiency


In pulsed mode the pulsed beam of the pump laser is divided into two pulses which are focused in the same volume of the test sample (Figure 67)


Figure 67 Diagram of the experimental setup with two beams of the pump-probe spectroscopy by reflection or transmission (A and B)

The possibility of having crystal doublerstriplers or a parametric amplifier and polarization optics or any device with a second laser on path B

The optical path of each component is adjusted so that the pump pulse arrives first to the sample followed by the probe pulse The probe laser cross-section is slightly smaller than that of the pump to limit edge effects The probe delay can be adjusted by increasing its optical path This method is termed the degenerated pump-probe spectroscopy [SHA 96] The temporal resolution is limited by the pulse width In the non-degenerated mode the pump laser and the probe laser have different frequencies [SHA 96] Either a second laser is used or the frequency modification is obtained by frequency doubling or by parametric conversion of the pump laser in nonlinear crystals

65 Theoretical model for the analysis of the experimental results

In the gas phase the vibrational relaxation studies aim to determine the mechanisms that govern the transfer of energy of a system subjected to an external constraint If we are interested only in the vibration excited molecules can lose energy radiatively either by spontaneous emission (stimulated if a FP cavity is used) or non-radiatively during collisions with other particles There are basically

Transmittedsignal

Reflected signal

Ultrafast Laser

AB


two relaxation processes by collisions by V-T transfers and by V-V transfers In both cases the default energy is transferred to the degrees of freedom of translation which is acts as an energy reservoir or thermostat that dissipates this energy The first theory developed by Bethe and Teller [BET 40] uses a model of binary collisions to describe V-T transfers Calculations show that the variation of the intrinsic relaxation time of a diatomic gas depends on the temperature and the density of the molecules The theory extended by Schwartz et al [SCH 52] to gas mixtures leads to relaxation times that are strongly shortened through a V-V transfer in the presence of a dopant or an impurity More elaborate semiclassical theories have been developed by considering the quantum nature of molecules to interpret the broadening andor the shift of vibration-rotation lines by collisions at different temperatures and pressures such as the AndersonndashTsaondashCurnutte theory that considers long-range electrostatic interactions and uses a ldquocut-offrdquo procedure [AND 49 TSA 62] or the impact theory of Fiutak and Van Kranendonck for Raman lines [FIU 62] The theory of Robert and Bonamy [ROB 79] which treats the degrees of translation classically and includes interactions at short and medium ranges to suppress the ldquocut-offrdquo procedure has been successfully confirmed by experimental results in the case of water vapor [LAB 86] or atmospheric carbon dioxide [DAH 88 ROS 88] Taking into account terms originating from an expansion to an order higher than 2 and complex terms in the diffusion matrix the Robert Bonamy Complex theory [LYN 96 GAM 98] allows us to evaluate transfers of energy with a better precision than the dispersion of the experimental results on H2O vapor [NGO 12] or the CO2 molecule [GAM 14]

In the condensed phase mechanisms valid for the gas phase are substantially modified When a molecule is isolated within a solid composed of atoms or molecules at a low temperature [LEG 77] its degrees of freedom in the gas phase are modified While the internal vibration modes are preserved the rotational movements are altered Depending on the size and the spatial configuration of the trapped molecule and following the expansion of the nanocage that is trapping the molecule the rotational movement is sometimes confined to an oscillation of small amplitude about its axis of symmetry (libration)


The trapped molecule interacts with its environment consisting partly of the matrix atoms and partly of the molecules themselves and impurities that cannot be eliminated during the sample preparation All these entities are involved in the relaxation of the energy of excited molecules through a number of processes [ZUM 78 BLU 78 LIN 80] The different associated relaxation pathways in the energy dissipation of small diatomic and polyatomic molecules can accordingly be classified into three groups [LIN 77]

1) V-V intra- and intermolecular transfers

2) transfers to the lattice modes or intrinsic relaxation (interactions between the molecule and the lattice)

3) radiative relaxation or fluorescence emission (Figure 68)

Figure 68 The possible pathways of relaxation of an excited molecule in the solid phase (vj mode) D donors D and A acceptor M matrix (a) V-V intra-

and intermolecular transfers (1 2 3) (b) transfer to lattice modes intrinsic relaxation (45) (c) radiative relaxation fluorescence emission (6)

Dvj=0

vj=1

2

AvA=0

vA=1

0

Jmax

3

5

M0

n

4

D

vj=0

vi=1

vi=2vj=1

ASELASER

1

6

kr

vi=0

6


The influence of the environment plays a key role in the competition between these different pathways that depend on the degree of interaction between the different degrees of freedom of the molecule as well as the intrinsic properties of the solid matrix

The different relaxation channels of molecules that have been transferred to an excited level vi when trapped in a matrix M (condensed phase) (Figure 68) are as follows

1) the intramolecular transfer from vj to vi

2) the resonant migration of the energy from an excited molecule (donor D(vj)) to another identical molecule that is not excited D (vj)

3) the V-V non-resonant intermolecular transfer between an excited donor and an acceptor A

4) the direct transfer from D to phonons of a lattice in a multiphonon process of order n

5) the transfer to phonons through the rotation of the donor D

6) the purely spontaneous radiative relaxation (kr) or amplified stimulated emission

The last three processes proceed through a relaxation mechanism that is termed ldquointrinsicrdquo and for which the molecule is considered to be isolated in the matrix whereas in cases 2 and 3 the energy dissipation requires an interaction with the other trapped molecules All these processes are in competition and some relaxation channels are often masked by the fastest mechanisms Experimental and theoretical studies nevertheless allow us to characterize them all as a function of the different parameters of the complex system

651 Radiative relaxation

When the coupling between the trapped molecule and the host matrix is weak the molecule emits intrinsic fluorescence that is not shortened by non-radiative phenomena The radiation of the molecules depends on the purely radiative lifetime of the emitting level In the case of an electric dipole transition between states i and f the


probability of a radiative transition is given by Einstein spontaneous emission coefficient

4 21 3643

iif if if

f

gA Rh gπτ νminus= = [65]

where gi and gf are the degeneracies of the levels and ifν is the frequency of the transitions involved and ifR is the matrix element of the dipole moment of the given transition In the condensed phase electric dipole moments are hardly perturbed by the crystal field but the refractive index effect plays a significant role in rare gas matrix as the local electric field in the vicinity of the trapped molecule is a function of the mediumrsquos polarization (depolarizing field) Hence a decrease in the radiative lifetime follows given by

2 29

( 2)s gn nτ τ=

+

where sτ corresponds to the radiative lifetime in the solid phase corrected for the effect of the refractive index n of the medium and gτ is the lifetime in the gas phase When non-radiative transfers generally faster than radiative transfers are also active the radiative emission is shortened The real lifetime of the excited state are thus measured in the presence of significant interactions Moreover if the thermodynamic conditions are favorable stimulated fluorescence or a superradiant emission substitute to spontaneous one It is necessary in this case to model the radiant system in order to access to relaxation parameters

652 Non-radiative relaxation

In the most general case the lifetime τ of the level in condensed phase is no longer due to purely radiative transfer and is shortened by coupling to the phonons of the matrix The phonon population is considered as a quantized thermal bath capable of absorbing the energy dissipated in the matrix during a vibrational transition of the molecule

i f


The vibrational energy of the molecule is thus transferred to this reservoir in the form of kinetic energy to the matrix The number of phonons involved in this process is proportional to the energy difference between the vibrational levels The characteristics of these transfers were described by many theoretical models There are two different approaches to model this phenomenon one that assumes that the short-range repulsive forces between the molecule and its environment are responsible for the transfer to phonons [NIT 73 NIT 74a NIT 74b MUK 75 BER 77 GER 77 BER 79] possibly assisted by the rotational motion [LEG 77] and one based on the binary collision model [SUN 68] developed to interpret vibration-translation (V-T) transfers in liquids [ZWA 61 LIT 67] and gases [SCH 52]

In the ldquotransfer to phononsrdquo approach the phonon spectrum is reduced to that of a mean frequency phonon (νm) defined according to the vibrational energy difference ΔE between the initial state (vi) and the final state (vf) ( ) ( )i f mE E v E v NhνΔ = minus = where N is an integer characterizing the order of the process Since the probability of multiphonon transfer decreases exponentially with the number of phonons (N) involved corresponding to the energy difference between

the states and (energy gap law ln( )m

k νν

prop minus ) and the phonon

population of frequency ν depending on the temperature as given by

the equation 1n1

hkTe

ν ν=minus

the dependence of the relaxation with

the temperature is determined by the following law

( )( ) n 1(0)

Nk Tk ν= + [66]

where nν is the thermal population of the mean or local phonon

In the presence of rotational sublevels an additional relaxation channel is activated for the energy transfer The energy difference between two rotational levels is compensated by the excitation of a single phonon This V-R transfer is then more likely to occur than the

i f


V-T transfer In models that include the participation of the rotation [LEG 77] the relaxation constant is expressed empirically as

mexp( J )k αprop minus [67]

where Jm ( mJBωcong ) is the rotational quantum number and B the

rotational constant of the level which is closest to the vibrational level that relaxes Models that are more elaborate take into account the competition between the various relaxation pathways and correlatively adjust the number of phonons to those involving a greater or lesser number of rotational quanta

Some models assume that the relaxation is monitored by the local phonons νL assisted by one or two phonons νph of the matrix [BER 77 GER 77 BER 79] corresponding to the difference in energy

f f i iE(v J ) E(v J ) L phNh nhν νminus = + [68]

where n = 1 or 2 In general Ji = 0 and Jf takes the maximum value Jm which is compatible with an exothermic transfer provided that the moment of inertia of the molecule is small resulting in large quanta of rotation and that the Jm value is not too high The larger ΔJ = Jf minus Ji the smaller the transfer probability (vi Ji) rarr (vf Jf) These models are suitable for hydrogenated molecules (NH3 HCl CH3F etc) and for the study of the isotopic effect

In the ldquomodel of binary collisionsrdquo approach the interaction between the molecule and the matrix is modeled as a collision between hard spheres The dominant factor at a low temperature is an exponentially decreasing function of the difference in translational momentum

f i f(v v ) v 2 Eμ μ μminus asymp asymp Δ

where vf and vi represent the relative speeds (initial and final) and μ represents the reduced moleculendashatom mass This model is valid when the vibrational energy of the molecule is transferred directly to the


matrix without involving the rotational or librational degrees of freedom The relaxation constant is then expressed as

22exp( 2 E)k π μ

μ αprop minus Δ

[69]

where 1α is the range of the repulsive atomndashmolecule potential

exp( )Mm jV Xαasymp minus

with Xj representing the distance that separates the jth atom-matrix (M) to the center of mass of the molecule (m)

Finally models that fully describe the interaction of the molecule with its environment were developed and numerically solved [KON 83 LAK 87 LAK 11] They are based on the inclusion model the Green functions of the perfect crystal for calculating the deformation of the solid host around the trapped molecule [MAR 65] and the adiabatic approximation to decouple the slow modes (phonons libration disturbed rotation etc) characterized by low frequencies and the vibrational modes which are at a higher frequency Vibrational relaxation on lattice phonons is determined by including the adiabatic constraint between the slow and fast modes in the model A cumulant expansion of the evolution operator of the moleculendashmatrix system is used for the calculations With these models the analysis of the competition between rotation and phonon relaxation paths is improved and the various contributions to the constant of relaxation of the different processes multimode multiphonon monomode multiphonon or mixed involving both orientation and phonon modes can be calculated The relative importance of the different channels depends on the temperature

When the molecules trapped in the crystal are subjected to long-range multipole interactions that depend on the distance between the interaction centers it is necessary to consider intermolecular transfers as a possible relaxation path mechanism If this effect is usually negligible in diluted samples this is not the case at high concentrations Because the molecules are closer they can interact


more easily and exchange their energy Under standard conditions of concentration (12000) in a rare gas matrix and at a low temperature (lt 60 K) the shift of the vibrational frequencies induced by multipolar interactions (dipolendashdipole etc) between trapped molecules is small compared to the one induced by the crystalline medium (local field) In intermolecular transfer processes the vibrational energy stored by the molecule D (called donor) as a result of excitation is transmitted to another molecule A (called acceptor) capable of absorbing the energy received Resonant vibrational energy transfers have been observed between two identical molecules in a fast time scales regime compared to the emission time durations of IR fluorescence observed in matrices [DUB 68 DUB 75] and concomitantly non-resonant transfers between molecules belonging to different species [GOO 76] If the acceptors are molecules of different species with slightly different vibrational levels compared to that of the donors D the exchange of energy must be accompanied by the excitation of a phonon of the solid medium The energy gap between the vibration levels should not be too high however Generally in non-resonant energy transfer processes between a donor and an acceptor the participation of one or more phonons is necessary to balance the loss of energy the transfer is then an exothermic process

There is no fundamental difference between electronic and vibrational energy transfers The various theories dealing with electronic energy transfer can be used to describe the intermolecular vibrational energy transfer From a microscopic point of view each trapped molecule is relaxing with the same transfer rate constant during vibrational intramolecular transfer processes However the probability of this transfer is a function of the distance and the corresponding macroscopic time constant is an overall value that takes into account the addition of pair interactions between each isolated molecule The difficulties to develop such models lie in the correspondence between the microscopic rate constants and the macroscopic rate constants that are the only ones that can be measured experimentally

Three types of transfer may compete between a donor D (excited molecule) and a donor D (ground state molecule) between a donor D


and an acceptor A (molecule capable of accepting vibrational energy) and between an acceptor A and another acceptor A through intra- and intersystem processes (Figure 68)

To distinguish between the relative contributions of the processes involved two limiting cases can be considered the one corresponding to a direct transfer of the energy of an excited molecule (donor) to an unexcited molecule (acceptor) and the other case corresponding to the transfer of the excitation energy to the acceptor by migration of energy within a group of donors The energy migrates in the sample from one donor to another donor until it is in the vicinity of an acceptor where it is absorbed Several theories have been proposed to simultaneously process the transfer of energy between donorndashdonor and donorndashacceptor namely Forster [FOR 49] Dexter [DEX 53] and Weber models [WEB 71] in which the dynamics of the transfer is modeled by diffusion equations When the donors D are weakly coupled to the surrounding environment the dipolendashdipole interactions and the radiative relaxation are the main channels for the exchange of energy with the environment In the case of a dipolendashdipole interaction between a donor D and an acceptor A the probability of the intermolecular transfer can be calculated by applying Fermirsquos golden rule [FOR 48] This probability is expressed in terms of an integral and given by the following expression

6 63 1 1 ( ) ( )

8 (2 )DA D ADA A D

P f f dc n R

ν ν νπ ν τ τ

= [610]

where n is the mediumrsquos index τD and τA are the donorrsquos and acceptorrsquos radiative lifetimes RDA is the distance between donors and acceptors and the functions f(ν) represent normalized lines shapes In the case where one of the molecules is inactive in the infrared but has a quadrupole moment the energy transfer can take place by the dipolendashquadrupole interaction The probability of the transition in this case is then inversely proportional to 8

DAR The overlap integral ( ) ( )D Af f dν ν ν is a measure of the degree of resonance between the

transitions of two molecules If the degree of resonance between the absorptions of donors and acceptors is zero the transfer cannot a


priori take place However in matrices the overlapping of frequencies can be achieved via phonons the probability of transfer decreasing with the order N of the multiphoton process The transfer is quasi-resonant when the D and A molecules are identical During this process there is no energy relaxation but simply a migration of the excitation throughout the sample To analyze the role of this resonant migration on the non-resonant intermolecular transfers the following three cases can be considered

ndash the migration between donors is fast the excitation is delocalizing among the donors The acceptors closest to the donors thereby behave as excitation traps It can then be considered that the RDA distance is the same for all donors The donor population then decreases exponentially and the relaxation constant is given by

61

1445 Ao

N MCk kR

= + where ko is the relaxation constant of the

donors NA is the acceptor concentration M is the number of atoms in the matrix per unit volume C is a constant derived from the expression of PDA (equation [610]) and R1 is the distance between nearest neighbors

ndash the migration between donors is negligible each excited molecule transfers its energy to the nearest acceptor The molecules are randomly distributed in the sample and the RDA distance varies statistically around a mean value The transfer probability is given by the sum of the individual probabilities (equation [610]) over the entire distribution of the distances The excited molecules decay is not exponential and follows the law ( ) exp( )k t tαprop minus so that the non-exponential decay of the excited molecules is given by

( ) (0)exp( ( ))D D oN t N k t tα= minus + where α is a constant depending on NA and CDA

ndash the diffusion and transfer times are of the same order of magnitude this hybrid case actually combines the two previous cases Initially the excitation remains spatially localized and the donor population evolves as in case 2 Then the diffusion mechanism distributes the excitation throughout the sample The distance between a donor and an acceptor is no longer critical and all donors are


equivalent as in the first case The tail of the population decay of excited donors is exponential

653 The theoretical model of induced fluorescence

In this section a theoretical model that can be applied to molecules that are trapped in matrices at low temperatures is presented The thermodynamic conditions of such systems lead to a population inversion when the molecules are directly pumped to energy levels that are higher than the first excited state since the intermediate levels are not populated at low temperatures In this case a new relaxation pathway is possible by stimulated emission [APK 84] and which because it is very fast dominates natural slower spontaneous fluorescence

Figure 69 Stimulated emission between two excited levels

By analogy with the theory of lasers the sample can be assimilated to a laser cavity characterized by a loss coefficient which takes into account the state of the samplersquos surface its low surface reflectance coefficient and a strong amplification coefficient (equations [63] and [64]) given the high density of emitting molecules in a solid medium A laser wave is built when the gain is greater than or equal to losses The gain is connected to the population inversion and the stimulated


emission coefficient losses are due to leakage of photons outside the pumped volume The problem can then be reduced to that of a two-level system between which the inversion operates (Figure 69) The evolution of the system can be modeled by a set of coupled equations that describe the evolution of populations N1 and N2 of the levels and an equation that expresses the variation of the number of photons Nph present in the cavity dN3dt This system of equations that can be solved numerically by the RungendashKutta method of order 4 for example is written as

21 2 12 2 21

12 1 12 2 21 1 10

32 1 12 2 21 3

( ) ( ( ) ( )) ( ) ( ) ( )

( ) ( ( ) ( )) ( ) ( ) ( )

( ) ( ( ) ( )) ( ) ( ) ( ) p

dN t N t N t B t N t A tdt

dN t N t N t B t N t A N t Adt

dN t N t N t B t N t A N t Kdt

ρ

ρ

ρ

= minus minus + Λ = minus + minus = minus + minus

[611]

where the coefficient Aij represents the spontaneous emission and the Einstein coefficient and Bij (Bij = Bji) represents the absorption and stimulated emission The value ρ(t) corresponds to the density of the radiation present in the sample and Kp is the loss coefficient of the cavity The pumping term Λ(t) is given by the form of the pump laser pulse It can be associated with a Gaussian function which is expressed as

2

2( ) exp( ) 2

tt δ σσσ π

minus Λ = minus

[612]

where σ is the width at 1e of the laser pulse (for a mid-height width of 5 ns then σ = 7 ns at 1e) and δ is the fraction of molecules carried by the pump laser on the excited level The density of photons ρ(t) present in the cavity at time t is expressed from N3 N and γ2-1 the homogeneous width of the transition source of the photons

213

21

( ) ht N Nνργ

= [613]


The numerical resolution of these equations allows us to highlight the existence of a threshold below which the stimulated emission disappears This threshold depends on the percentage of pumped molecules and the Einsteinrsquos spontaneous coefficient A rather long lifetime of the level 2 in relation to the non-radiative relaxation is a favorable situation for the observation of a stimulated emission Measuring the intrinsic lifetime of the excited state is then no longer possible Three characteristic times T1 the population relaxation T2 level 2 phase relaxation and Tf the photonrsquos residence time in the cavity must to be compared

The model can be extended to cases where intramolecular transfers occur [ZON 85] The simulation shows that the acceleration of the relaxation of the excited level by stimulated emission amplifies the pumping efficiency The form of the equations [611] infers an adiabatic variation of the resident field with the population difference The polarization of the medium is assumed to vary much faster than the population inversion so that the field variations depend only on the population inversion This approximation is justified for homogeneous mediums where the T2 dephasing time is very short compared to the lifetime Tf of a photon in the cavity and the radiative relaxation lifetime T1 of the level participating in the stimulated emission In the case when the propagation of the electric field of the light wave allows for the macroscopic polarization of the sample Bloch equations must instead be used They are based on the matrix density formalism to describe the population of the levels and on Maxwell equations to account for the propagation of the electric field which couples the stimulated photon field to the nonlinearity of the medium In this case two equations are necessary the equation of LiouvillendashVon Neumann (equation [614]) describing the evolution of the density matrix ρ(t) averaged over all molecules and the Maxwellrsquos wave equation of the electric field

The LiouvillendashVon Neumann equation is expressed as follows

[ ] ( ) 1 ( ) ( )2

d t i H t tdtρ ρ ρ= minus minus Γ + Λ

[614]


where in the quantum description of the isolated system H is the sum of the molecular Hamiltonian H0 and the interaction Hamiltonian between the field and polarization Hint The last two terms correspond to the phenomenological description of the relaxation (Γ matrix containing the phase relaxation terms and spontaneous terms A B = AB + BA being the anticommutator) and the population (Λ matrix containing intramolecular relaxation terms) respectively

Maxwellrsquos wave equation of the electric field is given by

2 2

0 0 0 02 2( ) E E PEt t t

μ σ ε μ μpart part partnabla and nabla and + + = minuspart part part

[615]

where σ represents the conductivity of the medium comprising the sources of energy loss and P

is the macroscopic polarization

Considering that ε and μ vary little within the medium the numerical resolution of the BlochndashMaxwell equations or the Bloch optics (equation [625] for a two-level system) allows for the simulation of the superradiant and chaotic stimulated emission which appears as unstable and periodic intense peaks as a function of the characteristics of the environment and therefore of the parameters included in the model

654 The theoretical model of the thermal energy transfer

In a typical experiment for time-resolved spectroscopy the aim is to study the temporal evolution of the population of a system andor concomitant dephasing processes and thus to determine the corresponding relaxation rate constants The theories developed to determine these relaxation constants lead in principle to the identification of the contributions of the different relaxation pathways to the relaxation rate constants and their calculation [LAK 87] The same theories can be used to calculate the spectral line shapes and shifts [ROB 67] in conventional frequency resolved spectroscopy (absorption emission Rayleigh scattering Raman scattering etc) Within the framework of frequency and time-resolved spectroscopy of systems trapped in matrices [DAH 97 CHA 00] the system that is


studied is subject to an electromagnetic radiation while it is simultaneously interacting with another system generally larger in volume (thermal bath in statistical physics) Since under these conditions the system is not in a pure state the time-dependent Schroumldinger equation can no longer be applied to determine its time evolution The system must indeed be described by a statistical ensemble and it is necessary to use the density matrix formalism (Chapter 3 section 341) to determine its temporal evolution (equation [336])

This section describes the general method that applies the master equation to determine the temporal evolution of the density operator and gives access to the relaxation rate constants when a system is interacting with a thermal bath As in Chapter 5 (equations [538] and [544]) it can be written that

Thermostat system intˆ ˆ ˆ ˆH H H H= + + [616]

where systemˆ ˆ ˆ ˆ

N e e NH H H H minus= + + with ˆNH the Hamiltonian of the

nuclei êH the Hamiltonian of electrons and ˆ

e NH minus the Hamiltonian of the interaction between the nuclei and the electrons and ThermostatH is the Hamiltonian of the thermal bath where the entire system is considered to be isolated

Theoretical models built on the method of the master equation can treat any relaxation process that depends on the perturbation such as the electronic vibrational rotational relaxations electron transfer or other thermal processes (multiphonon etc)

The master or Liouville equation is written (from equation [336]) in the form

ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ( )( ) ( ) i iH H H iLtρ ρ ρ ρ ρpart = minus minus = minus = minus part

[617]

where L represents the Liouville operator By developing the commutator of equation [617] L takes the form


system Thermostat int 0 intˆ ˆ ˆ ˆ ˆ ˆL L L L L L= + + = + In the Liouville space ρ is a

vector and L is a super operator and we write the Liouville equation in a matrix form

ˆ ˆ ˆ( )ik

ik jl jljl

i Ltρ ρpart = minuspart [618]

where L is a matrix with four indices that connects each element of ρ which is a matrix with two indexes In this expression

ˆ ˆ ˆ

ik jl ij kl kl ijL H Hδ δ= minus Although formally the Schroumldinger equation (equation [335] valid for pure states) the Von Neumann equation (equation [336] valid for mixed states and a set description) and the Liouville equation (equation [617]) are equivalent only the latter allows for the introduction of the physical phenomena such as the dephasing mechanism with the use of super operators

The systemrsquos density matrix is determined by taking the trace on the quantum states of the thermal bath such as

[ ]ˆ ˆ( ) ( )SBt Tr tρ ρ= [619]

where to simplify the S symbol is used for the system and B for the thermal bath and TrB for the trace To remove variables from the thermal bath the initial chaos is assumed that allows the decoupling of the density matrix operator with the initial conditions ˆ ˆ ˆ(0) (0) (0)S Bρ ρ ρ= As the bath does not evolve with respect to the

system anymore ˆ ˆ( ) (0)B Btρ ρ= can be factorized

By expanding the operator it can be shown that the master equation of the systemrsquos density matrix takes the following form

ˆˆ ˆ ˆ ˆˆ ˆ ˆ ˆ

SSS S S Sik

S ik jl jl ik jl jljl jl

iL i Lt t

ρρ ρ ρ ρ ρpartpart = minus minusΓ = minus minus Γpart part [620]


where Γ represents the damping or dephasing operator and where L and Γ are matrices with four indices that connect each element of ρ which is a matrix with two indices In this expression

ˆ ˆ ˆ

ik jl ij kl kl ijL H Hδ δ= minus

In the case of a system with two levels at the approximation level of order 2 equation [620] may be expressed in the following formal form

ˆ ( ) ˆ ˆˆ ˆ ( ) ( )

ˆ ( )ˆ( ) ( )

s s

s s s s s s s s s s s s

s s

s s s s s s s s

Sk k S S

k k k k k k k k k k k kk

Sk k S

k k k k k k k k

tt t

tt

i tt

ρρ ρ

ρω ρ

prime prime prime primeprime

primeprime prime prime prime

part= minus Γ minus Γ

partpart

= minus + Γpart

[621]

where

( )

ˆ ˆ

ˆ

1ˆ ˆ ˆ ˆ( )2

s s s s s s s s s ss

s s s s s s

s s s s s s s s ss s s s s s s s

k k k k k k k k k kk k

k k k k k k

dk k k k k k k k k k k k k k k k

k

k

prime prime primeprime prime

prime prime prime

prime prime prime prime prime prime prime prime

Γ = minus Γ = minus

Γ = minus

Γ = Γ + Γ + Γ

and

2int2 ˆ (0) ( )

s s B B k k k k s B s Bs B s BB B

Bk k k k k k k k

k kk H E Eπ ρ δ

prime primeprime prime primeprime

= minus

Because of the one-to-one correspondence between the quantum states of a system and the density matrix the latter can be used to model the two-level system (Jaynes Cummings) discussed in Chapter 5 (section 532) and define the coherent state The density operator has no uncertainty on the phase whereas it is inherent with the wave function for which the phase disappears when its square is

computed The matrix1 00 0

ρ =

is associated with the fundamental


state g and the matrix0 00 1

ρ =

is associated with the excited

state e The superposition of the two states ( )12

g eΨ = + is

then in correspondence with the associated density matrix which is

written as 1 111 12

ρ =

from the definition mn n mc cρ lowast= where ck are

the coefficients of basis states g and e In the case each state is given with a statistical ensemble average the probability Pk of finding the system in one of two states or e is equal to Pk = 12 so that the density matrix is given by

1 12 2

g g e eρ = +

In the matrix form the density matrix is then written as

1 010 12

ρ =

The two matrices are different because in the second case the coherent state does not exist but gives the same probability of finding the system in a pure state g or e

The evolution of the density matrix is given by the Von Neumann equation (equation [336]) in the eigenstates of the Hamiltonian H (the system is isolated with energy ε1 and ε2 and 1 2H g g e eε ε= + ) such that

11 12 1 11 12 11 12 1

21 22 2 21 22 21 22 2

1 2 12

2 1 21

0 00 0

0 ( )( ) 0

d idt

i

ρ ρ ε ρ ρ ρ ρ ερ ρ ε ρ ρ ρ ρ ε

ε ε ρε ε ρ

= minus minus minus = minus minus

[622]


11 22andρ ρ are constants and 12 21( ) and ( )t tρ ρ are given by

1 2 1 212 12 21 21( ) exp( ) (0) and ( ) exp( ) (0)t i t t i tε ε ε ερ ρ ρ ρminus minus

= minus = +

The system oscillates between two states at the frequency2 1

12 hε εν minus

=

In Liouville space the evolution equation (equations [617] and [618]) applies to vectors and if we classify the elements of the matrix of operator L so as to write the non-zero elements first then the equation can be expressed as

12 121 2

21 212 1

11 11

22 22

0 0 00 0 00 0 0 00 0 0 0

it

ρ ρε ερ ρε ερ ρρ ρ

minus minuspart = minus part

[623]

In this way a damping factor reflecting the dephasing mechanism can be simply added to the right-hand side of the equation giving the evolution of the elements 12 21( ) and ( )t tρ ρ such as

( ) ( ) ( 1 or 2)iki k ik ik

t i i kt

ρ ε ε ρ Γρpart= minus minus minus ne =

part

which lead to

1 212 12( ) exp( ) exp( ) (0)t i t tε ερ Γ ρminus

= minus minus

and

1 221 21( ) exp( )exp( ) (0)t i t tε ερ Γ ρminus

= + minus

If the system is in interaction with the electromagnetic field of a radiation the Hamiltonian is written as

1 2ˆ ˆ ( )( )H g g e e E t g e e gε ε μ= + + +


In the matrix form the Hamiltonian is expressed as

10

2

ˆ ( )ˆ ˆ ( )ˆ ( )

E tH H E t

E tε μ

μμ ε

minus = minus = minus

[624]

The evolution of the density matrix is then given by the following LiouvillendashVon Neumann equation

12 121 2

21 212 1

11 11

22 22

0 ( ) ( )0 ( ) ( )

( ) ( ) 0 0( ) ( ) 0 0

E t E tE t E ti

E t E ttE t E t

ρ ρε ε μ μρ ρε ε μ μρ ρμ μρ ρμ μ

minus minus minus minuspart = minus minuspart

minus

[625]

which are also called the Bloch optical equations

In the rotating frame (of frequency 2 112 h

ε εν minus= or angular

frequency ω12 asymp ω) to discard the rotating part and extract only the interesting part of ρ denoted ρ that varies slowly relative to ν12 the equation is written as

12 12

21 21

11 1122 22

0 ( ) ( )0 ( ) ( )

( ) ( ) 0 0

( ) ( ) 0 0

t tt t

it t t

t t

Δ Ω Ωρ ρΔ Ω Ωρ ρ

ρ ρΩ Ωρ ρΩ Ω

minus minus minus part = minus part minus minus

[626]

with

212 and ( ) ( ) (1 )i t i t i t i tt e e e eω ω ω ωΔ ω ω Ω Ω Ωminus= minus + = + = +

and where 0EμΩ =

is the Rabi frequency The angular frequencies

are shifted by ω in the reference frame of the rotating field with a


term ( )tΩ that is constant with respect to ω and a term rotating at 2ω The electric field therefore includes two terms one that oscillates in phase but at 2ω (the integral over time makes no contribution) and a term that oscillates in antiphase to ω In this case the constant term only can be kept in the equations within what is termed the rotating wave approximation At resonance 0Δ = and only the terms

11 22andρ ρ vary in time the crossed terms being constant and Rabi oscillations between the two energy levels are observed with a contrast that is less pronounced depending on Δ (close to resonance) If a dephasing term is introduced (condensed phase) the system loses in this case its coherence and Rabi oscillations are damped and disappear The terms 11 22andρ ρ tend toward the value of frac12 and the crossed terms are damped as well If the optical pumping is sufficiently strong the Rabi oscillations can be faster than the detuning and an inversion in a two-level system can be achieved

The model developed for a system with two levels can be extended to a multilevel system interacting with a thermal bath to get the relaxation constants that correspond to thermal phenomena in a given medium [BLU 78 LIN 77 LIN 80 KON 83 LAK 87 LAK 11] These models can be used to calculate the shape of the spectral lines [AND 49 TSA 62 FIU 62 ROB 67 ROB 79]

66 Conclusions

The experimental techniques of induced fluorescence and DR by laser (LIF and DR) are used to study energy transfers in a given environment at the nanoscale level Time-resolved spectroscopy leads to the intrinsic times of the processes involved in energy transfers between the different degrees of freedom of studied systems a defect or trapped system such as an atom or a molecule in a matrix The time characteristics of the physical processes can be obtained by means of theoretical models based on methods of statistical and quantum physics using the density matrix formalism If the main relaxation channels can be identified numerical methods can applied to simulate


through these models the dissipation of energy in a system subjected to an excitation The mechanisms underlying the intrinsic exchange of energy between the different modes of vibration libration or rotation and translation of the trapped system or between the trapped system and its environment can thus be analyzed

7

Defects in a Heterogeneous Medium

In time-resolved spectroscopy the intrinsic energy transfer rate is directly determined by the double IRndashIR resonance method (pumpndashprobe where IR is infrared) or by the analysis of induced fluorescence after the excitation by a pump laser The response to a laser excitation is analyzed to study heterogeneous environments comprising buried defects A laser is used to probe the variations in reflectivity or transmission as a function of time In ultrafast timescale experiments these properties depend on the effects of buried defects on ultrasonic waves generated by the pump laser In model systems consisting of a molecule with a low number of atoms and a host medium such as a rare gas (RG) matrix at low temperatures the laser probes the degrees of freedom states that are capable of exchanging energy The experimental setup requires an apparatus consisting of lasers a cryostat rapid detection systems and a data storage system with a computer The energy levels involved in the exchanges are associated with modes of vibration libration or hindered rotation of the molecular system and with the phonon modes corresponding to the collective vibrations of the environment or with local modes The redistribution of vibrational energy from an excited level depends on various parameters such as the energy mismatch the concentration of the sample the host medium or the temperature Determining this dependence leads to the rules governing the competition between the various transfer processes It also provides more specific information on energy dissipation channels which can be compared with those determined by other methods To access the characteristic time constants of heat evacuation in the medium and the characteristic time constants of local heat trapping at the level of defects or in-homogeneities experimental results are compared with the simulation performed by numerical methods based on theoretical models

71 Introduction

In this chapter the methods used in fundamental research to study and analyze the energy transfers in model systems using a nanosecond


First Edition


laser pump in the medium IR range are presented Model systems are composite materials solid matrices and trapped molecules RGs (Ne Ar Kr and Xe) and nitrogen constitute convenient condensed phase matrices At experimental temperatures these matrices are inert unreactive and transparent in the spectral absorption range of the molecules

The samples are composed of one polyatomic molecule (n le 7) (n = 2 diatomic CO [DUB 77] n = 3 triatomic CO2 N2O O3 [BRO 93 JAS 95 DAH 97 CHA 00] n = 4 NH3 [ABO 73] n = 5 CH3F [GAU 76] and n = 7 SF6 [BOI 85]) trapped in a RG matrix at very low temperatures between 5 and 30 K With a RG electrical interactions are limited to n-polar moments (dipolar quadruple-polar octuple-polar) and to the induced polarization of the noble gas atoms With an electronic configuration that is completely filled RGs are very stable and almost non-reactive In matrices small molecules are characterized by a low number of vibrational modes (2n ndash 5 for linear molecules or 2n ndash 6 for nonlinear ones) of rotational and constrained translation or libration modes At very low temperatures thermal effects are minimized and the thermodynamic equilibrium corresponds only to the population of the ground vibrational energy state of the probe molecules and to collective matrix vibrations population very few phonons modes The energy transfer pathways of the different degrees of freedom are reduced and it is easier to study the basic mechanisms of relaxation

Two types of energy transfer are studied from the pumped system to the matrix or non-excited systems (intersystem transfer) or from the excited energy level to other lower levels (intrasystem transfer) These transfers are characterized by different time constants Specific studies identify the different types of transfers described in Chapter 6 (section 64) The particular studies to be performed as a function of various parameters such as the sample concentration the host environment or the temperature are described The analysis of experimental results to determine the characteristic parameters that need to be considered in the exchange of energy between a triatomic molecule (CO2 N2O and O3) and a RG solid matrix at low temperatures is based on theoretical models presented in Chapter 6 The coupling between the

Defects in a Heterogeneous Medium 175

RG matrix the host medium and the trapped molecule is explicitly or implicitly taken into account in order to model and interpret these transfers

72 Experimental setup

The experiments of laser-induced fluorescence and double IR-IR resonance on O3 CO2 and N2O are achieved with a compound laser system (pump laser) (YAG dye doubler crystals (visible) tripler quadrupler quintupler (UV) and mixer (IR)) coupled with a continuous CO2 laser (probe laser) and a system containing a cryostat and an interferometer for preparing and characterizing the samples The diagram of the experimental device is given in Figure 71 [DAH 97]

Figure 71 Experimental setup for laser spectroscopy induced fluorescence and double IRndashIR resonance

monochromator

HUET

LASER CO2

Spectrometer

FTIR BRUKERIFS113v

(003 cm-1)

LASER

Nd YAG

YG 781C20

DYE

LDS 867

Li NbO3

1064 nm

532 nm

PULSED SOURCE

QUANTEL

870 nm5 ns 20 Hz

Δ σ=085cm-1

220-5000 nm

GeAu

HgCdTeor GeCu

COMPUTER

PRINTER

OSCILLOSCOPETEKTRONIX

Preamplifier

HgCdTe

PUMP

PROBE

CaF2

CaF2

GeAu

2100 cm- -1

100 μJfilter

D

D

D

D

E

Lenses

KBr

D Diaphragm

E Retractable

PHOTOCHEMISTRY

532 nm

355nm

266 nm Trigg

er

mon

ochr

omat

or


721 Pump laser

A YAG pulsed laser pumped by flash is used for the pumping of the studied systems This laser emits at a rate of 20 Hz at 1064 microm pulses of a duration τ = 5 ns via a Pockels cell This laser radiation is doubled in frequency (532 nm) and is used to pump a dye laser which emits IR pulses at around 870 nm The beam from the dye laser (35 mJpulse) and the YAG residual laser (40 mJpulse) are mixed in a LiNbO3 crystal which generates a beam in the mid-IR In an optimized configuration with a maximum dye yield the crystal delivers pulses of about 200 microJ with the same spectral and temporal characteristics as those of the dye pulse (Δν = 08 cmminus1 and τ = 5 ns) Its frequency is between 2085 cmminus1 and 5000 cmminus1 (λ between 2 and 48 microm) and its polarization is vertical In experimental configuration radiation of about 2200 cmminus1(λ cong 45 microm) in resonance with the ozone transition ν1 + ν3 (around 2100 cmminus1) or ν3 of CO2 and N2O (around 2300 cmminus1) in matrix is used as the pump source

722 Probe laser

The probe laser is a CO2 laser built on a cavity closed at one of its extremities by a golden metallic concave mirror of 3 m in curvature radius and at the other end by a diffraction grating of 150 lines per mm (blazing at 10 microm) The laser medium consists of a mixture of three gases (CO2 He and N2) in an average relative ratio of 13422 under a total pressure of about 155 torr The probe beam is extracted from the cavity by a skew plane mirror partially introduced into the laser cavity The less intense output of the zeroth-order grating is used to monitor the probe laser power variations Closing of the laser tube by NaCl plates at Brewster angle horizontally polarizes the laser beam In this configuration a large number of transitions of the rotationndashvibration bands of the CO2 molecule in the spectral range of 96 and 106 microm can be used to probe the transitions of the studied molecules A He-Ne laser beam is superimposed on the paths of the pump and probe lasers to facilitate optical adjustments by materializing IR beams The beam divergence is reduced by interposing at midway a telescope of radius of curvature of


2100 mm to reduce the cross-section of the laser beam on the sample to a diameter of between 1 mm (CO2 and N2O) and 2 mm (O3)

723 Detection system

The time-resolved spectroscopy requires a rapid detection system A set of photoconductive detectors sensitive in the IR range is used A GeCu detector sensitive in the range of 2 to 30 microm with a peak at 21 microm is used to analyze the short and intense signals It is cooled with liquid helium and its resistance ranges from 1 Ω at ambient temperature to 35 MΩ It works without a preamplifier polarized with a current of a few microamperes through a resistance of 220 Ω A very compact polarization box is welded at the cell output and a coaxial cable transmits the signal to the oscilloscope over a length of 20 cm The signal of the pump laser measured with this setup (Figure 72(a)) leads to a full-width half-maximum equal to that specified by the manufacturer of the laser (5 ns) The decrease in the signal is distorted by the effects of the measurement circuit on the detection of the fast signal

Figure 72 Shape of the pulsed laser signals (5 ns) based on the detection system (a) GeCu (b) MCT

For laser-induced fluorescence studies on 13CO2 and N2O a mercury cadmium telluride (MCT) detector sensitive from 25 to 18 microm with a peak at 14 microm and a detectivity of 30 times 1010 cmWradicHz in the region of 16 microm is used It has a preamplifier with a bandwidth of 10 MHz It is polarized with currents varying

-01

-008

-006

-004

-002

0

002

004

5 7 9 11 13 15 17 19 21 23 25Temps(ns)

Am

plitu

de (m

V)

FWMH = 48ns

-02

02

06

1

14

0 1 2 3 4 5 6

Temps (μs)

Ampl

itude

(UA

)

MCT Belova) GeCu b)

Time (micros)

Time (ns)FWHM 48 ns


between 10 and 40 mA With this detector the pump laser signal is characterized by a rise time up to the maximum of 38 plusmn 2 ns and an exponential decrease of 305 plusmn 5 ns (Figure 72(b))

A very sensitive HgCdTe or MCT detector cooled with liquid nitrogen is used to detect low intensity signals Its crystal has a detection range between 1 and 20 microm with a peak at approximately 10 microm It can be used alone with a polarization current of 10ndash15 mA via a resistance of 78 Ω (resistance when cooled of 64 Ω) or equipped with a preamplifier with a gain of 10000 and bandwidth of 350 kHz This detector analyzes the 5 ns laser pulse decrease of 500 ns with a maximum that is reached to 500 ns (Figure 73(a)) In a configuration without a preamplifier the amplitude of the signal is divided by four orders of magnitude and the same pulse is measured with a decrease of 120 ns and a maximum at 85 ns (Figure 73(b)) This detector is ideally suited for the study of signals which are observed on a time interval greater than 200 ns

The signals are recorded on a digital storage oscilloscope of adjustable input resistance of 50 Ω (short signals) or 1 MΩ The signals single or averaged are transferred to a computer for analysis A program using the least squares method reduces the curves to exponential sums A curve is reconstructed from the stored values and from the time constants of exponentials and their respective weights

Figure 73 Shape of the pulsed laser signals (5 ns) based on the detection system a) with preamplifier b) without preamplifier

-02

02

06

10

14

00 02 04 06 08 10 12 14

Temps (μs)

Ampl

itude

(UA

)

MCT SAT

-02

02

06

1

14

0 1 2 3 4 5 6

Temps (μs)

Ampl

itude

(UA

)

MCT SATa) b)

Time (micros)Time (micros)


A very rapid (rise time lt 1 ns) pyroelectric detector (lithium tantalate LiTaO3) calibrated and linear up to 3 V measures the power of the pump laser It is connected to the 50 Ω input of the oscilloscope (1 MΩ if the signal intensity is too weak) and continuously controls the power of the laser probe and the stability of the intensity of the pump beam

The MCT detector is placed behind the window of the cryostat at 45deg to the sample A germanium plate and an optical low-pass frequency filter (cutoff at νc = 1800 cmminus1) eliminates the parasitic residue of the laser radiation scattered by the sample Spectral analysis of the fluorescence signals is performed using a monochromator with interchangeable blazed grating For the signals at 10 microm the grating is made up of 75 lines per mm (with blaze at 12 microm) with a theoretical resolution of 02 cmminus1 at 10 microm in the first order when the input slits are closed to the minimum (01 mm) The grating is made up of 60 lines per mm (with blaze at 16 microm) for the analysis of signals at 16 microm The grating which is not fully covered by the laser beam has a resolution of about 08 cmminus1 After passing through the monochromator the light signal is focused by a ZnSe lens on the MCT

Figure 74 Diagram of the cryostat for sample preparation

He liquide N2

liquide

77 k 77 k

4 k Pompe reacuteglant le flux drsquoheacutelium

vide vide

Faisceau issu du

spectromegravetre Pompage Vide

Vanne micro fuite

Meacutelange gazeux

Filament chauffant

Jauge drsquoheacutelium

Reacutecupeacuteration recyclage de lrsquoheacutelium

Heacutelium gazeux

N2 liquide

Heliumgas

Exhaust Helium gas recycling

Pump regulatingHelium gas flux

HeliumJauge

Heating coil

Beam fromFTIR

spectrometer

Gasmixture

Pumpvacuum

Micro leakeagevalve

LiquidHe4Kva

cuum

vacu

um

LiquidN2

77 K

LiquidN2

77 K


724 Sample preparation setup

A cryostat (Figure 74) with a liquid helium tank is used to prepare a solid sample by the method of jet condensation on the gold-coated surface of a sample holder from a gas mixture in the concentrations defined by the ratio mM (m for molecule and M for matrix) Depending on the matrix the condensation of the mixture is performed at a deposition temperature (TD) set at about two-thirds of the sublimation temperature A microleakage valve is used to control the deposition rate through the pressure measured inside the cryostat The usual rate of deposition is 2 torrmin (a few millimoles per hour) A polycrystalline layer is obtained Within an hour the thickness (in torr) of the deposited polycrystal is proportional to the cube of the matrix lattice parameter [GAU 80] corresponding to 100 microm The thickness is measured by the technique of interference fringes using a He-Ne laser (near-normal incidence) while 10 torr of the gas mixture is deposited under the same conditions as during the experiments The ratio 3 1 3

0 117 μmtorr nme a minus minus= is a function of the geometry of the deposition system

A simultaneous deposition of two or three different gas mixtures is possible with the technique of condensation of a jet gas in contrast to pulse jet techniques or crystal growth from the liquid phase But the optical quality of the sample is then lower and it is necessary to optimize the deposition temperature TD A high temperature favors better growth of the crystal but also the migration of species to the surface of the sample during deposition which increases the formation of polymers (organic molecules) or aggregates of trapped molecules A low temperature limits the formation of aggregates by reducing the rate of migration at the expense of the size of the microcrystals which contributes to the inhomogeneous width of an absorption line and which favors the formation of trapping sites known as ldquounstablerdquo A compromise must be found between these two limits Optimum temperatures determined empirically correspond approximately to two-thirds of the sublimation temperature of the matrix gas To minimize the formation of aggregates it is sometimes necessary to use a lower value For the study as a function of the temperature the


sample holder is equipped with a heating resistor and a control system based on a diode sensor that determines the current to be used for heating in connection to the set temperature The diode can also control the temperature variations Two other sensors monitor or measure the temperature a platinum resistance for temperatures above 30 K and a carbon resistance which is more sensitive for temperatures below 30 K

The spectroscopic study is performed with a Fourier transform IR spectrometer operating under primary vacuum The maximum resolution available without apodization is 003 cmminus1 The spectra are recorded between 500 and 4000 cmminus1 with a resolution of 05 or 015 cmminus1 using a deuterated triglycine sulfate (DTGS) detector operating at room temperature An adjustable diaphragm controls the cross-section of the beam at the output of the compartment containing the IR source For a resolution of 015 cmminus1 it is necessary to reduce its diameter to 5 mm but the signal to noise ratio then increases With xenon and krypton matrices that strongly scatter the incident IR light the amount of energy transmitted is sometimes critical for a resolution of 015 cmminus1 For thick deposits the signal-to-noise ratio is too low and the samples have to be analyzed at a low resolution of 05 cmminus1 With the MCT detector that is cooled with liquid nitrogen and is thus more sensitive than the DTGS the spectra are recorded more rapidly But for the same resolution the signal to noise ratio is lower since the flux of globar source on the MCT detector must be limited to avoid saturation This detector is suitable for thick deposits The spectrometer is controlled by a computer using Bruker software which calculates from a choice of apodization functions the Fourier transform of the recorded interferogram A set of charts are used to optimize the choice of the electronic filters the scanning speed and the aperture of the source diaphragm for the desired resolution

The interferometer is provided with a beam extractor for studies by reflection at 30 cm from its outlet facing a cryostat window (Figure 71) The space between the spectrometer and the cryostat is used for the optical devices necessary to perform the time-resolved spectroscopic study The air through which the IR beam travels is


purged with nitrogen gas to reduce the absorption of carbon dioxide molecules or water moisture in the air Two reference spectra are recorded at 5 K with resolutions of 05 and 015 cmminus1 before the deposition They are used to calculate the absorbance spectra (logarithm of the ratio of the reference spectrum and the sample spectrum) A spectroscopic study is simultaneously conducted with the deposition in order to monitor the optical density and verify that the absorption spectra increases linearly with the amount of deposited gas mixture

73 Application to a model system

731 Inert noble gas matrix

RG matrices consist of weakly bound atoms Cohesion is due to weak electrostatic van der Waals type bonds and so they crystallize at low temperatures (25ndash165 K) In theoretical models the interatomic forces are represented by a 12-6 LennardndashJones type potential

( )12 6

4E rr rσ σε

= minus [71]

where σ and ε (Table 71) are defined by the equations E(σ) = 0 and ε = minusEmin (Figure 75) The repulsive term (in 1r12) reflects the overlapping of electronic clouds and the attractive term (in 1r6) represents the dipolendashdipole interaction induced dipolendashdipole and London interactions Noble gases generally crystallize in a face-cubic centered (fcc) system Impurities stabilize the hexagonal close-packed (hcp) structure The rigidity of the matrix increases with the size of its constituent atoms

Different trapping sites are possible in RG crystals (Figure 76) the molecule can take the place of one or more atoms of the lattice or if its size is small enough locate itself in an octahedral and tetrahedral interstitial site of the fcc lattice Table 72 gives the diameters of different trapping sites


Figure 75 Interaction potential for different rare gas atoms

Figure 76 Possible trapping sites in a fcc lattice

Ne Ar Kr Xe O N C ε(cm-1) 2433 8401 12500 15710 3990 2625 2470 σ(Aring) 276 345 365 397 288 338 400

Table 71 LennardndashJones parameters of rare gases and oxygen nitrogen and carbon atoms

2 3 4 5 6 7 8 9 10-200

-150

-100

-50

0

50

100

150

200

-ε

reσ

Neacuteon Argon Krypton Xeacutenon

V(r)

(cm

-1)

r (Aring)

NeonArgonKryptonXenon

Sites de substitution

Sites interstitiels octaeacutedriques

Sites interstitiels teacutetraeacutedriques

Substitutional site

Octahedral Intersticial

Tetrahedral Intersticial


Table 72 Diameters of possible trapping sites in rare gases

732 Molecular system trapped in an inert matrix

The energy of the single crystal is represented by the interaction between two atoms in the crystal by

( )6 6

4 1gr grjj jj gr

jj jj

V rr rσ σ

ε = minus

[72]

where rjjprime represents the distance between two atoms j and jprime of the matrix and σgr and εgr are the LennardndashJones parameters of the pure state RG atoms

Figure 77 Possible trapping sites of CO2 in a fcc lattice For a color version of this figure see wwwistecoukdahoonanometerzip

Rare gas substitutionalsingle(Aring)

intersticialoctahedral(Aring)

intersticialtetrahedral(Aring)

Neon 307 12 06Argon 375 148 078

Krypton 401 164 09Xenon 431 17 108

vacantOxygegravene

a Single substitutional site b Double substitutional site

Carbon OxygenArgon Void


A molecule inserted into a RG crystal is subjected to interaction forces with the latticersquos atoms Given their respective sizes O3 N2O and CO2 molecules are substituted to one or two atoms in a site of Oh and D2h symmetry respectively in the fcc lattice (Figures 77 and 78) and in some cases in a compact hexagonal structure (hcp) Trapped in a substitutional site the molecule is equivalent to a point defect of the pure crystal and this results in a local deformation (Figure 78) of the crystalline structure which can be determined by the technique of the lattice Greenrsquos functions of the crystal [LAK 87 DAH 97] (equation [75])

Figure 78 O3 trapping sites in a fcc lattice and atomic distortions of the noble gas Ar

The interaction VMj of the latticersquos atom j with the set of the three atoms i of the molecule is given by the equation

12 63

2

1

1( ) 42

ij ijMj ij ij j Mj

i ij ij

V r Er r

σ σε α

=

= minus minus [73]

where rij is the distance between the atom i of the molecule and the atom j of the crystal and αj is the polarizability of the atom j of


the crystal εij and σij are determined by the combination rules of LorentzndashBerthelot defined by the equations

1

2( )ij ii jjε ε ε=

and

2ii jj

ij

σ σσ

+=

By combining equations [72] and [73] VStat the static potential interaction of the doped crystal is obtained as follow

( )Stat

( ) ( )ij jj Mj ij jj jj

j jj j jV r r V r V rprime

lt

= + [74]

where j ne 0 in the case of a single site and j ne 1 in the case of a double site The stability of the sites is determined by a search for the equilibrium configuration of the doped crystal The displacement vectors of the atoms of the matrix are obtained by solving a system of equations that satisfy the equilibrium condition of the distorted crystal (setting the gradient of VStat to zero)

The energy of the distorted crystal is obtained from the Taylor series expansion of the static potential VStat up to the second order assuming that the displacement ξj of the crystal atom j is small compared to the other vector quantities By introducing the Green matrix as the inverse of the matrix of force constants (the Hessian matrix of VStat) the displacement ξj of an atom j in the direction α is expressed as

j jj jj

G Fα αβ β

βξ = [75]

where jjGαβprime are the matrix elements of the Green function and jF β is

the first derivative of the interaction potential Vstat in the direction defined by β


The intensity of the interaction Uij between the atoms i of the molecule and the atoms j of the RG crystal determines the relaxation pathways of the energy deposited in the molecule If the intersystem intensity is higher than the intrasystem intensity (Uii lt Uij) energy is conveyed from the trapped molecule to the host system which then must evacuate this energy Otherwise it is an intrasystem transfer (Uij lt Uii) that takes place before the transfer to the host matrix occurs

The inserted molecules are identified by their absorption spectra which are in the form of a Q branch without any rotational structure for O3 CO2 and N2O Two absorption lines are observed for each vibrational mode of O3 because it gets trapped in two different sites (Figure 78) In the case of CO2 two absorption lines are observed for each vibrational mode in argon due to trapping in a single substitutional site (unstable) and a double substitutional site (stable) (Figure 77) but only one line is observed for each vibrational mode in krypton and xenon matrices (one single substitutional site) Similarly only a single line is observed for N2O as a result of only a double substitutional trapping site (identified by the lifting of the degeneracy of mode ν2)

Although the three molecules are triatomic they differ (Table 73) by their structures (linear for CO2 and N2O and nonlinear for O3) dimensions and properties of symmetry and electrical characteristics (quadrupolar moment for CO2 and dipolar for O3 and N2O) Different coupling effects are expected with the different RG matrices (Ar Kr and Xe) leading to a variety of energy relaxation pathways in these matrices

Table 73 Parameters of the rigid molecule r0 bond length (Aring) β bond angle (degree) micro dipolar moment (Debye) and

Qii quadrupolar moment along the i axis of the molecule (Debye Aring)

r0(Aring) β (degree) μ (D) Qzz (DAring) Qxx(DAring) Qyy(DAring)

16O3 1278 1168 0532 -14 -07 21

13C16O2 116 180 0 -43 215 215

14N216O N-N1128 N-O1842

180 166 -30 15 15


733 Experimental results for the induced fluorescence

As described in Chapter 6 fluorescence can be spontaneous or stimulated Its temporal properties are determined from a number of characteristic times (Figure 79) such as t0 the time lag between the beginning of the rise of the signal and that of the pulse of the pump (the delay of excitation) rise of the signal characterized by τR the position of the signalrsquos maximum τM relative to the beginning of the rise time and the decay of the signal characterized by τ the time after which it is divided by e (considering it is exponential) In addition to complete the information that these time indicators provide on the interaction between the trapped molecule and its environment fluorescence must be characterized by its spectral range (the emitting vibrational transitions) its excitation spectrum (spectral range of fluorescence emission and its width according to the frequency of the pump laser scanning the absorption range of the pumped mode) its threshold that depends on the power of the laser pump its polarization and its directivity These characteristics are also studied as a function of annealing and concentration of the sample Finally the yield is determined as a function of all these parameters

Figure 79 Time characteristics of a signal t0 delayτR rise τM maximum τ1 and (τ2) decay


The fluorescence is observed after excitation at about 45 microm of the vibrational mode ν1 + ν3 in the electronic ground state of 16O3 [JAS 94 JAS 95 DAH 97 JAS 98 CHA 00b] and the vibrational mode ν3 of 13CO2 [BER 96 DAH 97 CHA 98 CHA 00a CHA 00c CHA 02 VAS 03] and 14N2O [BER 96 DAH 97 CHA 00b CHA 02] The fluorescence differs from the pump pulse by a delay and a lower intensity Correlated to the absorption zone it disappears completely if the frequency of the laser is displaced outside the absorption range Its intensity is optimized by displacing the laserrsquos impact on the sample

Many ldquosingle-shotrdquo signals or a signal averaged over 1000 shots in order to minimize laser instabilities can be recorded on the oscilloscope as well as the pump pulse reference signal Several ldquosingle-shotrdquo can be measured in order to obtain the envelope of the fluorescence signals during the measurement of time interval

Fluorescence is characterized by a threshold effect linked to the energy of the laser pump below which it disappears This phenomenon is observed for O3 in a krypton matrix in Figures 710(a) and (b) and for N2O in argon in Figure 715(b) showing the simultaneous recording of the observed fluorescence (MCT) and pump laser (GeAu) over a hundred successive shots In the neighborhood of the threshold an instability of the fluorescence signal greater than that of the pump laser is observed The instabilities trace an envelope that appears thicker on the observed signals Maxima and minima are visualized on a set of the numerous acquired signals Figures 710(a) and 715(b) give an example of what is observed when the laser is close to the threshold some fluorescence signals have a zero minimum while the envelope of the laser does not pass through zero Fluorescence is absent below a threshold connected to a low value of the excitation laser power when its intensity fluctuates For stronger pulses the fluorescence signal is always greater than zero because this threshold is never reached Above the threshold the fluorescence intensity increases linearly with laser power

In the vicinity of 1000 cmminus1 two peaks are observed in the fluorescence spectra upon excitation of each of the absorption maxima


of the O3 doublet (two trapping sites) of ν1 + ν3 for O3Xe (Figure 711) The frequency of the fluorescence matches that of the transition 2ν3rarrν3 (Figure 712) The two spectra have a structure with one peak with a gap between the centers of the two peaks which is the same as that between the two components of the transition doublet 2ν3rarrν3 The absence of a second emission peak in these spectra shows that there are no intersite transfers for the duration of the fluorescence When one of the sites is excited no corresponding radiation to the other site is detected The two trapping sites are well separated and the transfer of energy is of the intrasystem type

Figure 710 Envelope of the fluorescence signals and the laser pump as a function of the trigger level (O3Kr = 1200 width 130 microm) a) 15 V b) 41 V

-4

-3

-2

-1

0

1

2

3

4

5

6

0 50 100 150 200 250 300 350 400 450 500 550 600

Temps (ns)

Am

plitu

de(V

)

Fluorescence au MCT(50)

Laser pompe au GeAu

Niveau de deacuteclenchement -15Va)

Laser pump at GeAu Time (ns)

Fluorescence at MCT (50)

a) Trigger level -15V

-5

-4

-3

-2

-1

0

1

2

3

4

5

0 50 100 150 200 250 300 350 400 450 500 550 600


Laser pompe au GeAuTemps (ns)

Am

plitu

de (V

) Niveau de deacuteclenchement -41Vb)b) Trigger level -41V

Time (ns)Laser pump at GeAu

Fluorescence at MCT


Figure 711 Spectral analysis of the fluorescence for the two trapping sites (sample O3 Xe = 1200 thickness 85 microm T = 5 K)

Figure 712 Vibrational energy levels of 16O3 in Xe (single-site HF and double-site LF) pump (ν1 +ν3) fluorescence transition at

10 microm (2ν3 rarr ν3) and probe transitions (Cold and Hot band)

1000 10005 1001 10015 1002 10025 1003 10035 1004 10045 1005

freacutequence pompe20904


10027

10036

Freacutequence de lafluorescence (cm-1)

Am

plitu

de n

orm

aliseacute

e(SU

)N

orm

edA

mpl

itude

(AU

) Pump frequency 20904

Pump frequency 20901

Frequency of the

Vibrational levels of O3

1 0 1

2032120306

1381 0 2 0

6991

17114

0 1 0

0 0 0

1 1 0

ν1 +ν3

ν1+ ν2

ν2

10 μm

Frequency cm-1((Xenon)

1003310027cm-1

0 0 11 0 0

ν1

ν3

1028810279

1097310966

2ν2

17884

0 1 1 ν2+ ν3

0 0 20 3 0

2ν3

3ν2

20914209032067

PUM

P

PROB

EPR

OBE

FLUO

RESC

ENCE


In the case of the CO2Ar sample (dilution 12000) three emissions are analyzed (Figures 713(a) and (b)) for each site By pumping the ν3 mode of the unstable site at 227369 cmminus1 emissions are observed at 596 614 and 627 cmminus1 and in the stable site the pumping at 227966 cmminus1 leads to the observation of emissions at 595 615 and 626 cmminus1 (Figure 63 of Chapter 6)

Figure 713 Three fluorescence signals a) stable site b) unstable site Ar13CO2 = 2000 T = 5 K

The different decay times τ as well as the delay times t0 relative to the pump pulse are measured from ldquosingle shotrdquo signals or averaged ones and are reported in Table 74 In the case of the stable site two

Time (microS)

Time (microS)


short emission signals in the response timescale (τ = 305 plusmn 5 s) of the detection chain and a third somewhat longer are measured For the latter two exponentials are required to fit most signals (short with an average of 755 plusmn 28 ns and long averaging to 2 micros)

Table 74 Radiative transition frequencies observed in argon matrix and temporal characteristics (22791 cmminus1 (stable site) and 22734 cmminus1(unstable site) 13CO2Ar = 12000 T = 5 K)

The times are longer for the unstable site by a factor of 15 For the component at 627 cmminus1 the fit leads to an exponential with a τ of 508 plusmn 24 ns for the short component The ratio A1A2 of the amplitude of this component to that of the second when two exponentials are needed for the adjustment is approximately 4 For the component at 596 cmminus1 a signal which is fitted to an exponential with a τ of 512 plusmn 67 ns and also signals having a plateau at the maximum of the fluorescence signal are sometimes observed It is difficult to fit the long component of the unstable site with a single exponential Some signals are composed of a short exponential followed by a long exponential beginning later Others include a plateau at the maximum of the signal For the long component τ is 8 plusmn 1 μs Delays with respect to the pump pulse are also variable and are systematically longer with components of the unstable site They decrease in intensity when the optical density and the pumping power increase

The global fluorescence signal observed for both sites (Figures 714(a) and (b)) is more chaotic for the unstable site with signals consisting of superposed short components (one at 10 microm and two at 16 microm) and a long component (one at 16 microm)

Stable site Unstable sitecomponent short(1)

626cm-1

short(2) 595cm-1

long(3) 615cm-1

short(1) 627cm-1

short(2) 596cm-1

long(3) 614cm-1

delay 60ns 155 ns 155 ns 138 ns 474 ns asymp1 μsrise 100 ns 330 ns 330 ns 330 ns 354 ns 45 μs

decay 322plusmn8 ns 314 plusmn8 ns 755plusmn28 ns (asymp2μs)

508 plusmn24 ns 512plusmn67 ns 8plusmn1 μs


Figure 714 Global fluorescence a) stable site b) unstable site Ar13CO2 = 2000 T = 5 K

In the case of 14N2O one fluorescence at 10 μm represented by R1 (MCT detector) as shown in Figures 715(a) and (b) is observed The time decay constant τ is 330 plusmn 8 ns (Figure 715(b)) when that of the pump laser signal (Gaussian pulse of a time duration of 5 ns) is 305 plusmn 5 ns For the fluorescence signal τM is 38 plusmn 2 ns (Figure 715(a)) Figure 715(b) shows the envelope of the fluorescence signals (MCT R1) when the intensity of the pump laser (GeAu R2) varies which indicates the existence of a threshold value of the pump for the stimulated emission (fluorescence disappears for some non-zero values of the pump intensity)

a)

time (ns)

Am

plitu

de (V

))

b)

time (ns)

Am

plitu

de (m

V))


Figure 715 Fluorescence a) delay b) envelope ArN2O = 2000 T = 5 K

Studies of the amplitude of the fluorescence signal as a function of the power of the laser pump realized with a grid polarizer show different possible effects depending on the molecule the trapping site and the host matrix

Figure 716 Fluorescence at 10 microm as a function of the pump energy ArN2O = 2000 T = 5 K (square)

260 microm (rhombus) 180 microm (triangle) 90 microm

The amplitude of the fluorescence signal varies linearly with the power of the pump laser with a different threshold effect depending on the thickness in the case of N2O (Figure 716) In the case of CO2 fluorescence variation follows two different schemes (Figures 717 and


718(a) and (b)) with the energy of the pump laser In Figure 717 which gives the variation of the intensity of the fluorescence at 10 microm for 13CO2 in a concentrated sample the signal is observed to be very chaotic

Figure 717 Fluorescence at 10 microm as a function of the pump energy (unstable site Ar13CO2 = 520 T = 5 K)

In Figures 718(a) and (b) which show the variation of the intensity of each observed fluorescence at 16 microm per site (stable Figure 718(a) or unstable Figure 718(b)) of CO2 in argon the threshold effect is different for each line and each trapping site and sometimes two different types of linear intensity variations with pump energy after the threshold are identified

In the case of ozone the frequency of the only observed fluorescence corresponds to the 2ν3-ν3 transition (Figure 712) and no intersite transfer is revealed for the duration of the fluorescence Its appearance occurs beyond a power threshold of the excitation laser and this threshold varies from one matrix to another It is always higher in the single site (HF) than in the double site (LF) Above this threshold the amplitude of the fluorescence signal increases linearly with the power of the laser pulse the concentration or the thickness of the specimen With regard to the temporal aspect fluorescence is extinguished when the laser excitation ceases and its duration is less than 5 ns The fluorescence signal starts a few nanoseconds after the


start of the samplersquos excitation This delay increases as the threshold increases and when the laserrsquos power decreases It depends on the size of the site and the matrix The yield of the fluorescence emitted by the HF site (single site) is more sensitive to temperature than when it is emitted by the LF site (double site) It increases from neon to xenon and when the matrix is annealed

Figure 718 Fluorescence at 16 microm as a function of the pump energy a) stable site b) unstable site (Ar13CO2 = 2000 T = 5 K)


734 Experimental results for the double resonance

The results mainly relate to studies with the CO2 laser probe on ozone trapped in RG and nitrogen matrices [JAS 95 DAH 97 DAH 98] In argon matrix the relaxation of ozone in the HF and LF sites can be probed from the ground state (cold band) with lines P26 and P28 and in nitrogen with the P24 line The coincidences between the CO2 laser lines and O3 levels in the other matrices make it possible to probe the hot bands with level v2 = 1 as the initial transition one (Figure 712 Xe P38 and P40 Ar P42 and P44 Kr and Ne P46 and P48 N2 P40) As described in Chapter 6 a negative signal is expected in the case of hot bands (Figure 719(a) B-P42) when the level ν2 gets populated and a positive signal is observed (Figure 719(a) A-P26) when the ground state is pumped

For the response of the probe signal to be worthy of analysis the diameter of the probe beam is set at 6 mm on the sample for a pump beam of 2 mm in diameter The pump passes through the sample twice at zero angle incidence while the probe beam makes an angle of 45deg with the normal at the samplersquos surface In this configuration it is necessary to distinguish two types of probed sample volumes the volume pumped wherein the physical processes are correlated only to the radiative relaxation (stimulated fluorescence) on a timescale less than 1 μs and the volume outside the path of the laser pump wherein the physical processes are correlated with the radiative (stimulated and spontaneous fluorescence) and non-radiative relaxation on a timescale greater than 1 μs Thermal effects related to non-radiative relaxation and time constants that are long are easily identified in this configuration by shifting the probe beam relative to the pump beam To observe the intrinsic relaxation in the pumped volume it is necessary to optimize the coincidence of the pumping beam with the portion of the probe beam incident on the detector in order to minimize the effect of the deviation of the probe beam on the surface of the detector under thermal effects

A signal of double resonance (DR) is characterized by three temporal parameters τM τR and τ The time τM corresponds to the time at which the maximum of the signal is reached and corresponds to the average time for the depopulation and the repopulation of the initial


level of the transition being probed The time τR is the characteristic time of signal rise or decrease and corresponds to the dynamics of the interaction between the probe and the initial level The decay time constant τ is usually determined by adjusting an exponential s(t) = Aexp(minustτ) to a wisely selected portion of the signal It is sometimes necessary to use two exponentials s(t) = A1 exp(minustτ1) + A2 exp(minustτ2) for the fit when two independent phenomena are juxtaposed These times are associated with the recovery of its original population state by the initial level

A signal due to a thermal effect (Figure 719(b)) that is initiated when an excited molecule transfers its energy to the environment is generally superposed to the DR signal over a timescale greater than the measured times (τ1 and τ2) To analyze the DR signal on its timescale it is necessary to have a negligible amplitude of the thermal noise compared to that of the DR signal A second type of thermal effect (Figure 719(b)) known in the theory of lasers as the lens effect [CAS 73] should also be minimized in order to properly analyze the DR signal This effect is revealed by the superposition of a second signal positive or negative depending on the relative positions of the pump and probe beams Its decrease (or growth) occurs on the same timescale as the thermal signal of the first type This effect can be minimized by adjusting the relative positions of the beams so that the positive signal compensates the negative signal

Figure 719 Double resonance signals a) carried by the line P26 (A 42 micros) and the line P42 (B 41 micros) of the CO2 laser

(O3Ar = 12000 width 110 mm) b) the probe(P26) and pump superposed (A) and the probe (P26) and pump laterally displaced (B)

a) b)


Figure 720 Decay time constant of the DR signal as a function of the concentration at 5 K a) O3Ar b) O3Kr

To differentiate among the different pathways of energy transfer discussed in Chapter 6 the DR studies are performed as a function of the samplersquos concentration the excitation spectrum the signal rise time the temperature the matrix and the pumped site (LF and HF in the GR) The τi (τM τR τ1 and τ2) are determined as a function of these various parameters (matrix concentration excitation frequency site and temperature) [JAS 95 DAH 97 DAH 98]

The concentration effect leads to the determination of the intrinsic relaxation time constants in the matrices (Figures 720(a) and (b)) for argon and krypton)

0

1

2

3

4

5

6

0 1000 2000 3000 4000 5000 6000 7000

Tem

ps (micro

s)

Dilution (ArO3)

a)

Tim

e

Dilution (ArO3)

0

20

40

60

80

100

120

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Dilution O3Kr

Tem

ps(micro

s)

b)

Tim

e (micros

)

Dilution (KrO3)


Figure 721 Repopulation time constant of the ground state in xenon

With xenon (Figure 721) the diffusion of thick deposits limits the experiments to samples with a maximum dilution of 20001 for which the times measured are still significant At this dilution the intrinsic relaxation must be determined by extrapolation For neon and nitrogen measurements were made without a preamplifier since the time constants are within the range of nanoseconds The results are given in Table 2 in [DAH 98] In a typical experiment at 12000 for the single site the time constants measured on the hot bands and the fundamental band are identical (Figure 719(a)) The molecules of this site transfer 700 cmminus1 to the lattice phonons in 42 micros or at a constant rate of 24 times 105 sminus1

The excitation spectra in argon (the probe laser is fixed on P42 and the pump laser ldquosweepsrdquo the profile of ν1 + ν3) as well as the measure of τM (Figure 722) shows that in the concentrated samples the energy moves from one class of molecule to another within the absorption profile of the line (spectral diffusion) and even from one site to another For a 150 dilution the energy propagates between the two sites the amplitude variation then reproduces the absorption profile of mode ν1 + ν3 In contrast the spectrum of the sample diluted at 12000 leads to a line profile with a single peak only Since the lower

0

50

100

150

200

250

300

350

0 500 1000 1500 2000 2500

dilution XeO3=

Tem

ps(micro

s)Ti

me (

micros)

Dilution (XeO3)


limit for the relaxation time constant of ν2 level of ozone in xenon is determined at 320 μs the spectral distribution is determined in this matrix from the measurement of τM

By comparing the τM of the cold and hot bands in argon and krypton a shift between the relaxation constants of the levels above ν2 and that of ν2 (assuming that the ground level is repopulated mainly by the level just above which is ν2 Figure 712) is highlighted The rise times of signals from the hot bands (Figure 722) characterized by the position of the signalrsquos maximum are longer (slower rise) than that of the fundamental band and give some information on the timescale of the intramolecular relaxation rates of the ν3 level toward the ν2 level

Figure 722 Rise times of DR signals carried by the lines P-42-negative and P26-positive (O3Ar = 12000 thickness 130 mm MCT detector)

In the more concentrated samples (150) the position of the maximum of the signal connected to the P42 line is reduced to 14 μs This effect is due to the shortening of the lifetime of the level v2 = 1 (v2 mode) by the concentration The rise of the signal however does not seem to be influenced by the concentration which is the expected tendency for an intramolecular cascade The thermal effect helps to

-1

-08

-06

-04

-02

0

02

04

06

08

0 05 1 15 2 25 3 35 4 45 5

Temps (micros)

Am

plitu

de (U

A)

21micros

15micros

04micros

Laser Pompe

Signal P(26)

Signal P(42)

Time (micros)

15 micros

21 micros

04 micros

Pump Laser


slow the rise of signals observed with P26 and P42 lines and the maximum shift is a consequence of the onset of this process in the matrix A series of measurements for the temperatures between 5 and 30 K shows that in the single site (HF) the relaxation time changes little up to 20 K and slightly shortens above 25 K For the double site (LF) the same type of study was conducted by probing the population variations of the v2 level with the P44 line When the molecules of the double site are excited by the laser pump the decay of the observed signal is faster (35 micros) than that corresponding to the single site (42 micros) This is the only difference that is experimentally determined between single and double sites

The results obtained in the different matrices are summarized in [DAH 98] At a fixed probe frequency the behavior of the signal rise times shows that the spectral diffusion phenomenon occurs inside inhomogeneous profiles when the pump laser scans the spectral range of the absorption line At high concentrations of argon (O3Ar = 1250) the intermolecular energy transfer is observed between the two trapping sites In xenon matrices this transfer occurs at lower concentrations (12000) From the relaxation rate of the level v2 = 1 toward the ground state measured at different concentrations in the GR and N2 matrices a maximum relaxation time constant called the intrinsic relaxation time constant τi is determined in the different matrices it covers three orders of magnitude from a few hundred nanoseconds in neon to 320 micros in xenon

74 Analysis by means of theoretical models

741 Determination of experimental time constants

The time constants are associated with the physical processes involved in the transfer of energy in the IR region Depending on the physical conditions these processes may be spontaneous emission induced emission absorption phase relaxation non-radiative relaxation by collision transfer or by multiphonon processes and transfers to the degrees of freedom of the slow librational modes or


faster vibrational and rotational modes The results are first discussed with reference to relaxation mechanisms and the analysis is then carried further with a model developed from previous theoretical studies

To interpret the absence of emissions at 16 microm in the case of N2O compared to the case of CO2 it is assumed that the intermolecular transfer to dimers of N2O (inter-system transfers) shortens the lifetime of the emitting levels increasing the stimulated emission transition thresholds of N2O transitions that occurs at 16 microm For concentrations of 12000 N2O dimers are more numerous compared to the case of CO2 in an argon matrix Furthermore as shown by the potential energy calculations [BER 96 DAH 99 LAK 00] the argon-N2O coupling is stronger than that between argon and CO2 even in the more favorable conditions when CO2 is located in a single site (stable site) As a result in argon the T1 population relaxation time is smaller for N2O than for CO2 When the absorption lines corresponding to the transitions between energy levels (Figures 63 and 712) can be determined experimentally either by conventional spectroscopy resolved in frequency by means of an interferometer or by laser absorption by scanning the absorption spectral range of the line the time constants can be obtained from the spectra In this case the formula giving the intensity of a vibrationndashrotation absorption line in the gas phase taking into account the shape of the absorption line (Gaussian by Doppler effect Lorentzian when broadened by pressure) [DAH 88] is applied

max

min

328 exp( )( ) 1 exp d

3 ( )g i Bif if if if i if

B

hc hc k TS f N g Rhc k T Q T

σ

σ

π σσ σ σ σ σ minus= minus minus minus

[76]

where σif is the wavenumber (in cmminus1) of the particular rovibrational transition N is the number of molecules per unit volume hcσi is the energy of the ground state Q(T) designates the total partition function of the molecule at temperature T Rif is the transition moment and finally gi is the rovibrational degeneracy due to the nuclear spin of the ground state In this formula the line shape is a Voigt function given by the convolution of a Gaussian function with a Lorentzian one Since the Doppler effect and the broadening by pressure are


concomitant the integral of the line shape over the absorption region

is equal to 1 ( max

min1 ( ) diff

σ

σσ σ σ= minus )

A corrective term due to the refractive index n effect of the solid RG must be applied in the condensed phase If the rotationndashvibration interaction is neglected and if the ground state is not degenerate (g0 = 1) the intensity of a vibrational line of a molecule in condensed phase at a low temperature is written as

max

min

22 3 21 1 8 ( ) d3 3

mif if if if

nS f Rn hc

σ

σ

π σ σ σ σ += minus

[77]

where Rifprime refers to the vibrational transition moment Written in this form the relation [77] provides essential information on the dipole transition moments of vibrational modes in the RG matrix These values can be experimentally extracted from the integrated intensity of the absorption spectrum from

0 ( )1 12 ( ) 2

ifmif if

t if

IS Ln d I

lN I lNσ σ

σσ σ

minus= =

minus [78]

where N is the number of molecules per cubic centimeter 2l is the length of the path of the radiation in the sample and the integrand is the integrated absorption measured from the IR absorption spectrum

For example from the absorption spectrum of the mode ν3 of the ozone molecule the gas phase lifetime was calculated from its

integrated absorption coefficient 0 ifif

SS

P= as

03

28

2 032110( )gaz

ifc Sντ

σrarr= [79]

where σ is expressed in cmndash1 c is expressed in msndash1 and S0if is

expressed in cmndash2 atmndash1 The lifetime of the ν3 band in the gas phase is


948 ms [SEC 81] Its lifetime can then be calculated in the different RG matrices by introducing the crystal field correction from the refractive index of the medium (Chapter 6 section 651)

2 29

( 2)s gn nτ τ=

+ [710]

The lifetime of a level can also be determined from the line width It is the sum of a homogeneous and inhomogeneous contributions The homogeneous width results from interactions of the molecule with its environment which induce frequency fluctuations that are identical for all the molecules of the sample and attributed to mainly two causes The first cause related to the population relaxation (type T1 Chapter 6 section 653) is due to the spontaneous transfer of energy of some dipoles toward the radiation and non-radiative relaxation channels The spontaneous emission (Chapter 6 section 651) is characterized by the constant γrad = 1T1 The non-radiative transfers are due to the dissipative coupling of active dipoles with the environment or inelastic collisions with the atoms forming the cage The second cause (type T2 Chapter 6 section 653) is due to the phase relaxation through elastic collisions that unlike the population relaxation does not alter the population or hence the energy of the initial level In contrast there is a gradual loss of phase coherence and an increase in partial incoherence The energy diffusion in an inhomogeneous profile (spectral distribution) also contributes to the decoherence effect of the phase in fact a slight shift in frequency between the trapped molecules is responsible for a dephasing effect of type T2 during the intermolecular energy transfer and that depends on the concentration The width of the observed transition varies with the concentration and temperature The inhomogeneous width concatenates the shifts in frequency due to differences in the environment of each trapped molecule Because of the numerous tiny crystal defects the lattice sites are not all identical and as a result the interactions with the matrix vary from one molecule to another and the shift in frequency is not identical Static multipolar interactions between molecules randomly distributed in the matrix are also a factor of inhomogeneous broadening


Finally the individual response of each molecule is in the form of a Lorentz function whose profile depends on the temperature and concentration while the frequency distribution for all of the molecules is described by a Gaussian function whose profile depends only on the variations in the concentration of the sample The final line shape in the IR spectrum is given by their convolution product leading to a Voigt profile

The time constants T1 (population relaxation) and T2 (phase relaxation) are determined from the line widths that stem from the combined homogeneous and inhomogeneous contributions As a first approximation it is assumed that the contributions to the line widths are additive such that

measured homogeneous inhomogeneousγ γ γ= + [711]

Generally after annealing and the rearrangement of the matrix atoms the inhomogeneous component shrinks irreversibly and becomes insensitive to variations in temperature On the contrary homogeneous effects are mainly due to the interactions coupling the molecule to the matrix

With an increase in temperature the density of populated phonon states increases and inelastic and elastic collisions with the atoms of the cage are enhanced As a result times T1 and T2 are shortened Because elastic collisions do not change the populations of the excited states then in a first approximation the contributions of types T1 and T2 to the broadening are independent The evolution of the measured width is expressed in this case as a function of a constant term and the sum of the two homogeneous contributions which are temperature dependent such as

measured inhomogeneous dephasing relaxation( ) ( ) ( )T T Tγ γ γ γ= + + [712]

with

relaxation1

1( )2

TcT

γπ

= and dephasing2

1( )2

TcT

γπ

=


When these data are not available it is necessary to have recourse to modeling and simulation to determine in an initial step the order of magnitude of the time constants and then to optimize the calculated values by comparing the experimental data on the fluorescence to the results of simulation from equations [611] or [614] (Chapter 6) It is necessary in this case to consider the apparatus function of the instrumental device and the observational conditions to reproduce as closely as possible the data collected from the experiment

Regarding the results of the DR they are discussed and compared with data in the literature in the context of the isolated binary collision model or multiphonon process The different pathways of energy transfer can also be explicitly determined by first calculating the energies and quantum levels of the different degrees of freedom by applying an approximation similar to the Born Oppenheimer approximation (BO separation of slow and fast modes) and then by calculating the probabilities associated with the contributions of the different types of coupling between the trapped molecule and its environment The time constants of the various processes are evaluated by applying a perturbation method (equation [621]) after relaxing the BO approximation constraint as performed for HCl or CO [BLU 78 LIN 80 KON 83] NH3 or CH3F [GIR 85 LAK 87a LAK 87b] or O3 [LAK 93 BRO 95]

Another method is to compare the energies calculated in the BO approximation and from the differences in energy of the levels associated with the modes of the different degrees of freedom the use the isolated binary collision andor the multiphonon model to analyze experimental results [LAK 00 LAK 11 LAK 14 LAK 15] The relaxation time constants can also be determined numerically from relaxation models using the density matrix described in section 654 (Chapter 6) Moreover by using a least square procedure or by using minimization algorithms (Simplex LevenbergndashMarquardt BroydenndashFletcherndashGoldfarbndashShanno) [NEL 65 LEV 44 MAR 63 BRO 69 FLE 70 COL 70 SHA 70] as described in Chapter 8 the digital approach reduces the time necessary for analysis of experimental results from inverse methods Calculations are initially performed


with simplified models [CHA 00c CHA 02] which can then be improved to reproduce as faithfully as possible experimentation and observation conditions [DAH 16]

742 Theoretical model for the induced fluorescence

To analyze the experimental results [CHA 00] the thresholds necessary for a stimulated fluorescence observation were calculated and compared for the three molecules in the argon matrix using the model described in section 653 (Chapter 6) and section 741 (equations [77]ndash[710])

In the volume excited by the pump (a cylinder of diameter 2 mm and length 2l) the stimulated emission cross-section can be written as

22 3 21 1 83 3

ifeffif if

if

n Rn hc

σπσγ

+=

[713]

and from the average length l of a photon path in the volume evaluated to 26 cm [APK 84 CHA 00b] the photon leak rate from the

excited volume can be calculated from pcK

n l= The threshold

population inversion (Table 75) is then given by

1Sif eff

if

Nlσ

Δ =

for the transition from state i to state f When 2

S Tif

NNΔ le (NT is

the density of molecules) a stimulated emission can be expected to occur

These thresholds can be compared to experimental thresholds and the observations of fluorescence or not to classify relaxation pathways according to the timescale and identify the most significant


Table 75 Minimum population inversion estimation ΔNT for the onset of the stimulated emission in a 180 μm thick sample (ΔNTN0 le 50 ) (a) matrix values (b) gas phase (c) derived from (a) and (b) (d) experimental (e) deduced from (c) and (d)

Applying equations [611] of Chapter 6 the fluorescence simulations for ozone trapped in the nitrogen and the xenon matrix were performed using the values of the parameters given in Table 76 In Figure 720 which reproduces curves showing the fluorescence intensity versus time for different values of the pump power it is seen that the calculation provides a theoretical threshold of 15 μJ a value that is less than the measured value of 37 μJ

To find the experimental threshold the model is extended by introducing the intramolecular transfers in order to interpret the results obtained with ozone Intramolecular transfers that are most likely to occur in the population transfer from ν1+ν3 to 2ν3 and that from 2ν3 toward the ν1+ν2 ν2+ν3 or 2ν2 levels must be selected first The possible intramolecular process in the system of equations are then introduced by adding a linear term (KintrasdotNi) (equations [714]) and by considering an additional level to take into account the time delay in the transfer between the levels ν1+ν3 and 2ν3 From the experimental

A) N2O (ArN2O = 2000) B) O3 (ArO3=200)

10 μm ν3-ν1 ν3-2ν2

775 μm ν1+ν2-ν2

17 μm ν2 manifold 31-20 31-22

20-11

10 μm 2ν3- ν3 double site single site

νlu (cm-1) γlu(a) (cm-1) 103 |Rlu|2 (b) (D2)

9385 (a) 1051 (a) 014 014 288 026

1290 (a) 010 36

581 (b) 571 (b) 008 008 958 388

579 (b) 008 488

1011 (a) 1012 (a) 026 030 71

1016 σul (c) (cm2) 0092 0009 22 033 013 017 13 11 10-16 ΔNT (e) (cm-3) ΔNTN0(e) ()

280 (d) 2900 21 (d) 215

12 09

80 200 6 15

150 12

20 24 021 05

C) CO2 (ArCO2 = 2000)

10 μm ν3-ν1 single site double site

ν3-2ν2 double site

16 μm ν2 31-20 single site double site

manifold 31-22 double site

20-11

single site double site

νlu (cm-1) γlu(a) (cm-1) 103 |Rlu|2 (b) (D)2

913 (b) 036 012 177

1018 (b) 012 067

625 (a) 626 (a) 009 0032 506 101

596 (a) 32 41

614 (a) 6135 (a) 009 0032 217 43


1230 400 140 115

950 270

15 27 2 09

7 22

37 65 5 2


results it can be assumed that the population transfer from ν1+ν3 to the level 2ν3 occurs instantaneously

21 2 3 2 1 2 2 1 2 int 2

12 1 3 2 1 2 2 1 1 1 0 1 int 1

32 1 3 2 1 2 2 1 3 1

( ) ( )

( )

dN ( ) dt

ra

ra

dN N N N K N A N K tdt

dN N N N K N A N A N Kdt

N N N K N A N K

minus minus

minus minus minus

minus minus

= minus minus minus + Λ = minus + minus minus = minus + minus

[714]

Table 76 Parameters used in numerical simulations (concentration 1200)

In equations [714] the percentage δ of molecules pumped by the laser is simply given by the ratio between the number of molecules that absorb a photon of the laser (Nabs) and the number of molecules present in the portion of the sample travelled by the laser (NO3) δ = NabsNO3 where NO3 is the product of the density (N) and the


volume excited by the laser (V) The cross-section diameter of the laser on the sample is equal to 2 mm and the laser makes a round trip

in the sample The result is a volume equal to 2100

eV π= (cm3) where e

is the thickness of the sample traversed twice (under the experimental conditions a laser pulse excites at most a few percent of the molecules present in the ground state far from a saturation regime with δmax being equal to 45 times 10ndash2) The laser power effect is contained in the term δ

Figure 723 Simulation of the fluorescence signal as a function of time for different values of the energy of the pump laser (O3Xe = 1200 thickness 85 microm)

The shape of the signal and the experimentally observed delay are compatible with the numerical results obtained with the model described in Chapter 6 Figure 723 shows the change in the fluorescence signal over time for different values of intensity of the laser pump

The beginning of fluorescence is more and more delayed when the laser power is decreased and in parallel the signal shrinks since its amplitude decreases with that of the laser pulse In the experimental measurements the latter phenomenon is partly hidden by the apparatus function of the detection chain In the model the concentration effect is identical to that of the power of the laser pulse

000E+00 400E+00 800E+00 120E+01 160E+01 200E+01

Temps(ns)

Am

plitu

de (U

niteacute

s arb

itrai

res)

x10 5

x10

a) Plaser=200microJpulse

b) Plaser=40microJpulse

c) Plaser=4microJpulse

d) Plaser=3microJpulse

a)

b)c)

d)

Am

plitu

de (A

U)

Time (ns)


since when diluting the sample the absorption coefficient α that is correlated to the calculation of δ is diminished equivalent to a decrease in the pump power

The series of peaks (spikes during the pulse duration of the pump laser) that appears in the rising of each signal is likely integrated by the detection system which is not fast enough to track these oscillations This effect is attributed to the transient phenomena that precede the installation of a laser wave in a cavity at first the difference in population (ΔN) increases rapidly and exceeds the operating condition (threshold) This increase is stopped because of the saturation effect due to the repopulation of the lower level ΔN then decreases and passes the threshold in the opposite direction therefore the field in the cavity is at its maximum The latter begins to decrease and reaches its minimum at the same time as the population difference The stimulated emission having disappeared the laser rebuilds a population on the upper level and the cycle starts again

The theoretical model described in Chapter 6 extended by the inclusion of intramolecular transfers reproduces satisfactorily the variations and delay thresholds induced by the different parameters such as the site effect matrix temperature power and concentration The equations allow the determination of the order of magnitude of the intramolecular relaxation time constants of the excited vibrational levels of ozone trapped in the different matrices The energy relaxation of ozone after the ν1 + ν3 level is excited may be interpreted by the transfer of energy through different channels at different timescales An initial step corresponds to an ultrarapid intramolecular transfer (lt1 ns in all the matrices) of molecules from the energy level of the mode ν1 + ν3 to the mode 2ν3 A part of the molecules then relaxes to the level ν3 leading to the emission of stimulated fluorescence (lt10 ns) and the rest of the molecules are transferred to the level ν2 + ν3 during the duration of the pulse (7 ns) The level ν3 is then depopulated by intramolecular transfer toward level ν2 The molecules that transit through the level ν2 + ν3 must also relax on the level ν2 but on a slightly longer timescale The last step of the ozone relaxation is determined by the energy decay of level ν2 toward the ground state This step is much slower than the previous


ones because the energy gap (700 cmndash1) is twice as high compared to those involved in the transfers between levels ν3 and ν2

The comparison of the molecular properties of O3 CO2 and N2O [CHA 00b] has validated the theoretical model The conditions to observe a stimulated emission depend on the molecular parameters (widths transition moments of the absorption lines) The model includes all radiative and non-radiative transfers and can be applied to determine the magnitude of the intramolecular energy transfer constants

743 Theoretical model for the DR

Theoretical models have been described in section 652 (Chapter 6) to study processes by which an excited vibrational state relaxes to a ground state ( (v ) (v )f f i if i i e E J E Jrarr rarr ) to achieve thermal equilibrium In these models the matrix is a thermal bath and energy transfer involves either several phonons or binary collisions Direct transfer to phonons is determined by the energy gap law (energy mismatch corresponding to N phonons) (equation [66]

( ) (0) ( 1)Nk T k nν= + ) and for binary collisions by the momentum mismatch between the trapped molecule and a matrix atom the reduced mass of the colliders and the repulsive interaction potential

(equation [69] 2 2 exp( 2 )k Eπα

μ prop minus μΔ

) These models have been

extended by including other effects such as the indirect transfer through the rotation via an empirical model (equation [67]

exp( )mK Jαprop minus ) or through local phonons νL assisted by one or two phonons of the matrix νph (equation [68]

( ) ( )i i f f L phE v J E v J Nh nhν νminus = + ) In the process where the intramolecular transfers are not negligible relaxation constants can be determined from the probability of transfer from a donor D to an

acceptor A (equation [610] 6 63 1 1

8 (2 )DADA A D

Pc n Rπ ν τ τ

=

( ) ( )D Af f dν ν ν ) The models based on the formalism of the density


matrix are solved numerically in general and can be applied to both types of relaxation processes

Comparing the experimental results with the predictions of the theoretical models [DAH 98] shows that the transfer to the matrix from the highest energy level is best interpreted by the momentum mismatch included in the isolated binary collision model which is based on the repulsive portion of the interaction between the trapped molecule and the matrix rather than by the mismatch of energy by a direct process to several phonons The plot on a logarithmic scale of O3 relaxation times measured in HF and LF sites of the different matrices shows that the logarithm of kmicro2 (log(kmicro2) = A+B(2microΔE)12) clearly appears as a linear function of (2microΔE)12 (Figure 724 (log(kmicro2) = A+B(2microΔE)12) for the two sites

Figure 724 Intrinsic relaxation constant of the studied polyatomic molecules in noble gas matrices depending on the

momentum mismatch (between 5 and 9 K depending on the molecules)

Figure 724 compares the relaxation of molecules O3 SF6 NH3 CH3F and CD3F in a RG matrix when the first excited level (last level just above the ground state for relaxation process) drives the last stage of relaxation For O3 NH3 and SF6 the experimental results are


aligned but for CH3F and CD3F the points lie on a curve This comparison shows that the energy transfer is a V-T type when the points are aligned but the two V-T and V-R energy transfer mechanisms are comparable when the points are not aligned

However equation [69] implies that the slope of the straight line is proportional to 1α which is connected to the range of the repulsive potential (exp(ndashαr)) between the molecule and the RG It is therefore possible to obtain the magnitude of this coefficient from the slope of the line The value is 95 Aringndash1 for O3 It is comparable to that found for SF6 (α = 10 Aringndash1) [BOI 87] and is two times higher than that estimated from molecular beam experiments The relaxation of the level v2 = 1 of O3 may thus be explained by isolated binary collisions with atoms of the matrix Other triatomic molecules CO2 and N2O in different matrices can have a similar effect to that of O3 that behaves itself like SF6 The results show that from two different points or from one and the value of the coefficient α the rate constants for the other matrices andor isotopic compounds can be predicted

75 Conclusions

From the comparison of the theoretical predictions of the binary collisions model applied to small polyatomic molecules studied in matrices it is possible to determine some predictive rules on intrinsic relaxation time constants of the studied model systems The V-T or the rotation-assisted transfer is predominant in the relaxation of the energy and therefore of the thermal effects The use of a theoretical model adjusted to the particular physical system allows the interpretation of the observed stimulated emission phenomenon The power threshold at which this phenomenon occurs and the delay with which the signal appears with respect to the exciting pulse can thus be calculated Moreover the delayndashthreshold relationship can be evaluated and quantitatively explained and the relative efficiencies in the different matrices qualitatively interpreted The magnitudes of non-radiative relaxation constants connected to thermal effects can also be obtained These results show that the study of model systems


can be a starting point to develop or confirm models applicable to other systems in the case of energy transfers when different relaxation pathways may participate in relaxation processes and hence heat propagation

The results obtained with a laser pump in the nanosecond regime shows that the phenomena that contribute to thermal relaxation are characterized by time constants ranging from the nanosecond to the millisecond The experimental techniques described may be implemented with picosecond or femtosecond lasers Phenomena that are characterized by shorter time constants can then be studied Energy transfers occur locally on timescales lower than the nanosecond and cause heating that dissipates through different pathways This heat propagation is characterized macroscopically in the microsecond range

8

Defects at the Interfaces

The encapsulation by potting of an embedded mechatronic system serves to protect the electronic components and circuits against external conditions (vibration temperature changes humidity corrosion etc) As part of the design of an inserted metal leadframe (IML) type power module it is necessary to determine the effect of the potting on the encapsulated elements and identify the physicalndashchemical parameters that can vary at the interfaces of materials in contact Ellipsometry is an optical non-destructive probe technique sensitive to surfaces and interfaces which makes it possible to change the optical properties of materials under stress (mechanical thermal and chemical) in the electromagnetic spectrum (ranging from microwaves to infrared (IR) In the IR range the spectroscopic ellipsometry (SE) technique is applied to study the copperndashnickel polymer interfaces or aluminumndashresin siliconndashresin or quartzndashresin The ability of the encapsulation materials to withstand combined stresses of humidity and temperature can also be evaluated To identify changes in the physical and chemical properties that have an impact on the functionality of the encapsulation materials an inverse method based on an optimization algorithm is numerically performed to analyze the experimental results

81 Measurement techniques by ellipsometry

Ellipsometry is a non-destructive optical technique used to study the changes of multimaterials and assembly interfaces present in the devices of the mechatronic systems or embedded electronic systems under the influence of external stresses (humidity heat chemical) The technique being sensitive to surfaces and interfaces it may be applied to samples representative of surfaces and interfaces to


First Edition


measure the change in the optical properties of the materials before and after undergoing stresses For example to characterize the ability of the potting to withstand the stresses in operation mode polymerndashmetal type samples mimicking materials assembled in the modules are fabricated and then studied by SE in the near IR-IR Studies on various types of polymers that may be present are then performed to determine the range of variation of the interfacersquos properties Experimental data can thus be obtained with a measured dispersion enabling the application of the reliability-based design optimization (RBDO) model described in Chapter 2 to optimize the design of embedded electronic or mechatronic systems

The optical behavior of a material is characterized by a refractive index written in the complex form to take into account absorption and refraction phenomena As described in Chapter 5 this refractive index depends on the wavelength λ of the electromagnetic radiation passing through the medium Ellipsometry is a method of studying lightrsquos polarization state after it is reflected on a samplersquos surface in order to determine the samplersquos physical and optical characteristics particularly its refractive index but also its thickness and roughness

The reflected electric field Er can be split into two orthogonal polarized components Erp and Ers (Chapter 4 section 41) whose amplitudes determine the ellipticity of the reflected wave This ellipticity is characterized by the ratio [DAH 15] as

( ) with tanrp p ip ip p i

rs s is is s

E r E E re

E r E E rΔρ ρ ψ= = = = [81]

where Eip and Eis correspond to the amplitudes of the incident wave for each type of polarization

From the initial state of polarization of the incident wave Ei and the measurement of the polarization state after reflection an ellipsometer provides the ellipsometric parameter ρ (equation [81]) which can be expressed in terms of two ellipsometric parameters Ψ and Δ that are

Defects at the Interfaces 221

related to variations in the ellipsersquos shape (ratio between the minor and the major axis tilt) and leads to the determination of ρ

Three techniques (extinction rotation and phase modulation [PM]) can be used to measure ellipsometric parameters Ψ and Δ and determine the polarization state Whatever technique is used the apparatus comprises the optical devices shown in Figure 81 two tilting arms and a sample holder One of the arms is composed of a light source and an optical setup for obtaining an incident wave of known polarization The other arm consists of an analyzer and a detector for measuring the wave polarization state reflected by the sample

Figure 81 Elements of an ellipsometer Source S polarizer P λ4 quarter wave plate compensator C

phase modulator PM sample E analyzer A detector D

The measurement and analysis process comprises the following steps

ndash preparation of the incident lightrsquos initial polarization state

ndash lightndashmatter interaction (sample) and reflection

ndash measurement of the reflected waversquos polarization state


ndash calculation of Ψ and Δ (tan(Ψ) and cos(Δ))

ndash evaluation of random and systematic errors on Ψ and Δ

ndash determination of the physical properties of the material (n k d etc) from Ψ and Δ using a numerical optimization method (inverse problem)

An ideal ellipsometer does not exist because each of the three ellipsometry techniques presents advantages and disadvantages The choices of possible setups are limited if the ellipsometer must be fast accurate and spectroscopic at the same time

811 The extinction measurement technique

This is a manual method that is slow but very precise and rather suitable for a single wavelength (monochromatic) The measuring device is shown schematically in Figure 81 The optical setup generally consists of a monochromatic source (laser or lamp equipped with a filter) a polarizer a compensator (quarter-wave plate in general) an analyzer and a photomultiplier

Assuming that the sample is isotropic so that the polarized waves p (parallel to the incidence plane) and s (perpendicular to the incidence plane) do not mix after reflection on the sample the source arm provides a wave in a known polarization state A first optical device consisting of a quarter-wave plate and a polarizer gives a luminous flux of equal intensity in all directions of the electric field (circular polarization) An element P C or A (polarizer compensator or analyzer) is then fixed and the other two are rotated to reduce the intensity behind A to zero [ASP 74 ASP 75 COL 90]

The principle of the extinction ellipsometer is the following

ndash the light is linearly polarized after passing through P

ndash it is then converted into elliptically polarized light by C

ndash the latter is oriented so as to obtain a rectilinear polarization after reflection of light on the sample (the compensator has a function


opposite to that of the sample by offsetting the samplersquos impact on the polarization state of the incident light)

ndash A is then oriented perpendicularly to the obtained rectilinear polarization leading to the light beamrsquos extinction The orientations of the polarizer the quarter-wave plate and the analyzer determine the samplersquos ellipsometric parameters by

tan tan( )tan tan1 tan tan( )

i C P Ce Ai C P C

Δ minus minusΨ = minus+ minus

[82]

where P is the polarizerrsquos angle C is the compensatorrsquos angle and A is the analyzerrsquos angle measured relative to the incidence plane For a given compensatorrsquos angle to each pair of values (ψΔ) correspond two pairs of angles P and A The extinction ellipsometers present the advantage of a direct calculation of angles ψ and Δ and are more accurate than other types of ellipsometers

812 The measurement by rotating optical component technique

The measurement by rotating optical component technique lends itself well to automation of measurement as well as its use over a wide spectral range (SE) The optical setup is easy to perform and the measurement technique is relatively accurate However the acquisition of measurement data is slow because it is limited by the mechanics (speed of the rotating elements) The rotational frequency of the rotating element (P C A) is between 50 and 100 Hz The spectral range in wavelength is very wide (from UV to near IR) The light beam is modulated in polarization by the rotation of one of the optical components polarizer P analyzer A or compensator C This ellipsometry technique has a drawback which is linked to the indeterminacy on the sign of Δ The rotating polarizer setup is sensitive to the residual polarization of the source while in the case of a rotating analyzer setup the use of a detector that is insensitive to the polarization state is necessary [ASP 74 ASP 75]


The rotating compensator ellipsometer (RCE) overcomes all these constraints and calculates the ellipsometric parameters without indeterminacy in the sign of Δ However this type of ellipsometer is more suited to a fixed wavelength

A calibration procedure is necessary prior to measurements on samples in the case of a spectroscopic ellipsometer In the rotating polarizer setup the spectrometer is placed after the analyzer allowing dispersal and filtering of stray light generated at the sample level In the configuration of the rotating analyzer the spectrometer is located between the source and the polarizer which implies that the detector is much more sensitive to stray light

813 The PM measurement technique

The PM ellipsometer device comprises the source the polarizer the analyzer the detector and a phase modulator [DRE 82] The latter is located just after or just before the analyzer (Figure 81) With this technique a photoelastic modulator (photoelastic silica bar subjected to a stress produced by an oscillating piezoelectric transducer at a frequency of 50 kHz) introduces a phase difference between Es and Ep No special characteristic device is required to handle polarization effects at the level of the source and detector The PM ellipsometer has the advantage of having an excellent accuracy on Δ and a fast acquisition rate because of the very high modulation frequency (~50 kHz)

Another advantage is the absence of mechanical vibrations (except for the phase modulator) because the optical components are fixed during measurements which eliminates mechanical fluctuation problems that can occur in the case of the rotating elements However a high-performance electronic system that is capable of providing signal acquisition and processing at a frequency compatible with the modulation at 50 kHz is required The modulator must be calibrated as a function of the wavelength and the excitation voltage must be


locked to the wavelength as well [ACH 89] As the modulator is very sensitive to temperature fluctuations a device to stabilize temperature in the surroundings of the ellipsometer is necessary The detector used is a multichannel photomultiplier with a high sensitivity for UV-visible and near IR range The lamp various optical elements and transmission of air limit the spectrum to the range of 193ndash2000 nm

82 Analysis of results by inverse method

The analysis of ellipsometry data is a problem of deterministic optimization as described in Chapter 2 (section 22) The optimization is performed by comparing the experimental values to the theoretical ellipsometric parameters calculated from a model [DAH 04a DAH 04b NOU 07 LOU 08 DAH 15] To calculate the different coefficients of reflection and transmission [AZZ 77] at the interface of the different layers that make up the sample structure the matrix methods of Abeles [ABE 50] or Hayfield and White [HAY 64] are applied Each layer is characterized by optical properties depending on the type of the layerrsquos material (air polymer solid amorphous semiconductor ceramic etc)

By considering the layer stack of Figure 82 each layer thickness is denoted by Di and each complex index by in n ikα α= + nα and kα are the real and imaginary parts of the complex index θi is the angle of incidence of the incident ray in the i-layer Thus from these parameters the overall Fresnel coefficients for a sample can be calculated using Abeles matrix formalism to finally deduce the inversion of ellipsometry equations that leads to the ellipsometric parameters of the sample In the z-dimension the total electric field is the sum of two components one that propagates in the increasing z direction (that is to say toward the substrate) and the other in the decreasing z direction (toward the surface of the sample) denoted respectively as ( )E z+ and ( )E zminus It is assumed that the beam is linearly p or s polarized and that this property is preserved at the


crossing of the layer boundaries The field at two depths z and z is connected by the matrix transformation

11 12

21 22

( ) ( )( ) ( )

S SE z E zS SE z E z

+ +

minus minus

prime = prime

[83]

Figure 82 Stack of N isotropic layers of thickness Di and index ni of the sample of total thickness D on a substrate

Taking z and zprime on each side of the interface i(i+1) the interface crossing matrix Iii+1 can be written as

1 1

1 1

111

i ii i

i ii i

rI

rt+

+++

=

[84]

where r and t are the Fresnel coefficients whose polarization-dependent expressions of type p or s are given by

1 1 1 1 1 1

1 1 1 1

cos cos cos coscos cos cos cos

i i i i i i i ipi i si i

i i i i i i i i

n n n nr rn n n n

θ θ θ θθ θ θ θ

+ + + ++ +

+ + + +

minus minus= =+ +

[85]

and

1 11 1 1 1

2 cos 2 coscos cos cos cos

i i i ipi i si i

i i i i i i i i

n nt tn n n n

θ θθ θ θ θ+ +

+ + + +

= =+ +

[86]

Substrate

Ambient Medium

Di


Thus two matrices Iii+1 are defined according to the considered polarization mode Assuming z and z are in the same layer i separated by a distance Di the propagation matrix in the layer can be written as

0 2with cos 2 cos0

i

i

i

i i i i i i i ii

eT D n D n

e

δ

δ

πδ θ πσ θλminus

= = =

[87]

The matrices T are independent of the polarization mode but their expressions assume that the crossed layer is isotropic The response of the total stack to the beamrsquos crossing is governed by a formula such as [85] and [86] with z at the substrate level and z at the ambient level Successive angles in each layer are defined by SnellndashDescartes law The matrix is then written in the productrsquos form

( )1

01 1 12 2 1 1 1 1 10

N

i i i N N N i i ii

S I T T T I T T I I Tminus

+ + minus minus +=

= = prod [88]

By developing the matrix product for the two polarization modes that is for the two types of matrices IiI + 1 pseudo-Fresnel coefficients can be defined for the multilayer system in the form

21 21

11 11

andp sp s

p s

S Sr rS S

= = [89]

leading to the following expression of the ellipsometric parameter

( )21 11

11 21

tanp s iP

S p s

S Sr er S S

ρ ψ Δ= = = [810]

The ellipsometric parameters ψ and Δ or (tan (ψ) and cos (Δ)) are obtained numerically from a computer connected to the detection chain For an air layer substrate system the ψ and Δ dependence as a function of all the parameters to be determined can be symbolically written as

( ) a s s f f ff n n k n k eΨ = [811]


and

( )a s s f f fg n n k n k eΔ = [812]

where the indices a s and f correspond to the ambient medium substrate and film respectively The analysis consists of comparing the values of the parameters tan(ψth) or ψth and cos(Δth) or Δth calculated from a model with the measured values tan(ψexp) or ψexp and cos(Δexp) or Δexp

It is necessary in this case to define a cost or objective function (Chapter 2 section 241) to optimize the comparison From the ellipsometric parameters ρth and ρexp a cost function σ is defined (mean square error) that determines the difference between the calculated and measured values by

2

exp2

1 exp

( ) ( 11 ( )

nj th j j

j j

X

n m

ρ θ ρ θσ

ρ θ=

minus=

minus minus part

[813]

If the signal to noise ratio which is different in different zones of the measurementrsquos spectral range is taken into account the experimental data affected by the noise is better adjusted by the Jellison formula As a function of ellipsometric angles it is written as

2 2

exp exp

2 21 exp exp

( ) ( ( ) ( 1 1 ( ) ( )

n j th j j j th j j

j j j

X X

n m

θ θ θ θσ

θ θ=

Ψ minus Ψ Δ minus Δ = + minus minus partΨ partΔ

[814]

where n represents the number of data points that is two times the number of wavelengths andor angles of incidence chosen for measurement m is the number of unknown parameters to be adjusted and partΨ or partΔ is the standard deviations of the experimental data This equation has n values for n2 wavelengths (or angles) because there are two measured values ψ and Δ per wavelength (or angle) Data weighting by the inverse of the standard deviations allows the


reduction of the contribution to the adjustment of the measurements affected by noise In this expression 1 2( )mX x x x=

is a vector

whose components are the different parameters to adjust For example in the case of a transparent isotropic film on an absorbent substrate 1 2 3 4 ( )f f s sX x n x d x n x k= = = = =

and thus m = 4 the

indices f and s corresponding respectively to the film and the substrate In the case of an absorbant film on substrate the ambient medium is usually air (nair = 1) of known index and it leaves only five unknowns to be determined (m = 5) with

1 2 3 4 5 ( )f f f s sX x n x k x d x n x k= = = = = =

It should be noted finally that there is an interdependence between certain parameters of

1 2 ( )mX x x x=

Thus there is a strong correlation between the refractive index and the thickness of a film To reduce this interdependence effect a minimum set of measured parameters is necessary for the adjustment In this perspective data for several incident angles can be combined if there are many parameters to adjust For example for three parameters to be adjusted (df n and kf) it takes at least six sets of measurements of which three are angles of incidence (Ψ and Δ are measured each time)

A more complex layer model can take into account the presence of roughness on the surface of the thin layer or at the interface between two layers To determine the roughness of the film surface the surface is stratified into two flat and homogeneous layers of different media and indices (Figure 83) The first layer is composed exclusively of the studied material whose index and thickness is to be determined (medium 1) Above the second layer is composed of an effective medium corresponding to an air-material ldquocompositerdquo characterized by an effective index and a percentage of inclusion between air and the material so as to take into account the presence of roughness (medium 2) It is necessary to limit the thickness to a value that is physically acceptable that is to say the average thickness of roughness as well as the fraction of inclusion of air in the layer (MaxwellndashGarnett model)


Figure 83 Rough surface modeled by an effective medium

The following formula is used

2 2 2 2

2 2 2 2 2 2

e h i hi

ie h i h

n n n nfn n n n

minus minus=

+ + [815]

where ni is the inclusion medium index nh is the medium host index ne is the effective medium index and fi is the inclusion fraction All these parameters must also be fitted by the adjustment algorithm to minimize the differences between the calculated and experimental values Optimization is a search problem of the maximum or minimum value of a function f(x) (Chapter 2 section 22) which may have local minima or maxima (Figure 84) It is always possible to arrive at a minimization problem (in this case - f (x) is optimized) and the algorithm used must be able to reach the minimum value by avoiding local minima

Figure 84 Maxima and minima of a function

Substrate

Thin layer (medium1)

Effective medium (medium 2)

Local Maxima

Local MinimumGlobal Minimum

Global MaximumF(x)

x


Generally because models depend nonlinearly on variables to be adjusted the cost function σ is a nonlinear function of the parameters of 1 2 ( ) mX x x x=

The minimum can be achieved only by an

iterative method one starts with an initial estimate of the parameters which is refined at each step until the parameters no longer vary Inversion algorithms that can be classified into two categories are generally used those that require the gradient calculation such as the descent method with the gradient or conjugate gradient the method of Newton the method of LevenbergndashMarquardt (LM) [LEV 44 MAR 63] the method of DavidonndashFletcherndashPowell [DAV 59 FLE 63] or the method of BroydenndashFletcherndashGoldfarbndashShanno (BFGS) [BRO 69 FLE 70 GOL 70 SHA 70] and those that minimize the function without calculating the gradient as the simplex method [DAN 90] or the bisection method [POW 64 BRE 73] With methods based on the computation of the gradient a series of vectors is constructed which converge to the point that minimizes the multivariable function At the order of iteration k the vector

0 0 0 01 2 ( ) mX x x x=

at the order 0 (initial starting point) has been

submitted to a sequence of transformations consisting of adding a small variation ( )kXnabla

around the current point to each component of

X(k) such that

( 1) ( ) ( ) 01 2k k kX X X k+ = + nabla =

[816]

The calculation of ( )kXnabla

depends on the criterion function σ and its derivatives

In ellipsometry the algorithms that are commonly used are the nonlinear simplex method of Nelder and Mead [NEL 65 PRE 86] a zeroth-order method that does not require the calculation of the gradient and the LM and BFGS method which are methods of the second order that require the computation of the gradient of the cost function for evaluating the derivatives up to the second order Methods of order two consist of approaching the cost function by a Taylor expansion up to the second order LM and BFGS use a quasi-Newtonian method based on Newtonrsquos method to evaluate the


Hessian matrix (section 822 equation [817]) for faster convergence than the methods of steepest descent With most nonlinear optimization methods a local minimum is obtained in the vicinity of the initial estimate The latter must therefore be determined carefully Parameter validation is a minimization problem with constraints (Chapter 2 section 22) The methods used are recalled in the following sections The algorithms can be easily obtained by a search on the internet as well as programs in the language used (Fortran C language Matlab Maple Python etc)

821 The simplex method

The simplex method was developed by Danzig [DAN 90] in the United States in 1947 The simplex algorithm is used to solve linear problems in canonical or standard forms Its main advantage is its low computational time for relatively large problems The simplex method is to minimize the scalar multivariable function σ by a trial and error method starting from a number of potential solutions that are probed until convergence toward an adequate solution

Figure 85 Geometric transformations of a simplex

The nonlinear optimization algorithm of Nelder and Mead [NEL 65] is based on the simplex model of Danzig a geometrical method that aims to construct a simplex of N + 1 vertices from N parameters to which a number of operations is applied If the function to be minimized σ is of two variables the simplex is a triangle if it is

d) Contraction NDa) Reflection b) Expansion c) Contraction 1D


of three variables then the simplex is a tetrahedron An algorithm that compares the values of the function to be determined at each vertex of the simplex is applied The vertex that gives the least adequate value (the largest) is replaced by a new vertex The values of σ are recalculated and the search for the minimum is sequentially sought by applying the withdrawal and replacement procedure for a better vertex This process generates a sequence of simplexes for which the values of the function at the vertices become increasingly small In principle the hypersurface of simplexes decreases progressively thus converging toward the coordinates of the minimum This criterion is not necessarily feasible at each replacement step In this case the simplex is subject to other operations (Figure 85) reflection (maintaining the volume) or expansion (maintaining the volume) or contraction or multidimensional contraction (reduction in the volume close to the solution) At each transformation the criterion function is evaluated to find a better point than that calculated in the previous step The procedure is continued until the stage corresponding to the stopping criteria (less than or equal to the threshold)

If the number of variables is n a general simplex of n +1 vertices is constructed from n initial values starting by successively giving an increase at each starting value These operations can be implemented on the simplex for example by sorting the vertices of the simplex according to the value of the cost function to select three vertices having respectively the best score the worst score and the one just before the worst score The vertex of the worst score is replaced by the vertex on the new simplex obtained by reflection with respect to the hyperplane defined by the other vertices which requires doing a search in the direction opposite to that which gave the worst score If the cost function is improved the simplex is expanded otherwise a one-dimensional contraction is performed If despite this the cost function does not improve the algorithm performs a multidimensional contraction in several directions The vertex corresponding to the maximum value of σ is replaced by a point in the n variables space which leads to a correspondingly lower value of σ

In the case of a function of two variables f (x y) for example the simplex is a triangle Let S1 (x1 y1) S2 (x2 y2) and S3 (x3 y3) be the


vertices of the simplex and zk = f (Sk) the calculated values to vertices Sk k = 1 2 3 Let m be the (minimum) μ the (average) and M the (max) values calculated at vertices Sk k = 1 2 3 in ascending order The vertex corresponding to M must be replaced The simplex algorithm is performed in the following manner by calculating

1) the mid-point

SM = (fminus1(m) + fminus1(micro))2

2) the vertex reached by reflection

SR = SM + (SM minus (fminus1(M)) = 2SM minus (fminus1(M))

3) the vertex reached by expansion

SD = SR + (SR ndash SM) = 2 SR ndash SM

4) the vertex reached by contraction SC from the two possible vertices for contraction C1 = ((fminus1(M) + SM)2) and C2 = (SM + SR)2) and from the condition if f(C1) lt f(C2) then SC = C1 or else SC = C2

5) the best vertex

SF = (fminus1(M) + fminus1(m))2

The following algorithm is then applied if f (SR) ltμ then apply case I or else apply case II

ndash Case I if (m lt f(SR)) then replace fminus1(M) by SR or else calculate SD and if f(SD) lt micro) replace fminus1(M) by SD or else replace fminus1(M) by SR

ndash Case II if (f(SR) lt M) then replace fminus1(M) by SR If f(SR) ge M then calculate SC and if f(SC) lt M then replace fminus1(M) by SC or else

calculate SF replace fminus1(M) by SF and replace fminus1(micro) by SM

822 The LM method

The LM method [MAR 63] is an improvement to the NewtonndashRaphson method It consists of replacing in the vicinity of the current


point xk (xk is a vector of m parameters to be adjusted) the function σ by its Taylor expansion up to second order (quadratic approximation) that is to say

21( ) ( ) ( )( ) ( ) ( )( )2

k T k k k T k kx x x x x x x x x xσ σ σ σ= + nabla minus + minus nabla minus [817]

where x represents a vector of m parameters to adjust ( )T kxσnabla is the transpose of the function σrsquos gradient at the current point xk

2 ( )kkH xσ= nabla is the Hessian matrix of σ at the current point xk and

(x minus xk) the difference vector between vectors x and xk (the exponent T denotes the transpose of the difference vector)

A new vector xk + 1 corresponding to the minimum of σ(x) at the harmonic approximation if it exists is considered The Hessian matrix is then defined as positive and the function σ(x) is strictly convex The minimum being unique is defined by the condition

1( ) 0kxσ +nabla = leading to

1 2 1 1( ( )) ( ) ( )k k k k k kkx x x x x H xσ σ σ+ minus minus= minus nabla nabla = minus nabla [818]

If the function σ is quadratic the convergence is ensured in one iteration If the function is of any order Newtonrsquos method turns into an iterative method with a quadratic convergence order in the neighborhood of the minimum To fix the poor global convergence property of the Newtonrsquos method a linear search method along the search direction in steps of βk is introduced in order to minimize the function σ(xk + βkdk) or arrive at σ(xk + βkdk) lt σ(xk) where

2 1 1( ( )) ( ) ( )k k k kkd x x H xσ σ σminus minus= minus nabla nabla = minus nabla [819]

This method requires the calculation of the Hessian or of its inverse It is not always possible to do so and especially when it is not positive the displacement direction dk is not a direction of descent and the overall convergence is not assured The quasi-Newtonian


methods are implemented to generalize the iterative Newton formula without explicit calculation of the Hessian They generate a sequence of symmetric positive definite matrices that are approximations always better of the real Hessian matrix or of its inverse and toward which they converge

The algorithm of the method must take into account the inequality and equality constraints that correspond to physically acceptable values From a starting vector x0 representative of a vector of the parameters m initialized to physically acceptable values the coefficients of the matrix H0 are initialized at 1 At the iteration k the direction of descent is calculated by the equation 1 ( )k k

kd H xσminus= minus nabla The search direction is carried out in steps of βk corresponding to the minimization of the function σ(xk + βkdk) or to the inequality σ (xk + βkdk) lt σ (xk) which leads to the iterative formula

1 1 ( )k k kk kx x H xβ σ+ minus= minus nabla [820]

The matrices Hk are calculated according to the equation Hk + 1 = Hk + f(δkγk) with 1k k

k x xδ += minus and 1( ) ( )k kk x xγ σ σ+= nabla minus nabla

and where f is a function defined in the LM algorithm

The LM method requires the knowledge of the value of the objective function and its gradient The Hessian is approximated as

( ) ( )k kkH x x Iσ σ λΤ= nabla nabla + [821]

where λ ge 0 From equation [820] xk + 1 can be determined from

( ) ( ) ( ) ( )k k k kk k kH x x x I xδ σ σ σ λ δ σΤ = minusnabla nabla nabla + = minusnabla [822]

If σ (xk + δk) lt σ (xk) the solution xk + δk is accepted and a new

iteration is switched on if the stop criterion is not met with xk + δk and

λ2 If σ (xk + δk) gt σ (xk) another iteration is started from xk and 2λ to determine 1k k

k x xδ += minus


823 The quasi-Newton BFGS method

With the BFGS algorithm the same procedure as with the LM algorithm is followed At the iteration k the direction of descent is calculated by the equation 1 ( )k k

kd H xσminus= minus nabla and the search direction is performed in steps of βk corresponding to the minimization of the function σ(xk + βkdk) or to the inequality σ(xk + βkdk) lt σ(xk) which leads to the iterative formula 1 1 ( )k k k k

kx x H xβ σ+ minus= minus nabla

With the BFGS algorithm the inverse of the Hessian of the order k + 1 is replaced by the following formula

1 1 11 11 (1 )

T T T Tk k k k k k k k k k k

k k T T Tk k k k k k

H H HH H γ γ γ γ δ γ γ δδ γ γ γ δ γ

minus minus minusminus minus+

+= + + minus [823]

where as previously 1k kk x xδ += minus and 1( ) ( )k k

k x xγ σ σ+= nabla minus nabla

83 Characterization of encapsulating material interfaces of mechatronic assemblies

A mechatronic power module is typically made up of microchips which are assembled on a substrate by soldering and electrically connected by wires To produce high-performance power modules able to operate without failure under conditions of use materials assembly processes and interconnection techniques are developed for the power module so that they can withstand thermal mechanical thermomechanical electrical and chemical stresses generated by the conditions of use Substrates that are both electrically insulating and thermally conductive are utilized The thermal expansion coefficients of the substrate chips and assembly materials are adjusted To protect the module from chemical damage (corrosion) and environmental damage (mechanical stress) the module components are embedded in a polymer-based material (potting) The polymer must be electrically insulating resistant to temperature changes and must ensure in operating conditions a hermetic function resistance to chemicals and


mechanical protection to vibrations The potting materials used in mechatronics modules are silicone gels epoxy resins and polyurethane resins

Figure 86 Schematic of IML module before and after the potting operation

A power module from an IML technology is shown in Figure 86 In this module the resin coating covers the silicon-based power chips (IGBT and diodes) the thin aluminum connecting wires (bondings) and the nickelndashcopper metallization of the substrate

To characterize the ability of the resin coating to resist stresses in operating mode the defects in the polymerndashmetal interface of the modulersquos components are studied by SE before and after the application of an external stress Different silicon gels and an epoxy resin are considered as well as substrates made of quartz silicon aluminum and nickel-plated copper to represent coated metallic surfaces to be protected

To determine the intrinsic properties of the investigated resins bulk samples are fabricated and characterized To study the interface defects samples of resin films deposited on substrates made of quartz silicon aluminum or nickel-plated copper are characterized by ellipsometry

IML Module

Without resin With resin


831 Coating materials studied and experimental protocol

The encapsulating silicone gels are polysiloxanes These polymers are characterized by the presence of siliconndashoxygen bonds and SindashC bonds They are different from organic polymers by the oxygen content of the polymer skeleton (SindashO) compared to the CndashC skeleton of natural organic polymer The SindashC bond is responsible for the thermal and chemical resistance The absence of unsaturated double bonds in the silicon skeleton unlike the primary organic skeleton provides a high resistance to oxidation of silicones Silicon gels are very stable at elevated temperatures ge180 degC and support an electrical insulation of 20 kVmm They are more resistant to chemicals and are sufficiently moisture proof

Three groups of samples are considered

ndash single-component silicon gels (denoted Mi i = 1 2 3)

ndash bicomponent silicon gels (denoted Bi i = 1 2 3 4) consisting of two parts (A) and (B) mixed in equal proportions

ndash an epoxy resin (E1)

The study of defects at the interfaces by SE is performed on seven silicon gel samples and on an epoxy resin Table 81 shows the curing temperatures and viscosities of these polymers

The dimensions and shapes of the quartz substrates (QZ) silicon (Si) aluminum (Al) or nickel-plated copper (Cu-Ni) are presented in Table 82

The thickness of the thin layers on the copperndashnickel substrate is shown in Table 83

The bulk samples are fabricated using a potting mold consisting of two parts made of Teflon A pellet about 3-mm thick is thus obtained after the passage through an oven

The thin polymer film deposits are formed on the four types of substrates in Table 82 by the following method


ndash chemical cleaning and drying

ndash depositing of a few drops of polymer on the substrate and centrifugation (spin coating)

ndash vacuum degassing under 30 mbar and then annealing at the curing temperature

The samples are characterized in the near and mid-IR by a RCE with a tungsten lamp as a light source They are characterized from 17 to 30 microm (333ndash5900 cmminus1) by a variable angle ellipsometer (VASE-IR) and the RCE using a FTIR spectrometer for the spectral characterization of the light source [WOO 00 DAH 10 ALA 11 SCI 12 KHE 14]

Curing temperature (degC) Viscosity (Pamiddots)

Bicomponent silicon gels

B1 150 05

B2 70 1

B3 150 1

B4 150 02

Single-component silicon gels

M1 120 04

M2 120 095

M3 150 07

Epoxy E1 125 4

Table 81 Characteristics of the silicone gels and epoxy samples

Material Shape Dimensions (cm)

Copper (Cu) Square 2 times 2

Silicon (Si) Circle Diameter 2

Aluminum (Al) Square 2 times 2

Quartz (Qz) Square 2 times 2

Table 82 Substrate characteristics


Copperndashnickel substrate Layer thickness (microm)

Copper 1000

Ni electrochemical 4

Silver 05

Table 83 Characteristics of the thin layers of the Cu-Ni substrate

The optical path of the IR-VASE ellipsometer is composed of a polarizer the sample a compensator and an analyzer (PSCA mode) A deuterium triglycine sulfate detector is used to collect the light after the analyzer The IR-VASE can also be used for measuring the reflectance (R) and the transmittance (T) of samples The instrument is automated for alignment data acquisition and calibration procedures The resolution is adjustable (64ndash1 cmminus1) The measurements are taken at an incident angle of 70

The protocol to characterize the optical property of a sample from measurements is as follows For bulk samples the refractive index n and the extinction coefficient k are determined by an inverse method For thin films a single structural model consisting of a 1 mm thick substrate (QZ Si Al Cu-Ni) with a layer of polymer is used The thickness values determined by the ellipsometric study in the UVndashvisible range [KHE 14 DAH 15] are taken as the starting values to adjust the model The general oscillator (Osc-Gen) optical model that fits the optical properties of samples (Figure 87) provides a choice of different types of oscillators (harmonic Gaussian Tauc and Cody Lorentz Drude and Lorentz etc) The Lorentz model is mainly used

832 Study of bulk coatings

Studies of bulk coatings are used to characterize the optical properties of each type of polymer Figures 88(a) and 88(b) show the ellipsometric experimental parameters Ψ and Δ the real and imaginary parts of the refractive index and dielectric constants of the B1 silicon gel sample Figure 89 gives the absorption coefficient of the B2 silicon gel sample


Figure 87 Optical model ldquoGen Oscrdquo of the software WVASE

a)

b)

Figure 88 a) Ellipsometric parameters of the B1 silicon gel b) the real part n and imaginary part k of the complex refractive

index and dielectric constant εprime and εprimeprime of the B1 silicon gel

Generated and Experimental

Wavelength (microm)0 3 6 9 12 15 18 21

Y in

deg

rees

12

15

18

21

24

27

30

33Model Fit Exp E 70deg

Ψde

gree

s



D in

deg

rees

-10

0

10

20

30


Δde

gree

s

genosc_ir (gels silicones) Optical Constants


Inde

x of

Ref

ract

ion

n Extinction C

oefficient k

10

12

14

16

18

000

010

020

030

040

050

060nk

n r

eal p

art

k imaginary

part



Rea

l(Die

lect

ric C

onst

ant)

e1

Imag(D

ielectric Constant) e

2

10

15

20

25

30

35

00

03

06

09

12

15

18e1e2

εrsquo re

al p

art

imaginary

part εrsquorsquo


Figure 89 Comparison of absorption coefficients of the B2 silicon gel minus ATR minus SE

For bulk samples in silicon gels the simulated and experimental curves are adjusted by nine oscillators For the epoxy resin they are adjusted using 11 oscillators

The evolution of the index n with λ is given by a function that decreases with the wavelength λ The presence of absorption zones (k ne 0) modifies this feature and regions where n increases with λ are observed that is to say regions of anomalous dispersion The n values are below 18 in the mid-IR region from 17 to 18 microm

In the case of the silicon gel sample M1 beyond the wavelength of 18 microm the index n increases with the wavelength λ probably due to an absorption zone

For silicon gels the number of peaks observed on the curve of the absorption coefficient k as a function of the wavelength coefficient varies between 6 and 8 There are 12 peaks for the epoxy resin These peaks correspond to the absorption areas that are measured by the attenuated total reflection (ATR) method When the results obtained in the case of the two-component B2 (Figure 89) with the techniques of SE and ATR are compared a good agreement on the position of the lines and the absorption bands is noted

500 1000 1500 2000 2500 3000 3500 4000

00

05

10

Abso

rptio

n

Longueur dpnde en (cm-1)

Absorption du B2 par ATR coefficient k du 2 par Ellipso

Wavenumber(cm-1)

Abso

rptio

n


Different absorption regions are depicted on the absorption curves of the samples of silicon gels B1 E1 and M1 and epoxy resin All samples absorb strongly between 800 and 1500 cmminus1 in the ratio M1B1E1 of 40203 Between 1500 and 3500 cmminus1 absorption of M1 is negligible compared to that of B1 and E1 with intensity ratios of B1E1 = 31 between 1500 and 2800 cmminus1 and 11 between 2800 and 3500 cmminus1 respectively

Figure 810 Comparison of absorption spectra of polymers B1 M1 and E1

833 Study of defects at the interfaces

The characterization of the samples by ellipsometry allows the determination of the effects of external stresses (humidity thermal effect) on the substratendashpolymer interfaces from the change in optical properties of the polymer

After placing samples of encapsulated substrates in the ldquoSuper HATrdquo equipment described in [POU 15] very fast temperature variations between ndash45degC and 95degC are activated When the sample temperature has stabilized at 95degC moist air is introduced This

genosc_ir (gels silicones) m1 Optical Constants

Wave Number (cm -1)0 1000 2000 3000 4000 5000 6000

Abs

orpt

ion

Coe

ffici

ent i

n 1

cm

0

3000

6000

9000

12000

15000

M1

Wavenumber (cm-1)


Wave Number (cm -1)500 1000 1500 2000 2500 3000 3500

Abso

rptio

n C

oeffi

cien

t in

1cm

0

2000

4000

6000

8000

B1

Wavenumber (cm-1)e1 Optical Constants

Wave Number (cm -1)0 1000 2000 3000 4000 5000 6000

Abso

rptio

n C

oeffi

cien

t in

1cm

0

300

600

900

1200

1500

E1

Wavenumber (cm-1)


temperature variation stress in a wet atmosphere is applied in a cyclic mode to reproduce the operating conditions of a power module

8331 Aluminum polymer and copperndashnickel polymer interfaces

The effect of cumulative stresses of temperature and humidity on the interfaces is studied on the (Al and CuNi) substrates The Al-polymer interface simulates the encapsulation of the bonding wires The CuNi interface simulates the encapsulation of the IML power modulersquos substrate

The ellipsometric parameters cos (Δ) measured before and after the application of stress are shown in Figures 811ndash816

Figure 811 Comparison of cos (Δ) spectra of polymer interfaces B1 B2 on Al before and after humidity and heat stresses For a color

version of this figure see wwwistecoukdahoonanometerzip



0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ

Longueur donde (cm-1)

B1Al AVANT B1Al APRES

Wavenumber (cm-1)0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

del

ta



Wavenumber (cm-1)

0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ


B4AlAVANT B4Al APRES

Wavenumber (cm-1)


Figure 813 Comparison of cos (Δ) spectra of polymer interfaces M1 M3 on Al before and after humidity and heat stresses For a color


Figure 814 Comparison of cos (Δ) spectra of polymer interfaces B1 B2 on CuNi before and after humidity and heat stresses For a color




0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ


M1Al AVANT M1Al APRES

Wavenumber (cm-1)0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)

0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ


B1CuNi (T0) B1CuNi (H+T)

Wavenumber (cm-1)0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ


B2CuNi (t0) B2CuNi (H+T)

Wavenumber (cm-1)

0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ


B3CuNi (T0) B3CuNi (T+H)

Wavenumber (cm-1)0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ

Loongueur donde (cm-1)


Wavenumber (cm-1)


Figure 816 Comparison of cos (Δ) spectra of polymer interfaces M1 M3 on CuNi before and after humidity and heat stresses For a color


8332 Quartzndashpolymer interfaces

In the UVndashvisible range for an incidence angle of 70deg the curves of the parameter tan (Ψ) as a function of the wavelength are located around the value of 044 and those of cos (Δ) are around the value of 1 for all polymers (B1 B2 B3 B4 M1 M2 M3) deposited on quartz substrates copperndashnickel and aluminum [KHE 14 DAH 15]

Figure 817 Comparison of spectra cos (Δ) of polymer interfaces B1 (left) and B2 (right) on quartz and Al For a color version

of this figure see wwwistecoukdahoonanometerzip

The temperature stability and the chemical inertness of the quartz substrate relative to the polymers lead to the determination of the values of the refractive index n which are found to be comparable with those determined on the bulk sample The quartzndashpolymer interface is then used as a reference for comparing the results obtained with other substrates that are characteristic of the materials present in

0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

C

os Δ


M1CuNi (T0) M1CuNi (H+T)

Wavenumber (cm-1)0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)

0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0

-1 0

-0 5

0 0

0 5

1 0

Cos

Δ

L o n g u e u r d o n d e e n (cm -1)

B 1 Q U A R T Z B 1 A l

Wavenumber (cm-1)0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0

- 1 0

- 0 5

0 0

0 5

1 0

Cos

Δ

L o n g u e u r d o n d e e n ( c m -1 )

B 2 Q u a r t z B 2 A l

Wavenumber (cm-1)


an IML-type power module by analyzing the ellipsometric parameter cos (Δ) The curves shown in Figures 817ndash822 (Bi (i = 1 4) Mi (i = 1 3) and E1) show the differences between the spectra obtained with quartz and metal substrates

Figure 818 Comparison of spectra cos (Δ) of polymer interfaces B3 (left) and B4 (right) on quartz and Al For a color version of this figure see wwwistecoukdahoonanometerzip

Figure 819 Comparison of spectra cos (Δ) of polymer interfaces M1 (left) M2 (center) and M3 (right) on quartz and Al For a color version of this figure see wwwistecoukdahoonanometerzip

Figure 820 Comparison of spectra cos (Δ) of polymer interfaces B1 (left) and B2 (right) on quartz and CuNi For a color version of this figure see wwwistecoukdahoonanometerzip

0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0

-1 0

-0 5

0 0

0 5

1 0

Cos

Δ

L o n g u e u r d o n d e e n ( c m - 1 )


Wavenumber (cm-1)0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0

-1 0

-0 5

0 0

0 5

1 0

Cos

Δ

L o n g u e u r d o n d e (c m -1)

B 4 Q u a r tz B 4 A l

Wavenumber (cm-1)

0 1000 2000 3000 4000 5000

-10

-05

00

05

10

Cos

Δ

Longueur (cm-1)

M1QUARTZ M1Al

Wavenumber (cm-1)0 1000 2000 3000 4000 5000

-10

-05

00

05

10

Cos

Δ

Lo ngu eur d ond e en (cm -1)

M 2 Q u artz M 2 A l

Wavenumber (cm-1)0 1000 2000 3000 4000 5000

-10

-05

00

05

10

Cos

Δ

Longueur d onde en (cm -1)

M 3Q uartz M 3A l

Wavenumber (cm-1)

1000 2000 3000 4000

-10

-05

00

05

10

cosΔ


B1Quartz B1CuNi

Wavenumber (cm-1)2000 4000

-10

-05

00

05

10

Cos

Δ


B2QUARTZ B2CuNi

Wavenumber (cm-1)



Figure 822 Comparison of spectra cos (Δ) of polymer interfaces M1 (left) and M3 (right) on quartz and CuNi For a color


8333 Silicon polymer interfaces

In the near UV to near IR region the variations of tan (Ψ) and cos (Δ) strongly depend on the polymer deposited on the silicon substrate unlike the case observed with quartz copper-nickel or aluminum substrates The spectra recorded with the silicon substrate are characterized by oscillations beyond 500 nm This difference is due to a chemical interaction between the silicon and the polymer that alters the optical properties of the resulting material [DAH 15] Figures 823ndash826 show the differences in the spectra of cos (Δ) for a deposit on quartz and a deposit on Si in the near IR to mid-IR region

0 1000 2000 3000 4000 5000

-10

-05

00

05

10

Cos

Δ


B3Quarz B3CuNi

Wavenumber (cm-1)0 1000 2000 3000 4000 5000

-10

-05

00

05

10

Cos

Δ


B4QUARTZ B4CuNi

Wavenumber (cm-1)

0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

del

ta


M1QUARTZ M1CuNi

Wavenumber (cm-1)0 1000 2000 3000 4000 5000

-10

-05

00

05

10

Cos

Δ


M3QUARTZ M3CuNi

Wavenumber (cm-1)


Figure 823 Comparison of cos (Δ) spectra of polymer interfaces B1 and B2 on quartz and Si For a color version of

this figure see wwwistecoukdahoonanometerzip

Figure 824 Comparison of cos (Δ) spectra of polymer interfaces B3 and B4 on quartz and Si For a color version


Figure 825 Comparison of cos (Δ) spectra of polymer interfaces M1 M2 and M3 on quartz and Si For a color version


1000 2000 3000 4000 5000

00

05

10

Cos

Δ


B2 QUARTZ B2Si

Wavenumber (cm-1)1000 2000 3000 4000 5000

-05

00

05

10

Cos

Δ


B1Quartz B1Si

Wavenumber (cm-1)

0 1000 2000 3000 4000 5000

-05

00

05

10

Cos

Δ


B3QUARTZ B3Si

Wavenumber (cm-1)0 1000 2000 3000 4000 5000

-10

-05

00

05

10

Cos

Δ


B4QUARTZ B4Si

Wavenumber (cm-1)

1000 2000 3000 4000 5000

01

02

03

04

05

06

07

08

09

10

Cos

Δ

Longueur donde(cm-1)

M2QUARTZ M2Si

Wavenumber (cm-1)0 2000 4000

-10

-05

00

05

10

Cos

Δ


M1QUARTZ M1Si

Wavenumber (cm-1)


Figure 826 Comparison of cos (Δ) spectra of polymer interfaces M3 and E1 on quartz and Si For a color version



Silicon gel samples are of the polydimethylsiloxane family They are characterized by the absorption spectra as shown in Figure 827 The bands corresponding to SindashC bond vibrations appear as a very strong peak at around 784 cmminus1 due to the deformation of the Si-CH3 followed by a stretching mode at 864 and 1258 cmminus1 The antisymmetric stretching mode of siloxane Si-O functions are around 1008 and 1082 cmminus1 and are in the form of an intense peak accompanied by a shoulder

Figure 827 Absorption spectra by ATR of silicon gels B (1ndash4) and M (1ndash3) For a color version of this figure see wwwistecoukdahoonanometerzip

1000 2000 3000 4000

0994

0996

0998

1000

Cos

Δ


E1Quartz E1Si

Wavenumber (cm-1)0 1000 2000 3000 4000 5000 6000

070

075

080

085

090

095

100

Cos

Δ


M3QUARTZ M3Si

Wavenumber (cm-1)

1 0 0 0 1 5 0 0 2 00 0 2 5 0 0 3 0 0 0 3 5 0 0

0 0

0 2

0 4

0 6

0 8

1 0

1 2

7 00 80 0 90 00 0

0 2

0 4

0 6

0 8

1 0

1 2

Ab

sor

ban

ce

W av en u m b er (c m -1)

B 1 B 2 B 3 B 4 M 1 M 2 M 3

Ab

sorb

an

ce

W ave n u m b e r ( c m -1)


The main absorption lines and vibration modes of the methyl groups have valence vibration bands at around 2962 and 2904 cmminus1 and deformation vibration bands at around 1413 cmminus1 followed by a small band at 1440 cmminus1 and a significant band at 1258 cmminus1 The material M2 distinguishes itself from the other polymers with absorption peaks at 755 840 and 1257 cmminus1 which is due to the presence of SiO-(CH3)3 groups in the polymer

To facilitate the comparison between the curves recorded on a given substrate before and after a stress the spectral range is divided into four zones as follows

ndash zone 1 before 1400 cmminus1

ndash zone 2 between 1400 and 2200 cmminus1

ndash zone 3 between 2200 and 2500 cmminus1 (absorption region of atmospheric CO2)

ndash zone 4 beyond 2500 cmminus1


In zone 1 oscillations are observed for all polymers The amplitudes of these oscillations are variable different from those obtained with the quartz except possibly for B2 and M3

In zone 2 cos (Δ) = 1 for B2 and M3 For other polymers cos (Δ) is different from 1 The separation from 1 depends on the polymer In zone 3 there is a significant effect except for M3 and E1

In zone 4 in the case of B2 B3 M2 and M3 cos (Δ) are slightly different from 1 For E1 there is a variation beyond 3500 cmminus1

8342 Aluminum polymer interfaces

In zone 1 the observations made are the same as for the silicon except for M3 In zone 2 the same values of cos (Δ) are obtained for all materials In zone 3 cos (Δ) is close to minus1 for all materials changes in cos (Δ) depend on the material In zone 4 cos (Δ) increases from minus1 to 1 this variation depends on the material


8343 Copperndashnickel polymer interfaces

The same variations as for aluminum are observed except in zone 4 where no difference is observed on the curves obtained with quartz For the material E1 the parameter cos (Δ) do not change The experimental curves are adjusted using nine oscillators for silicon gels The analysis is performed using the thicknesses determined by the ellipsometric study carried in the UVndashVis region [DAH 15] In the absorption zone the refractive index decreases and then increases in the vicinity where the absorption starts in the region of abnormal dispersion when n increases with λ Six to eight absorption peaks corresponding to the anomalous dispersion region of n are found The values of n are below 26 for all materials except for M2 for which n varies up to 45

84 Conclusions

The relevance and interest in the application of SE is that this technique is sensitive to surfaces and interfaces while being non-destructive The analyses by SE show that materials in contact with the coating polymer undergo modifications Physical evidence of reactions between the polymer and the coated material is seen in the modifications of the refractive index of the polymers which is indicated by the curves of the parameters cos (Δ) For example silicon reacts with all polymers studied except for the M3 polymer

The SE characterization technique is used to study the effects of an environmental stress on the interfaces Samples mimicking the interfaces of the metalndashpolymer coating of a power module have been subjected to thermal stress in the presence of humidity Variations in optical properties are interpreted Changes in the optical properties of the polymers that depend on the nature of the coated metal are highlighted These changes are important in the case of aluminum and less so in the case of copperndashnickel However no changes are observed after the polymer has reacted with the (Silicon) substrate


during the samplersquos preparation The results presented in this chapter show that the interfaces between the resin coating and materials constituting a power module can be modified as a result of temperature variations in a humid medium Defects that appear at the metalndashpolymer interfaces may be the cause of failure of a mechatronic module Finally these property changes must be taken into account in the design of mechatronic modules by the RBDO method

9

Application to Nanomaterials

At the nanoscale carbon nanotubes (CNTs) are characterized by a diversity of structures and morphologies that induce a very high variability of their mechanical properties and therefore considerable uncertainty on measurements In this context this chapter describes a method for determining the mechanical properties of nanomaterials combining experimentation and modeling by applying the reliability-based optimization method (RBDO) The mechanical properties of nanomaterials are investigated by a model based on the finite element method and are characterized by the nanoindentation technique The RBDO method is used to estimate elasticndashplastic properties of CNTs The numerical results show that this is an effective method to assess the properties of CNTs

91 Introduction

Nanotechnology and nanomaterials are subject to extensive academic and industrial research Since their discovery in 1991 CNTs attracted considerable interest due to their exceptional physical electrical mechanical and chemical properties [LIJ 91] Thus they are studied in the fields of microelectronics mechanical and electronic systems and composite materials

The macroscopic structures of CNTs can be classified into two categories single-walled CNTs (SWCNT) or single sheet CNTs and multiwalled CNTs (MWCNTs) or multisheet CNTs


First Edition


SWCNT structures are found in three forms armchair zigzag and chiral They have various radii and lengths The structure of a SWCNT may be regarded as a monatomic graphite sheet (1ndash2 nm) rolled to form a cylindrical tube made up of rings of six carbon atoms These cylindrical tubes may have one or two closed ends per hemisphere SWCNT properties have many applications in medicine electronics and environmental science MWCNTs consist of graphite multilayers disposed in concentric or spiral cylinders

CNT properties have been the subject of constant research in recent years A bibliographical review is given in the thesis of Tang [TAN 15] Most of anticipated CNT properties are based on perfect atomic structure models However there is a great variability in the predictions of the results from CNT characterization The uncertainty analysis of CNT properties is therefore necessary

92 Mechanical properties of SWCNT structures by MEF

By using finite element methods Youngrsquos modulus and the shear modulus of SWCNT structures corresponding to armchair zigzag and chiral forms are evaluated For this equivalence equations between the molecular mechanics and the calculation of SWCNT structures are established The interactions between the carbon atoms (CndashC) are modeled by finite elements of linear spring type and torsion type

In structural mechanics the construction of an individual nanotube is achieved from models based on point masses connected by elastic springs in a regular assembly The potential energy of this assembly is the sum of the electrostatic interactions and the potential energy that depends on the structurersquos characteristics The force field is derived from the potential energy and is expressed as [GHA 12]

r vdw elU U U U U U Uθ φ ω= + + + + + [91]

where the different terms represent the interaction energy related to

ndash Ur the bond strength

Application to Nanomaterials 257

ndash Uθ the bond angle bending

ndash Uφ the dihedral torsion angle

ndash Uω the out-of-plane torsion

ndash Uvdw the Van der Waals forces

ndash Uel the electrostatic forces

In covalent systems the contributions that are most significant to the total steric energy come from the first four terms of equation [91] Assuming small deformations the energy is given by [TAN 15]

2 20

1 1( ) ( )2 2r r rU k r r k r= minus = Δ [92]

2 20

1 1( ) ( )2 2

U k kθ θ θθ θ θ= minus = Δ [93]

21 ( )2

U U U kτ φ ω τ φ= + = Δ [94]

where kr is the bond stretching force constant Δr is the bond stretch kθ is the bond bending force constant Δθ is the bending bond angle kτ is the torsional resistance and Δφ is the twisting angle bond variations

Since the structure of a nanotube is considered as an assembly of elastic springs the expression for the potential energy of the bonding strength is obtained by applying Hookersquos law It is equivalent to the potential energy of compressiontension of a spring of stiffness kr The terms pertaining to the potential energy connected to the bending and torsion angles are equivalent to the potential energy of a spring in torsion with a stiffness of kθ and kτ respectively With this approximation the parameters kr kθ and kτ are estimated for the different bonds (stretching bending angle torsion angle) The resistance force constants are estimated by [GHA 12]

kr = 652 times 10ndash7 Nsdotnmndash1 kθ = 876 times 10ndash10 N nm radndash2 and kτ = 287 times 10-10 N nm radndash2


921 Youngs modulus of SWCNT structures

Figure 91 shows the variation of the Youngrsquos modulus of SWCNT structures of armchair zigzag and chiral forms as a function of the nanotubes radii The curves show that the radius has a significant effect on the value of Youngrsquos modulus For the three SWCNT structures the Youngrsquos modulus increases with radius up to a critical value Beyond this value the Youngrsquos moduli converge to a constant value This limit is expected as it is due to the effect of the CNTrsquos curvature Indeed when the diameter of the nanotubes increases the deformation of the CndashC bond becomes less important

Figure 91 Variation of Youngrsquos moduli of SWCNT structures of armchair zigzag and chiral forms as a function of the radius

For a given radius the SWCNTs of armchair form have a Youngrsquos modulus slightly greater than that of zigzag SWCNTs the moduli of zigzag SWCNTs are slightly higher than those of chiral SWCNTs This result is consistent with results from the literature [GIA 08 MAH 12 LU 12]

The numerical results in Figure 92 show how the Youngrsquos modulus varies with the length of SWCNT structures Depending on the shape armchair (8 8) zigzag (14 0) or chiral (9 6) Youngrsquos


moduli are different When varying the length of the forms from 357 to 824 nm 327 to 838 nm and 356 to 928 nm the maximum variations of Youngrsquos modulus are 2 5 and 14 respectively

Figure 92 Youngrsquos moduli of SWCNT structures of armchair zigzag and chiral forms as a function of length

922 Shear modulus of SWCNT structures

The variation in the shear modulus of the SWCNT structures of armchair zigzag and chiral forms is shown in Figure 93

Figure 93 Shear moduli of SWCNT structures of armchair zigzag and chiral forms as a function of the radius


For small values of the radius the shear modulus of the SWCNT structures of zigzag and armchair forms increases with the radius Beyond a certain value the shear moduli tend to stabilize at a certain limit The results also show that the shear moduli of SWCNT structures of the zigzag form are higher than those of the chiral and armchair forms This difference is due to the atomic structure Indeed in an armchair type SWCNT structure a third of CndashC bonds are aligned with the direction of the radial load In SWCNT of zigzag and chiral forms all CndashC bonds form an angle with the radial loading direction In addition for chiral SWCNTs the shear modulus strongly depends on variations of chirality as shown in Figure 93

923 Conclusion on the modeling results

The results of these numerical studies show that elastic moduli of SWCNT structures (shear modulus and Youngrsquos modulus) strongly depend on the radius and the chirality of the nanotubes For smaller radius values the modulus of elasticity increases with the increment radius When the radius becomes larger all the elastic moduli converge to a constant value In addition moduli of SWCNT of zigzag and chiral forms are more susceptible to the variation in radius than that of the armchair form

These results show the large variability of CNTs mechanical properties Consequently a large dispersion in the experimental data is expected

93 Characterization of the elastic properties of SWCNT thin films

The elastic properties of SWCNT structures are measured by nanoindentation of thin films approximately 200 nm thick The technique of nanoindentation consists of measuring the displacement of a diamond probe in contact with the materialrsquos surface For indentation measurements the probe penetrates the material when a load is applied up to a maximum load value or displacement Then the load is gradually reduced and the probe returns to its original


position in the course of the mechanical relaxation of the material which may be plastic or elastic

During the indentation process the load and displacement are continuously measured The curves of the charge and discharge of the indenter as a function of its displacement are thus obtained Analysis of this curve determines the hardness and the elastic modulus of SWCNT films

931 Preparation of SWCNT structures

Thin SWCNT films are prepared by centrifugal induction [TAN 15] To spread the nanotubes a dilute SWCNT suspension in ethanol is exposed to ultrasound for 20 min Then the following preparation protocol is applied

ndash mounting the silicon chips on the spinner

ndash depositing of the SWCNT solution on the silicon chips

ndash starting of the spin to spread the fluid over the entire surface by centrifugation until the layer reaches the desired thickness (Figure 94)

a) b)

Figure 94 The process of centrifugal induction for thin SWCNT films a) static distribution process b) stages of centrifugal induction

After centrifugation the ethanol solvent is evaporated at room temperature The samples are then heated for 2 h at a temperature of


300degC and then cooled slowly to room temperature After this heat treatment the nanotubes are arbitrarily oriented on the silicon substrate by Van der Waals forces Figure 95 presents clusters of nanotubes distributed on silicon substrates

932 Nanoindentation

The system to measure hardness by nanoindentation has a resolution of 1 nN for the load and 02 pm for the displacement The measuring range of the film thickness is 200 nm The tip used is a Berkovich tip (pyramidal geometry with a triangular base) The room temperature is stabilized at 25 plusmn 1ordmC The nanoindentation system (Figure 96) is isolated from vibrations The tests are performed at 18 different points on the film Table 91 gives the different control parameters

Figure 95 Optical microscope image of nanotubes distributed on silicon substrates

Maximum load (mN) 3 Limit stop load (mN) 015 Initial loading (mN) 005 Loading rate (mNs) 01

Unloading rate (mNs) 01 Indentations 18

Rest time at maximum load (s ) 5

Table 91 The measurement parameters in the indentation procedure


Figure 96 System to measure hardness by nanoindentation

933 Experimental results

The experimental results of the 18 loadndashdisplacement curves are shown in Figure 97 Two sets of curves are obtained upon withdrawal This dispersion in the measurements can be attributed to the existence of defects in the SWCNTs

Figure 97 Loadndashdisplacement experimental results


For the analysis of the curves an adjustment to a power law is used between 100 and 20 of the discharge power such that

( )mfp h hα= minus [95]

where α and m are parameters that depend on the material and hf is a parameter that is related to the withdrawal of the indentor and which is determined by adjustment

The first portions of the discharge curves are linear [DOE 86] for certain materials as shown in Figure 97 In this zone the discharge stiffness S is related to the contact area by the equation

2 rdPS E Adh π

= = [96]

where S = dPdh is the discharge stiffness of initial discharge data Er is the reduced elastic modulus defined by equation [99] and A is the projected area of the elastic contact Knowing ldquoArdquo the area of contact Er is deduced from S (equation [96]) and hence the Youngrsquos modulus of material (equation [99])

The mean values of the 18 indentations provided by the test system according to this method are given in Table 92 This table shows that there is a very great dispersion of mean values The dispersions of the hf and α parameters follow a statistical log-normal distribution whereas the dispersion parameter m follows a normal distribution These statistical distributions are checked for a small-sized sample and for a confidence level of 95

Maximum load Pmax (mN) 3054 plusmn 20007

Maximum depth hmax (nm) 7768 plusmn 206Hardness (GPa) 1257719 plusmn 0759

Reduced modulus Er (GPa) 16981778plusmn 4911

Youngs modulus Et (GPa) 19283plusmn 13922

Table 92 Results of nanoindentation


Given these distributions the theoretical loadndashdisplacement curves for the top 70 of the discharging process is obtained from the power law (equation [95]) by applying the Monte Carlo method for a sample of 1000 pieces Figure 98 shows the experimental and theoretical loadndashdisplacement curves of the upper part of the discharging process The experimental curves are strictly within the 95 confidence interval of the results of numerical simulation

Figure 98 Load versus displacement test and modeling results For a color version of this figure see wwwistecoukdahoonanometerzip

The uncertainties concerning the hardness and Youngrsquos modulus of a SWCNT film structure are also determined For a 95 confidence level the standard uncertainty for the hardness is 1207 and the uncertainty for the Youngrsquos modulus is 1064

94 Bilinear model of thin film SWCNT structure

Measurements of instrumented nanoindentation in Figure 99 show that the material of the thin film SWCNT nanotube structure undergoes elastic and plastic deformation that is independent of time


thus revealing that this material has an elastoplastic behavior These elastoplastic properties of the SWCNT thin film structure can be studied by combining results calculated with the finite element technique with those of nanoindentation tests Various uncertainties are associated with the process of nanoindentation and contribute to the dispersion of the loadndashdischarge curves The RBDO method discussed in Chapter 2 is applied to take into account the uncertainties of the parameters of the nanoindentation process and to optimize the finite element model of the loadndashdischarge curve In order to analyze the reliability of the estimate provided by the model the distribution of the loadndashdisplacement curve is used

Figure 99 Load on a SWCNT film structure as a function of the displacement of the indenter while measuring nanoindentation

941 SWCNT thin film structure

The elastoplastic behavior of the SWCNT film based on a linear expression is described using the complete cycles of loadndashdischarge of the indentation of SWCNT thin film structures


To simulate the stressndashstrain behavior the following bilinear model is used

for

( ) forY

Y t Y Y

EE

ε σ σσ

σ ε ε σ σle

= + minus ge [97]

where σY and εY are the elasticity and deformation limits respectively and with εY = σYE where E is the Youngrsquos modulus and Et is the tangent modulus

Based on the linear elastoplastic model the reduced modulus Er in the discharge process is modified (Figure 99) The modulus of phase 2 is expressed by

phase2

12r

dPEdhA

π= [98]

where rE is the reduced modulus of phase 2 and

phase2

dPdh

is the slope

of the last part of the discharge curve

According to contact mechanics the tangent modulus Et can be deduced from

22

(1 )1 (1 ) i

r t iE E Eνν minusminus= + [99]

Because of the complexity introduced by the phase change the relationship of load versus displacement (pndashh) of a SWCNT thin film structure during the indentation is given by

( )r y tP P h E Eσ θ= [910]

where θ is the apex angle of indenter

The program assessing the properties of material use uncertainty analysis based on finite element calculations The curve loadndashdisplacement is obtained by simulation

min itp p= minus [911]


under

max max1

max

ih hh

minus le Δ [912]

2

iS SSminus le Δ [913]

where pi is the load vector of the ith iteration tp is the vector of the average load in the indentation test maxh and S are the average values of the maximum displacement and the contact stiffness and Δ1 and Δ2 are the limits of variation of the maximum displacement (hmax) and the contact stiffness (S)

942 Numerical models of thin film SWCNT structures

9421 Initial properties of the materials

For the indentor Youngrsquos modulus is 1143 GPa and the Poissonrsquos ratio is 007 For the silicon substrate the Youngrsquos modulus is 180 GPa and the Poissonrsquos ratio is 0278 [TAN 15] The Youngrsquos modulus of SWCNT thin film is 19283 plusmn 13922 GPa the initial value of the limit of elasticity Y0 is equal to 42 GPa and the Poisson coefficient v is 018 according to study in [TAN 15]

9422 Construction of the model by finite elements

The material behavior model uses the criterion of ldquoVon Mises with isotropic bilinear hardeningrdquo The indenter the SWCNT thin film structure and the substrate are meshed with three-dimensional solids of 20 nodes The interaction of the indenter and the sample is modeled as a frictionless surface to surface contact The interface between the film and the substrate is assumed to be bonded The mesh around the indenter is refined to describe the deformation and the stress gradient accurately (Figure 910) An average force is continuously applied to the top surface of the indenter in the z direction All degrees of


freedom of the lower nodes of the substrate are fixed The predetermined maximum value of the force is 3 mN

Figure 910 Model by finite elements of the indenter-film system

943 Numerical results

Figure 911 compares experimental results with those of the simulation

Figure 911 Results of testing and modeling of the load of a thin film SWCNT structure as a function of the displacement For a color



The dispersion of results is caused by the approximations of the model the uncertainties of the properties of the test material and the quality of the contact surface of the indenter

Figure 912 shows the distribution of deformations and stresses of the SWCNT film substrate system

Figure 912 Distribution of deformations and stresses of the SWCNT film substrate system For a color version of


Figure 913 compares the loadndashdisplacement property characteristics of SWCNT thin film structures for various forms of indenter and the same maximum load

A defect in the shape of the indenter directly affects the contact area To compensate for errors in the form of the indenter in finite element simulation the angle of the apex of the indenter is changed by using the following approximations

2 2 22 1 0 3 3 tanproj c c cA c h c h c h θ= + + = [914]

Table 93 presents the results of simulation iterations It shows that after five iterations the simulation data are close to the experimental results (Figures 913) Figures 914 915 and 916 show the effect on the loadndashdischarge curve as a function of the penetration depth of respectively the shape of the indenter the thickness of the thin film of the SWCNT structure and the silicon substrate Youngrsquos modulus


Figure 917 shows the distribution of discharge data from the experiment a Monte Carlo simulation and finite element simulations The RBDO method optimizes the model parameters of the nanoindentation process and brings the numerical results closer to the experimental results

Parameters (degC)

(Gpa)

(Gpa)

ℎ ℎ ℎ nm ∆ Mnnm ∆ Mean

experimental value 7768 00963

FE simulation

Iteration 1 653 42 42 108881 4017 00867 997

Iteration 2 70 42 42 8759 1276 010845 1262

Iteration 3 70 21 42 904407 1643 0114 1838

Iteration 4 70 8385 21 85903 106 010256 65

Iteration 5 70 8385 315 846446 897 010098 486

Iteration 6 70 8385 42 828116 661 009858 237

Table 93 Simulation results by finite element

Figure 913 Curves of loadndashdisplacement from testing and modeling For a color version of this figure see wwwistecoukdahoonanometerzip


Figure 914 Effects on the loadndashdisplacement curve of different indentor forms For a color version of this figure see

wwwistecoukdahoonanometerzip

Figure 915 Effect of the thickness of the SWCNT structure on the loadndashdisplacement curve For a color version



Figure 916 Effect of the substrates Youngs modulus on the loadndashdisplacement curve For a color version


Figure 917 Experimental and simulated discharge curves For a color version of this figure see wwwistecoukdahoonanometerzip


95 Conclusions

To use CNTs in industrial applications (sensors microchips etc) their mechanical and electromechanical properties must be well known and mastered

The characterization of mechanical properties of SWCNT structures is carried out by nanoindentation testing These measurements are characterized by a high dispersion To determine the spread of the statistical dispersion in the numerical model and uncertainties in the testing data RBDO presented in Chapter 2 is applied to the finite element models and experimental results A good correlation between the experimental and the numerical parts is obtained for the mechanical properties of SWCNT structures Tests and simulations show that the mechanical properties of SWCNTs are highly dependent on test conditions and their structural parameters

Bibliography

[ABE 50] ABELES F ldquoLa theacuteorie geacuteneacuterale des couches mincesrdquo Journal de Physique et Le Radium vol 11 no 7 p 307 1950

[ABO 73a] ABOUAF-MARGUIN L Etude du mouvement et de la relaxation vibrationnelle de lrsquoammoniac isoleacute en matrice agrave basse tempeacuterature PhD Thesis UPMC Paris 1973

[ABO 73b] ABOUAF-MARGUIN L DUBOST H LEGAY F Chemical Physics Letters vol 22 p 603 1973

[ACH 89] ACHER O BIGAN E DREVILLON B ldquoImprovements of phase‐modulated ellipsometryrdquo Rev Sci Instr vol 60 no 7 p 65 1989

[ALA 11] ALAYLI N Frittage de pacircte de nano et micro grains drsquoargent pour lrsquointerconnexion dans un module de meacutecatronique de puissance Elaboration caracteacuterisation et mise en œuvre PhD Thesis University of Versailles St Quentin en Yvelines 2011

[AMA 53] AMAT G Contribution agrave lrsquoeacutetude de lrsquointensiteacute des Bandes drsquoAbsorption Infrarouge Publications scientifiques et techniques du Ministegravere de lrsquoair Paris France 1953

[APK 84] APKARIAN VA Chem Phys Lett vol 110 p 168 1984

[AND 49] ANDERSON P W Phys Rev vol 76 p 647 1949

[ARE 72] ARECCHI FT COURTENS E GILMORE R et al ldquoAtomic coherent states in quantum opticsrdquo Physical Review A vol 6 no 6 pp 2221ndash2237 1972


First Edition


[ASK 85] ASKEY R WILSON J ldquoSome basic hypergeometric orthogonal polynomials that generalize Jacobi polynomialsrdquo Memories of the Americal Mathematical Society vol 54 no 319 1985

[ASP 74] ASPNES DE Journal of the Optical Society of America vol 64 no 5 pp 639ndash646 1974

[ASP 75] ASPNES DE STUDNA AA Appl Opt vol 14 pp 220ndash228 1975

[AZZ 77] AZZAM RMA BASHARA NM Ellipsometry and Polarized Light North Holland Co Amsterdam 1977

[BAR 61] BARCHEWITZ P Spectroscopie infrarouge 1 Vibrations moleacuteculaires Gauthier-Villars Paris France 1961

[BAR 66] BARCHEWITZ P Spectroscopie infrarouge 2 Fonction potentielle Moment dipolaire Gauthier-Villars Paris France 1966

[BAR 73] BARNETT JD BLOCK S PIERMARINI GJ Rev of Scientific Instruments vol 44 pp 1ndash9 1973

[BAS 11] BASAVAPOORNIMA C JAYASANKAR CK TROumlSTER T et al High Pressure Research vol 31 pp 121ndash125 2011

[BER 77] BERKOWITZ M GERBER RB Chem Phys Lett vol 49 p 260 1977

[BER 79] BERKOWITZ M GERBER RB Chem Phys vol 37 p 369 1979

[BER 96] BERRODIER I Recherche de la configuration drsquoeacutequilibre des moleacutecules de N2O et CO2 isoleacutees en matrices drsquoargon agrave basse tempeacuterature et calcul des deacuteplacements de freacutequence et eacutetude de lrsquoeacutemission stimuleacutee de N2O et 13CO2 pieacutegeacutees en matrice drsquoargon Thesis Marne la Valleacutee University Paris 1996

[BER 05] BERTOLOTTI M Masers and Lasers A Historical Approach CRC Press New York 2005

[BET 40] BETHE H TELLER E Ballistic Laboratory Aberdeen Proving Ground Report X-117 1940

[BLO 46] BLOCH F ldquoNuclear inductionrdquo Phys Rev vol 70 nos 7ndash8 pp 460ndash474 1946

[BLU 78] BLUMEN A LIN SH J Chem Phys vol 69 p 881 1978

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[BOC 93] BOCCARA AC PICKERING C RIVORY J ldquoSpectroscopic ellipsometryrdquo Proceedings of the 1st ICSE Paris 1993

[BOI 85] BOISSEL P Relaxation vibrationnelle en matrice SF6 et NH3 en matrice de gaz rare et drsquoazote PhD Thesis University of Paris-Sud Orsay 1985

[BOR 99] BORN M WOLF E Principles of Optics Harvard University Press Cambridge 1999

[BRE 73] BRENT R Section 73 Powellrsquos algorithm Algorithms for minimization without derivatives Prentice Hall Englewood Cliffs NJ 1973

[BRO 49] BROSSEL J ET KASTLER A CR Acad Sc vol 229 p 1213 1949

[BRO 68] DE BROGLIE L Ondes eacutelectromagneacutetiques et Photons Gauthier-Villars Paris 1968

[BRO 69] BROYDEN CG ldquoA new double-rank minimization algorithmrdquo Appl Phys A Mat Sci and Process vol 16 p 670 1969

[BRO 93] BROSSET P Position et mouvement de la moleacutecule drsquoozone isoleacutee en matrice inerte a basse tempeacuterature spectre drsquoabsorption infrarouge et modegravele theacuteorique PhD Thesis Pierre and Maris Curie University Paris 1993

[BRU 65] BRUHAT G Cours de physique geacuteneacuterale Optique Masson amp Cie Paris 1965

[BUB 74] BUBE RH Electronic Properties of Crystalline Solids Academic Press NY 1972

[BUC 81] BUCK B SUKUMAR CV Phys Lett A vol 81 p 132 1981

[BUZ 90] BUZEK VJ J Mod Opt vol 37 p 1033 1990

[CHA 50] CHANDRASEKHAR S Radiative Transfer Clarendon Press Oxford 1950

[CHA 56] CHANDRASEKHAR S ldquoThe illumination and polarization of the sunlight sky on Rayleigh scatteringrdquo Trans Am Phil Soc vol 44 p 6 1956

[CHA 98] CHABBI H DAHOO P GAUTHIER RB et al Chem Phys Lett vol 285 p 252 1998


[CHA 00a] CHABBI H Dynamique moleacuteculaire en matrice de gaz rare mouvements et relaxation vibrationnelle de 13CO2 et drsquoautres moleacutecules triatomiques PhD Thesis Pierre and Marie Curie University Paris 2000

[CHA 00b] CHABBI H DAHOO PR DUBOST H et al Low Temp Phys vol 26 p 972 2000

[CHA 00c] CHABBI H DAHOO PR GAUTHIER RB et al J Phys Chem A vol 104 2000 p1670

[CHA 02] CHABBI H GAUTHIER RB VASSEROT A et al J Chem Phys vol 117 2002 p4436

[COD 84] CODY GD ldquoThe optical absorption edge of a-Si Hrdquo in PANKOVE JI (ed) Semiconductors and Semimetals Academic Press New York vol 21 1984

[COH 73] COHEN-TANNOUDJI C DIU B LALOEuml F Meacutecanique Quantique Hermann Paris 1973

[COH 87] COHEN-TANNOUDJI C DUPONT-ROC J GRYNBERG G Photons et atomes Introduction agrave lrsquoeacutelectrodynamique quantique Inter-Editions Paris 1987

[COL 90] COLLINS RW Rev Sci Instrum vol 61 p 2029 1990

[CSE 04] CSELE M Fundamentals of Light Sources and Lasers John Wiley and Sons New York 2004

[DAH 88] DAHOO P Sur lrsquointensiteacute et lrsquoeacutelargissement par la pression des raies de vibration-rotation des bandes ν3 ν1+ ν3 et ν1+ ν3-2ν2

0 de 12C16O2 et ν3 de 14N2

16O Etude expeacuterimentale et interpreacutetation des paramegravetres drsquoeacutelargissement au moyen de modegraveles theacuteoriques semi- classiques PhD Thesis Pierre and Marie Curie University Paris 1988

[DAH 97] DAHOO PR Dynamique moleacuteculaire en phase condenseacutee agrave basse tempeacuterature Moleacutecules drsquointeacuterecirct atmospheacuterique pieacutegeacutees en matrice inerte- Spectroscopie reacutesolue en temps et en freacutequence et eacutetude de complexes faiblement lieacutes Habilitation agrave diriger des recherches University of Versailles St Quentin en Yvelines 1997

[DAH 98] DAHOO PR JASMIN D BROSSET P et al J Chem Phys vol 108 p 8541 1998

[DAH 99] DAHOO PR BERRODIER I RADUCU V et al Eur Phys J D vol 5 p 71 1999

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[DAH 03] DAHOO PR HAMON T SCHNEIDER M et al ldquoEllipsometry principles signal processing and applications to metrologyrdquo Proceedings of CIMNA Lebanon 2003

[DAH 04a] DAHOO PR HAMON T NEGULESCU B et al ldquoEvidence by spectroscopic ellipsometry of optical property change in pulsed laser deposited NiO films when heated in air at Neel temperaturerdquo Appl Phys A Mat Sci and Process vol 79 pp 1439ndash1443 2004

[DAH 04b] DAHOO PR GIRARD A TESSEIR M et al ldquoCharacterizaton of pulsed laser deposited SmFeO3 morphology effect of fluence substrate temperature and oxygen pressurerdquo Appl Phys A Mat Sc and Process vol79 pp 1399ndash1403 2004

[DAH 10] DAHOO PR ALAYLI N GIRARD A et al ldquoReliabilty in Mechatronic systems from TEM SEM and SE Material Analysisrdquo in UEDA O FUKUDA M PEARTON S et al (eds) Reliability and Materials Issues of Semiconductor Optical and Electrical Devices and Materials Warrendale PA 2010

[DAH 15] DAHOO PR KHETTAB M LINARES J et al ldquoNon-destructive characterization by spectroscopic ellipsometry of interfaces in mechatronic devicesrdquo in EL HAMI A POUGNET P (eds) Embedded Mechatronic Systems ISTE Press London and Elsevier Oxford 2015

[DAH 16] DAHOO PR PUIG R LAKHLIFI A et al ldquoSimulation of relaxation channels of CO2 in clathrate nanocagesrdquo Journal of Physics Conference Series 2016

[DAN 90] DANTZIG GB ldquoOrigins of the simplex methodrdquo in NASH G (ed) History of Scientific Computing ACM Press Reading MA 1990

[DAR 02] DARMO J BRATSCHITSCH R MULLER T et al Phys Med Biol vol 47 no 21 pp 3691ndash3697 2002

[DAV 59] DAVIDON NC Variable metric methods for minimization A E C Research and Development Argonne Lab Lemont Illinois 1959

[DAV 96] DAVIDOVICH L ldquoSub-Poissonian processesrdquo Rev Mod Phys vol 68 no 1 pp 127ndash173 1996

[DAV 02] DAVIES AG LINFIELD EH JOHNSTON MB Phys Med Biol vol 47 no 7 pp 3679ndash3689 2002

[DEM 96] DEMTROumlDER W Laser Spectroscopy Basic Concepts and Instrumentation 2nd ed Springer-Verlag BerlinHeidelberg 1996


[DEX 53] DEXTER DL J Chem Phys vol 21 p 836 1953

[DOE 86] DOERNER M NIX W ldquoA method for interpreting the data from depth-sensing indentation instrumentsrdquo Journal of Materials Research vol 1 pp 601ndash609 1986

[DRE 82] DREVILLON B PERRIN J MAROT R et al Rev Sci Instrum vol 53 p 969 1982

[DRE 03] DRESSEL M GRUNER G Optical Properties of Electrons in Matter 2nd ed Cambridge University Press 2003

[DRU 87] DRUDE P Ann Phys vol 32 p 584 1887

[DUB 76] DUBOST H CHARNEAU R Chem Phys vol 12 p 407 1976

[DUB 75] DUBOS H Etude des mouvements moleacuteculaires de la relaxation et des transferts drsquoeacutenergie vibrationnelle de lrsquooxyde de carbone isoleacute en matrice de gaz rare agrave basse temperature PhD Thesis University Paris Sud Orsay 1975

[ELH 13] EL HAMI A RADI B Uncertainty and Optimization in Structural Mechanics ISTE London and John Wiley amp Sons New York 2013

[FER 02a] FERGUSON B ZHANG XC ldquoMaterials for terahertz science and technologyrdquo Nat Mater vol 1 pp 26ndash33 2002

[FER 02b] FERLAUTO AS FERREIRA GM PEARCE JM et al J of App Phys vol 92 p 2424 2002

[FEY 57] FEYNMAN RP VERNON F HELLWARTH R ldquoGeometrical Representation of the Schroumldinger Equation for Solving Maser Problemsrdquo J App Phys vol 28 no 1 pp 49ndash52 1957

[FEY 65] FEYNMAN RP LEIGHTON RB SANDS M The Feynman Lectures on Physics Vol II the Electromagnetic Field Addison-Wesley Longman Reading MA 1965

[FEY 85] FEYNMAN RP QED The Strange Theory of Light and Matter Princeton University Press NJ 1985

[FEY 98] FEYNMAN RP Quantum Electrodynamics Perseus Publishing Westview Press CO 1998

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[FIU 62] FIUTAK J VAN KRANENDONCK J Can J of Phys vol 40 no 9 pp 1085ndash1100 1962

[FLE 63] FLETCHER R POWELL MJD ldquoA rapidly convergent descent method for minimizationrdquo Computer Journal vol 6 pp 163ndash168 1963

[FLE 70] FLETCHER R ldquoA new approach to variable metric algorithmsrdquo Computer Journal vol13 p 371 1970

[FLU 67] FLUGGE S GENTZEL L Encyclopedia of Physics Light and Matter Springer-Verlag Berlin 1967

[FOR 48] FORSTER T Ann Physik vol 2 p 55 1948

[FOR 49] FORSTER T Naturforsch vol A4 p 321 1949

[FOR 72] FORMANN RA PIERMARINI GJ BARNETT JD et al Science vol 176 pp 284ndash285 1972

[FOR 86] FOROUHI AR BLOOMER I Phys Rev B vol 34 p 7018 1986

[FOX 10] FOX M Optical Properties of Solids Oxford University Press 2010

[FUJ 07] FUJIWARA H Spectroscopic Ellipsometry Principles and Applications Wiley 2007

[GAM 98] GAMACHE R LYNCH R NESHYBA SP J Quant Spectr Rad Transfer vol 59 pp 319ndash335 1998

[GAM 14] GAMACHE R LAMOUROUX J BLOT-LAFON V J Quant Spectr Rad Transfer vol 135 pp 30ndash43 2014

[GAU 80] GAUTHIER RB Relaxation de lrsquoeacutenergie vibrationnelle du fluorure de meacutethyle en matrice inerte agrave basse temperature PhD Thesis University of Paris Sud Orsay 1980

[GER 77] GERBER RB BERKOWITZ M Phys Rev Lett vol 39 p 1000 1977

[GHA 91] GHANEM RG SPANGOS PD Stochastic Finite Elements A Spectral Approach Springer Verlag Berlin 1991

[GHA 12] GHADERI SH HAJIESMAILI E ldquoMolecular structural mechanics applied to coiled carbon nanotubesrdquo Computational Materials Science vol 55 pp 344ndash349 2012


[GIA 08] GIANNOPOULOS G KAKAVAS P ANIFANTIS N ldquoEvaluation of the effective mechanical properties of single walled carbon nanotubes using a spring based finite element approachrdquo Computational Materials Science vol 41 no 4 pp 561ndash569 2008

[GIN 78] GINZBURG V Physique Theacuteorique et Astrophysique Mir Moscow 1978

[GIR 85] GIRARDET C LAKHLIFI A J Chem Phys vol 88 p 126 1985

[GLA 67] GLAUBER RJ ldquoPhoton fields and classical fieldsrdquo Proceedings of the Symposium on Modern Optics vol 47 no 11 pp 1ndash18 1967

[GLA 63] GLAUBER RJ ldquoThe quantum theory of optical coherencerdquo Phys Rev A vol 130 p 2529 1963

[GOL 70] GOLDFARB D ldquoA family of variable metric algorithmsrdquo Mathematical computations vol 24 pp 24ndash26 1970

[GOL 03] GOLDSTEIN D Polarized Light Marcel Dekker NY 2003

[GOO 76] GOODMAN L BRUS LE J Chem Phys vol 65 p 1156 1976

[GOU 77] GOULD G Optically Pumped Laser Amplifiers Light Amplifiers Employing Collisions to Produce a Population Inversion US Patents 4053845 and 4704583 1977

[GUE 15a] GUERINE A EL HAMI A WALHA L et al ldquoA perturbation approach for the dynamic analysis of one stage gear system with uncertain parametersrdquo Mechanism and Machine Theory vol 92 pp 113ndash126 2015

[GUE 15b] GUERINE A EL HAMI A FAKHFAKH T et al ldquoA polynomial chaos method to the analysis of the dynamic behavior of spur gear systemrdquo Structural Engineering and Mechanics An International Journal vol 53 pp 819ndash831 2015

[HAY 64] HAYFIELD PCS WHITE GWT ldquoEllipsometry in the measurements of surfaces and Thin filmsrdquo in PASSAGLIA E STROMBERG RR KRUGER J (eds) National Bureau of Standards Miscellaneous Publication 256 US GPO Washington DC 1964

[HEC 02] HECHT E Optics 4th ed Pearson Education Inc Berlin 2002

[HOC 66] HOCKER O KOVACS MA RHODES CK et al Phys Rev Lett vol 17 p 233 1966

Bibliography 283

[HUA 97] HUARD S Polarization of light Masson Paris 1997

[HUM 05] HUMBERT C PEREMANS A ET SILIEN C Revue des questions scientifiques vol 176 no 2 pp 97ndash162 2005

[HUA 05] HUANG-CHANG L ldquoUsing N-D method to solve multi-response problem in Taguchirdquo Journal of Intelligent Manufacturing vol 16 pp 331ndash347 2005

[JAC 98] JACKSON JD Classical Electrodynamics 3rd ed John Wiley and Sons New York 1998

[JAS 94] JASMIN D DAHOO PR BROSSET P et al J Chem Phys vol 101 p 7337 1994

[JAS 95] JASMIN D Relaxation vibrationnelle de lrsquoozone en matrice inerte eacutemission stimuleacutee et transferts non-radiatifs PhD Thesis Pierre and Marie Curie University Paris 1995


[JAU 01] JAULIN L KIEFFER M DITRIT O et al Applied Interval Analysis With Examples in Parameter and State Springer-Verlag London 2001

[JAV 61] JAVAN A BENNETT WR HENRIOTT DR Phys Rev Lett vol 63 pp 106ndash110 1961

[JAY 63] JAYNES ET CUMMINGS FW ldquoComparison of quantum and semiclassical radiation theories with application to the beam maserrdquo Proceedings of IEEE vol 51 p 81 1963

[JAY 05] JAYASANKAR CK VENKATRAMU V BABU P et al J of Appl Phys vol 97 p 093523 2005

[JEL 66a] JELLISON JR GE Thin Solid Films vol 40 pp 290ndash291 1966

[JEL 66b] JELLISON JR GE MODINE FA Applied Physics Letters vol 69 pp 371ndash373 1996

[JON 41] JONES R ldquoA new calculus for the treatment of optical systems I Description and discussion of the calculusrdquo J Opt Soc Am vol 31 p 488 1941

[KAS 50] KASTLER A J Phys Rad vol 11 p 255 1950


[KAY 04] KAYMAZ I MCMAHON CA ldquoA probabilistic design system for reliability-based design optimizationrdquo Structural and Multidisciplinary Optimization vol 28 no 6 pp 416ndash426 2004

[KHA 04] KHARMANDA G EL HAMI A OLHOFF N ldquoGlobal reliability- based design optimizationrdquo in FLOUDAS CA (ed) Frontiers on Global Optimization Kluwer Academic Publishers vol 255 Kluwer Academic Publishers Netherlands 2004

[KHA 14] KHARMANDA G IBRAHIM M-H ABO AL-KHEER A et al ldquoReliability-based design optimization of shank chisel plough using optimum safety factor strategyrdquo Computers and Electronics in Agriculture vol 109 pp 162ndash171 2014

[KHE 14] KHETTAB M Etude de lrsquoinfluence du reacutesinage au niveau de LrsquoIML (Insulated Metal Leadframe) dans le packaging de module commutateur de courant meacutecatronique PhD Thesis University of Versailles St Quentin en Yvelines 2014

[KIT 96] KITTEL C Introduction to Solid State Physics John Wiley amp Sons New York 1996

[KLE 92] KLEIBER M HIEN TD The Stochastic Finite Element Method John Wiley and Sons New York 1992

[KOH 65] KOHN WS SHAM LJ Phys Rev A vol 140 pp 1133ndash1138 1965

[KON 83] KONO A LIN C J Chem Phys vol 78 no 5 pp 2607ndash2620 1983

[LAB 86] LABANI B Elargissement collisionnel des raies de vibration-rotation de moleacutecules toupies asymeacutetriques application agrave H2O Thesis University of Franche-Comteacute Besanccedilon 1986

[LAK 87a] LAKHLIFI A Etude theacuteorique de la moleacutecule drsquoammoniac pieacutegeacutee en matrice interactions mouvements et pheacutenomegravenes relaxationnels Thesis University of Franche-Comteacute Besanccedilon 1987

[LAK 87b] LAKHLIFI A GIRARDET C J Chem Phys vol 87 p 4559 1987

[LAK 93] LAKHLIFI A GIRARDET C DAHOO PR et al Chem Phys vol 177 p 31 1993

[LAK 00] LAKHLIFI A CHABBI H DAHOO PR et al Eur Phys J D vol 12 p 435 2000

Bibliography 285

[LAK 11] LAKHLIFI A DAHOO PR Chem Phys vol 386 pp 73ndash80 2011

[LAK 12] LAKHLIFI A DAHOO PR DARTOIS E et al ldquoModeling IR spectra of CO2 isotopologues and CH4 trapped In type I clathraterdquo EPOV From Planets to Life ndash Colloquium of the CNRS Interdisciplinary Initiative ldquoPlanetary Environments and Origins of Liferdquo Paris France November 29ndash30 2012

[LAK 15] LAKHLIFI A DAHOO PR PICAUD S et al Chem Phys vol 448 pp 53ndash60 2015

[LAN 66] LANDAU L LIFCHITZ E Theacuteorie des Champs Mir Moscow 1966

[LAN 89] LANDAU L LIFCHITZ E PITAEVSKI L et al Electrodynamique Quantique Mir Moscow 1989

[LEG 77] LEGAY F ldquoVibrational relaxation in matricesrdquo in MOORE CB (ed) Chemical and Biochemical Applications of Lasers Academic Press New York vol 2 1977

[LEV 44] LEVENBERG K ldquoA Method for the solution of certain Non-Linear Problems in Least Squaresrdquo The Quarterly of Applied Mathematics vol 2 pp 164ndash168 1944

[LIN 80] LIN SH Radiationless Transitions Academic Press New York 1980

[LIT 57] LITOVITZ TA ldquoTheory of ultrasonic thermal relaxation times in liquidsrdquo Journal of Chemical Physics vol 26 pp 469ndash473 1957

[LOU 64] LOUISELL WH Radiation and Noise in Quantum Electronics 4th ed Mc Graw-Hill New York 1964

[LU 12] LU X HU Z ldquoMechanical property evaluation of single-walled carbon nanotubes by finite element modelingrdquo Composites Part B Engineering vol 43 no 4 pp 1902ndash1913 2012

[LYN 96] LYNCH R GAMACHE R NESHYBA SP J Chem Phys vol 105 pp 5711ndash5721 1996

[MAH 12] MAHMOUDINEZHAD E ANSARI R BASTI A et al ldquoAn accurate spring-mass model for predicting mechanical properties of single-walled carbon nanotubesrdquo Computational Materials Science vol 62 pp 6ndash11 2012

[MAI 60] MAIMAN T Nature vol 187 no 4736 pp 493ndash494 1960

[MAR 65] MARADUDIN A Rep Prog Phys vol 28 p 331 1965


[MAR 63] MARQUARDT DW ldquoAn algorithm for least squares estimation of nonlinear parametersrdquo SIAM J Appl Math vol11 p 431 1963

[MAX 54] MAXWELL JC A Treatise on Electricity and Magnetism 3rd ed Dover New York 1954

[MEI 15] MEIS C Light and Vacuum World Scientific Publishing Co Singapore 2015

[MES 64] MESSIAH A Meacutecanique Quantique vols 1 ndash 2 Dunod Paris 1964

[MES 04] MESCHEDE D Optics Light and Lasers Wiley-VCH Verlag GmbH amp Co Weinheim Germany 2004

[MIR 13] MIR FA BANDAY JA CHONG C et al ldquoOptical and electrical characterization of Ni-doped orthoferrites thin films prepared by sol-gel processrdquo Euro Phys J App Phy vol 61 pp 10302ndash10305 2013

[MIZ 72] MIZUSHIMA M Theoretical Physics From Classical Mechanics to Group Theory of Microparticles John Wiley and Sons New York 1972

[MOH 10] MOHSINE A EL HAMI A ldquoA Robust Study of Reliability-Based Optimisation Methods under Eigen-frequencyrdquo International Journal of Computer Methods in Applied Mechanics and Engineering vol 199 nos 17ndash20 pp 1006ndash1018 2010

[MOO 66] MOORE R BAKER R Introduction to Interval Analysis SIAM Englewood Cliffs NJ 1966

[MOR 77] MORE J ldquoThe Levenberg-Marquardt algorithm Implementation and theoryrdquo Lecture Notes in Mathematics in WATSON GA (ed) Numerical Analysis Springer-Verlag Springer 1977

[MUK 75] MUKAMEL S JORTNER J J Chem Phys vol 63 p 63 1975

[MUL 48] MULLER H ldquoThe foundations of opticsrdquo J Opt Soc Am vol 38 p 661 1948

[MUR 02] MURAKI N MATOBA N HIRANO T et al Polymer vol 43 pp 1277ndash1285 2002

[MUS 99] MUSCOLINO G RICCIARDI N IMPOLLONIA N ldquoImproved dynamic analysis of structures with mechanical uncertainties under deterministic inputrdquo Probabilistic Engineering Mechanics vol 15 pp 199ndash212 1999

Bibliography 287

[NEL 65] NELDER J A MEAD R Computer Journal vol 7 p 308 1965

[NIT 74a] NITZAN A MUKAMEL S J Chem Phys vol 60 p 3929 1974

[NIT 74b] NITZAN A ILBEY RJ J Chem Phys vol 60 p 4070 1974

[NIT 73] NITZAN A JORTNER J Mol Phys vol 25 p 25 1973

[NGO 12] NGO NH TRAN H GAMACHE RR et al J Phil Trans R Soc A vol 370 pp 2495ndash2508 2012

[NOU 07] NOUN W BERINI B DUMONT Y et al ldquoCorrelation between electrical and ellipsometric properties on high-quality epitaxial thin films of the conductive oxide LaNiO3 on STO (001)rdquo Journal of Applied Physics vol102 pp 063709-1ndash063709-7 2007

[OHL 00] OHLIDAL I FRANTA D Progress in Optics Elsevier Amsterdam 2000

[PAP 97] PAPOUŠEK D ldquoVibrational-rotational spectroscopy and molecular dynamicsrdquo Advanced series in Physical Chemistry vol 9 1997

[PEN 99] PENDRY JB HOLDEN AJ ROBINS J et al ldquoMagnetism from conductors and enhanced non linear phenomenardquo IEEE Transactions on Microwave Theory and Techniques vol 47 no 11 p 2075 1999

[POW 64] POWELL M J D ldquoAn efficient method for finding the minimum of a function of several variables without calculating derivativerdquo Computer Journal vol 7 no 2 pp 155ndash162 1964

[POI 92] POINCARE H Theacuteorie matheacutematique de la lumiegravere Georges Carreacute Paris 1892

[POU 15] POUGNET P DAHOO PR ALVAREZ JL ldquoHighly Accelerated Testingrdquo in EL HAMI A POUGNET P (eds) Embedded Mechatronic Systems 2 ISTE Press London and Elsevier Oxford 2015

[POR 50] PORTER G Proceedings of the Royal Society of London Series A Mathematical and Physical Sciences vol 200 no 1061 pp 284ndash300 1950

[POR 68] PORTER G TOPP MR Nature vol 220 pp 1228ndash1229 1968


[PIC 95] PICKERING C Photonic Probes of Surfaces Elsevier Science BV 1995

[PRE 86] PRESS WH FLANNERY BP TEUKOLSKY SA et al Numerical Recipes Cambridge University Press 1986

[ROB 67] ROBERT D Contribution agrave la theacuteorie du spectre infrarouge des solutions liquide Thesis University of Franche-Comteacute Besanccedilon 1967

[ROB 79] ROBERT D BONAMY J J PhysParis vol 40 no 10 pp 923ndash943 1979

[ROS 88] ROSENMANN L Etudes theacuteorique et expeacuterimentale de lrsquoeacutelargissement par collisions des raies de CO2 perturbeacute par O2 H2O N2 et O2 constitution drsquoune base de donneacutees infrarouge et Raman appliqueacutee aux transferts thermiques et agrave la combustion Thesis Ecole Centrale Paris 1988

[ROS 98] ROSENBERG E ET VINTER B Optoeacutelectronique Masson SA Paris France 1998

[ROT 45] ROTHEN A Rev Sci Instrum 16 26 1945

[RHO 68] RHODES CK KELLY MJ JAVAN A J Chem Phys vol 48 p 5730 1968

[SAK 11] SAKURAI JJ NAPOLITANO J Modern Quantum Mechanics 2nd ed Addison-Wesley Pearson 2011

[SHA 96] SHAH J The Ultrafast Spectroscopy of Semiconductors and Semiconductor Nanostructurerdquo Springer-Verlag New York 1996

[SCH 58] SCHAWLOW AL TOWNES CH Phys Rev vol 112 p 1940 1958

[SCH 52] SCHWARTZ RN SLAWSKY ZI HERZFELD KF J Chem Phys vol 20 p 1591 1952

[SCI 12] SCIAMMA-OrsquoBRIEN E DAHOO PR HADAMCIK E et al ldquoOptical constant from 370nm to 900 nm of Titan tholinsproducted in a low pressure RF plasam dischargerdquo Icarus vol 218 pp 356ndash363 2012

[SEC 81] SECROUN C BARBE A JOUVE P et al J Mol Spec vol 85 pp 8ndash15 1981

Bibliography 289

[SHA 70] SHANNO DF A ldquoConditioning of quasi-Newton methods for function minimizationrdquo Math Comp vol 111 pp 647ndash656 1970

[SHU 09] SHUN LIEN C Physics of Photonic Devices John Wiley and Sons New York 2009

[SIE 86] SIEGMAN AE Lasers Publisher University Science Books Sausalito 1986

[SIM 00] SIMMONS JH POTTER KS Optical Materials Academic Press NY 2000

[STE 78] STENFIELD JI Laser and Coherence Spectroscopy Plenum Press New York 1978

[STO 52] STOKES GG ldquoThe Illumination and polarization of the sunlight sky on Rayleigh scatteringrdquo Trans Cambridge Phil Soc vol 9 no III p 399 1852

[SUD 63] SUDARSHAN ECG ldquoEquivalence of semiclassical and quantum mechanical descriptions of statistical light beamsrdquo Phys Rev Lett vol 10 nos 19ndash63 pp 277ndash279 1852

[SUN 68] SUN HY RICE SA J Chem Phys vol 42 p 3826 1968

[TAG 86] TAGUCHI G Introduction to Quality Engineering Designing Quality into Products and Processes ARRB Group Edition Melbourne 1986

[TAN 68] TANGO WJ LINK JK ZARE RN J Chem Phys vol 49 pp 4264ndash4268 1968

[TAN 15] TANG X Contribution agrave la simulation et lrsquoexpeacuterimentation des nanotubes de carbones avec prise en compte des incertitudes PhD Thesis Ecole doctorale Sciences Physiques Matheacutematiques et de lrsquoInformation pour lrsquoingeacutenieur 2015

[TAU 66a] TAUC J GRIGOROVICI R VANCU A Phys Stat Sol vol 15 p 627 1966

[TAU 66b] TAUC J GRIGOROVICI R VANCU A ldquoOptical properties and electronic structure of amorphous germaniumrdquo Physica Status Solidi vol 15 pp 627ndash637 1966

[TIF 67] TIFFANY WB MOOS HW SCHAWLOW AL Science vol 157 no 3784 pp 40ndash43 1967


[TOM 99] TOMPKINS HG MCGAHAN WA Spectroscopic Ellipsometry and Reflectometry Wiley New York 1999

[TOM 05] TOMPKINS HG IRENE EA Handbook of Ellipsometry William Andrew IncSpringer New York 2005

[TRO 13] TROTS DM KURNOSOV A BALLARAN TB et al Solid Earth vol 118 p 118 2013

[TSA 62] TSAO PJ CURNUTTE B J Quant Spectr Rad Transfer vol 2 pp 41ndash91 1962

[URB 53] URBACH F Phys Rev vol 92 p 1324 1953

[VAS 03] VASSEROT AM GAUTHIER RB CHABBI H et al J Mol Spec vol 220 p 201 2003

[VED 98] VEDAM K Thin Solid Films vol 313 p 1 1998

[VES 68] VESELAGO VG ldquoElectrodynamics of substances with simultaneously negative values of ε and micrordquo Sov Phy Uspekhi vol 10 no 4 1968 pp 509ndash518 1968

[VIA 07] VIAL A LAROCHE T J Phys D Appl Phys vol 40 p 7152 2007

[WAL 94] WALTER E JAULIN L ldquoGuaranteed characterization domains via set inversionrdquo Automatic Control Transaction vol 39 no 4 pp 886ndash889 1994

[WEB 71] WEBER MJ Phys Rev vol B4 p 2932 1971

[WOL 06] WOLFRAM T ELLIALTIOGLU S Electronic and Optical Properties of d-Band Perovskites Cambridge University Press 2006

[WOO 72] WOOTEN F Optical Properties of Solids Academic Press New York 1972

[WOO 00] WOOLLAM JA ldquoEllipsometry variable angle spectroscopicrdquo in WEBSTER JG (ed) Encyclopedia of Electrical and Electronics Engineering John Wiley and Sons New York 2000

[YAR 84] YARIV A YEH P Optical Waves in Crystals John Wiley and Sons New York 1984

[ZAD 65] ZADEH L ldquoFuzzy sets and systemsrdquo in FOX J (ed) System Theory J Polytechnic Press Brooklyn New York 1965

Bibliography 291

[ZEW 00] ZEWAIL A J Phys Chem vol 104 no 24 pp 5660ndash5694 2000

[ZON 85] ZONDY JJ GALAUP JP DUBOST H J of Luminescence vol 38 p 255 1985

[ZUM 78] ZUMOFEN J J Chem Phys vol 69 p 69 1978

[ZWA 61] ZWANZIG RW J Chem Phys vol 34 p 1931 1961

Index

A B C

absorption spectrum 205 acceptor 151 152 157ndash159 214 aluminum polymer interface 219

245ndash247 amplified stimulated emission

152 analysis in principal components

22ndash23 annihilation and creation

operators 57ndash61 assemblies 118 133 237ndash238 ATR 243 251 bandgap 116 BeerndashLambert law 126 bilinear model 265ndash273 Brillouin zone 115 125 126 carbon nanotubes 3 255 chaos polynomial 8 coating material 13 239ndash241 CodyndashLorentz model 130 coherent state 47 67 68 122

166 167 conduction band 95 97 105

106 116 123 125 126 130 continuous laser 136 143 148

Coulomb gauge 62 63 71 88 critical point of Van Hove 125

126 131

D E

defects 173 at the interfaces 219 density matrix 59 60 69 162

164ndash167 169 170 208 dephasing 162 163 165 166

168 170 206 design of experiments 2 9ndash14 detection system 147 173 177ndash

179 213 deterministic optimization 26

225 direct transition 126 double resonance signals 133 147

198ndash203 Drude model 95 103ndash105 Einstein coefficient 145 161 elastic properties 260ndash265 electromagnetic wave 47 48 52ndash

57 74 88 949798 103 104 146


First Edition


ellipsometry 72 73 96 131 219ndash222 225 231 238 244

encapsulation 219 245 energy gap 116 123 130 154

157 214 ensembles 164 167 excitation spectrum 188 200 experimental protocol 9 10

239ndash241 experimentation 133

F G

Fermirsquos golden rule 158 finite elements 29 256 268 269 fluorescence

signal 145 179 189 190 192ndash196 212

transition 191 Fock space 67 Fourouhi model 129 Fresnel 49ndash52 fundamental band147 201 202 fuzzy logic 15 18ndash20 Glauber 48 69 122 Greenrsquos matrix 186

H I

Hamiltonian 58 60 65 95 97 111 112 119 120 123 124 163 164 167ndash169

harmonic oscillator 47 57ndash61 65 67

Hilbert space 58 59 83 91 124 hot band 147 148 191198 201

202 Huyghens 49ndash52 IML 219 238 245 248 induced emission 136 203 inert noble gas matrix 182ndash184 inhomogeneous width 180 206

interaction Hamiltonian 95 119 124 163

intermolecular transfer 151 152 156ndash159 204

intersite transfer 190 196 intervals 15ndash18 intramolecular transfer 152 157

162 210 213 214 intrinsic relaxation 150 151198ndash

203 216 inverse method 133 135 208

219 225ndash232 241 isolated binary collisions 216

J L M

JaynesndashCummings model 118ndash123

Jellison model 228 Lagrangian field 64 laser induced fluorescence 124

133 143ndash145 146 175 177 LevenbergndashMarquardt method

231 libration 150 156 171 173 174 lifetime 31 143 145 152 153

158 162 202 204ndash206 light energy 47 limit states 33 Liouville equation 164 165 operator 164 local phonon 154 155 214 Lorentz gauge 57 model 95 105ndash111 131 241 master equations 122 matrix method 73ndash86 225 Maxwell equations 47 52ndash56

96 123 162 163 measurment by extinction 222ndash223

Index 295

optically rotating element 223ndash224

phase modulation 221 mechatronics 238 mid-infrared (MIR) 102 114

118 123 model system143 173 174 182ndash

203 216 monochromatic wave 51ndash52 Monte Carlo 223 265 271

N O P

nanoindentation 262ndash263 nanomaterials 255 nanoseconds 133ndash135146 173

196 201 203 217 nanotubes 3 255 258 260 261

262 near-infrared (NIR) 102 123

131 136 nickel-plated copper polymer

interface 238 non-radiative relaxation 133

153ndash160 162 198 203 206 216

optimal conditions 30 optimization 25 perturbation 3ndash7 124 164 208 polarization 71 Poynting vector 47 79 126 probe laser 148 149 175ndash177

201 prohibited direct transition 126 propagation equation 47 49 51ndash

53 62 pulsed laser125 136 143 146

176ndash178 pump laser 144 145 148 149

161 173 175ndash179 188 189 194ndash196 201ndash203 212 213

pump-probe 134 173

Q R

quantization of the electromagnetic field 61ndash66

quantum description of matter 111ndash118 electrodynamics 61 118 theory of light 57ndash69 quartz polymer interface 219

247ndash249 quasi-Newton method of

BroydenndashFletcherndashGoldfarbndashShanno 231

Rabi oscillation 118122 170 radiative relaxation 133 145

151ndash153 158 162 198 rapid detector 177 rare gas 153 157 173 183 184 RBDO 255 relaxation constant 155 156

159 163 170 202 214ndash216 relaxation time 139 150 200

202ndash204 208 213 215 216 reliability 2ndash9 based optimization 255 indices 27ndash29 34 RungendashKutta method 161

S

Schroumldinger equation 113ndash115 164 165

second quantization 60 66ndash69 Sellmeir model 108ndash111 silicon polymer interface 219

249ndash251 simplex method 231ndash234 spin transition 116


spontaneous emission 118 136ndash138 145 149 153 161 203 206

states of vibration 112 SWCNT 255

T U V W

t1 162 204 206 207 t2 162 206 207 Taguchi 9ndash14 TaucndashLorentz model 127ndash129 theoretical model of induced

fluorescence 160ndash163 theoretical model 133 thermal bath 153 164 165 170

214

thin film 114 241 260 265ndash270 time scale 81 157 transfer of thermal energy 163ndash

170 uncertainties 1 valence band 95105 116 123

126 130 vector potential 55 56 62 63

95 97 112 von Neumann equation 60 165

167 VV transfer150 wave optics 49ndash51 71 72

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Cover

Title Page

Copyright13

Contents

Preface

1 Uncertainties

11 Introduction


121 The MC method




131 Principle


14 The set approach







16 Conclusions

2 Reliability-based Design Optimization

21 Introduction









245 Limit states






26 Conclusions

3 The WavendashParticle Nature of Light

31 Introduction













4 The Polarization States of Light

41 Introduction







44 Conclusions

5 Interaction of Light and Matter

51 Introduction

52 Classical models

521 The Drude model








55 Conclusions

6 Experimentation and Theoretical Models

61 Introduction







64 The DR method








66 Conclusions

7 Defects in a Heterogeneous Medium

71 Introduction


721 Pump laser

722 Probe laser












75 Conclusions

8 Defects at the Interfaces







822 The LM method







84 Conclusions

9 Application to Nanomaterials

91 Introduction







932 Nanoindentation






95 Conclusions

Bibliography

Index

Other titles from iSTE in Mechanical Engineering and Solid Mechanics

EULA13

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Nanometer-scale Defect Detection Using Polarized Light


Abdelkhalak El Hami

Volume 2


Polarized Light


Abdelkhalak El Hami







Contents

Preface xi







16 Conclusions 23








26 Conclusions 44










Contents vii


44 Conclusions 93






55 Conclusions 130








66 Conclusions 170






75 Conclusions 216






84 Conclusions 253




Contents ix





95 Conclusions 274

Bibliography 275

Index 293

Preface








Preface xiii











1

Uncertainties


11 Introduction






First Edition







121 The MC method

1211 Origin


1212 Principle


Uncertainties 3




1214 Remark



1221 Principle





1222 Applications





Uncertainties 5


1223 Remark







[11]

x O

y F

[ ]2 2

2 2

156 22l 54 13l22l 4l 13l 3lmM54 13l 156 22l42013l 3l 22l 4l


[ ]2 2

3

2 2

12 6l 12 6l6l 4l 6l 2lEIK12 6l 12 6ll

6l 2l 6l 4l





[K] = b4middot[A]



[M]= b2middot[B]






Uncertainties 7









[12]


1232 Remark


( ) ( )j jj 0

X xinfin

=

ξ = φ ξ

Uncertainties 9




131 Principle













Uncertainties 11

















Uncertainties 13










Constraint

Den

sity





14 The set approach


Den

sity

Constraint

Constraint

Den

sity

Uncertainties 15



1411 Principle







( ) ( ) ( )( )( ) ( ) ( ) ( ) ( )( )

s s

u u

3

s s s u s s u s s u

s

3 3


u

1

1

x vx v





[13]



[ ] [ ]0 0000s u s u tx x v v = =


Uncertainties 17



Mean value 375 1000 15000 200000





1413 Conclusion


temps (s)

Deacutepla

cement

(m)

time (s)

Disp

lace

men

t(m

)D

ispla

cem

ent (

m)

Time (s)




1421 Principle





F(t) = 20 sin(80t)


Uncertainties 19



1422 Conclusion



stemps

Deacutep

lacem

ent (

m)

( )time

Dis

plac

emen

t(m

)


Dis

plac

emen

t (m

)

Time ( ) s






stemps

Deacutep

lacem

ent (

m)

( )time

Disp

lace

men

t(m


Disp

lace

men

t (m

)

Time ( ) s

Uncertainties 21





2 2 21 2 p + + + 1c c c = [15]



2 12 1 22 2 p2 p+ + +Y c X c X c X= [16]


2 2 212 22 p2 + + + 1c c c = [17]

REMARKndash








1 2 r

1 2 p

( +V + +V ) 90 ( +V + +V )VV

asymp [19]




Uncertainties 23





16 Conclusions


2



21 Introduction



First Edition






( )( ) ( )( )( ) ( )( )

1 1 1

2 2 2

min Under 0

0

t

t



[21]











2nd Limit state

1st Limit state

Feasable Region




u= T(x y)



( )min( ) 0

Tu uwith H x u

β =le

[23]


Normed space

ReliabiltyDomain

FailureDomain




NO

YES







Verify convergence

End




( ) ( ) TH HL u u u H x yλ λ= + [24]



[25]

( ) 0H

L H x uλ

part = =part

[26]



Hu

partpart

is not



Gy

partpart

[27]


02 =partpart+=

partpart

jHj

j uHu

uL λ

( )jkj u

uxTyG

uH

partpart

partpart=

partpart minus

1










[28]



+++=s

MMr

IIffcT ssrrPCPCPCCC

Cost

Pf

CT

Cf Pf

Cc







245 Limit states
















Bonding sheet

C-stage (Component)


C-stage (Component)


C-stage (Component)

Layer 1

Layer 2

Layer 3

Layer 4

Layer 5

Layer 6


Matrix layer

a) b)


Matrix layer

Warp fibers

Fill fibers

(a) (b)








a) b)

Layers

Stratified

Stratified

Designation

(a) (b)






[29]


Design Optimization




Structural Analyzis






4

4

(1 )(1 )

x y Fr x y f f m f

xy Fr xy f f m f


= + minus

= + minus




4 1

1fFr

m f

VMM V

ξηη

+=

minus [211]


1

f

m

f

m

MMMM

ηξ

minus

=

+

[212]

whereby






Copper






ρcu (kgm3) 8930

ρf (kgm3) 2750

ρm (kgm3) 1200

Vf () 10

Exf (Gpa) 725

Eyf (Gpa) 725

Gxyf (Gpa) 30

Em (Gpa) 26

Gm (Gpa) 0985

nf 02

nm 032






( )

4 4

4

4

1( )( )

objCu Cu FR f FR

FR pl pl

f FR Cu

F Xh V h

h N h

X V h h

ρ ρ

θ

= + = =

[213]




4 _ 4 4 _

_ _

01 04

0 90

f

FR m FR FR M

Cu m Cu Cu M

Vh h hh h h

θ


[214]










( )

( )

1

1 p

ip sel i N

ii

f Xi N P

f X=

forall isin =

[216]


max

max max

max

1 05 0 105

i b sel i s s


i f sel i s s


Chr C P P

λλ λ λ

λ


[217]







Np 50

λs 07

Iter 125

Pm 006

f1b 200




Values 1038 07542 200135 003736



a) b)



26 Conclusions



deg V f


h FR4

hcu




3




First Edition


31 Introduction









22

2 21 0uuc t


[31]

where 2 2 2

22 2 2x y z







cvn

= [32]












SurfaceA

( ) ( )P t I r t dA= [35]








2

2 0E EcωΔ + =

[37]


22


[38]


22

2 0kcωminus + =

[39]




0

E divE ρε

nabla = =

[310]

BE rot Et


[311]

0B divBnabla = =

[312]

0 0EB rotB jt


[313]





= minus

[314]


2 2

0 0 2 2 21 0F FF F


part part

[315]























divE ρε

=

and ErotB jt

μ ε part= +part

[317]



2 2

2 2 2 0cr rF FF F


part part

[318]




[319]

and


[320]


B rotA=

[321]

and

AE gradVt


[322]



V Vt

ψpart= minuspart

[323]

and by

A A divψ= +

[324]


0

AV divt

ρε


[325]

and 2



part part

[326]



0div A =

[327]


0

V ρε

Δ = minus [328]

and

2

0 0 02 21A VA j div


part part

[329]






21div 0VAc t

part+ =part

[330]


2

2 20

1 VVc t

ρε


[331]

and

2

0 0 02AA j


part

[332]








22 2 2 21 1 1





Lp mxx

part= =part

[334]


k k kH Eψ ψ= [335]


that 0

k kk

CΨ ψinfin

=



k kk

ψ ψinfin

=

Ψ = Ψ




H it


ˆ( ) (0)



iHtU t eminus=


ˆ ˆ ˆˆ( )k k kk



pρ ψ ψ= is the







22 2 2 2 2ˆ 1 1 1ˆ ˆ( )





ˆ ˆ2x ipa ω

ω+=

and ˆ ˆ2x ipa ω

ω+ minus=

[338]


1a a+ = [339]


ˆ ( )2

x a aω

+= + and ˆ ( )2



1( )2

H a aω += + [341]


1 1( ) ( )2 2

H n a a n nω ω+= + = + [342]





1 1a n n n+ = + + [343]


0 0a = [345]

( ) 0

nann

+

= [346]


and in









AE gradV E Et perp


[347]




[348]



0

( ) ( ( ) ( ) ( ) ( ))2 n n n nn

n


[349]


with


minus= and

3


Lα α=

[350]


2 3 2 2 2 30

1( ) ( ) ( )

2 2

Ni

ii





2 3 2 2 2 301

1( ) ( ) (( ) ( ) )

2 2

Ni

i coul Ni


= + + + minus [352]




2 2 2 30 (( ) ( ) )2









22 3 30 03

0

1( ) (2 ) ( )2 2 2 n

n n

A d r L aL α

α

ε εε ω

= [354]

and

22 2 3 3 20 03

0

1( ) ) (2 ) ( ) ( )2 2 2 n n

n n

c rotA d r L aL α

α

ε ε ωε ω

= minus [355]


[352] is written as

21

12 22

1

( ) ( )2

( ) ( )( )2

Ni

i coul NiN

i i i n n ni n n

mL x U x x


ωω

=

=

= + +

bull + minus

[356]


2 22field

( )( )2 n n n

n n

L a aα αα

ωω

= minus [357]



field

1( )2n n n

nH a aα α

αω += + [358]







21

1

1 ( ( )) ( )2

1( )2

N

i i i coul Ni i

n n nn

H p q A x t U x xm

a aα αα

ω

=

+

= minus +

+ +

[359]






2n

n α

ω






[360]

where

[361]

and

[362]






( ) ( )( ) ( ) ( )E r t E r t E r t+ minus= +


( ) ( ) ( ) ki tk k




[363]




0k

ki C k

infin

=

= [364]



( ) ( )( ) ( ) ( )moy




[365]



0

1 1k

d k kα α α απ

infin

=

= = [366]

a α α α= [367]



2 2

( )

k

P k ek

α αminus= [368]

2 2

0

ki

ke k re

kα θαα

infinminus

== equiv




2 2( )N kα α α=


meanρ α α= [369]




4



41 Introduction





First Edition



Type p wave 1 0

1 0

cos coscos cos

rp i rp

ri i r

E n nr

E n nθ θθ θ

minus= =

+

[41]

Type s wave 0 1

0 1

cos coscos cos

rs i rs

ri i r

E n nrE n n

θ θθ θ

minus= =+

[42]












( ) ( )0 0ˆ ˆ( ) yx i kz ti kz t


[45]



( )0 0ˆ ˆ( ) ( )yx i i kz ti


[46]



222

0 0 0 0


x y x y

E EE EE E E E

φ φ

+ minus = [47]








βα







00 0 0

0

ˆ ˆx

yx

y

ixii

x y iy

E eE e x E e y E

E e

φφφ

φ

+ = =

[49]


00 0 1 0 2

0

x

yx

y

ix ii

x yiy

E eE E e J E e J

E e

φφφ

φ

= = +

[410]


1

10

J =

and 2

01

J =

[411]


00 0 0 1

100

xx x

ii ix

x xE e

E E e E e Jφ

φ φ = = =

[412]



0

2 200 0

1 x

y

ixx

iy yx y

E eJJ

J E eE E

φ

φ

= = +

[413]




[414]



Jαα α αα α α

minus = =

[415]


112DJ

i

=

and

112GJ

i

= minus

[416]


cos( )

siniJe φ

αφ α

α

=

[417]





[418]



[419]


2 22 2cos ( ) sin ( ) 1a bE E

t ta b


[420]


( ) ( )1 12 22 2 2 2

0 0 0 0 0 0 0 02 sin 2 sin



= [421]


0 02 20 0

2 costan(2 ) x y

x y

E E

E E

ϕ Ψ = minus

[422]



( )2 2 22 2

0 0

0 0 0 0

1 ˆ ˆ ˆ2 2 2 2



[423]


0 02 20 0

0 02 20 0



tan 2 sin 2 tan

x y

x y

x y

x y

E EE E

E EE E

β α ϕ ϕ

α ϕ ϕ

β ϕ

= =+

Ψ = =minus

= Ψ

[424]





can be viewed as




5yE i= or 25xE =




with



10

01

1112

11

2 i

11

2 i minus

[425]





1 2 1 22 2 2 ( )n e n e e n nπ π πφλ λ λ










4 20 0 1 0 1 0 1 0 00 1 0 0 0 0 1 0

ii i

i

ee e

i eπ π

φ

φplusmn plusmn

minus minus [427]






0

1

2

3 ( )

x x y y

x x y y

x y y x

x y y x


lowast lowast

lowast lowast

lowast lowast

lowast lowast

+ minus = = + minus

[428]


0

1

2

3 ( )

x x y y

x x y y

x y y x

x y y x

E E E ESE E E ES


lowast lowast

lowast lowast

lowast lowast

lowast lowast

+ minus = + minus

[429]





0 0 90

1 0 90

2 45 45

3 D G


minus

+ minus = minus

minus

[430]


0

1

2

3


sin 2

S IS IS IS I

ββ

β

Ψ = Ψ

[431]



1 2 320

S S SVS

+ += )






0 1 2 3

1 0 1 0 0 1 0

0 1 0 1 1 0 0i

iσ σ σ σ

minus = = = = minus

[433]



2 2 2 20 1 2 3S S S S= + + [434]


2 2 2 21 2 3 00 S S S Slt + + lt [435]


0 00 01 02 03 0

1 10 11 12 13 1

2 20 21 22 23 2

3 30 31 32 33 3

s e

s es e

s e

s e



= =

[436]








4

1 1 1 1 1 10 0 0 0 0 01 0 0 02 2 2 2 2 21 1 0 0 0 0 0 1 0 0 0 0 0 00 0 2 2 1 1 0 0 0 1 0 0 0 00 00 0 0 0 2 2 0 0 1 0 1 10 00 0 0 0 0 0 0 0 2 2

ie

π

minus minus minus


1

n

ii

M M=

= prod [437]





1 0 0 11 0 0 10 1 1 00 0

P

i i

minus =

minus

[439]





1 0

2 0

3 0

cos 2 cos2sin 2 cos

sin 2

S SS S S

S S

ββ

β

Ψ = = Ψ

[440]


2

1

32 2 2

1 2 3

1 arctan( )2

1 arcsin( )2

SSS

S S Sβ

Ψ = + +

[441]






2β

2Ψ

Fixed ellipticity

Fixed

orientation









1

ˆ( )i ii

x=






21 22 23

31 32 33

R R RR R R R

R R R

=



( )i S u

uR e αα minus=

[442]


cos sin 0sin cos 0

0 0 1

izSR e α

α αα α minus

minus = =

where 0 0

0 00 0 0

z

iS i

minus =

[443]





2 ( 1)S s s= +

where s =1


and (1)1









parallel to the














S i x y zσ = =




2i j ij Iσ σ δ= [448]



x= plusmn and ( )0 1 0 1 2

yi= plusmn



i

x y z i

ee

ϕ

ϕ


minus = + + =

minus



0 cos( 2) 0 sin(( 2) 1

1 cos( 2) 0 sin(( 2) 1

i

i

e

e

ϕσ

ϕσ

θ θ

θ θ

= +

= minus [449]




1

ϕ

θ

x

z

y

0

σ

0x

1x

1y0y

σx

σz

σy





2

2

cos 2( )sin 2

i

i

eJ

e

φ

φ

αφ αα

minus

+

=

[450]

44 Conclusions






5



51 Introduction



First Edition








Reflected wave

Transmitted wave






52 Classical models



= + + [51]





22 22 31 2

2 2 2 2 2 21 2 3



[52]



2 2i

λλ λminus

(i = 1 2 3)



2202

( ) ( ) ( )( ) ( ) ( )e e ee e e e






2e

ed rmdt

is the force of

Newton 20e em rω



drmdt

γminus

is the



minus minus and

is the Lorentz



2 20

(0)( )( )

i t

ee

eE er tm i

ω

ω ω γω

--

=- -

[54]

2 20

(0)( )( )

i t

ee

i eE ev tm i

ωωω ω γω

minus

=minus minus

[55]




0( ) 1 ( ) 1 ( ) 1 ( ) ( ) ( )k ke k e ek k


[56]

and as a result

220

02 2 2 20 0

(0)(0)( ) ( ) (0)( ) ( )

i ti tp i te

e ee


ωωωε ω




[57]






220

2 2 2 20 0

(0)(0)( ) ( ) (0)( ) ( )

i ti tp i te

e ee


m i i

ωωωωε ωω σ




[58]



(0) (0)locE E=



0 0 (1 )D E P E Eε ε χ ε= + = + =

[59]


0 0Erot B micro Et

σ ε part= +part

[510]



2

0

n εε

= where 0

iε ε εε

= + and 2 2( )n n ik= minus [511]



or

2 2 2

n ε ε ε+ += and [512]

2 2 2




2 20

2 ( )( ) 1 ( )



0

2 ( ) 1( )( )



2 20

2 ( ) ( ) 1kn P dω ωω ω

π ω ω


0

2 ( )( )


infin

prime=minus [514]




2 20

( )2( ) ( )( )



prime minus and

02 2

0

4 2 ( ) 1( )( )




2 20

2 ( ) ( ) 1( )



0

2 2 ( ) ( )( ) ( ) low




521 The Drude model







2 2

2 20


ωεΔ Δ

= - - = = + = [517]


2 220

2 2 20

1 1 1 1 p pe

D D

N e mii i i



[518]




2

2 2 p

D

ωε ε

ω γinfin= minus+

and 2

2 2 pD

D

ωγεω ω γ

=+

[519]














220

2 2 2 21 1 pe

i ii iL i iL

N e mi i




20

2 2 2 2

epe

i ii iL i iL

N e mi i



++-

_

a

b




22

2 20

1 1 pr

L

ni

ωε


minus minus [522]


2 22 0

2 2 2 20 0

22 2 2 20 0

1( ) ( )

( ) ( )

pL

Lp

L



minus= +minus +

=minus +

[523]


ε(ω)

ω

εrsquorsquoεrsquo

ω0



2

2 20 0

1 11 2( )

pωε

ω ω ω= +

minus and

2

4 20 0

11 2( )

p Lω γ ωε

ω ω ω=

minus [524]

and for ω gtgt ω0

22 2

1 1 pL

ε ωω γ

= ++

and 22 2 1 L

pL

γ ωε ωω γ

= ++

[525]




20

220 0

12

12

p

L

ω ω ωεω ω ω

γ

minus= + minus+

and 2

220 0

22

12

p L

L

ω γεω ω ω

γ

= minus+

[526]





22 0

2 2 2 21 1

1N N

k e k

k kok k ok k

N e m fni i





22 0

2 2 2 21 1

1N N

k e k

k kok ok




2 2 2 22 22

2 2 2 20 01 1

11 ( )2 2N N

ok k okk k

e ek kok ok






n2 = A + B λ2 + Cλ4 [530]



n2 = Aprime λ2 + A + B λ2 + C λ4 [531]




0 0


lt gt= = + [532]


can be written that

0

0

1( ) ( ) ( ( ) ( )) ( )3

( )(1 3 )


N E r tN

α αε

αα ε

= = +

=minus

[533]


0

0

1 11 )2 31 3

rr

r

N NNα ε εε α


[534]






22 1 2

0 0 1

( ) 11 12 ( )r

e L

N N fenm i


minus minus [535]





0

( )T















4s3d4p

ATOME

E

EA EB

El

Eal

ΨA ΨBΨs

Ψas


4s

3d 4p

E







1 2 3( )a a aΩ = and


2

( ) ( ) ( ) ( )2nk nk nk nk

pH r V r r E rm

Ψ = + Ψ = Ψ

[536]



0 0 1 1 2 2 3 3k k G k h b h b h b= + = + + +











E

r

Ec

Ev

BC

BV

ELECTRONSLIBRES

GAP Eg

SOLIDE ATOMEISOLE

METAL

Ef

qK

εε K

Free electrons




2-y





Fen+ Fen+

T hν P

Etat BS

eg

t2g

S=12

Δeacutel

Etat HS

S=52

eg

t2g

ΔeacutelFe3+

3d6

eg

t2g

S=0

Etat BS

Δeacuteleg

t2g

S=2

Etat HS

ΔeacutelFe2+

3d6T hν P

BS State HS State

BS State HS State










ˆˆ ( )2 2




intˆ ˆ ˆ ˆ




ˆˆ ˆ2

H ESΩ= is





Ω= + + + + [540]


dagger dagger0 1


Ω= + = + + + + [541]

where dagger

0



ˆˆ ˆ ˆ ˆ ˆ( )2 2




12 2ˆ1 ( 1)

2 2

ac

i ja

c

n nH

n n

ωω

ωω

Ω + + Ψ Ψ =

Ω + + minus

[542]




cos( ) sin( ) 1

sin( ) cos() 1n n

n

n n e n g

n n e n g

α αα

+ = + +



and




( )1(0) 2



( ) cos( 1 ) sin( 1 ) 12 2



( )2 1( ) 1 ( ) 1 cos( 1 )2






0(0)champ n

nC nα

infin

=

Ψ = = (2

2

n

nC en




C n nα αinfin







( )0

1( ) ( ) 1 cos( 1 )2g

nP t p n n t

infin

=


ngP t =








intˆ ˆ ˆ ˆ



NH is the









22

intˆ ( ) ( ) ( )

2k k kk



0ˆ ˆ ˆ



2


[546]


2

int



[547]




2


[548]


v v32( ) ( ( ) ( ) )


= minus minus


)



v 32( )

8cρ ωπ

=v

v ( ) ( )( )


dSE E


At critical points



32

v 0 01

( ( ) ( )) ( ) ( )c g i i ii

E k E k E k a k k=




v2 ( ( ) ( ))i ci







intensity ( 20







gC Eα ω ωω

= minus









2

( ) gi T g T

E EE E A

Eε

minus gt =

[549]



0 2 2 2 2

0

( )( )

Li L

A E CEEE E C E

ε =minus + [550]




2

20

2 2 2 2 2 20

( ) ( ) ( )for( )

( ) ( )( )

for0

TL i T i L

gg L

g


E E E C EE E

ε ε ε= times


le

[551]



21 1 2 2

( )2( ) ( )g

TLTL

E

E P dE


infin




2

2

( )( ) g

FB

A E Ek E

E BE Cminus

=minus +

[553]








20

2 2 2 2 20

2

( )1( )

( )0exp

L gc

UTLu

cu

A E C E Efor E E

E E E C EE

A E for E EE E

ε


lt ltminus

[554]



2 2 20

2 2 2 2 20

20 0

2 2 2 2 20

2( )( ) 2 2 ( )( )

( )exp

( )

cu c g c c g

c c

gcu

u c c


AE C E EEAE E E C E



[555]




20

2 2 2 2 2 2 20

2

1

( )( ) ( ) for( ) ( )

( )for 0exp

gt

g pCL

tt

u


E E E E EE E

ε

minus= times ge


[556]


55 Conclusions





6




First Edition


61 Introduction


















(frequency nn

cνλ



22(1 )

lQr

πλ

=minus

R=100

PUMPING

AMPLIFYING MEDIUM

R=98 T=2

2L = nλ

Photons

ν = nc2L




between modes

150MHz2ncl

νΔ = =





and


dE


and is




[61]

where0

1 1 12 2

Nn χ αχε



lt iE


gt iE






( ) 2

0

1 1ee gg

a

e p gi

α ρ ρω ω τ

= minusminus minus

[62]


α centcent

e



( )2

2

2 20

1( ) 1ee gg

a



[63]


2 20

2 2i i

time

E cEdpP Edt


= minus sdot = =

[64]



=




0




( )0

0ee ggρ ρ- gt






se p g E e p gτ =

where Es is the


22

0 02

12 2

ss

cE cPe p g

ε ετ = =




















1 213

0

10 VmPEcSε

=











227951 227366

0 00 1 (1)

1 11 0 (1)

125710125841 125801

1 00 0 (1)

0 22 0 (1)

64310 64491 64451

0 11 0 (1)

0 00 0 (0)

1 00 0 (2)

0 33 0 (1)

1 11 0 (2)

ν3

(ν1+ ν23ν2)

Fermi resonance

(ν12ν2)

Fermi resonance

ν2

1 2

3

10 μm


128610128841 128801

203482 203398 203322

193034 193569 193440

188210 188 441 188401

137302136954 136981






64 The DR method







BEAMABSORBENT


COLLECTING LENS



LENS


LIGHT DETECTION








probepump

pumpprobe

Transmitted probe

pump

time

Transmitted probe

probeprobe

pump pumppump

time













Transmittedsignal

Reflected signal

Ultrafast Laser

AB











Dvj=0

vj=1

2

AvA=0

vA=1

0

Jmax

3

5

M0

n

4

D

vj=0

vi=1

vi=2vj=1

ASELASER

1

6

kr

vi=0

6















4 21 3643

iif if if

f



2 29

( 2)s gn nτ τ=

+




i f





k νν



the equation 1n1

hkTe

ν ν=minus



( )( ) n 1(0)

Nk Tk ν= + [66]



i f














22exp( 2 E)k π μ

μ αprop minus Δ

[69]













6 63 1 1 ( ) ( )

8 (2 )DA D ADA A D

P f f dc n R

ν ν νπ ν τ τ

= [610]







61

1445 Ao

N MCk kR














21 2 12 2 21

12 1 12 2 21 1 10

32 1 12 2 21 3

( ) ( ( ) ( )) ( ) ( ) ( )

( ) ( ( ) ( )) ( ) ( ) ( )

( ) ( ( ) ( )) ( ) ( ) ( ) p




ρ

ρ

ρ


[611]


2

2( ) exp( ) 2

tt δ σσσ π

minus Λ = minus

[612]


213

21

( ) ht N Nνργ

= [613]





[ ] ( ) 1 ( ) ( )2


[614]




2 2

0 0 0 02 2( ) E E PEt t t


[615]

















[617]





ˆ ˆ ˆ( )ik

ik jl jljl



ˆ ˆ ˆ



[ ]ˆ ˆ( ) ( )SBt Tr tρ ρ= [619]





SSS S S Sik


iL i Lt t




ˆ ˆ ˆ



ˆ ( ) ˆ ˆˆ ˆ ( ) ( )

ˆ ( )ˆ( ) ( )

s s


s s

s s s s s s s s

Sk k S S


Sk k S

k k k k k k k k

tt t

tt

i tt

ρρ ρ

ρω ρ




partpart

= minus + Γpart

[621]

where

( )

ˆ ˆ

ˆ

1ˆ ˆ ˆ ˆ( )2


s s s s s s



k k k k k k


k

k


prime prime prime



Γ = minus

Γ = Γ + Γ + Γ

and

2int2 ˆ (0) ( )


Bk k k k k k k k

k kk H E Eπ ρ δ


= minus



ρ =




ρ =



g eΨ = + is


written as 1 111 12

ρ =



1 12 2

g g e eρ = +


1 010 12

ρ =



11 12 1 11 12 11 12 1

21 22 2 21 22 21 22 2

1 2 12

2 1 21

0 00 0

0 ( )( ) 0

d idt

i


ε ε ρε ε ρ


[622]




= minus = +


12 hε εν minus

=


12 121 2

21 212 1

11 11

22 22

0 0 00 0 00 0 0 00 0 0 0

it



[623]


( ) ( ) ( 1 or 2)iki k ik ik

t i i kt


part

which lead to


= minus minus

and


= + minus





10

2

ˆ ( )ˆ ˆ ( )ˆ ( )

E tH H E t

E tε μ

μμ ε


[624]


12 121 2

21 212 1

11 11

22 22

0 ( ) ( )0 ( ) ( )

( ) ( ) 0 0( ) ( ) 0 0

E t E tE t E ti

E t E ttE t E t



minus

[625]





12 12

21 21

11 1122 22

0 ( ) ( )0 ( ) ( )

( ) ( ) 0 0

( ) ( ) 0 0

t tt t

it t t

t t




[626]

with


and where 0EμΩ =







66 Conclusions




7



71 Introduction



First Edition










monochromator

HUET

LASER CO2

Spectrometer

FTIR BRUKERIFS113v

(003 cm-1)

LASER

Nd YAG

YG 781C20

DYE

LDS 867

Li NbO3

1064 nm

532 nm

PULSED SOURCE

QUANTEL

870 nm5 ns 20 Hz

Δ σ=085cm-1

220-5000 nm

GeAu

HgCdTeor GeCu

COMPUTER

PRINTER


Preamplifier

HgCdTe

PUMP

PROBE

CaF2

CaF2

GeAu

2100 cm- -1

100 μJfilter

D

D

D

D

E

Lenses

KBr

D Diaphragm

E Retractable

PHOTOCHEMISTRY

532 nm

355nm

266 nm Trigg

er

mon

ochr

omat

or


721 Pump laser


722 Probe laser








-01

-008

-006

-004

-002

0

002

004

5 7 9 11 13 15 17 19 21 23 25Temps(ns)

Am

plitu

de (m

V)

FWMH = 48ns

-02

02

06

1

14

0 1 2 3 4 5 6

Temps (μs)

Ampl

itude

(UA

)

MCT Belova) GeCu b)

Time (micros)

Time (ns)FWHM 48 ns






-02

02

06

10

14

00 02 04 06 08 10 12 14

Temps (μs)

Ampl

itude

(UA

)

MCT SAT

-02

02

06

1

14

0 1 2 3 4 5 6

Temps (μs)

Ampl

itude

(UA

)

MCT SATa) b)






He liquide N2

liquide

77 k 77 k


vide vide

Faisceau issu du


Vanne micro fuite

Meacutelange gazeux

Filament chauffant



Heacutelium gazeux

N2 liquide

Heliumgas



HeliumJauge

Heating coil

Beam fromFTIR

spectrometer

Gasmixture

Pumpvacuum

Micro leakeagevalve

LiquidHe4Kva

cuum

vacu

um

LiquidN2

77 K

LiquidN2

77 K















( )12 6

4E rr rσ σε

= minus [71]








2 3 4 5 6 7 8 9 10-200

-150

-100

-50

0

50

100

150

200

-ε

reσ


V(r)

(cm

-1)

r (Aring)





Substitutional site







( )6 6

4 1gr grjj jj gr

jj jj

V rr rσ σ

ε = minus

[72]






Neon 307 12 06Argon 375 148 078

Krypton 401 164 09Xenon 431 17 108

vacantOxygegravene







12 63

2

1

1( ) 42

ij ijMj ij ij j Mj

i ij ij

V r Er r

σ σε α

=

= minus minus [73]




1


and

2ii jj

ij

σ σσ

+=


( )Stat



lt

= + [74]



j jj jj

G Fα αβ β

βξ = [75]










16O3 1278 1168 0532 -14 -07 21

13C16O2 116 180 0 -43 215 215

14N216O N-N1128 N-O1842

180 166 -30 15 15













-4

-3

-2

-1

0

1

2

3

4

5

6

0 50 100 150 200 250 300 350 400 450 500 550 600

Temps (ns)

Am

plitu

de(V

)


Laser pompe au GeAu





-5

-4

-3

-2

-1

0

1

2

3

4

5

0 50 100 150 200 250 300 350 400 450 500 550 600



Am

plitu

de (V



Fluorescence at MCT





1000 10005 1001 10015 1002 10025 1003 10035 1004 10045 1005



10027

10036


Am

plitu

de n

orm

aliseacute

e(SU

)N

orm

edA

mpl

itude

(AU



Frequency of the


1 0 1

2032120306

1381 0 2 0

6991

17114

0 1 0

0 0 0

1 1 0

ν1 +ν3

ν1+ ν2

ν2

10 μm


1003310027cm-1

0 0 11 0 0

ν1

ν3

1028810279

1097310966

2ν2

17884

0 1 1 ν2+ ν3

0 0 20 3 0

2ν3

3ν2

20914209032067

PUM

P

PROB

EPR

OBE

FLUO

RESC

ENCE





Time (microS)

Time (microS)







626cm-1

short(2) 595cm-1

long(3) 615cm-1

short(1) 627cm-1

short(2) 596cm-1

long(3) 614cm-1







a)

time (ns)

Am

plitu

de (V

))

b)

time (ns)

Am

plitu

de (m

V))

























a) b)





0

1

2

3

4

5

6

0 1000 2000 3000 4000 5000 6000 7000

Tem

ps (micro

s)

Dilution (ArO3)

a)

Tim

e

Dilution (ArO3)

0

20

40

60

80

100

120

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Dilution O3Kr

Tem

ps(micro

s)

b)

Tim

e (micros

)

Dilution (KrO3)





0

50

100

150

200

250

300

350

0 500 1000 1500 2000 2500

dilution XeO3=

Tem

ps(micro

s)Ti

me (

micros)

Dilution (XeO3)






-1

-08

-06

-04

-02

0

02

04

06

08

0 05 1 15 2 25 3 35 4 45 5

Temps (micros)

Am

plitu

de (U

A)

21micros

15micros

04micros

Laser Pompe

Signal P(26)

Signal P(42)

Time (micros)

15 micros

21 micros

04 micros

Pump Laser










max

min

328 exp( )( ) 1 exp d


B


σ

σ


[76]




is equal to 1 ( max

min1 ( ) diff

σ

σσ σ σ= minus )


max

min

22 3 21 1 8 ( ) d3 3

mif if if if

nS f Rn hc

σ

σ


[77]


0 ( )1 12 ( ) 2

ifmif if

t if

IS Ln d I

lN I lNσ σ

σσ σ

minus= =

minus [78]




SS

P= as

03

28

2 032110( )gaz

ifc Sντ

σrarr= [79]





2 29

( 2)s gn nτ τ=

+ [710]









with

relaxation1

1( )2

TcT

γπ

= and dephasing2

1( )2

TcT

γπ

=










22 3 21 1 83 3

ifeffif if

if

n Rn hc

σπσγ

+=

[713]



n l= The threshold


1Sif eff

if

Nlσ

Δ =


S Tif

NNΔ le (NT is







A) N2O (ArN2O = 2000) B) O3 (ArO3=200)

10 μm ν3-ν1 ν3-2ν2

775 μm ν1+ν2-ν2

17 μm ν2 manifold 31-20 31-22

20-11



9385 (a) 1051 (a) 014 014 288 026

1290 (a) 010 36

581 (b) 571 (b) 008 008 958 388

579 (b) 008 488

1011 (a) 1012 (a) 026 030 71


280 (d) 2900 21 (d) 215

12 09

80 200 6 15

150 12

20 24 021 05

C) CO2 (ArCO2 = 2000)





20-11



913 (b) 036 012 177

1018 (b) 012 067

625 (a) 626 (a) 009 0032 506 101

596 (a) 32 41

614 (a) 6135 (a) 009 0032 217 43


1230 400 140 115

950 270

15 27 2 09

7 22

37 65 5 2



21 2 3 2 1 2 2 1 2 int 2

12 1 3 2 1 2 2 1 1 1 0 1 int 1

32 1 3 2 1 2 2 1 3 1

( ) ( )

( )

dN ( ) dt

ra

ra



N N N K N A N K

minus minus

minus minus minus

minus minus


[714]











000E+00 400E+00 800E+00 120E+01 160E+01 200E+01

Temps(ns)

Am

plitu

de (U

niteacute

s arb

itrai

res)

x10 5

x10





a)

b)c)

d)

Am

plitu

de (A

U)

Time (ns)












μ prop minus μΔ






8 (2 )DADA A D

Pc n Rπ ν τ τ

=











75 Conclusions





8






First Edition






rs s is is s

E r E E re

E r E E rΔρ ρ ψ= = = = [81]




























i C P Ce Ai C P C


[82]

















11 12

21 22

( ) ( )( ) ( )


+ +

minus minus

prime = prime

[83]



1 1

1 1

111

i ii i

i ii i

rI

rt+

+++

=

[84]


1 1 1 1 1 1

1 1 1 1



i i i i i i i i

n n n nr rn n n n


+ + + ++ +

+ + + +

minus minus= =+ +

[85]

and

1 11 1 1 1


i i i ipi i si i

i i i i i i i i

n nt tn n n n

θ θθ θ θ θ+ +

+ + + +

= =+ +

[86]

Substrate

Ambient Medium

Di



0 2with cos 2 cos0

i

i

i

i i i i i i i ii

eT D n D n

e

δ

δ


= = =

[87]


( )1

01 1 12 2 1 1 1 1 10

N

i i i N N N i i ii


+ + minus minus +=

= = prod [88]


21 21

11 11

andp sp s

p s

S Sr rS S

= = [89]


( )21 11

11 21

tanp s iP

S p s

S Sr er S S

ρ ψ Δ= = = [810]


( ) a s s f f ff n n k n k eΨ = [811]


and

( )a s s f f fg n n k n k eΔ = [812]



2

exp2

1 exp

( ) ( 11 ( )

nj th j j

j j

X

n m

ρ θ ρ θσ

ρ θ=

minus=

minus minus part

[813]


2 2

exp exp

2 21 exp exp

( ) ( ( ) ( 1 1 ( ) ( )

n j th j j j th j j

j j j

X X

n m

θ θ θ θσ

θ θ=


[814]




is a vector


and thus m = 4 the




1 2 ( )mX x x x=






2 2 2 2

2 2 2 2 2 2

e h i hi

ie h i h

n n n nfn n n n

minus minus=

+ + [815]



Substrate



Local Maxima


Global MaximumF(x)

x





0 0 0 01 2 ( ) mX x x x=




X(k) such that

( 1) ( ) ( ) 01 2k k kX X X k+ = + nabla =

[816]

















1) the mid-point







5) the best vertex






822 The LM method




21( ) ( ) ( )( ) ( ) ( )( )2


























k x xδ += minus







1 1 11 11 (1 )





+= + + minus [823]











IML Module





















B1 150 05

B2 70 1

B3 150 1

B4 150 02


M1 120 04

M2 120 095

M3 150 07

Epoxy E1 125 4










Copper 1000


Silver 05








a)

b)





Y in

deg

rees

12

15

18

21

24

27

30


Ψde

gree

s



D in

deg

rees

-10

0

10

20

30


Δde

gree

s



Inde

x of

Ref

ract

ion

n Extinction C

oefficient k

10

12

14

16

18

000

010

020

030

040

050

060nk

n r

eal p

art

k imaginary

part



Rea

l(Die

lect

ric C

onst

ant)

e1

Imag(D


2

10

15

20

25

30

35

00

03

06

09

12

15

18e1e2

εrsquo re

al p

art

imaginary

part εrsquorsquo







500 1000 1500 2000 2500 3000 3500 4000

00

05

10

Abso

rptio

n



Wavenumber(cm-1)

Abso

rptio

n








Wave Number (cm -1)0 1000 2000 3000 4000 5000 6000

Abs

orpt

ion

Coe

ffici

ent i

n 1

cm

0

3000

6000

9000

12000

15000

M1

Wavenumber (cm-1)


Wave Number (cm -1)500 1000 1500 2000 2500 3000 3500

Abso

rptio

n C

oeffi

cien

t in

1cm

0

2000

4000

6000

8000

B1


Wave Number (cm -1)0 1000 2000 3000 4000 5000 6000

Abso

rptio

n C

oeffi

cien

t in

1cm

0

300

600

900

1200

1500

E1

Wavenumber (cm-1)










0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

del

ta



Wavenumber (cm-1)

0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)








0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)

0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)

0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)









0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

C

os Δ



Wavenumber (cm-1)0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)

0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0

-1 0

-0 5

0 0

0 5

1 0

Cos

Δ



Wavenumber (cm-1)0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0

- 1 0

- 0 5

0 0

0 5

1 0

Cos

Δ



Wavenumber (cm-1)






0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0

-1 0

-0 5

0 0

0 5

1 0

Cos

Δ



Wavenumber (cm-1)0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0

-1 0

-0 5

0 0

0 5

1 0

Cos

Δ



Wavenumber (cm-1)

0 1000 2000 3000 4000 5000

-10

-05

00

05

10

Cos

Δ

Longueur (cm-1)

M1QUARTZ M1Al

Wavenumber (cm-1)0 1000 2000 3000 4000 5000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)0 1000 2000 3000 4000 5000

-10

-05

00

05

10

Cos

Δ


M 3Q uartz M 3A l

Wavenumber (cm-1)

1000 2000 3000 4000

-10

-05

00

05

10

cosΔ


B1Quartz B1CuNi


-10

-05

00

05

10

Cos

Δ


B2QUARTZ B2CuNi

Wavenumber (cm-1)







0 1000 2000 3000 4000 5000

-10

-05

00

05

10

Cos

Δ


B3Quarz B3CuNi

Wavenumber (cm-1)0 1000 2000 3000 4000 5000

-10

-05

00

05

10

Cos

Δ


B4QUARTZ B4CuNi

Wavenumber (cm-1)

0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

del

ta


M1QUARTZ M1CuNi

Wavenumber (cm-1)0 1000 2000 3000 4000 5000

-10

-05

00

05

10

Cos

Δ


M3QUARTZ M3CuNi

Wavenumber (cm-1)








1000 2000 3000 4000 5000

00

05

10

Cos

Δ


B2 QUARTZ B2Si

Wavenumber (cm-1)1000 2000 3000 4000 5000

-05

00

05

10

Cos

Δ


B1Quartz B1Si

Wavenumber (cm-1)

0 1000 2000 3000 4000 5000

-05

00

05

10

Cos

Δ


B3QUARTZ B3Si

Wavenumber (cm-1)0 1000 2000 3000 4000 5000

-10

-05

00

05

10

Cos

Δ


B4QUARTZ B4Si

Wavenumber (cm-1)

1000 2000 3000 4000 5000

01

02

03

04

05

06

07

08

09

10

Cos

Δ


M2QUARTZ M2Si


-10

-05

00

05

10

Cos

Δ


M1QUARTZ M1Si

Wavenumber (cm-1)







1000 2000 3000 4000

0994

0996

0998

1000

Cos

Δ


E1Quartz E1Si

Wavenumber (cm-1)0 1000 2000 3000 4000 5000 6000

070

075

080

085

090

095

100

Cos

Δ


M3QUARTZ M3Si

Wavenumber (cm-1)

1 0 0 0 1 5 0 0 2 00 0 2 5 0 0 3 0 0 0 3 5 0 0

0 0

0 2

0 4

0 6

0 8

1 0

1 2

7 00 80 0 90 00 0

0 2

0 4

0 6

0 8

1 0

1 2

Ab

sor

ban

ce


B 1 B 2 B 3 B 4 M 1 M 2 M 3

Ab

sorb

an

ce


















84 Conclusions





9



91 Introduction




First Edition

















2 20


2 20

1 1( ) ( )2 2


21 ( )2

U U U kτ φ ω τ φ= + = Δ [94]































a) b)





932 Nanoindentation

















2 rdPS E Adh π

= = [96]





















for

( ) forY

Y t Y Y

EE

ε σ σσ

σ ε ε σ σle

= + minus ge [97]



phase2

12r

dPEdhA

π= [98]


phase2

dPdh

is the slope



22

(1 )1 (1 ) i



( )r y tP P h E Eσ θ= [910]





under

max max1

max

ih hh

minus le Δ [912]

2


























Parameters (degC)

(Gpa)

(Gpa)



FE simulation

Iteration 1 653 42 42 108881 4017 00867 997

Iteration 2 70 42 42 8759 1276 010845 1262

Iteration 3 70 21 42 904407 1643 0114 1838

Iteration 4 70 8385 21 85903 106 010256 65

Iteration 5 70 8385 315 846446 897 010098 486

Iteration 6 70 8385 42 828116 661 009858 237













95 Conclusions



Bibliography











First Edition

















Bibliography 277



























0 de 12C16O2 et ν3 de 14N2





Bibliography 279




























Bibliography 281
































Bibliography 283






























Bibliography 285































Bibliography 287






























Bibliography 289
































Bibliography 291





Index

A B C









126 131

D E


164ndash167 169 170 208 dephasing 162 163 165 166





57 74 88 949798 103 104 146


First Edition






F G


signal 145 179 189 190 192ndash196 212


H I









203 216 inverse method 133 135 208


J L M







Index 295





N O P


196 201 203 217 nanotubes 3 255 258 260 261




153ndash160 162 198 203 206 216



53 62 pulsed laser125 136 143 146

176ndash178 pump laser 144 145 148 149


pump-probe 134 173

Q R









S







T U V W



214






Other titles from

in










































Cover

Title Page

Copyright13

Contents

Preface

1 Uncertainties

11 Introduction


121 The MC method




131 Principle


14 The set approach







16 Conclusions


21 Introduction









245 Limit states






26 Conclusions


31 Introduction














41 Introduction







44 Conclusions


51 Introduction

52 Classical models

521 The Drude model








55 Conclusions


61 Introduction







64 The DR method








66 Conclusions


71 Introduction


721 Pump laser

722 Probe laser












75 Conclusions








822 The LM method







84 Conclusions


91 Introduction







932 Nanoindentation






95 Conclusions

Bibliography

Index


EULA13

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ltFEFF4f7f752890194e9b8a2d7f6e5efa7acb7684002000410064006f006200650020005000440046002065874ef653ef5728684c9762537088686a5f548c002000700072006f006f00660065007200204e0a73725f979ad854c18cea7684521753706548679c300260a853ef4ee54f7f75280020004100630072006f0062006100740020548c002000410064006f00620065002000520065006100640065007200200035002e003000204ee553ca66f49ad87248672c4f86958b555f5df25efa7acb76840020005000440046002065874ef63002gt DAN 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 DEU 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 ESP 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 FRA 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Abdelkhalak El Hami

Volume 2


Polarized Light


Abdelkhalak El Hami







Contents

Preface xi







16 Conclusions 23








26 Conclusions 44










Contents vii


44 Conclusions 93






55 Conclusions 130








66 Conclusions 170






75 Conclusions 216






84 Conclusions 253




Contents ix





95 Conclusions 274

Bibliography 275

Index 293

Preface








Preface xiii











1

Uncertainties


11 Introduction






First Edition







121 The MC method

1211 Origin


1212 Principle


Uncertainties 3




1214 Remark



1221 Principle





1222 Applications





Uncertainties 5


1223 Remark







[11]

x O

y F

[ ]2 2

2 2

156 22l 54 13l22l 4l 13l 3lmM54 13l 156 22l42013l 3l 22l 4l


[ ]2 2

3

2 2

12 6l 12 6l6l 4l 6l 2lEIK12 6l 12 6ll

6l 2l 6l 4l





[K] = b4middot[A]



[M]= b2middot[B]






Uncertainties 7









[12]


1232 Remark


( ) ( )j jj 0

X xinfin

=

ξ = φ ξ

Uncertainties 9




131 Principle













Uncertainties 11

















Uncertainties 13










Constraint

Den

sity





14 The set approach


Den

sity

Constraint

Constraint

Den

sity

Uncertainties 15



1411 Principle







( ) ( ) ( )( )( ) ( ) ( ) ( ) ( )( )

s s

u u

3

s s s u s s u s s u

s

3 3


u

1

1

x vx v





[13]



[ ] [ ]0 0000s u s u tx x v v = =


Uncertainties 17



Mean value 375 1000 15000 200000





1413 Conclusion


temps (s)

Deacutepla

cement

(m)

time (s)

Disp

lace

men

t(m

)D

ispla

cem

ent (

m)

Time (s)




1421 Principle





F(t) = 20 sin(80t)


Uncertainties 19



1422 Conclusion



stemps

Deacutep

lacem

ent (

m)

( )time

Dis

plac

emen

t(m

)


Dis

plac

emen

t (m

)

Time ( ) s






stemps

Deacutep

lacem

ent (

m)

( )time

Disp

lace

men

t(m


Disp

lace

men

t (m

)

Time ( ) s

Uncertainties 21





2 2 21 2 p + + + 1c c c = [15]



2 12 1 22 2 p2 p+ + +Y c X c X c X= [16]


2 2 212 22 p2 + + + 1c c c = [17]

REMARKndash








1 2 r

1 2 p

( +V + +V ) 90 ( +V + +V )VV

asymp [19]




Uncertainties 23





16 Conclusions


2



21 Introduction



First Edition






( )( ) ( )( )( ) ( )( )

1 1 1

2 2 2

min Under 0

0

t

t



[21]











2nd Limit state

1st Limit state

Feasable Region




u= T(x y)



( )min( ) 0

Tu uwith H x u

β =le

[23]


Normed space

ReliabiltyDomain

FailureDomain




NO

YES







Verify convergence

End




( ) ( ) TH HL u u u H x yλ λ= + [24]



[25]

( ) 0H

L H x uλ

part = =part

[26]



Hu

partpart

is not



Gy

partpart

[27]


02 =partpart+=

partpart

jHj

j uHu

uL λ

( )jkj u

uxTyG

uH

partpart

partpart=

partpart minus

1










[28]



+++=s

MMr

IIffcT ssrrPCPCPCCC

Cost

Pf

CT

Cf Pf

Cc







245 Limit states
















Bonding sheet

C-stage (Component)


C-stage (Component)


C-stage (Component)

Layer 1

Layer 2

Layer 3

Layer 4

Layer 5

Layer 6


Matrix layer

a) b)


Matrix layer

Warp fibers

Fill fibers

(a) (b)








a) b)

Layers

Stratified

Stratified

Designation

(a) (b)






[29]


Design Optimization




Structural Analyzis






4

4

(1 )(1 )

x y Fr x y f f m f

xy Fr xy f f m f


= + minus

= + minus




4 1

1fFr

m f

VMM V

ξηη

+=

minus [211]


1

f

m

f

m

MMMM

ηξ

minus

=

+

[212]

whereby






Copper






ρcu (kgm3) 8930

ρf (kgm3) 2750

ρm (kgm3) 1200

Vf () 10

Exf (Gpa) 725

Eyf (Gpa) 725

Gxyf (Gpa) 30

Em (Gpa) 26

Gm (Gpa) 0985

nf 02

nm 032






( )

4 4

4

4

1( )( )

objCu Cu FR f FR

FR pl pl

f FR Cu

F Xh V h

h N h

X V h h

ρ ρ

θ

= + = =

[213]




4 _ 4 4 _

_ _

01 04

0 90

f

FR m FR FR M

Cu m Cu Cu M

Vh h hh h h

θ


[214]










( )

( )

1

1 p

ip sel i N

ii

f Xi N P

f X=

forall isin =

[216]


max

max max

max

1 05 0 105

i b sel i s s


i f sel i s s


Chr C P P

λλ λ λ

λ


[217]







Np 50

λs 07

Iter 125

Pm 006

f1b 200




Values 1038 07542 200135 003736



a) b)



26 Conclusions



deg V f


h FR4

hcu




3




First Edition


31 Introduction









22

2 21 0uuc t


[31]

where 2 2 2

22 2 2x y z







cvn

= [32]












SurfaceA

( ) ( )P t I r t dA= [35]








2

2 0E EcωΔ + =

[37]


22


[38]


22

2 0kcωminus + =

[39]




0

E divE ρε

nabla = =

[310]

BE rot Et


[311]

0B divBnabla = =

[312]

0 0EB rotB jt


[313]





= minus

[314]


2 2

0 0 2 2 21 0F FF F


part part

[315]























divE ρε

=

and ErotB jt

μ ε part= +part

[317]



2 2

2 2 2 0cr rF FF F


part part

[318]




[319]

and


[320]


B rotA=

[321]

and

AE gradVt


[322]



V Vt

ψpart= minuspart

[323]

and by

A A divψ= +

[324]


0

AV divt

ρε


[325]

and 2



part part

[326]



0div A =

[327]


0

V ρε

Δ = minus [328]

and

2

0 0 02 21A VA j div


part part

[329]






21div 0VAc t

part+ =part

[330]


2

2 20

1 VVc t

ρε


[331]

and

2

0 0 02AA j


part

[332]








22 2 2 21 1 1





Lp mxx

part= =part

[334]


k k kH Eψ ψ= [335]


that 0

k kk

CΨ ψinfin

=



k kk

ψ ψinfin

=

Ψ = Ψ




H it


ˆ( ) (0)



iHtU t eminus=


ˆ ˆ ˆˆ( )k k kk



pρ ψ ψ= is the







22 2 2 2 2ˆ 1 1 1ˆ ˆ( )





ˆ ˆ2x ipa ω

ω+=

and ˆ ˆ2x ipa ω

ω+ minus=

[338]


1a a+ = [339]


ˆ ( )2

x a aω

+= + and ˆ ( )2



1( )2

H a aω += + [341]


1 1( ) ( )2 2

H n a a n nω ω+= + = + [342]





1 1a n n n+ = + + [343]


0 0a = [345]

( ) 0

nann

+

= [346]


and in









AE gradV E Et perp


[347]




[348]



0

( ) ( ( ) ( ) ( ) ( ))2 n n n nn

n


[349]


with


minus= and

3


Lα α=

[350]


2 3 2 2 2 30

1( ) ( ) ( )

2 2

Ni

ii





2 3 2 2 2 301

1( ) ( ) (( ) ( ) )

2 2

Ni

i coul Ni


= + + + minus [352]




2 2 2 30 (( ) ( ) )2









22 3 30 03

0

1( ) (2 ) ( )2 2 2 n

n n

A d r L aL α

α

ε εε ω

= [354]

and

22 2 3 3 20 03

0

1( ) ) (2 ) ( ) ( )2 2 2 n n

n n

c rotA d r L aL α

α

ε ε ωε ω

= minus [355]


[352] is written as

21

12 22

1

( ) ( )2

( ) ( )( )2

Ni

i coul NiN

i i i n n ni n n

mL x U x x


ωω

=

=

= + +

bull + minus

[356]


2 22field

( )( )2 n n n

n n

L a aα αα

ωω

= minus [357]



field

1( )2n n n

nH a aα α

αω += + [358]







21

1

1 ( ( )) ( )2

1( )2

N

i i i coul Ni i

n n nn

H p q A x t U x xm

a aα αα

ω

=

+

= minus +

+ +

[359]






2n

n α

ω






[360]

where

[361]

and

[362]






( ) ( )( ) ( ) ( )E r t E r t E r t+ minus= +


( ) ( ) ( ) ki tk k




[363]




0k

ki C k

infin

=

= [364]



( ) ( )( ) ( ) ( )moy




[365]



0

1 1k

d k kα α α απ

infin

=

= = [366]

a α α α= [367]



2 2

( )

k

P k ek

α αminus= [368]

2 2

0

ki

ke k re

kα θαα

infinminus

== equiv




2 2( )N kα α α=


meanρ α α= [369]




4



41 Introduction





First Edition



Type p wave 1 0

1 0

cos coscos cos

rp i rp

ri i r

E n nr

E n nθ θθ θ

minus= =

+

[41]

Type s wave 0 1

0 1

cos coscos cos

rs i rs

ri i r

E n nrE n n

θ θθ θ

minus= =+

[42]












( ) ( )0 0ˆ ˆ( ) yx i kz ti kz t


[45]



( )0 0ˆ ˆ( ) ( )yx i i kz ti


[46]



222

0 0 0 0


x y x y

E EE EE E E E

φ φ

+ minus = [47]








βα







00 0 0

0

ˆ ˆx

yx

y

ixii

x y iy

E eE e x E e y E

E e

φφφ

φ

+ = =

[49]


00 0 1 0 2

0

x

yx

y

ix ii

x yiy

E eE E e J E e J

E e

φφφ

φ

= = +

[410]


1

10

J =

and 2

01

J =

[411]


00 0 0 1

100

xx x

ii ix

x xE e

E E e E e Jφ

φ φ = = =

[412]



0

2 200 0

1 x

y

ixx

iy yx y

E eJJ

J E eE E

φ

φ

= = +

[413]




[414]



Jαα α αα α α

minus = =

[415]


112DJ

i

=

and

112GJ

i

= minus

[416]


cos( )

siniJe φ

αφ α

α

=

[417]





[418]



[419]


2 22 2cos ( ) sin ( ) 1a bE E

t ta b


[420]


( ) ( )1 12 22 2 2 2

0 0 0 0 0 0 0 02 sin 2 sin



= [421]


0 02 20 0

2 costan(2 ) x y

x y

E E

E E

ϕ Ψ = minus

[422]



( )2 2 22 2

0 0

0 0 0 0

1 ˆ ˆ ˆ2 2 2 2



[423]


0 02 20 0

0 02 20 0



tan 2 sin 2 tan

x y

x y

x y

x y

E EE E

E EE E

β α ϕ ϕ

α ϕ ϕ

β ϕ

= =+

Ψ = =minus

= Ψ

[424]





can be viewed as




5yE i= or 25xE =




with



10

01

1112

11

2 i

11

2 i minus

[425]





1 2 1 22 2 2 ( )n e n e e n nπ π πφλ λ λ










4 20 0 1 0 1 0 1 0 00 1 0 0 0 0 1 0

ii i

i

ee e

i eπ π

φ

φplusmn plusmn

minus minus [427]






0

1

2

3 ( )

x x y y

x x y y

x y y x

x y y x


lowast lowast

lowast lowast

lowast lowast

lowast lowast

+ minus = = + minus

[428]


0

1

2

3 ( )

x x y y

x x y y

x y y x

x y y x

E E E ESE E E ES


lowast lowast

lowast lowast

lowast lowast

lowast lowast

+ minus = + minus

[429]





0 0 90

1 0 90

2 45 45

3 D G


minus

+ minus = minus

minus

[430]


0

1

2

3


sin 2

S IS IS IS I

ββ

β

Ψ = Ψ

[431]



1 2 320

S S SVS

+ += )






0 1 2 3

1 0 1 0 0 1 0

0 1 0 1 1 0 0i

iσ σ σ σ

minus = = = = minus

[433]



2 2 2 20 1 2 3S S S S= + + [434]


2 2 2 21 2 3 00 S S S Slt + + lt [435]


0 00 01 02 03 0

1 10 11 12 13 1

2 20 21 22 23 2

3 30 31 32 33 3

s e

s es e

s e

s e



= =

[436]








4

1 1 1 1 1 10 0 0 0 0 01 0 0 02 2 2 2 2 21 1 0 0 0 0 0 1 0 0 0 0 0 00 0 2 2 1 1 0 0 0 1 0 0 0 00 00 0 0 0 2 2 0 0 1 0 1 10 00 0 0 0 0 0 0 0 2 2

ie

π

minus minus minus


1

n

ii

M M=

= prod [437]





1 0 0 11 0 0 10 1 1 00 0

P

i i

minus =

minus

[439]





1 0

2 0

3 0

cos 2 cos2sin 2 cos

sin 2

S SS S S

S S

ββ

β

Ψ = = Ψ

[440]


2

1

32 2 2

1 2 3

1 arctan( )2

1 arcsin( )2

SSS

S S Sβ

Ψ = + +

[441]






2β

2Ψ

Fixed ellipticity

Fixed

orientation









1

ˆ( )i ii

x=






21 22 23

31 32 33

R R RR R R R

R R R

=



( )i S u

uR e αα minus=

[442]


cos sin 0sin cos 0

0 0 1

izSR e α

α αα α minus

minus = =

where 0 0

0 00 0 0

z

iS i

minus =

[443]





2 ( 1)S s s= +

where s =1


and (1)1









parallel to the














S i x y zσ = =




2i j ij Iσ σ δ= [448]



x= plusmn and ( )0 1 0 1 2

yi= plusmn



i

x y z i

ee

ϕ

ϕ


minus = + + =

minus



0 cos( 2) 0 sin(( 2) 1

1 cos( 2) 0 sin(( 2) 1

i

i

e

e

ϕσ

ϕσ

θ θ

θ θ

= +

= minus [449]




1

ϕ

θ

x

z

y

0

σ

0x

1x

1y0y

σx

σz

σy





2

2

cos 2( )sin 2

i

i

eJ

e

φ

φ

αφ αα

minus

+

=

[450]

44 Conclusions






5



51 Introduction



First Edition








Reflected wave

Transmitted wave






52 Classical models



= + + [51]





22 22 31 2

2 2 2 2 2 21 2 3



[52]



2 2i

λλ λminus

(i = 1 2 3)



2202

( ) ( ) ( )( ) ( ) ( )e e ee e e e






2e

ed rmdt

is the force of

Newton 20e em rω



drmdt

γminus

is the



minus minus and

is the Lorentz



2 20

(0)( )( )

i t

ee

eE er tm i

ω

ω ω γω

--

=- -

[54]

2 20

(0)( )( )

i t

ee

i eE ev tm i

ωωω ω γω

minus

=minus minus

[55]




0( ) 1 ( ) 1 ( ) 1 ( ) ( ) ( )k ke k e ek k


[56]

and as a result

220

02 2 2 20 0

(0)(0)( ) ( ) (0)( ) ( )

i ti tp i te

e ee


ωωωε ω




[57]






220

2 2 2 20 0

(0)(0)( ) ( ) (0)( ) ( )

i ti tp i te

e ee


m i i

ωωωωε ωω σ




[58]



(0) (0)locE E=



0 0 (1 )D E P E Eε ε χ ε= + = + =

[59]


0 0Erot B micro Et

σ ε part= +part

[510]



2

0

n εε

= where 0

iε ε εε

= + and 2 2( )n n ik= minus [511]



or

2 2 2

n ε ε ε+ += and [512]

2 2 2




2 20

2 ( )( ) 1 ( )



0

2 ( ) 1( )( )



2 20

2 ( ) ( ) 1kn P dω ωω ω

π ω ω


0

2 ( )( )


infin

prime=minus [514]




2 20

( )2( ) ( )( )



prime minus and

02 2

0

4 2 ( ) 1( )( )




2 20

2 ( ) ( ) 1( )



0

2 2 ( ) ( )( ) ( ) low




521 The Drude model







2 2

2 20


ωεΔ Δ

= - - = = + = [517]


2 220

2 2 20

1 1 1 1 p pe

D D

N e mii i i



[518]




2

2 2 p

D

ωε ε

ω γinfin= minus+

and 2

2 2 pD

D

ωγεω ω γ

=+

[519]














220

2 2 2 21 1 pe

i ii iL i iL

N e mi i




20

2 2 2 2

epe

i ii iL i iL

N e mi i



++-

_

a

b




22

2 20

1 1 pr

L

ni

ωε


minus minus [522]


2 22 0

2 2 2 20 0

22 2 2 20 0

1( ) ( )

( ) ( )

pL

Lp

L



minus= +minus +

=minus +

[523]


ε(ω)

ω

εrsquorsquoεrsquo

ω0



2

2 20 0

1 11 2( )

pωε

ω ω ω= +

minus and

2

4 20 0

11 2( )

p Lω γ ωε

ω ω ω=

minus [524]

and for ω gtgt ω0

22 2

1 1 pL

ε ωω γ

= ++

and 22 2 1 L

pL

γ ωε ωω γ

= ++

[525]




20

220 0

12

12

p

L

ω ω ωεω ω ω

γ

minus= + minus+

and 2

220 0

22

12

p L

L

ω γεω ω ω

γ

= minus+

[526]





22 0

2 2 2 21 1

1N N

k e k

k kok k ok k

N e m fni i





22 0

2 2 2 21 1

1N N

k e k

k kok ok




2 2 2 22 22

2 2 2 20 01 1

11 ( )2 2N N

ok k okk k

e ek kok ok






n2 = A + B λ2 + Cλ4 [530]



n2 = Aprime λ2 + A + B λ2 + C λ4 [531]




0 0


lt gt= = + [532]


can be written that

0

0

1( ) ( ) ( ( ) ( )) ( )3

( )(1 3 )


N E r tN

α αε

αα ε

= = +

=minus

[533]


0

0

1 11 )2 31 3

rr

r

N NNα ε εε α


[534]






22 1 2

0 0 1

( ) 11 12 ( )r

e L

N N fenm i


minus minus [535]





0

( )T















4s3d4p

ATOME

E

EA EB

El

Eal

ΨA ΨBΨs

Ψas


4s

3d 4p

E







1 2 3( )a a aΩ = and


2

( ) ( ) ( ) ( )2nk nk nk nk

pH r V r r E rm

Ψ = + Ψ = Ψ

[536]



0 0 1 1 2 2 3 3k k G k h b h b h b= + = + + +











E

r

Ec

Ev

BC

BV

ELECTRONSLIBRES

GAP Eg

SOLIDE ATOMEISOLE

METAL

Ef

qK

εε K

Free electrons




2-y





Fen+ Fen+

T hν P

Etat BS

eg

t2g

S=12

Δeacutel

Etat HS

S=52

eg

t2g

ΔeacutelFe3+

3d6

eg

t2g

S=0

Etat BS

Δeacuteleg

t2g

S=2

Etat HS

ΔeacutelFe2+

3d6T hν P

BS State HS State

BS State HS State










ˆˆ ( )2 2




intˆ ˆ ˆ ˆ




ˆˆ ˆ2

H ESΩ= is





Ω= + + + + [540]


dagger dagger0 1


Ω= + = + + + + [541]

where dagger

0



ˆˆ ˆ ˆ ˆ ˆ( )2 2




12 2ˆ1 ( 1)

2 2

ac

i ja

c

n nH

n n

ωω

ωω

Ω + + Ψ Ψ =

Ω + + minus

[542]




cos( ) sin( ) 1

sin( ) cos() 1n n

n

n n e n g

n n e n g

α αα

+ = + +



and




( )1(0) 2



( ) cos( 1 ) sin( 1 ) 12 2



( )2 1( ) 1 ( ) 1 cos( 1 )2






0(0)champ n

nC nα

infin

=

Ψ = = (2

2

n

nC en




C n nα αinfin







( )0

1( ) ( ) 1 cos( 1 )2g

nP t p n n t

infin

=


ngP t =








intˆ ˆ ˆ ˆ



NH is the









22

intˆ ( ) ( ) ( )

2k k kk



0ˆ ˆ ˆ



2


[546]


2

int



[547]




2


[548]


v v32( ) ( ( ) ( ) )


= minus minus


)



v 32( )

8cρ ωπ

=v

v ( ) ( )( )


dSE E


At critical points



32

v 0 01

( ( ) ( )) ( ) ( )c g i i ii

E k E k E k a k k=




v2 ( ( ) ( ))i ci







intensity ( 20







gC Eα ω ωω

= minus









2

( ) gi T g T

E EE E A

Eε

minus gt =

[549]



0 2 2 2 2

0

( )( )

Li L

A E CEEE E C E

ε =minus + [550]




2

20

2 2 2 2 2 20

( ) ( ) ( )for( )

( ) ( )( )

for0

TL i T i L

gg L

g


E E E C EE E

ε ε ε= times


le

[551]



21 1 2 2

( )2( ) ( )g

TLTL

E

E P dE


infin




2

2

( )( ) g

FB

A E Ek E

E BE Cminus

=minus +

[553]








20

2 2 2 2 20

2

( )1( )

( )0exp

L gc

UTLu

cu

A E C E Efor E E

E E E C EE

A E for E EE E

ε


lt ltminus

[554]



2 2 20

2 2 2 2 20

20 0

2 2 2 2 20

2( )( ) 2 2 ( )( )

( )exp

( )

cu c g c c g

c c

gcu

u c c


AE C E EEAE E E C E



[555]




20

2 2 2 2 2 2 20

2

1

( )( ) ( ) for( ) ( )

( )for 0exp

gt

g pCL

tt

u


E E E E EE E

ε

minus= times ge


[556]


55 Conclusions





6




First Edition


61 Introduction


















(frequency nn

cνλ



22(1 )

lQr

πλ

=minus

R=100

PUMPING

AMPLIFYING MEDIUM

R=98 T=2

2L = nλ

Photons

ν = nc2L




between modes

150MHz2ncl

νΔ = =





and


dE


and is




[61]

where0

1 1 12 2

Nn χ αχε



lt iE


gt iE






( ) 2

0

1 1ee gg

a

e p gi

α ρ ρω ω τ

= minusminus minus

[62]


α centcent

e



( )2

2

2 20

1( ) 1ee gg

a



[63]


2 20

2 2i i

time

E cEdpP Edt


= minus sdot = =

[64]



=




0




( )0

0ee ggρ ρ- gt






se p g E e p gτ =

where Es is the


22

0 02

12 2

ss

cE cPe p g

ε ετ = =




















1 213

0

10 VmPEcSε

=











227951 227366

0 00 1 (1)

1 11 0 (1)

125710125841 125801

1 00 0 (1)

0 22 0 (1)

64310 64491 64451

0 11 0 (1)

0 00 0 (0)

1 00 0 (2)

0 33 0 (1)

1 11 0 (2)

ν3

(ν1+ ν23ν2)

Fermi resonance

(ν12ν2)

Fermi resonance

ν2

1 2

3

10 μm


128610128841 128801

203482 203398 203322

193034 193569 193440

188210 188 441 188401

137302136954 136981






64 The DR method







BEAMABSORBENT


COLLECTING LENS



LENS


LIGHT DETECTION








probepump

pumpprobe

Transmitted probe

pump

time

Transmitted probe

probeprobe

pump pumppump

time













Transmittedsignal

Reflected signal

Ultrafast Laser

AB











Dvj=0

vj=1

2

AvA=0

vA=1

0

Jmax

3

5

M0

n

4

D

vj=0

vi=1

vi=2vj=1

ASELASER

1

6

kr

vi=0

6















4 21 3643

iif if if

f



2 29

( 2)s gn nτ τ=

+




i f





k νν



the equation 1n1

hkTe

ν ν=minus



( )( ) n 1(0)

Nk Tk ν= + [66]



i f














22exp( 2 E)k π μ

μ αprop minus Δ

[69]













6 63 1 1 ( ) ( )

8 (2 )DA D ADA A D

P f f dc n R

ν ν νπ ν τ τ

= [610]







61

1445 Ao

N MCk kR














21 2 12 2 21

12 1 12 2 21 1 10

32 1 12 2 21 3

( ) ( ( ) ( )) ( ) ( ) ( )

( ) ( ( ) ( )) ( ) ( ) ( )

( ) ( ( ) ( )) ( ) ( ) ( ) p




ρ

ρ

ρ


[611]


2

2( ) exp( ) 2

tt δ σσσ π

minus Λ = minus

[612]


213

21

( ) ht N Nνργ

= [613]





[ ] ( ) 1 ( ) ( )2


[614]




2 2

0 0 0 02 2( ) E E PEt t t


[615]

















[617]





ˆ ˆ ˆ( )ik

ik jl jljl



ˆ ˆ ˆ



[ ]ˆ ˆ( ) ( )SBt Tr tρ ρ= [619]





SSS S S Sik


iL i Lt t




ˆ ˆ ˆ



ˆ ( ) ˆ ˆˆ ˆ ( ) ( )

ˆ ( )ˆ( ) ( )

s s


s s

s s s s s s s s

Sk k S S


Sk k S

k k k k k k k k

tt t

tt

i tt

ρρ ρ

ρω ρ




partpart

= minus + Γpart

[621]

where

( )

ˆ ˆ

ˆ

1ˆ ˆ ˆ ˆ( )2


s s s s s s



k k k k k k


k

k


prime prime prime



Γ = minus

Γ = Γ + Γ + Γ

and

2int2 ˆ (0) ( )


Bk k k k k k k k

k kk H E Eπ ρ δ


= minus



ρ =




ρ =



g eΨ = + is


written as 1 111 12

ρ =



1 12 2

g g e eρ = +


1 010 12

ρ =



11 12 1 11 12 11 12 1

21 22 2 21 22 21 22 2

1 2 12

2 1 21

0 00 0

0 ( )( ) 0

d idt

i


ε ε ρε ε ρ


[622]




= minus = +


12 hε εν minus

=


12 121 2

21 212 1

11 11

22 22

0 0 00 0 00 0 0 00 0 0 0

it



[623]


( ) ( ) ( 1 or 2)iki k ik ik

t i i kt


part

which lead to


= minus minus

and


= + minus





10

2

ˆ ( )ˆ ˆ ( )ˆ ( )

E tH H E t

E tε μ

μμ ε


[624]


12 121 2

21 212 1

11 11

22 22

0 ( ) ( )0 ( ) ( )

( ) ( ) 0 0( ) ( ) 0 0

E t E tE t E ti

E t E ttE t E t



minus

[625]





12 12

21 21

11 1122 22

0 ( ) ( )0 ( ) ( )

( ) ( ) 0 0

( ) ( ) 0 0

t tt t

it t t

t t




[626]

with


and where 0EμΩ =







66 Conclusions




7



71 Introduction



First Edition










monochromator

HUET

LASER CO2

Spectrometer

FTIR BRUKERIFS113v

(003 cm-1)

LASER

Nd YAG

YG 781C20

DYE

LDS 867

Li NbO3

1064 nm

532 nm

PULSED SOURCE

QUANTEL

870 nm5 ns 20 Hz

Δ σ=085cm-1

220-5000 nm

GeAu

HgCdTeor GeCu

COMPUTER

PRINTER


Preamplifier

HgCdTe

PUMP

PROBE

CaF2

CaF2

GeAu

2100 cm- -1

100 μJfilter

D

D

D

D

E

Lenses

KBr

D Diaphragm

E Retractable

PHOTOCHEMISTRY

532 nm

355nm

266 nm Trigg

er

mon

ochr

omat

or


721 Pump laser


722 Probe laser








-01

-008

-006

-004

-002

0

002

004

5 7 9 11 13 15 17 19 21 23 25Temps(ns)

Am

plitu

de (m

V)

FWMH = 48ns

-02

02

06

1

14

0 1 2 3 4 5 6

Temps (μs)

Ampl

itude

(UA

)

MCT Belova) GeCu b)

Time (micros)

Time (ns)FWHM 48 ns






-02

02

06

10

14

00 02 04 06 08 10 12 14

Temps (μs)

Ampl

itude

(UA

)

MCT SAT

-02

02

06

1

14

0 1 2 3 4 5 6

Temps (μs)

Ampl

itude

(UA

)

MCT SATa) b)






He liquide N2

liquide

77 k 77 k


vide vide

Faisceau issu du


Vanne micro fuite

Meacutelange gazeux

Filament chauffant



Heacutelium gazeux

N2 liquide

Heliumgas



HeliumJauge

Heating coil

Beam fromFTIR

spectrometer

Gasmixture

Pumpvacuum

Micro leakeagevalve

LiquidHe4Kva

cuum

vacu

um

LiquidN2

77 K

LiquidN2

77 K















( )12 6

4E rr rσ σε

= minus [71]








2 3 4 5 6 7 8 9 10-200

-150

-100

-50

0

50

100

150

200

-ε

reσ


V(r)

(cm

-1)

r (Aring)





Substitutional site







( )6 6

4 1gr grjj jj gr

jj jj

V rr rσ σ

ε = minus

[72]






Neon 307 12 06Argon 375 148 078

Krypton 401 164 09Xenon 431 17 108

vacantOxygegravene







12 63

2

1

1( ) 42

ij ijMj ij ij j Mj

i ij ij

V r Er r

σ σε α

=

= minus minus [73]




1


and

2ii jj

ij

σ σσ

+=


( )Stat



lt

= + [74]



j jj jj

G Fα αβ β

βξ = [75]










16O3 1278 1168 0532 -14 -07 21

13C16O2 116 180 0 -43 215 215

14N216O N-N1128 N-O1842

180 166 -30 15 15













-4

-3

-2

-1

0

1

2

3

4

5

6

0 50 100 150 200 250 300 350 400 450 500 550 600

Temps (ns)

Am

plitu

de(V

)


Laser pompe au GeAu





-5

-4

-3

-2

-1

0

1

2

3

4

5

0 50 100 150 200 250 300 350 400 450 500 550 600



Am

plitu

de (V



Fluorescence at MCT





1000 10005 1001 10015 1002 10025 1003 10035 1004 10045 1005



10027

10036


Am

plitu

de n

orm

aliseacute

e(SU

)N

orm

edA

mpl

itude

(AU



Frequency of the


1 0 1

2032120306

1381 0 2 0

6991

17114

0 1 0

0 0 0

1 1 0

ν1 +ν3

ν1+ ν2

ν2

10 μm


1003310027cm-1

0 0 11 0 0

ν1

ν3

1028810279

1097310966

2ν2

17884

0 1 1 ν2+ ν3

0 0 20 3 0

2ν3

3ν2

20914209032067

PUM

P

PROB

EPR

OBE

FLUO

RESC

ENCE





Time (microS)

Time (microS)







626cm-1

short(2) 595cm-1

long(3) 615cm-1

short(1) 627cm-1

short(2) 596cm-1

long(3) 614cm-1







a)

time (ns)

Am

plitu

de (V

))

b)

time (ns)

Am

plitu

de (m

V))

























a) b)





0

1

2

3

4

5

6

0 1000 2000 3000 4000 5000 6000 7000

Tem

ps (micro

s)

Dilution (ArO3)

a)

Tim

e

Dilution (ArO3)

0

20

40

60

80

100

120

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Dilution O3Kr

Tem

ps(micro

s)

b)

Tim

e (micros

)

Dilution (KrO3)





0

50

100

150

200

250

300

350

0 500 1000 1500 2000 2500

dilution XeO3=

Tem

ps(micro

s)Ti

me (

micros)

Dilution (XeO3)






-1

-08

-06

-04

-02

0

02

04

06

08

0 05 1 15 2 25 3 35 4 45 5

Temps (micros)

Am

plitu

de (U

A)

21micros

15micros

04micros

Laser Pompe

Signal P(26)

Signal P(42)

Time (micros)

15 micros

21 micros

04 micros

Pump Laser










max

min

328 exp( )( ) 1 exp d


B


σ

σ


[76]




is equal to 1 ( max

min1 ( ) diff

σ

σσ σ σ= minus )


max

min

22 3 21 1 8 ( ) d3 3

mif if if if

nS f Rn hc

σ

σ


[77]


0 ( )1 12 ( ) 2

ifmif if

t if

IS Ln d I

lN I lNσ σ

σσ σ

minus= =

minus [78]




SS

P= as

03

28

2 032110( )gaz

ifc Sντ

σrarr= [79]





2 29

( 2)s gn nτ τ=

+ [710]









with

relaxation1

1( )2

TcT

γπ

= and dephasing2

1( )2

TcT

γπ

=










22 3 21 1 83 3

ifeffif if

if

n Rn hc

σπσγ

+=

[713]



n l= The threshold


1Sif eff

if

Nlσ

Δ =


S Tif

NNΔ le (NT is







A) N2O (ArN2O = 2000) B) O3 (ArO3=200)

10 μm ν3-ν1 ν3-2ν2

775 μm ν1+ν2-ν2

17 μm ν2 manifold 31-20 31-22

20-11



9385 (a) 1051 (a) 014 014 288 026

1290 (a) 010 36

581 (b) 571 (b) 008 008 958 388

579 (b) 008 488

1011 (a) 1012 (a) 026 030 71


280 (d) 2900 21 (d) 215

12 09

80 200 6 15

150 12

20 24 021 05

C) CO2 (ArCO2 = 2000)





20-11



913 (b) 036 012 177

1018 (b) 012 067

625 (a) 626 (a) 009 0032 506 101

596 (a) 32 41

614 (a) 6135 (a) 009 0032 217 43


1230 400 140 115

950 270

15 27 2 09

7 22

37 65 5 2



21 2 3 2 1 2 2 1 2 int 2

12 1 3 2 1 2 2 1 1 1 0 1 int 1

32 1 3 2 1 2 2 1 3 1

( ) ( )

( )

dN ( ) dt

ra

ra



N N N K N A N K

minus minus

minus minus minus

minus minus


[714]











000E+00 400E+00 800E+00 120E+01 160E+01 200E+01

Temps(ns)

Am

plitu

de (U

niteacute

s arb

itrai

res)

x10 5

x10





a)

b)c)

d)

Am

plitu

de (A

U)

Time (ns)












μ prop minus μΔ






8 (2 )DADA A D

Pc n Rπ ν τ τ

=











75 Conclusions





8






First Edition






rs s is is s

E r E E re

E r E E rΔρ ρ ψ= = = = [81]




























i C P Ce Ai C P C


[82]

















11 12

21 22

( ) ( )( ) ( )


+ +

minus minus

prime = prime

[83]



1 1

1 1

111

i ii i

i ii i

rI

rt+

+++

=

[84]


1 1 1 1 1 1

1 1 1 1



i i i i i i i i

n n n nr rn n n n


+ + + ++ +

+ + + +

minus minus= =+ +

[85]

and

1 11 1 1 1


i i i ipi i si i

i i i i i i i i

n nt tn n n n

θ θθ θ θ θ+ +

+ + + +

= =+ +

[86]

Substrate

Ambient Medium

Di



0 2with cos 2 cos0

i

i

i

i i i i i i i ii

eT D n D n

e

δ

δ


= = =

[87]


( )1

01 1 12 2 1 1 1 1 10

N

i i i N N N i i ii


+ + minus minus +=

= = prod [88]


21 21

11 11

andp sp s

p s

S Sr rS S

= = [89]


( )21 11

11 21

tanp s iP

S p s

S Sr er S S

ρ ψ Δ= = = [810]


( ) a s s f f ff n n k n k eΨ = [811]


and

( )a s s f f fg n n k n k eΔ = [812]



2

exp2

1 exp

( ) ( 11 ( )

nj th j j

j j

X

n m

ρ θ ρ θσ

ρ θ=

minus=

minus minus part

[813]


2 2

exp exp

2 21 exp exp

( ) ( ( ) ( 1 1 ( ) ( )

n j th j j j th j j

j j j

X X

n m

θ θ θ θσ

θ θ=


[814]




is a vector


and thus m = 4 the




1 2 ( )mX x x x=






2 2 2 2

2 2 2 2 2 2

e h i hi

ie h i h

n n n nfn n n n

minus minus=

+ + [815]



Substrate



Local Maxima


Global MaximumF(x)

x





0 0 0 01 2 ( ) mX x x x=




X(k) such that

( 1) ( ) ( ) 01 2k k kX X X k+ = + nabla =

[816]

















1) the mid-point







5) the best vertex






822 The LM method




21( ) ( ) ( )( ) ( ) ( )( )2


























k x xδ += minus







1 1 11 11 (1 )





+= + + minus [823]











IML Module





















B1 150 05

B2 70 1

B3 150 1

B4 150 02


M1 120 04

M2 120 095

M3 150 07

Epoxy E1 125 4










Copper 1000


Silver 05








a)

b)





Y in

deg

rees

12

15

18

21

24

27

30


Ψde

gree

s



D in

deg

rees

-10

0

10

20

30


Δde

gree

s



Inde

x of

Ref

ract

ion

n Extinction C

oefficient k

10

12

14

16

18

000

010

020

030

040

050

060nk

n r

eal p

art

k imaginary

part



Rea

l(Die

lect

ric C

onst

ant)

e1

Imag(D


2

10

15

20

25

30

35

00

03

06

09

12

15

18e1e2

εrsquo re

al p

art

imaginary

part εrsquorsquo







500 1000 1500 2000 2500 3000 3500 4000

00

05

10

Abso

rptio

n



Wavenumber(cm-1)

Abso

rptio

n








Wave Number (cm -1)0 1000 2000 3000 4000 5000 6000

Abs

orpt

ion

Coe

ffici

ent i

n 1

cm

0

3000

6000

9000

12000

15000

M1

Wavenumber (cm-1)


Wave Number (cm -1)500 1000 1500 2000 2500 3000 3500

Abso

rptio

n C

oeffi

cien

t in

1cm

0

2000

4000

6000

8000

B1


Wave Number (cm -1)0 1000 2000 3000 4000 5000 6000

Abso

rptio

n C

oeffi

cien

t in

1cm

0

300

600

900

1200

1500

E1

Wavenumber (cm-1)










0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

del

ta



Wavenumber (cm-1)

0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)








0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)

0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)

0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)









0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

C

os Δ



Wavenumber (cm-1)0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)

0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0

-1 0

-0 5

0 0

0 5

1 0

Cos

Δ



Wavenumber (cm-1)0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0

- 1 0

- 0 5

0 0

0 5

1 0

Cos

Δ



Wavenumber (cm-1)






0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0

-1 0

-0 5

0 0

0 5

1 0

Cos

Δ



Wavenumber (cm-1)0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0

-1 0

-0 5

0 0

0 5

1 0

Cos

Δ



Wavenumber (cm-1)

0 1000 2000 3000 4000 5000

-10

-05

00

05

10

Cos

Δ

Longueur (cm-1)

M1QUARTZ M1Al

Wavenumber (cm-1)0 1000 2000 3000 4000 5000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)0 1000 2000 3000 4000 5000

-10

-05

00

05

10

Cos

Δ


M 3Q uartz M 3A l

Wavenumber (cm-1)

1000 2000 3000 4000

-10

-05

00

05

10

cosΔ


B1Quartz B1CuNi


-10

-05

00

05

10

Cos

Δ


B2QUARTZ B2CuNi

Wavenumber (cm-1)







0 1000 2000 3000 4000 5000

-10

-05

00

05

10

Cos

Δ


B3Quarz B3CuNi

Wavenumber (cm-1)0 1000 2000 3000 4000 5000

-10

-05

00

05

10

Cos

Δ


B4QUARTZ B4CuNi

Wavenumber (cm-1)

0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

del

ta


M1QUARTZ M1CuNi

Wavenumber (cm-1)0 1000 2000 3000 4000 5000

-10

-05

00

05

10

Cos

Δ


M3QUARTZ M3CuNi

Wavenumber (cm-1)








1000 2000 3000 4000 5000

00

05

10

Cos

Δ


B2 QUARTZ B2Si

Wavenumber (cm-1)1000 2000 3000 4000 5000

-05

00

05

10

Cos

Δ


B1Quartz B1Si

Wavenumber (cm-1)

0 1000 2000 3000 4000 5000

-05

00

05

10

Cos

Δ


B3QUARTZ B3Si

Wavenumber (cm-1)0 1000 2000 3000 4000 5000

-10

-05

00

05

10

Cos

Δ


B4QUARTZ B4Si

Wavenumber (cm-1)

1000 2000 3000 4000 5000

01

02

03

04

05

06

07

08

09

10

Cos

Δ


M2QUARTZ M2Si


-10

-05

00

05

10

Cos

Δ


M1QUARTZ M1Si

Wavenumber (cm-1)







1000 2000 3000 4000

0994

0996

0998

1000

Cos

Δ


E1Quartz E1Si

Wavenumber (cm-1)0 1000 2000 3000 4000 5000 6000

070

075

080

085

090

095

100

Cos

Δ


M3QUARTZ M3Si

Wavenumber (cm-1)

1 0 0 0 1 5 0 0 2 00 0 2 5 0 0 3 0 0 0 3 5 0 0

0 0

0 2

0 4

0 6

0 8

1 0

1 2

7 00 80 0 90 00 0

0 2

0 4

0 6

0 8

1 0

1 2

Ab

sor

ban

ce


B 1 B 2 B 3 B 4 M 1 M 2 M 3

Ab

sorb

an

ce


















84 Conclusions





9



91 Introduction




First Edition

















2 20


2 20

1 1( ) ( )2 2


21 ( )2

U U U kτ φ ω τ φ= + = Δ [94]































a) b)





932 Nanoindentation

















2 rdPS E Adh π

= = [96]





















for

( ) forY

Y t Y Y

EE

ε σ σσ

σ ε ε σ σle

= + minus ge [97]



phase2

12r

dPEdhA

π= [98]


phase2

dPdh

is the slope



22

(1 )1 (1 ) i



( )r y tP P h E Eσ θ= [910]





under

max max1

max

ih hh

minus le Δ [912]

2


























Parameters (degC)

(Gpa)

(Gpa)



FE simulation

Iteration 1 653 42 42 108881 4017 00867 997

Iteration 2 70 42 42 8759 1276 010845 1262

Iteration 3 70 21 42 904407 1643 0114 1838

Iteration 4 70 8385 21 85903 106 010256 65

Iteration 5 70 8385 315 846446 897 010098 486

Iteration 6 70 8385 42 828116 661 009858 237













95 Conclusions



Bibliography











First Edition

















Bibliography 277



























0 de 12C16O2 et ν3 de 14N2





Bibliography 279




























Bibliography 281
































Bibliography 283






























Bibliography 285































Bibliography 287






























Bibliography 289
































Bibliography 291





Index

A B C









126 131

D E


164ndash167 169 170 208 dephasing 162 163 165 166





57 74 88 949798 103 104 146


First Edition






F G


signal 145 179 189 190 192ndash196 212


H I









203 216 inverse method 133 135 208


J L M







Index 295





N O P


196 201 203 217 nanotubes 3 255 258 260 261




153ndash160 162 198 203 206 216



53 62 pulsed laser125 136 143 146

176ndash178 pump laser 144 145 148 149


pump-probe 134 173

Q R









S







T U V W



214






Other titles from

in










































Cover

Title Page

Copyright13

Contents

Preface

1 Uncertainties

11 Introduction


121 The MC method




131 Principle


14 The set approach







16 Conclusions


21 Introduction









245 Limit states






26 Conclusions


31 Introduction














41 Introduction







44 Conclusions


51 Introduction

52 Classical models

521 The Drude model








55 Conclusions


61 Introduction







64 The DR method








66 Conclusions


71 Introduction


721 Pump laser

722 Probe laser












75 Conclusions








822 The LM method







84 Conclusions


91 Introduction







932 Nanoindentation






95 Conclusions

Bibliography

Index


EULA13








Contents

Preface xi







16 Conclusions 23








26 Conclusions 44










Contents vii


44 Conclusions 93






55 Conclusions 130








66 Conclusions 170






75 Conclusions 216






84 Conclusions 253




Contents ix





95 Conclusions 274

Bibliography 275

Index 293

Preface








Preface xiii











1

Uncertainties


11 Introduction






First Edition







121 The MC method

1211 Origin


1212 Principle


Uncertainties 3




1214 Remark



1221 Principle





1222 Applications





Uncertainties 5


1223 Remark







[11]

x O

y F

[ ]2 2

2 2

156 22l 54 13l22l 4l 13l 3lmM54 13l 156 22l42013l 3l 22l 4l


[ ]2 2

3

2 2

12 6l 12 6l6l 4l 6l 2lEIK12 6l 12 6ll

6l 2l 6l 4l





[K] = b4middot[A]



[M]= b2middot[B]






Uncertainties 7









[12]


1232 Remark


( ) ( )j jj 0

X xinfin

=

ξ = φ ξ

Uncertainties 9




131 Principle













Uncertainties 11

















Uncertainties 13










Constraint

Den

sity





14 The set approach


Den

sity

Constraint

Constraint

Den

sity

Uncertainties 15



1411 Principle







( ) ( ) ( )( )( ) ( ) ( ) ( ) ( )( )

s s

u u

3

s s s u s s u s s u

s

3 3


u

1

1

x vx v





[13]



[ ] [ ]0 0000s u s u tx x v v = =


Uncertainties 17



Mean value 375 1000 15000 200000





1413 Conclusion


temps (s)

Deacutepla

cement

(m)

time (s)

Disp

lace

men

t(m

)D

ispla

cem

ent (

m)

Time (s)




1421 Principle





F(t) = 20 sin(80t)


Uncertainties 19



1422 Conclusion



stemps

Deacutep

lacem

ent (

m)

( )time

Dis

plac

emen

t(m

)


Dis

plac

emen

t (m

)

Time ( ) s






stemps

Deacutep

lacem

ent (

m)

( )time

Disp

lace

men

t(m


Disp

lace

men

t (m

)

Time ( ) s

Uncertainties 21





2 2 21 2 p + + + 1c c c = [15]



2 12 1 22 2 p2 p+ + +Y c X c X c X= [16]


2 2 212 22 p2 + + + 1c c c = [17]

REMARKndash








1 2 r

1 2 p

( +V + +V ) 90 ( +V + +V )VV

asymp [19]




Uncertainties 23





16 Conclusions


2



21 Introduction



First Edition






( )( ) ( )( )( ) ( )( )

1 1 1

2 2 2

min Under 0

0

t

t



[21]











2nd Limit state

1st Limit state

Feasable Region




u= T(x y)



( )min( ) 0

Tu uwith H x u

β =le

[23]


Normed space

ReliabiltyDomain

FailureDomain




NO

YES







Verify convergence

End




( ) ( ) TH HL u u u H x yλ λ= + [24]



[25]

( ) 0H

L H x uλ

part = =part

[26]



Hu

partpart

is not



Gy

partpart

[27]


02 =partpart+=

partpart

jHj

j uHu

uL λ

( )jkj u

uxTyG

uH

partpart

partpart=

partpart minus

1










[28]



+++=s

MMr

IIffcT ssrrPCPCPCCC

Cost

Pf

CT

Cf Pf

Cc







245 Limit states
















Bonding sheet

C-stage (Component)


C-stage (Component)


C-stage (Component)

Layer 1

Layer 2

Layer 3

Layer 4

Layer 5

Layer 6


Matrix layer

a) b)


Matrix layer

Warp fibers

Fill fibers

(a) (b)








a) b)

Layers

Stratified

Stratified

Designation

(a) (b)






[29]


Design Optimization




Structural Analyzis






4

4

(1 )(1 )

x y Fr x y f f m f

xy Fr xy f f m f


= + minus

= + minus




4 1

1fFr

m f

VMM V

ξηη

+=

minus [211]


1

f

m

f

m

MMMM

ηξ

minus

=

+

[212]

whereby






Copper






ρcu (kgm3) 8930

ρf (kgm3) 2750

ρm (kgm3) 1200

Vf () 10

Exf (Gpa) 725

Eyf (Gpa) 725

Gxyf (Gpa) 30

Em (Gpa) 26

Gm (Gpa) 0985

nf 02

nm 032






( )

4 4

4

4

1( )( )

objCu Cu FR f FR

FR pl pl

f FR Cu

F Xh V h

h N h

X V h h

ρ ρ

θ

= + = =

[213]




4 _ 4 4 _

_ _

01 04

0 90

f

FR m FR FR M

Cu m Cu Cu M

Vh h hh h h

θ


[214]










( )

( )

1

1 p

ip sel i N

ii

f Xi N P

f X=

forall isin =

[216]


max

max max

max

1 05 0 105

i b sel i s s


i f sel i s s


Chr C P P

λλ λ λ

λ


[217]







Np 50

λs 07

Iter 125

Pm 006

f1b 200




Values 1038 07542 200135 003736



a) b)



26 Conclusions



deg V f


h FR4

hcu




3




First Edition


31 Introduction









22

2 21 0uuc t


[31]

where 2 2 2

22 2 2x y z







cvn

= [32]












SurfaceA

( ) ( )P t I r t dA= [35]








2

2 0E EcωΔ + =

[37]


22


[38]


22

2 0kcωminus + =

[39]




0

E divE ρε

nabla = =

[310]

BE rot Et


[311]

0B divBnabla = =

[312]

0 0EB rotB jt


[313]





= minus

[314]


2 2

0 0 2 2 21 0F FF F


part part

[315]























divE ρε

=

and ErotB jt

μ ε part= +part

[317]



2 2

2 2 2 0cr rF FF F


part part

[318]




[319]

and


[320]


B rotA=

[321]

and

AE gradVt


[322]



V Vt

ψpart= minuspart

[323]

and by

A A divψ= +

[324]


0

AV divt

ρε


[325]

and 2



part part

[326]



0div A =

[327]


0

V ρε

Δ = minus [328]

and

2

0 0 02 21A VA j div


part part

[329]






21div 0VAc t

part+ =part

[330]


2

2 20

1 VVc t

ρε


[331]

and

2

0 0 02AA j


part

[332]








22 2 2 21 1 1





Lp mxx

part= =part

[334]


k k kH Eψ ψ= [335]


that 0

k kk

CΨ ψinfin

=



k kk

ψ ψinfin

=

Ψ = Ψ




H it


ˆ( ) (0)



iHtU t eminus=


ˆ ˆ ˆˆ( )k k kk



pρ ψ ψ= is the







22 2 2 2 2ˆ 1 1 1ˆ ˆ( )





ˆ ˆ2x ipa ω

ω+=

and ˆ ˆ2x ipa ω

ω+ minus=

[338]


1a a+ = [339]


ˆ ( )2

x a aω

+= + and ˆ ( )2



1( )2

H a aω += + [341]


1 1( ) ( )2 2

H n a a n nω ω+= + = + [342]





1 1a n n n+ = + + [343]


0 0a = [345]

( ) 0

nann

+

= [346]


and in









AE gradV E Et perp


[347]




[348]



0

( ) ( ( ) ( ) ( ) ( ))2 n n n nn

n


[349]


with


minus= and

3


Lα α=

[350]


2 3 2 2 2 30

1( ) ( ) ( )

2 2

Ni

ii





2 3 2 2 2 301

1( ) ( ) (( ) ( ) )

2 2

Ni

i coul Ni


= + + + minus [352]




2 2 2 30 (( ) ( ) )2









22 3 30 03

0

1( ) (2 ) ( )2 2 2 n

n n

A d r L aL α

α

ε εε ω

= [354]

and

22 2 3 3 20 03

0

1( ) ) (2 ) ( ) ( )2 2 2 n n

n n

c rotA d r L aL α

α

ε ε ωε ω

= minus [355]


[352] is written as

21

12 22

1

( ) ( )2

( ) ( )( )2

Ni

i coul NiN

i i i n n ni n n

mL x U x x


ωω

=

=

= + +

bull + minus

[356]


2 22field

( )( )2 n n n

n n

L a aα αα

ωω

= minus [357]



field

1( )2n n n

nH a aα α

αω += + [358]







21

1

1 ( ( )) ( )2

1( )2

N

i i i coul Ni i

n n nn

H p q A x t U x xm

a aα αα

ω

=

+

= minus +

+ +

[359]






2n

n α

ω






[360]

where

[361]

and

[362]






( ) ( )( ) ( ) ( )E r t E r t E r t+ minus= +


( ) ( ) ( ) ki tk k




[363]




0k

ki C k

infin

=

= [364]



( ) ( )( ) ( ) ( )moy




[365]



0

1 1k

d k kα α α απ

infin

=

= = [366]

a α α α= [367]



2 2

( )

k

P k ek

α αminus= [368]

2 2

0

ki

ke k re

kα θαα

infinminus

== equiv




2 2( )N kα α α=


meanρ α α= [369]




4



41 Introduction





First Edition



Type p wave 1 0

1 0

cos coscos cos

rp i rp

ri i r

E n nr

E n nθ θθ θ

minus= =

+

[41]

Type s wave 0 1

0 1

cos coscos cos

rs i rs

ri i r

E n nrE n n

θ θθ θ

minus= =+

[42]












( ) ( )0 0ˆ ˆ( ) yx i kz ti kz t


[45]



( )0 0ˆ ˆ( ) ( )yx i i kz ti


[46]



222

0 0 0 0


x y x y

E EE EE E E E

φ φ

+ minus = [47]








βα







00 0 0

0

ˆ ˆx

yx

y

ixii

x y iy

E eE e x E e y E

E e

φφφ

φ

+ = =

[49]


00 0 1 0 2

0

x

yx

y

ix ii

x yiy

E eE E e J E e J

E e

φφφ

φ

= = +

[410]


1

10

J =

and 2

01

J =

[411]


00 0 0 1

100

xx x

ii ix

x xE e

E E e E e Jφ

φ φ = = =

[412]



0

2 200 0

1 x

y

ixx

iy yx y

E eJJ

J E eE E

φ

φ

= = +

[413]




[414]



Jαα α αα α α

minus = =

[415]


112DJ

i

=

and

112GJ

i

= minus

[416]


cos( )

siniJe φ

αφ α

α

=

[417]





[418]



[419]


2 22 2cos ( ) sin ( ) 1a bE E

t ta b


[420]


( ) ( )1 12 22 2 2 2

0 0 0 0 0 0 0 02 sin 2 sin



= [421]


0 02 20 0

2 costan(2 ) x y

x y

E E

E E

ϕ Ψ = minus

[422]



( )2 2 22 2

0 0

0 0 0 0

1 ˆ ˆ ˆ2 2 2 2



[423]


0 02 20 0

0 02 20 0



tan 2 sin 2 tan

x y

x y

x y

x y

E EE E

E EE E

β α ϕ ϕ

α ϕ ϕ

β ϕ

= =+

Ψ = =minus

= Ψ

[424]





can be viewed as




5yE i= or 25xE =




with



10

01

1112

11

2 i

11

2 i minus

[425]





1 2 1 22 2 2 ( )n e n e e n nπ π πφλ λ λ










4 20 0 1 0 1 0 1 0 00 1 0 0 0 0 1 0

ii i

i

ee e

i eπ π

φ

φplusmn plusmn

minus minus [427]






0

1

2

3 ( )

x x y y

x x y y

x y y x

x y y x


lowast lowast

lowast lowast

lowast lowast

lowast lowast

+ minus = = + minus

[428]


0

1

2

3 ( )

x x y y

x x y y

x y y x

x y y x

E E E ESE E E ES


lowast lowast

lowast lowast

lowast lowast

lowast lowast

+ minus = + minus

[429]





0 0 90

1 0 90

2 45 45

3 D G


minus

+ minus = minus

minus

[430]


0

1

2

3


sin 2

S IS IS IS I

ββ

β

Ψ = Ψ

[431]



1 2 320

S S SVS

+ += )






0 1 2 3

1 0 1 0 0 1 0

0 1 0 1 1 0 0i

iσ σ σ σ

minus = = = = minus

[433]



2 2 2 20 1 2 3S S S S= + + [434]


2 2 2 21 2 3 00 S S S Slt + + lt [435]


0 00 01 02 03 0

1 10 11 12 13 1

2 20 21 22 23 2

3 30 31 32 33 3

s e

s es e

s e

s e



= =

[436]








4

1 1 1 1 1 10 0 0 0 0 01 0 0 02 2 2 2 2 21 1 0 0 0 0 0 1 0 0 0 0 0 00 0 2 2 1 1 0 0 0 1 0 0 0 00 00 0 0 0 2 2 0 0 1 0 1 10 00 0 0 0 0 0 0 0 2 2

ie

π

minus minus minus


1

n

ii

M M=

= prod [437]





1 0 0 11 0 0 10 1 1 00 0

P

i i

minus =

minus

[439]





1 0

2 0

3 0

cos 2 cos2sin 2 cos

sin 2

S SS S S

S S

ββ

β

Ψ = = Ψ

[440]


2

1

32 2 2

1 2 3

1 arctan( )2

1 arcsin( )2

SSS

S S Sβ

Ψ = + +

[441]






2β

2Ψ

Fixed ellipticity

Fixed

orientation









1

ˆ( )i ii

x=






21 22 23

31 32 33

R R RR R R R

R R R

=



( )i S u

uR e αα minus=

[442]


cos sin 0sin cos 0

0 0 1

izSR e α

α αα α minus

minus = =

where 0 0

0 00 0 0

z

iS i

minus =

[443]





2 ( 1)S s s= +

where s =1


and (1)1









parallel to the














S i x y zσ = =




2i j ij Iσ σ δ= [448]



x= plusmn and ( )0 1 0 1 2

yi= plusmn



i

x y z i

ee

ϕ

ϕ


minus = + + =

minus



0 cos( 2) 0 sin(( 2) 1

1 cos( 2) 0 sin(( 2) 1

i

i

e

e

ϕσ

ϕσ

θ θ

θ θ

= +

= minus [449]




1

ϕ

θ

x

z

y

0

σ

0x

1x

1y0y

σx

σz

σy





2

2

cos 2( )sin 2

i

i

eJ

e

φ

φ

αφ αα

minus

+

=

[450]

44 Conclusions






5



51 Introduction



First Edition








Reflected wave

Transmitted wave






52 Classical models



= + + [51]





22 22 31 2

2 2 2 2 2 21 2 3



[52]



2 2i

λλ λminus

(i = 1 2 3)



2202

( ) ( ) ( )( ) ( ) ( )e e ee e e e






2e

ed rmdt

is the force of

Newton 20e em rω



drmdt

γminus

is the



minus minus and

is the Lorentz



2 20

(0)( )( )

i t

ee

eE er tm i

ω

ω ω γω

--

=- -

[54]

2 20

(0)( )( )

i t

ee

i eE ev tm i

ωωω ω γω

minus

=minus minus

[55]




0( ) 1 ( ) 1 ( ) 1 ( ) ( ) ( )k ke k e ek k


[56]

and as a result

220

02 2 2 20 0

(0)(0)( ) ( ) (0)( ) ( )

i ti tp i te

e ee


ωωωε ω




[57]






220

2 2 2 20 0

(0)(0)( ) ( ) (0)( ) ( )

i ti tp i te

e ee


m i i

ωωωωε ωω σ




[58]



(0) (0)locE E=



0 0 (1 )D E P E Eε ε χ ε= + = + =

[59]


0 0Erot B micro Et

σ ε part= +part

[510]



2

0

n εε

= where 0

iε ε εε

= + and 2 2( )n n ik= minus [511]



or

2 2 2

n ε ε ε+ += and [512]

2 2 2




2 20

2 ( )( ) 1 ( )



0

2 ( ) 1( )( )



2 20

2 ( ) ( ) 1kn P dω ωω ω

π ω ω


0

2 ( )( )


infin

prime=minus [514]




2 20

( )2( ) ( )( )



prime minus and

02 2

0

4 2 ( ) 1( )( )




2 20

2 ( ) ( ) 1( )



0

2 2 ( ) ( )( ) ( ) low




521 The Drude model







2 2

2 20


ωεΔ Δ

= - - = = + = [517]


2 220

2 2 20

1 1 1 1 p pe

D D

N e mii i i



[518]




2

2 2 p

D

ωε ε

ω γinfin= minus+

and 2

2 2 pD

D

ωγεω ω γ

=+

[519]














220

2 2 2 21 1 pe

i ii iL i iL

N e mi i




20

2 2 2 2

epe

i ii iL i iL

N e mi i



++-

_

a

b




22

2 20

1 1 pr

L

ni

ωε


minus minus [522]


2 22 0

2 2 2 20 0

22 2 2 20 0

1( ) ( )

( ) ( )

pL

Lp

L



minus= +minus +

=minus +

[523]


ε(ω)

ω

εrsquorsquoεrsquo

ω0



2

2 20 0

1 11 2( )

pωε

ω ω ω= +

minus and

2

4 20 0

11 2( )

p Lω γ ωε

ω ω ω=

minus [524]

and for ω gtgt ω0

22 2

1 1 pL

ε ωω γ

= ++

and 22 2 1 L

pL

γ ωε ωω γ

= ++

[525]




20

220 0

12

12

p

L

ω ω ωεω ω ω

γ

minus= + minus+

and 2

220 0

22

12

p L

L

ω γεω ω ω

γ

= minus+

[526]





22 0

2 2 2 21 1

1N N

k e k

k kok k ok k

N e m fni i





22 0

2 2 2 21 1

1N N

k e k

k kok ok




2 2 2 22 22

2 2 2 20 01 1

11 ( )2 2N N

ok k okk k

e ek kok ok






n2 = A + B λ2 + Cλ4 [530]



n2 = Aprime λ2 + A + B λ2 + C λ4 [531]




0 0


lt gt= = + [532]


can be written that

0

0

1( ) ( ) ( ( ) ( )) ( )3

( )(1 3 )


N E r tN

α αε

αα ε

= = +

=minus

[533]


0

0

1 11 )2 31 3

rr

r

N NNα ε εε α


[534]






22 1 2

0 0 1

( ) 11 12 ( )r

e L

N N fenm i


minus minus [535]





0

( )T















4s3d4p

ATOME

E

EA EB

El

Eal

ΨA ΨBΨs

Ψas


4s

3d 4p

E







1 2 3( )a a aΩ = and


2

( ) ( ) ( ) ( )2nk nk nk nk

pH r V r r E rm

Ψ = + Ψ = Ψ

[536]



0 0 1 1 2 2 3 3k k G k h b h b h b= + = + + +











E

r

Ec

Ev

BC

BV

ELECTRONSLIBRES

GAP Eg

SOLIDE ATOMEISOLE

METAL

Ef

qK

εε K

Free electrons




2-y





Fen+ Fen+

T hν P

Etat BS

eg

t2g

S=12

Δeacutel

Etat HS

S=52

eg

t2g

ΔeacutelFe3+

3d6

eg

t2g

S=0

Etat BS

Δeacuteleg

t2g

S=2

Etat HS

ΔeacutelFe2+

3d6T hν P

BS State HS State

BS State HS State










ˆˆ ( )2 2




intˆ ˆ ˆ ˆ




ˆˆ ˆ2

H ESΩ= is





Ω= + + + + [540]


dagger dagger0 1


Ω= + = + + + + [541]

where dagger

0



ˆˆ ˆ ˆ ˆ ˆ( )2 2




12 2ˆ1 ( 1)

2 2

ac

i ja

c

n nH

n n

ωω

ωω

Ω + + Ψ Ψ =

Ω + + minus

[542]




cos( ) sin( ) 1

sin( ) cos() 1n n

n

n n e n g

n n e n g

α αα

+ = + +



and




( )1(0) 2



( ) cos( 1 ) sin( 1 ) 12 2



( )2 1( ) 1 ( ) 1 cos( 1 )2






0(0)champ n

nC nα

infin

=

Ψ = = (2

2

n

nC en




C n nα αinfin







( )0

1( ) ( ) 1 cos( 1 )2g

nP t p n n t

infin

=


ngP t =








intˆ ˆ ˆ ˆ



NH is the









22

intˆ ( ) ( ) ( )

2k k kk



0ˆ ˆ ˆ



2


[546]


2

int



[547]




2


[548]


v v32( ) ( ( ) ( ) )


= minus minus


)



v 32( )

8cρ ωπ

=v

v ( ) ( )( )


dSE E


At critical points



32

v 0 01

( ( ) ( )) ( ) ( )c g i i ii

E k E k E k a k k=




v2 ( ( ) ( ))i ci







intensity ( 20







gC Eα ω ωω

= minus









2

( ) gi T g T

E EE E A

Eε

minus gt =

[549]



0 2 2 2 2

0

( )( )

Li L

A E CEEE E C E

ε =minus + [550]




2

20

2 2 2 2 2 20

( ) ( ) ( )for( )

( ) ( )( )

for0

TL i T i L

gg L

g


E E E C EE E

ε ε ε= times


le

[551]



21 1 2 2

( )2( ) ( )g

TLTL

E

E P dE


infin




2

2

( )( ) g

FB

A E Ek E

E BE Cminus

=minus +

[553]








20

2 2 2 2 20

2

( )1( )

( )0exp

L gc

UTLu

cu

A E C E Efor E E

E E E C EE

A E for E EE E

ε


lt ltminus

[554]



2 2 20

2 2 2 2 20

20 0

2 2 2 2 20

2( )( ) 2 2 ( )( )

( )exp

( )

cu c g c c g

c c

gcu

u c c


AE C E EEAE E E C E



[555]




20

2 2 2 2 2 2 20

2

1

( )( ) ( ) for( ) ( )

( )for 0exp

gt

g pCL

tt

u


E E E E EE E

ε

minus= times ge


[556]


55 Conclusions





6




First Edition


61 Introduction


















(frequency nn

cνλ



22(1 )

lQr

πλ

=minus

R=100

PUMPING

AMPLIFYING MEDIUM

R=98 T=2

2L = nλ

Photons

ν = nc2L




between modes

150MHz2ncl

νΔ = =





and


dE


and is




[61]

where0

1 1 12 2

Nn χ αχε



lt iE


gt iE






( ) 2

0

1 1ee gg

a

e p gi

α ρ ρω ω τ

= minusminus minus

[62]


α centcent

e



( )2

2

2 20

1( ) 1ee gg

a



[63]


2 20

2 2i i

time

E cEdpP Edt


= minus sdot = =

[64]



=




0




( )0

0ee ggρ ρ- gt






se p g E e p gτ =

where Es is the


22

0 02

12 2

ss

cE cPe p g

ε ετ = =




















1 213

0

10 VmPEcSε

=











227951 227366

0 00 1 (1)

1 11 0 (1)

125710125841 125801

1 00 0 (1)

0 22 0 (1)

64310 64491 64451

0 11 0 (1)

0 00 0 (0)

1 00 0 (2)

0 33 0 (1)

1 11 0 (2)

ν3

(ν1+ ν23ν2)

Fermi resonance

(ν12ν2)

Fermi resonance

ν2

1 2

3

10 μm


128610128841 128801

203482 203398 203322

193034 193569 193440

188210 188 441 188401

137302136954 136981






64 The DR method







BEAMABSORBENT


COLLECTING LENS



LENS


LIGHT DETECTION








probepump

pumpprobe

Transmitted probe

pump

time

Transmitted probe

probeprobe

pump pumppump

time













Transmittedsignal

Reflected signal

Ultrafast Laser

AB











Dvj=0

vj=1

2

AvA=0

vA=1

0

Jmax

3

5

M0

n

4

D

vj=0

vi=1

vi=2vj=1

ASELASER

1

6

kr

vi=0

6















4 21 3643

iif if if

f



2 29

( 2)s gn nτ τ=

+




i f





k νν



the equation 1n1

hkTe

ν ν=minus



( )( ) n 1(0)

Nk Tk ν= + [66]



i f














22exp( 2 E)k π μ

μ αprop minus Δ

[69]













6 63 1 1 ( ) ( )

8 (2 )DA D ADA A D

P f f dc n R

ν ν νπ ν τ τ

= [610]







61

1445 Ao

N MCk kR














21 2 12 2 21

12 1 12 2 21 1 10

32 1 12 2 21 3

( ) ( ( ) ( )) ( ) ( ) ( )

( ) ( ( ) ( )) ( ) ( ) ( )

( ) ( ( ) ( )) ( ) ( ) ( ) p




ρ

ρ

ρ


[611]


2

2( ) exp( ) 2

tt δ σσσ π

minus Λ = minus

[612]


213

21

( ) ht N Nνργ

= [613]





[ ] ( ) 1 ( ) ( )2


[614]




2 2

0 0 0 02 2( ) E E PEt t t


[615]

















[617]





ˆ ˆ ˆ( )ik

ik jl jljl



ˆ ˆ ˆ



[ ]ˆ ˆ( ) ( )SBt Tr tρ ρ= [619]





SSS S S Sik


iL i Lt t




ˆ ˆ ˆ



ˆ ( ) ˆ ˆˆ ˆ ( ) ( )

ˆ ( )ˆ( ) ( )

s s


s s

s s s s s s s s

Sk k S S


Sk k S

k k k k k k k k

tt t

tt

i tt

ρρ ρ

ρω ρ




partpart

= minus + Γpart

[621]

where

( )

ˆ ˆ

ˆ

1ˆ ˆ ˆ ˆ( )2


s s s s s s



k k k k k k


k

k


prime prime prime



Γ = minus

Γ = Γ + Γ + Γ

and

2int2 ˆ (0) ( )


Bk k k k k k k k

k kk H E Eπ ρ δ


= minus



ρ =




ρ =



g eΨ = + is


written as 1 111 12

ρ =



1 12 2

g g e eρ = +


1 010 12

ρ =



11 12 1 11 12 11 12 1

21 22 2 21 22 21 22 2

1 2 12

2 1 21

0 00 0

0 ( )( ) 0

d idt

i


ε ε ρε ε ρ


[622]




= minus = +


12 hε εν minus

=


12 121 2

21 212 1

11 11

22 22

0 0 00 0 00 0 0 00 0 0 0

it



[623]


( ) ( ) ( 1 or 2)iki k ik ik

t i i kt


part

which lead to


= minus minus

and


= + minus





10

2

ˆ ( )ˆ ˆ ( )ˆ ( )

E tH H E t

E tε μ

μμ ε


[624]


12 121 2

21 212 1

11 11

22 22

0 ( ) ( )0 ( ) ( )

( ) ( ) 0 0( ) ( ) 0 0

E t E tE t E ti

E t E ttE t E t



minus

[625]





12 12

21 21

11 1122 22

0 ( ) ( )0 ( ) ( )

( ) ( ) 0 0

( ) ( ) 0 0

t tt t

it t t

t t




[626]

with


and where 0EμΩ =







66 Conclusions




7



71 Introduction



First Edition










monochromator

HUET

LASER CO2

Spectrometer

FTIR BRUKERIFS113v

(003 cm-1)

LASER

Nd YAG

YG 781C20

DYE

LDS 867

Li NbO3

1064 nm

532 nm

PULSED SOURCE

QUANTEL

870 nm5 ns 20 Hz

Δ σ=085cm-1

220-5000 nm

GeAu

HgCdTeor GeCu

COMPUTER

PRINTER


Preamplifier

HgCdTe

PUMP

PROBE

CaF2

CaF2

GeAu

2100 cm- -1

100 μJfilter

D

D

D

D

E

Lenses

KBr

D Diaphragm

E Retractable

PHOTOCHEMISTRY

532 nm

355nm

266 nm Trigg

er

mon

ochr

omat

or


721 Pump laser


722 Probe laser








-01

-008

-006

-004

-002

0

002

004

5 7 9 11 13 15 17 19 21 23 25Temps(ns)

Am

plitu

de (m

V)

FWMH = 48ns

-02

02

06

1

14

0 1 2 3 4 5 6

Temps (μs)

Ampl

itude

(UA

)

MCT Belova) GeCu b)

Time (micros)

Time (ns)FWHM 48 ns






-02

02

06

10

14

00 02 04 06 08 10 12 14

Temps (μs)

Ampl

itude

(UA

)

MCT SAT

-02

02

06

1

14

0 1 2 3 4 5 6

Temps (μs)

Ampl

itude

(UA

)

MCT SATa) b)






He liquide N2

liquide

77 k 77 k


vide vide

Faisceau issu du


Vanne micro fuite

Meacutelange gazeux

Filament chauffant



Heacutelium gazeux

N2 liquide

Heliumgas



HeliumJauge

Heating coil

Beam fromFTIR

spectrometer

Gasmixture

Pumpvacuum

Micro leakeagevalve

LiquidHe4Kva

cuum

vacu

um

LiquidN2

77 K

LiquidN2

77 K















( )12 6

4E rr rσ σε

= minus [71]








2 3 4 5 6 7 8 9 10-200

-150

-100

-50

0

50

100

150

200

-ε

reσ


V(r)

(cm

-1)

r (Aring)





Substitutional site







( )6 6

4 1gr grjj jj gr

jj jj

V rr rσ σ

ε = minus

[72]






Neon 307 12 06Argon 375 148 078

Krypton 401 164 09Xenon 431 17 108

vacantOxygegravene







12 63

2

1

1( ) 42

ij ijMj ij ij j Mj

i ij ij

V r Er r

σ σε α

=

= minus minus [73]




1


and

2ii jj

ij

σ σσ

+=


( )Stat



lt

= + [74]



j jj jj

G Fα αβ β

βξ = [75]










16O3 1278 1168 0532 -14 -07 21

13C16O2 116 180 0 -43 215 215

14N216O N-N1128 N-O1842

180 166 -30 15 15













-4

-3

-2

-1

0

1

2

3

4

5

6

0 50 100 150 200 250 300 350 400 450 500 550 600

Temps (ns)

Am

plitu

de(V

)


Laser pompe au GeAu





-5

-4

-3

-2

-1

0

1

2

3

4

5

0 50 100 150 200 250 300 350 400 450 500 550 600



Am

plitu

de (V



Fluorescence at MCT





1000 10005 1001 10015 1002 10025 1003 10035 1004 10045 1005



10027

10036


Am

plitu

de n

orm

aliseacute

e(SU

)N

orm

edA

mpl

itude

(AU



Frequency of the


1 0 1

2032120306

1381 0 2 0

6991

17114

0 1 0

0 0 0

1 1 0

ν1 +ν3

ν1+ ν2

ν2

10 μm


1003310027cm-1

0 0 11 0 0

ν1

ν3

1028810279

1097310966

2ν2

17884

0 1 1 ν2+ ν3

0 0 20 3 0

2ν3

3ν2

20914209032067

PUM

P

PROB

EPR

OBE

FLUO

RESC

ENCE





Time (microS)

Time (microS)







626cm-1

short(2) 595cm-1

long(3) 615cm-1

short(1) 627cm-1

short(2) 596cm-1

long(3) 614cm-1







a)

time (ns)

Am

plitu

de (V

))

b)

time (ns)

Am

plitu

de (m

V))

























a) b)





0

1

2

3

4

5

6

0 1000 2000 3000 4000 5000 6000 7000

Tem

ps (micro

s)

Dilution (ArO3)

a)

Tim

e

Dilution (ArO3)

0

20

40

60

80

100

120

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Dilution O3Kr

Tem

ps(micro

s)

b)

Tim

e (micros

)

Dilution (KrO3)





0

50

100

150

200

250

300

350

0 500 1000 1500 2000 2500

dilution XeO3=

Tem

ps(micro

s)Ti

me (

micros)

Dilution (XeO3)






-1

-08

-06

-04

-02

0

02

04

06

08

0 05 1 15 2 25 3 35 4 45 5

Temps (micros)

Am

plitu

de (U

A)

21micros

15micros

04micros

Laser Pompe

Signal P(26)

Signal P(42)

Time (micros)

15 micros

21 micros

04 micros

Pump Laser










max

min

328 exp( )( ) 1 exp d


B


σ

σ


[76]




is equal to 1 ( max

min1 ( ) diff

σ

σσ σ σ= minus )


max

min

22 3 21 1 8 ( ) d3 3

mif if if if

nS f Rn hc

σ

σ


[77]


0 ( )1 12 ( ) 2

ifmif if

t if

IS Ln d I

lN I lNσ σ

σσ σ

minus= =

minus [78]




SS

P= as

03

28

2 032110( )gaz

ifc Sντ

σrarr= [79]





2 29

( 2)s gn nτ τ=

+ [710]









with

relaxation1

1( )2

TcT

γπ

= and dephasing2

1( )2

TcT

γπ

=










22 3 21 1 83 3

ifeffif if

if

n Rn hc

σπσγ

+=

[713]



n l= The threshold


1Sif eff

if

Nlσ

Δ =


S Tif

NNΔ le (NT is







A) N2O (ArN2O = 2000) B) O3 (ArO3=200)

10 μm ν3-ν1 ν3-2ν2

775 μm ν1+ν2-ν2

17 μm ν2 manifold 31-20 31-22

20-11



9385 (a) 1051 (a) 014 014 288 026

1290 (a) 010 36

581 (b) 571 (b) 008 008 958 388

579 (b) 008 488

1011 (a) 1012 (a) 026 030 71


280 (d) 2900 21 (d) 215

12 09

80 200 6 15

150 12

20 24 021 05

C) CO2 (ArCO2 = 2000)





20-11



913 (b) 036 012 177

1018 (b) 012 067

625 (a) 626 (a) 009 0032 506 101

596 (a) 32 41

614 (a) 6135 (a) 009 0032 217 43


1230 400 140 115

950 270

15 27 2 09

7 22

37 65 5 2



21 2 3 2 1 2 2 1 2 int 2

12 1 3 2 1 2 2 1 1 1 0 1 int 1

32 1 3 2 1 2 2 1 3 1

( ) ( )

( )

dN ( ) dt

ra

ra



N N N K N A N K

minus minus

minus minus minus

minus minus


[714]











000E+00 400E+00 800E+00 120E+01 160E+01 200E+01

Temps(ns)

Am

plitu

de (U

niteacute

s arb

itrai

res)

x10 5

x10





a)

b)c)

d)

Am

plitu

de (A

U)

Time (ns)












μ prop minus μΔ






8 (2 )DADA A D

Pc n Rπ ν τ τ

=











75 Conclusions





8






First Edition






rs s is is s

E r E E re

E r E E rΔρ ρ ψ= = = = [81]




























i C P Ce Ai C P C


[82]

















11 12

21 22

( ) ( )( ) ( )


+ +

minus minus

prime = prime

[83]



1 1

1 1

111

i ii i

i ii i

rI

rt+

+++

=

[84]


1 1 1 1 1 1

1 1 1 1



i i i i i i i i

n n n nr rn n n n


+ + + ++ +

+ + + +

minus minus= =+ +

[85]

and

1 11 1 1 1


i i i ipi i si i

i i i i i i i i

n nt tn n n n

θ θθ θ θ θ+ +

+ + + +

= =+ +

[86]

Substrate

Ambient Medium

Di



0 2with cos 2 cos0

i

i

i

i i i i i i i ii

eT D n D n

e

δ

δ


= = =

[87]


( )1

01 1 12 2 1 1 1 1 10

N

i i i N N N i i ii


+ + minus minus +=

= = prod [88]


21 21

11 11

andp sp s

p s

S Sr rS S

= = [89]


( )21 11

11 21

tanp s iP

S p s

S Sr er S S

ρ ψ Δ= = = [810]


( ) a s s f f ff n n k n k eΨ = [811]


and

( )a s s f f fg n n k n k eΔ = [812]



2

exp2

1 exp

( ) ( 11 ( )

nj th j j

j j

X

n m

ρ θ ρ θσ

ρ θ=

minus=

minus minus part

[813]


2 2

exp exp

2 21 exp exp

( ) ( ( ) ( 1 1 ( ) ( )

n j th j j j th j j

j j j

X X

n m

θ θ θ θσ

θ θ=


[814]




is a vector


and thus m = 4 the




1 2 ( )mX x x x=






2 2 2 2

2 2 2 2 2 2

e h i hi

ie h i h

n n n nfn n n n

minus minus=

+ + [815]



Substrate



Local Maxima


Global MaximumF(x)

x





0 0 0 01 2 ( ) mX x x x=




X(k) such that

( 1) ( ) ( ) 01 2k k kX X X k+ = + nabla =

[816]

















1) the mid-point







5) the best vertex






822 The LM method




21( ) ( ) ( )( ) ( ) ( )( )2


























k x xδ += minus







1 1 11 11 (1 )





+= + + minus [823]











IML Module





















B1 150 05

B2 70 1

B3 150 1

B4 150 02


M1 120 04

M2 120 095

M3 150 07

Epoxy E1 125 4










Copper 1000


Silver 05








a)

b)





Y in

deg

rees

12

15

18

21

24

27

30


Ψde

gree

s



D in

deg

rees

-10

0

10

20

30


Δde

gree

s



Inde

x of

Ref

ract

ion

n Extinction C

oefficient k

10

12

14

16

18

000

010

020

030

040

050

060nk

n r

eal p

art

k imaginary

part



Rea

l(Die

lect

ric C

onst

ant)

e1

Imag(D


2

10

15

20

25

30

35

00

03

06

09

12

15

18e1e2

εrsquo re

al p

art

imaginary

part εrsquorsquo







500 1000 1500 2000 2500 3000 3500 4000

00

05

10

Abso

rptio

n



Wavenumber(cm-1)

Abso

rptio

n








Wave Number (cm -1)0 1000 2000 3000 4000 5000 6000

Abs

orpt

ion

Coe

ffici

ent i

n 1

cm

0

3000

6000

9000

12000

15000

M1

Wavenumber (cm-1)


Wave Number (cm -1)500 1000 1500 2000 2500 3000 3500

Abso

rptio

n C

oeffi

cien

t in

1cm

0

2000

4000

6000

8000

B1


Wave Number (cm -1)0 1000 2000 3000 4000 5000 6000

Abso

rptio

n C

oeffi

cien

t in

1cm

0

300

600

900

1200

1500

E1

Wavenumber (cm-1)










0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

del

ta



Wavenumber (cm-1)

0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)








0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)

0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)

0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)









0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

C

os Δ



Wavenumber (cm-1)0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)

0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0

-1 0

-0 5

0 0

0 5

1 0

Cos

Δ



Wavenumber (cm-1)0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0

- 1 0

- 0 5

0 0

0 5

1 0

Cos

Δ



Wavenumber (cm-1)






0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0

-1 0

-0 5

0 0

0 5

1 0

Cos

Δ



Wavenumber (cm-1)0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0

-1 0

-0 5

0 0

0 5

1 0

Cos

Δ



Wavenumber (cm-1)

0 1000 2000 3000 4000 5000

-10

-05

00

05

10

Cos

Δ

Longueur (cm-1)

M1QUARTZ M1Al

Wavenumber (cm-1)0 1000 2000 3000 4000 5000

-10

-05

00

05

10

Cos

Δ



Wavenumber (cm-1)0 1000 2000 3000 4000 5000

-10

-05

00

05

10

Cos

Δ


M 3Q uartz M 3A l

Wavenumber (cm-1)

1000 2000 3000 4000

-10

-05

00

05

10

cosΔ


B1Quartz B1CuNi


-10

-05

00

05

10

Cos

Δ


B2QUARTZ B2CuNi

Wavenumber (cm-1)







0 1000 2000 3000 4000 5000

-10

-05

00

05

10

Cos

Δ


B3Quarz B3CuNi

Wavenumber (cm-1)0 1000 2000 3000 4000 5000

-10

-05

00

05

10

Cos

Δ


B4QUARTZ B4CuNi

Wavenumber (cm-1)

0 1000 2000 3000 4000 5000 6000

-10

-05

00

05

10

Cos

del

ta


M1QUARTZ M1CuNi

Wavenumber (cm-1)0 1000 2000 3000 4000 5000

-10

-05

00

05

10

Cos

Δ


M3QUARTZ M3CuNi

Wavenumber (cm-1)








1000 2000 3000 4000 5000

00

05

10

Cos

Δ


B2 QUARTZ B2Si

Wavenumber (cm-1)1000 2000 3000 4000 5000

-05

00

05

10

Cos

Δ


B1Quartz B1Si

Wavenumber (cm-1)

0 1000 2000 3000 4000 5000

-05

00

05

10

Cos

Δ


B3QUARTZ B3Si

Wavenumber (cm-1)0 1000 2000 3000 4000 5000

-10

-05

00

05

10

Cos

Δ


B4QUARTZ B4Si

Wavenumber (cm-1)

1000 2000 3000 4000 5000

01

02

03

04

05

06

07

08

09

10

Cos

Δ


M2QUARTZ M2Si


-10

-05

00

05

10

Cos

Δ


M1QUARTZ M1Si

Wavenumber (cm-1)







1000 2000 3000 4000

0994

0996

0998

1000

Cos

Δ


E1Quartz E1Si

Wavenumber (cm-1)0 1000 2000 3000 4000 5000 6000

070

075

080

085

090

095

100

Cos

Δ


M3QUARTZ M3Si

Wavenumber (cm-1)

1 0 0 0 1 5 0 0 2 00 0 2 5 0 0 3 0 0 0 3 5 0 0

0 0

0 2

0 4

0 6

0 8

1 0

1 2

7 00 80 0 90 00 0

0 2

0 4

0 6

0 8

1 0

1 2

Ab

sor

ban

ce


B 1 B 2 B 3 B 4 M 1 M 2 M 3

Ab

sorb

an

ce


















84 Conclusions





9



91 Introduction




First Edition

















2 20


2 20

1 1( ) ( )2 2


21 ( )2

U U U kτ φ ω τ φ= + = Δ [94]































a) b)





932 Nanoindentation

















2 rdPS E Adh π

= = [96]





















for

( ) forY

Y t Y Y

EE

ε σ σσ

σ ε ε σ σle

= + minus ge [97]



phase2

12r

dPEdhA

π= [98]


phase2

dPdh

is the slope



22

(1 )1 (1 ) i



( )r y tP P h E Eσ θ= [910]





under

max max1

max

ih hh

minus le Δ [912]

2


























Parameters (degC)

(Gpa)

(Gpa)



FE simulation

Iteration 1 653 42 42 108881 4017 00867 997

Iteration 2 70 42 42 8759 1276 010845 1262

Iteration 3 70 21 42 904407 1643 0114 1838

Iteration 4 70 8385 21 85903 106 010256 65

Iteration 5 70 8385 315 846446 897 010098 486

Iteration 6 70 8385 42 828116 661 009858 237













95 Conclusions



Bibliography











First Edition

















Bibliography 277



























0 de 12C16O2 et ν3 de 14N2





Bibliography 279




























Bibliography 281
































Bibliography 283






























Bibliography 285































Bibliography 287






























Bibliography 289
































Bibliography 291





Index

A B C









126 131

D E


164ndash167 169 170 208 dephasing 162 163 165 166





57 74 88 949798 103 104 146


First Edition






F G


signal 145 179 189 190 192ndash196 212


H I









203 216 inverse method 133 135 208


J L M







Index 295





N O P


196 201 203 217 nanotubes 3 255 258 260 261




153ndash160 162 198 203 206 216



53 62 pulsed laser125 136 143 146

176ndash178 pump laser 144 145 148 149


pump-probe 134 173

Q R









S







T U V W



214






Other titles from

in










































Cover

Title Page

Copyright13

Contents

Preface

1 Uncertainties

11 Introduction


121 The MC method




131 Principle


14 The set approach







16 Conclusions


21 Introduction









245 Limit states






26 Conclusions


31 Introduction














41 Introduction







44 Conclusions


51 Introduction

52 Classical models

521 The Drude model








55 Conclusions


61 Introduction







64 The DR method








66 Conclusions


71 Introduction


721 Pump laser

722 Probe laser












75 Conclusions








822 The LM method







84 Conclusions


91 Introduction







932 Nanoindentation






95 Conclusions

Bibliography

Index


EULA13


Reliability of multiphysical systems set. Volume 2, Nanometer-scale defect detection using polarized...

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Transcript of Reliability of multiphysical systems set. Volume 2, Nanometer-scale defect detection using polarized...