Knittel Optics Of Thin Films Text

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KNITTL

OPTICS OFu

Z. KNITTL

WILEY


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Optics of Thin Films


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WILEY SERIES IN PURE AND APPLIED OPTICS

Advisory Editor

Stanley S. Ballard, University of Florida

allen and eberly Optical Resonance and Two-Level Atoms

cathey Optical Information Processing and Holography

caulfield and lu The Applications of Holography

francon and mallick Polarization Interferometers

GERRARD and burch Introduction to Matrix Methods in Optics

Hudson Infrared System Engineering

judd and wyszecki Color in Business, Science, and Industry, Third

knittl Optics of Thin Films

lengyel Introduction to Laser Physics

lengyel Lasers, Second Edition

LEVI Applied Optics, A Guide to Optical System Design, Volume I

louisell Quantum Statistical Properties of Radiation

moller and rothschild Far-Infrared Spectroscopy

pratt Laser Communication Systems

shulman Optical Data Processing

WILLIAMS AND BECKLUND Optics

zernike and midwinter Applied Nonlinear Optics


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Optics

of Thin Films(An Optical Multilayer Theory)

DR. ZDENfiK KNITTLMeopta Optics & Precision Mechanics Works, Prerov, Czechoslovakia

A Wiley-Interscience Publication


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96281CCESSION No.

CLASS No.

2 9 DEC 1978.

Published in co-edition with SNTL, Publishers of Technical Literature, Prague

Copyright 1976, by John Wiley & Sons, LtdAll rights reserved.

Nopart of this

book may bereproduced

byany means, nor transmitted,

nor translated into a machine language without

the written permission of the publisher

Library of Congress Cataloging in Publication Data:

Xnittl, Zdengk.

Optics of thin films.

(Wiley series in pure and applied optics)

A Wiley-Interscience publication.


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Contents

1. INTRODUCTION II

1.1 Historical and prefatory notes 11

1.2 The working hypotheses and the practical bearing of the theory 17

1.3 Wave equations of the electromagnetic field 201.4 The plane wave in a homogeneous medium 251.5 General types of solutions in non-homogeneous media .... 281.6 Transport of energy 29

1.7 Polarization states 32

2. OPTICS OF DIELECTRIC LAYERS 35

2.1 Notation and preparatory remarks 35

2.2 Boundary conditions in admittance notation 37

2.3 Fresnel's formulae 39

2.4 Input admittance at a boundary 40

2.5 Matrix treatment of reflection and transmission 41

2.5.1 Admittance matrix 41

2.5.2 Phase matrix 42

2.5.3 System transfer matrix 42

2.5.4 Evaluating the transfer coefficients from S 42

2.5.5 Interference matrix and evaluation of the transfer coeffi-

cients 45

2.5.6 Reduced matrix work R-coefficients only 462.6 Recursion formulae for multiple reflections 47

2.6.1 Generalized Airy summation 47

A l i 49


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CONTE

2.7 A recursion pattern for the input admittance. The Smith Chart2.8 Notes on numerical computations

3. BASIC DIELECTRIC DESIGN UNITS

3.1 The single layer

3.1.1 Amplitude and energy coefficients

3.1.2 Analysis of the energy formulae for normal and oblique

incidence

3.1.3 Reducing and increasing reflection by one layer. Observa-

tion in white light

3. 1 .4 The single layer as a prototype of monochromatic filtration

3.1.5 Theoretical principles of measuring the optical parameters

of thin films

3.2 Periodic structures

3.2.1 Fundamentals of typology and notations

3.2.2 A rough orientation in a HLH. . system3.2.3 Reflection at cardinal points


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CONTENTS

4. OPTICS OF METALS 182

4.1 Generalizing the Snell law. The inhomogeneous wave .... 1834.2 The invariants and Ketteler's relations 187

4.3 Orientation of the vectors E and H after refraction 1894.4 Fresnel's formulae on a dielectric/metal boundary 190

4.5 Poynting's vector in the refracted wave 192

4.6 The single metallic film I 93

4.6.1 Computing the outer field 195

4.6.2 Computing the inner field 200

5 GENERAL SYSTEMS OF LAYERS 2135.1 Absorption in a layer system 214

5.2 Potential transmittance and absorptance 218

5.3 Distribution of absorption across a layer system for the case of

weakly dissipative media 223

5.4 Total reflection 229

5.5 Frustrated total reflection233

6. GENERAL THEOREMS ON STRATIFIED MEDIA 2406.1 Left-and-right-incidence theorem 240

6.2 Reversibility theorem 241

6.2.1 Some conclusions from the reversibility theorem ... 245

6.3 Equivalence theorems - 4 ?

6.3.1 Computing the equivalent parameters for the basic period

(0.5B) A (0.5B) 2496.3.2 Analysing the spectral behaviour of an equivalent layer . 251

6.3.3 Stacking the basic period 255

6.3.4 Use of matching layers to manipulate pass-band ripple . 257

6.3.5 The stop-band 267

6.4 Theorems concerning induced transmission 270

6.4.1 Front-extension theorem . 270

6.4.2 Theorems on maximum potential transmittance .... 270

7. BASIC METAL/DIELECTRIC DESIGN UNITS 283


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CONT

7.1.4 The reflection etalon

7.1.5 Laser cavity

7.2 Dielectric-coated metallic mirrors

7.3 Some induced-transmission systems7.3.1 Pohlack's beam-splitters . . .

7.3.2 The induced-transmission monochromatic filter ....

8. SPECIFIC COMPUTATIONAL PROCEDURES8.1 Vlasov-Kard recursion formulae

8.2 Kard's dual expansions for r/d and 1/d

8.3 Kard's theory of translumination

8.4 Circle diagrams

8.5 Notes on non-polarizing beam-splitters

9. EXACT SYNTHESIS OF TUNED MULTILAYERS9.1 Principles of the procedure

9.1.1 Transformation of the formulas

9.1.2 Energy relations

9.1.3 Finding the a v ,b v9.1.4 Inversion of the N*\ NJ ?9.1.5 Some remarks about the matching methods

9.2 The antireflection problem

9.2.1 Butterworth and Chebyshev approximations of the low-

pass filter

9.2.2 Frequency transformations

9.2.3 Antireflecting a germanium substrate

9.3 The semireflection problem

9.3.1 Pohlack's expansion and its Chebyshev matching ....9.3.2 Tables of synthetized semireflectors

9.3.3 Oblique incidence

9.3.4 Simplified achromatism by Ak2 =09.4 Other cases of matching 4

9.4. The sloping edge in Lagrangian approximation

9.4.2 The pole-and-zero plot9.5 Phase-change-upon-reflection in terms of the zero distribution


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CONTENTS

9.7.2 The monochromatic LC-network analogy 413

9.7.3 The vanishing film analogy 416

9.7.4 The rationalized-thin-film analogy 4189.7.5 Insertion loss the analogy to optical transmission . . . 4219.7.6 Darlington's insertion-loss synthesis. Notes on the condi-

tions of realizability 424

10. INHOMOGENEOUS LAYERS 429

10.1 The WKBJ (geometrical optics) approximation 430

10.2 Some exact solutions 43310.2.1 Exponential layer 433

10.2.2 Linear layer 436

10.2.3 Notes on other solutions. Computer-aided solutions . . 437

10.3 The inhomogeneous layer as part of a layer system .... 43810.3.1 Generalizing the Fresnel coefficient for inhomogeneous

media in the approximation of geometrical optics . . . 438

10.3.2 The interference matrix of an inhomogeneous film . . 440

10.4 Transfer coefficients of single inhomogeneous layers .... 44410.4.1 Exponential layer 445

10.4.2 WKBJ layer 44510.4.3 Transition layer 446

10.5 A general method of integration 44810.5.1 Drude's formulae a first approximation ...... 453

10.6 Approximation by simple reflections. The hyperbolic profile 455

10.6.1 A relationship between the hyperbolic and exponentialprofiles. The method of simple reflections as a Fouriertransform 457

10.6.2 A reciprocity theorem on associated functions .... 45810.7 Numerical examples 459

10.8 Notes on the preparation of inhomogeneous layers 471

11. THICK LAYERS 480

11 1 Planeparallel glass blank with coatings 480


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CONTE

12. PERTURBATIONS ,

12.1 Dispersion of dielectrics

12.2 Spurious absorption

12.3 Surface roughness

12.4 Cumulative tint in lenses

12.5 Obliquity effects in lenses

12.6 Production errors

INDEX


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1

Introduction

1.1 Historical and prefatory notes

The anomalous behaviour of matter if it is made to exist in the form

of a thin film is an established phenomenon of modern physics which is

being exploited both in basic research into the properties of matter and

for industrial applications, once the anomalies are under control.Optics enjoys the historical priority of being the first branch of physics

to have been confronted with a thin film anomaly, in the form of coloured

reflection from soap bubbles, air wedges between glass surfaces and from

fatty patches on water. The earliest descriptions of these phenomena were

ndependently published in the 17th century by several natural philosophers:

Joannes Marcus Marci (1648), Robert Boyle (1663), F. M. Grimaldi (1665),Robert Hooke (1665). No explanation was given of these colours apart

from some vague speculations in terms of reflection and refraction.Thin film colours did not escape the attention of Isaac Newton (1704),

yet they eluded an explanation in terms of the corpuscular theory of light.

In fact, they remained the touchstone of optics until the beginning of the

19th century, when the foundations of physical optics were laid by Youngand Fresnel.

Interference of light waves was discovered to be the cause of the colours

of thin films, and the double-beam treatment of it on a plane-parallel layer


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12 OPTICS OF THIN

From the theoretical ideas of that time let us also recall Stokes' prin

of reversion (1849), used to account for the perfect blackness of the

spot in Newton's rings.However old, all these ideas have kept their importance in m

optical thin film theory. In fact, it was only lack of practical hints

prevented 19th-century physics from developing the concept of the

layer, and the corresponding methods of analysis. No technical meansknown of producing multilayers artificially and no multilayers were regis

in nature. The Fabry-Perot etalon of the 1890's was the first eleme

require the use of exact Airy summation to describe the profile of it

dinger fringes, but this was still a silvered monolayer rather thanthree-layer.

Not until 1917 can one discover a paper concerning a genuine

of plates of alternatively low and high refractivity. The object of the

was not a technical product, but a work of nature. This paper, by

a person than Lord Rayleigh, [1], gave an explanation of the spectral c

observed in the reflection of light on the covers of some coleopter

beetles, known to have laminar structure. We have essentially here

band theoryof dielectric multilayers, but, since no artificial stratification

of this kind could be produced at that time, the paper fell into obl

Stop-band theory was later redeveloped by various authors in various

The time was hardly ripe then for the application of single

An occasional observation (actually an unconscious duplication offindings by J. Fraunhofer and by Lord Rayleigh) of some atmosphericall

tarnished lenses led the optics manufacturer Dennis Taylor to the elabora

of artificial ageing of glass surfaces by etching, with the effect of re

unwanted reflection from the refracting boundaries. The interpretatio

of this effect was in terms of an intermediate refractivity step being intr

ed between air and the compact glass, as if the layer were acting as

one, with no interference involved. Strangely enough, it was noted

same time that the reflected light changed colour with the thickness

and that maximum effect was obtained with a slatey bronzereflection.

The British Patent No 29561, granted for this process in 1904,fi t trace of a would be commercial antireflection technique A


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INTRODUCTION 13

The real birth of thin film applications in optics had to wait until the

middle 1930's, when it was derived from the advances of vacuum technology

achieved within the framework of the electronics industry. The vacuum

evaporation process in a well-developed plant with an efficient pumping

system was discovered to be the best method of producing an interference

layer of specified optical properties. A. Smakula and /. Strong are cited

as independent fathers of the single antireflection layer in Germany and

the United States, respectively. Parallel to this, A. H. Pfund discovered

the reflection-increasing potentialities of evaporated high-index layers.

It was subsequently realized that one may evaporate more differentlayers in succession. This led, in the period 1937 -r- 1947, to the elaboration

of the first multilayer theories for antireflecting and reflecting systems as

well as for monochromatic interference filters. Pierre Rouard, Antonin

VaSicek and A. W. Crook, [2-4], may be regarded as the main contributorsto the generalization by recurrence of the Airy formulae to multilayer

systems, although a number of other authors have made various ad hoccomputations as well. Important design work of deep foresight, unfortunat-ely confined to patent literature, is due to Walter Geffcken, [5], who alsopioneered wet and gaseous-reaction deposition processes. A. G. Vlasov

seems to be the Russian classic of this era, [6].

Exacting technological requirements for the production of optically

effective and physically stable multilayers have caused the practical develop-

ments to be mostly confined to the laboratory phase. The manufacture ofthe Fabry-Perot type filter, which spread after 1945, may be regarded asa real commercial hit.

The Airy summation was also used by S. Tolansky in 1942 whenanalysing the Fizeau fringes in a silvered-wedge film.

Post-war development is characterized by increased confidence in therdle of optical multilayers, with corresponding efforts in the experimental

and theoretical spheres. Special evaporation plants were developed foroptical multilayers (with only a minority of firms adhering to additive wet

processes). Large scale research was undertaken into materials suitable forvarious spectral regions and satisfying the requirements of economical

manufacture and prolonged use of the films. Methods of controlled deposi-tion were developed, optical constants, porosity, structure, adhesion, etc.


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14 OPTICS OF THIN

predominant use of interference films in optical applications. Theory

also expected to help in interpreting measurements of optical const

in setting the production tolerances and in guiding the development

or modified systems.

Early milestone papers of this second era are, for example, the H

Dennison electromagnetic treatment of the reflection and transmiss

metallic interference filter, [7], the published thesis of F. Abeles, [8],

the first general treatment of stratified media in terms of the electromagnet

theory of light, and a paper by E. T. Welford, [9], introducing matrix c

tations different from those of Abeles. Pioneering design work was

undertaken by F. A. Turner. The first conference on Thin Films summ

by the Marseille University in 1949 showed how widespread and livelexperimental and theoretical activities actually were at that time, [10].

- In the subsequent years four basic text-books on thin films appe

each having its own method of approach and choice of material,supplementing each other in the demonstration that a self-sustained

discipline was being born. These were the works of H. Mayer, [11],

Heavens, [12], A. VaMcek, [13] and the advanced though less accessible

tise by G. V. Rozenberg, [22], (to be followed only much later by th

well-known books by H. Anders, [23] and H. A. Macleod, [24]).

From then on it would hardly be possible to follow the stormy de

ment in any balanced proportions. Reference to literature in the

book is kept at a moderate level, the goal being neither monograph-l

completeness, nor a historical look at every stage of the developm

At this point the author would like to make an apology to the effect

any shortcomings of documentation should be interpreted as his

ficient absorption of all the existing material rather than anything

discrimination.In the early stages, use of the theory was prevalently analy

interesting combinations, mostly the result of straightforward reaso

were mathematically studied and possibly improved in detail. Vector

mation often helped where general theory denied direct insight. Alt

thin film theory is a more compact piece of science than, for ex

geometrical optics, the final formulae to which it eventually leads

lend themselves to direct analysis in terms of the many design parame


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INTRODUCTION 15

made it easier to integrate the minority of the theorists with experimental

teams).

In the course of time real design theory began to grow, developing

new concepts and special procedures for various purposes. Analogies with

network theory were established, which made it possible to design some

analog computers and elaborate exact numerical syntheses. Large computer

programs were also written for automatic differential corrections.

The theory of optical interference films has by now developed into aspecific chapter of physical optics as regards its principles, but the design

aspects have formed it into a clean-cut technical discipline of about the same

importance for the science of optical filters as network theory is for electrical

filters. The author is tempted to apply a quotation from a network theory

classic to the effect that the science . . . is a beautiful blend of mathematics,

physics and engineering. This book on thin film theory will try to contribute

to the substantiation of the analogous belief that there is a beauty in only

skin deep.

It should be admitted that the interdependence between theory and

practice is much stronger in optics than in electronics, starting with thefact that the thin-film theorist cannot assemble his systems from commerc-

ially available separate parts with technical data printed on them. Theroad to a practical realization of theoretical ideas is more arduous here,which makes some of the practitionists sometimes think that enough has

already been done by the theorists to keep technology busy over the next

decade or so. This view can be justified if we admit that the discovery offascinating ideas may be postponed in harmony with the needs of practice.Alternatively, we may tolerate the thriving of some pure theory if the restof the science is flying at an operational height.

This is the philosophy adopted in the present work which is to beregarded as a text-book meant for advanced university courses or as areference book for those working in the field. The aim is to acquaint thereader with modern concepts and the procedures of that part of the electro-magnetic theory of light which is today applied to stratified media in optical

applications. A rough knowledge of Maxwell theory is assumed.Multilayers are regarded as elements affecting the propagation of

infinitely extended plane waves, none of the recent developments of wave


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16 OPTICS OF THIN

is presented in more or less established notations. There was more pedag

scope for the author in working through- therhwblved formalism of me

films, providing at the same time for a maximum of insight into thephysical situations occurring there. Some less well known results mfound in the sections dealing with the inner field. The fundamentals

inhomogeneous layers are presented in close connection with the forma

established for homogeneous multilayers.

Wherever possible in the intermediate stages of the presentation

applications are introduced in special chapters written from the view-p

of design. It was not the authors ambition to present anything like a com

design theory for thin films, which would call for a special volume, pos

written by several authors. It is hoped though that enough of this ma

has been incorporated to make the book well balanced between basicsapplications. The chapters on thick layers and on perturbations should

to this balance.

A substantial chapter from the other side of the balance is deto a type of exact synthesis, which has haunted the author's mind for

years. A number of unpublished results are presented here in full awareof the fact that their elegance surpasses their utility, at least in the fo

able future. Nonetheless, the general views obtained here, and the net

physical analogies, may deepen the understanding of thin film interferenas well as supporting the educational value of the book.

In addition to this, the equivalent of several papers or notes,

withheld from separate publication, is dispersed over other parts

book, (particularly in 2.6.5, 3.3, 3.4, 5.3, 6.3.4, 7.1.4, 11.3 and in Ch

it being left for the conversant reader to assess their usefulness

presentation of the particular subjects.

Finally a few words on how this book came into being. As isthe case, the nucleus was provided by personal notes piling up ove

years in industrial research, some of the results having been published

scientific journals. The occasion to lecture on thin films at the unive

later initiated a process of arrangement and growth resulting in two

mediate stages of condensation: Lecture Notes for the 1966 Czechoslo

Summer School and an enlarged version of these in English for mystay as guest lecturer at the Kungliga Tekniska Hogskolan, Stockholm.


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18 OPTICS OF THIN F

(i) the mass of the film is an optically isotropic medium, characterized

by its refractive index n, real for dielectrics and complex for m

(ii) two adjacent media are separated by a mathematical dividing p

where a discontinuous jump of refractive index occurs(iii) apart from this, a continuous dependence of refractivity on

coordinates is admitted. For practical purposes it is sufficient to con

er n to be dependent only on depth in the layer (so called no

inhomogeneity)

(iv) a layer is defined in space by two parallel dividing planes, as by

its lateral dimensions being practically infinite. The thickness o

layer is of the order of the wavelength of light

(v) the incident wave is considered as being plane, monochromatic

linearly polarized in one of the two basic azimuths, p and s,

respect to the plane of incidence

Let us at once mention some existing physical factors which are m

neglected, or considered additionally as a possible perturbation on

adopted simple model of stratified media.

These are:

a) polycrystalline structure of evaporated films, possibly causing

scattering or absorption of light

b) roughness of substrate and dividing planes, again leading to e

as by a)

c) anisotropy of film material due to structure or internal stress

d) dependence of structure and optical constants of films on their a

thickness (critical with some metals)

e) time-dependence of refractivity and thickness (ageing)

f) diffusion between adjacent materials, giving rise to internal transiti

layers

g) subsequent adsorption or oxidation, giving rise to external transiti

layers

h) inhomogeneity of films due non-stationary evaporation conditions

i) dispersion of optical constants of dielectrics

(always taken into account with metals)

To take account of a), b) calls for more work on the standard m

matical model with a corresponding increase of numerical work i


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INTRODUCTION 19

reflection is either treated in terms of geometrical optics or as diffraction

on very fine surface irregularities. The latter approach is of importance

for thin films, but so farit

has only been used forsingle

boundaries,to

account for anomalies in measured Fresnel reflectivities, or for single films,

to estimate measuring errors of ellipsometry. These problems are discussed

in 12.3. Neither surface nor volume scattering have been incorporated

into a complete multilayer theory.

In a simplified approach, volume scattering may be taken as the causeof slight energy losses in the collimated beams taking part in multiple

interference. Accordingly, an empirical damping factor may be introducedfor the amplitudes in the Airy summation formulae, leaving the refractive

indices real. This method will be used in sections 5.3 and 12.2 of this bookto account for some phenomena of practical bearing.

Although anisotropic media (point c) have long had their own matrixcaluculus [15], with a substantial recent generalization in [16], it can only

be used to compute the transformation of the amplitudes upon one tra-versal of the system, no account being taken of multiple reflections due to

internal Fresnel coefficients.

Items d) i) could be taken into account by the present theory ifthe actual optical parameters and the actual inhomogeneities of the systemwere available. Since a surgical look into a completed system is still an

exception (see [17, 18]), recourse is made to assumptions, and a best fit sithen sought between computed and measured results, [19].

The interpretation of the phenomenon listed under d) is beyond thescope of the macroscopic theory, being essentially a generalized theory

of dispersion. It has been intensely studied by the Marseille group of thinfilms and others, [20, 21].

Having established the working principles of the theory of thin films,

we may ask to what extent it may give valuable answers in physical researchand in practical applications. The situation differs between dielectrics andmetals.

In the optics of dielectrics, multilayer systems are analysed with anaccuracy which is adequate for subsequent photometric measurements.

Reliable predictions of stop-band or pass-band widths and positions aremade, as well as means being devised to correct long-wave or short-wave


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20 OPTICS OF THIN FI

square-topped) are designed and optimized with respect to losses. M

types of antiref lection layers have been computed for various band-width

and ripples. In addition, unorthodox demands of individual customersbe met, the development always starting with a computer analysis.

Some critical situations arise when the resulting reflectivity is m

to be zero or 100 percent (high-performance antireflection films,

mirrors). Photometry then raises higher demands than theory is able

meet with the underlying data. The problem is rather a technological

In the optics of metals, possibly combined with dielectrics, structura

effects cause discrepancies between theory and experiment due to uncer

inty of optical parameters (see above).Nevertheless, computations

useful at least to give a fairly good qualitative theory of the system u

study, which then guides the final refinement in the experimental ph

Also, some advanced design concepts in the optics of combined m

dielectric layers are due to theory alone (e.g. all the phenomena associate

with the concept of induced transmission).

Last but not least, the macroscopic theory -sometimes also c

phenomenological- defines measurable optical constants n, or r\ an

and yields formulae for their determination from suitably conduct

measurements.

1.3 Wave equations of the electromagnetic field*)

The theory of stratified media is concerned with the propagation

linearly polarized harmonic waves in systems of media which are

partwise homogeneous and/or normally inhomogeneous, i.e. their op

constants depend on the depth measured along the normal to the

In these cases the form of the waves is planar, but generally this is no

We therefore first assume the components of the electromagneticto have general phasor functions:

E(x, y, z) eJa * H(x, y, z) e

jtot

j = V( D

*) Systematic development of ideas called for this and the following se


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INTRODUCTION 21

The problem of finding various types of electromagnetic (linearly

polarized, harmonic) waves in various types of media is identical with

that of finding the phasorfunctions as solutions of differential equations

which result from Maxwell's equations.

Assuming the media to be at rest and without any electrical charge,

Maxwell's equations for the harmonic waves are (in Gaussian units)

div (eE) = div (/iH) = (1-la, b)

cud = -j H, curl H = j ^ IE (l-2a, b)

where c is the velocity of light and the complex dielectric constant

e = s - j (1-3)CO

is the well known formal expedient accounting for the conduction current

in (l-2b). (Any time we need to stress that a quantity is complex the horizon-

tal bar will be used. However, other quantities not marked in this way

may also be complex numbers. Complexity due to cumulation of layers

is occasionally marked by bold face types.)

For reasons of generality we shall first assume the material constants

to depend on all three variables:

e(x, y, z), n(x, v, z), a{x, y, z), e(x, y, z)

We then havediv (eE) = e div E +- E grad

Setting the left-hand side equal to zero and introducing the logarithmic

derivative leads to the relationship

divE = -Egrad(lne) (1-4)

Taking the curl of (l-2a) we have

curl. curl E = grad div E - AE = -j curl (juH) (1-5)c

where the right-hand side, with fi variable, must be performed as

( H) l H + d H


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22 OPTICS OF THIN F

In a similar way we obtain, quite symmetrically,

2

AH + ifiH + grad (H. grad In ^) + grad lni x curl H =c

These are the differential equations for the phasor functions in the

general case of inhomogeneity. It is only in this case that there is comp

symmetry.

The symmetry is in fact almost complete, because in the latterterms In fi alternates with In s or with In i. This symmetry vanishes

specialize for normal inhomogeneity:

Introducing the coordinate system in such a way that the layer surfare parallel with Oxy, the material constants may be functions of zAssuming in addition n to be constant (and possibly equal to unity)optical applications we have

(* )-E + ^-i + grad(,^^-) =2

A H + \ ipH + grad (In i) x curl H =c

Now let a linearly polarized E-wave be incident on the layer suwith its vector oriented normal to the plane of incidence, this plane

identified with Oxz.

(l-7a) then assumes the form

d2 E v 8

2 E V co2

_, . _-ri- + -rir + -r ^( z ) E y = ox az c

Ex , Ez being zero.Turning to the third term in (l-7b), it is evaluated for a norm

inhomogeneous layer as

JdH x dH z \ dln$ JdH x dH y \ din's'

\ Sz s 8x f ) ^ dz MJ \dy dz ) Mdz h ,

The second bracketed term vanishes, however, if we take E to be p


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INTRODUCTION 23

Writing (l-2a) out in full also,

we can infer from both equations

(l-7b) may therefore be written in component form as

a2

dx

d2 HI . 8 2 HT . co 2

5x2 dz

2 te \d z 8x J c 2

dx 2 dz 2 c2

pi(z)J7,-0 (l-8c)

Eqs. (l-8a, b, c) describe the harmonic field for a linearly polarized

wave of the so-called transverse electric (TE) type, traditionally also denoted

as s-polarized. We shall adhere to the latter term.

Considering the other basic polarizationstate, the /^-polarized, or

transverse magnetic (TM), wave, we have H = jH y and by the same argum-ent as above find

7 dy dy

The third term in (l-7b) is now evaluated as

Sine 8H

-J dz dz

and the equation reduces to

^ aX _ aini(z) 8H^ jgl Mz)H .dx 2 dz 2 Sz dz c

2 * w y

The third term in (l-7a) written out in full yields

(l-9a)


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26 OPTICS OF THIN

Introducing by analogy with dielectric media a complex refra

index n = t] jx such that the intuitive relationship k = k n (n

tive index as the ratio of phase velocities) is formally replaced by k(with no implication so far for the phase velocities), we see that

is fulfilled if

n = y/Ou)

which implies

ri

2 -x 2 = fis, t\x = JHfJL = af ii (1-1

where T is the period of the oscillations.From (l-16a,b) one can deduce

J}= y [V(* 2 + 4 * 2 T 2 ) + ] 0-17

This is a relationship between the optical parameters r\, x of a d

tive medium and its electro-magnetic material constants e, n, a. For

trics we simply have

n = n = VO0

The explicit importance of these relationships is very limited, b

they have only been verified to some approximation in the far infr

The problem is that one uses the constants e, \i, a measured with m

less static fields in phenomena involving very high frequencies.

The only inference we shall make from (1-16) and (1-17) is th

optical constants of metals explicitly depend on frequency T _1 ,with dielectrics this dependence is only implicit through e. The pra

consequence is that the measured values r\ and x are much more dispe

than those of n.

We thus come to another point of this discussion: adaptingby a condition of the type (1-13) leads to the introduction of som

parameters r\, x, measurable by purely wave-optical means, as will tu

in due course.


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INTRODUCTION 27

where d = x sin + z cos (9 is the distance covered by the wave in thedirection of its normal at a phase velocity

v = c\n

there being some amplitude attenuation exp ( 2nk _1 xd).The components of the complex refractive index t], x may thus naturally

be interpreted as refractive index proper and absorption index.

In order to make (1-1 3a, b) compatible with Maxwell's equations,some further conditions must be fulfilled in addition to (1-14). Taking (1-4)

with constant s, (l-13a) yields

div = - jk fi = n/2

In addition to Hs = we have also Es = 0, which is the well-known condi-tion of transversality for electromagnetic waves (established here for the

plane wave within the formalism developed for the />-polarization).

Taking now (l-2b) for (l-13a, b) we have

dH y dH y . ca __dz ox c

which in both components leads to

\JP = k if

depending on ft = + nil. Referring to (1-15) we may write

tf = l(\ . / - nS (1-20)

the latter version being important for optics where n = 1. Js is practicallyequal to n due to (1-15).

The sign problem is obviously settled if we express our result in the

form

H =n[sx ] (1-21)

We have thus shown all the essentials of a plane monochromaticwave in a homogeneous medium. It is described by the expression (1-19),


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28 OPTICS OF THIN

inhomogeneous waves, where such a parallelism is lost. Situations

kind are encountered when an absorbing medium is entered through a

surface at oblique incidence. We shall develop the corresponding generations in the section dealing with metallic films.

1.5 General types of solutions in non-homogeneous media

We shall now approach the solution of (1-8) and (1-9) in terms

method known as separation of variables. Taking ahint

from (1-12)can be written as a product of the type as#U(x) . V(z) exp (jcot ),

of course, U and V are the remaining exponentials, we try to writes-polarized wave in the normally inhomogeneous medium as

=jy =/ ; UWV(z)

Inserting into (l-8a) we obtain

/a 2 v a 2 -,,A tI a2 u w .+

rfis(z) v

J U+

r V =

\8z 2 c 2 7~ 8x2

which can be reconciled with the following two ordinary differential

tions

^ + J Az) V = G2 V, ^ = -G 2 U (1.2dz 2 c z dx

G being a constant.

Obviously, (l-22b) has asolution

U(x) = const . exp (jGx)so that

E, = *,V(z) exp (+jGx)

will be a correct phasor of the s-polarized wave, if V(z) is a solutio

^ + f (z) . V = 0, f (z) = ^- /i8(z) - G2


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INTRODUCTION 29

incidence angle of the wave being 6> . Equalizing the corresponding jc-func-

tions for a fixed z yields

G = n sin6> (1-26)cWith y (x, z) known, the magnetic components for the s-case are

determined from (l-2a) as

H. = ;--, HT = -- ^- (l-27a, b)JC0/I dz jCOfl ex

Turning now to the /^-polarization, we write the transverse magneticphasor as

*

H = JHy = WMX) V(2)Inserting into (l-9a), separation of variables is again obtained and

Hy = jf,v(z) - exp (+ jGx) (1-28)is a solution if v(z) satisfies the equation

The electric component is then determined from (l-2b) as

*.--A^. *. A^L 0-30a,b)jcoe oz jcoe ox

The second order differential equations (1-25) and (1-29) are thecardinal problems of the theory of inhomogeneous media. Their solution

will be presented in Chapter 10 for several types of inhomogeneity.

1.6 Transport of energy

The measurable quantity in optics is the radiation flux rather than theamplitude or phase of electromagnetic waves. Interference films are expected

to modify the energy fluxes rather than the wavefronts. We shall nowbriefly treat the corresponding concepts.

Electromagnetic theor d fi th energ fl th t d t


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30 OPTICS OF THIN

A theorem then states that for any kind of electromagnetic disturbaa balance exists between the rate of change of electromagnetic

stored inside a closed surface and the flux of the vector P across that suP is therefore interpreted as the radiation flux per unit of area normal

With quickly oscillating fields it is only the time-averaged values

are meaningful. We shall now deduce a formula for computing theaverage of (1-31) given a plane monochromatic wave in complex not

of the type (1-12). This is of course a symbolic replacement for th

part of itself, allowing easier manipulations provided all the operati

are linear. Such is not the case with (1-31) and we must write it outin order to see the correct result. *

Taking the field components generally as E(x, y, z) exp Qcot)

H(x, y, z) exp (jcot), we are interested in the time average at a fixed

in space of

P = _L [Re ( e j * + E* x H + E x H*)

16re

On the time average the first two terms vanish and we have

PAV

= ^Re(ExH*) '

which is a general formula for computing the average flux of energy

electromagnetic harmonic wave with given phasor functions E(x, y,H(x, y, z).

Let us first use this to compute the energy flux carried by a

travelling wave of the type (1-12), possibly propagating as a damped

homogeneous wave of the type (1-19). Using (1-21) we obtain


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32 OPTICS OF THIN

This formula indicates that the total (in other words: net) flux of

across a surface is given by the difference of the individual fluxes carri

by the travelling waves plus a third term which is due to interference

two waves.

In dielectrics with real n this term vanishes and we have the inturesult that the net flux is simply the difference (or algebraic sum)

two partial fluxes. One can always reason in this way in outer mediaare never metallic, and in dielectric films generally. The implications

metallic films will again be deferred to a later occasion, where the

of inhomogeneity of waves will also be taken into consideration.

1.7 Polarization states

So far we have only considered linearly polarized waves oscillatingthe same direction. This is actually the only case of importance as

energy detection is concerned. In this section we shall pay some atten

to this aspect of interference.

Let us consider two travelling waves E t and 2 differing in az

in the Oxy plane by /?. If E l is directed along the x-axis, we may writaverage Poynting vector of the two waves superposed as

- Re {[iY^ + S 2 cos P) + }S 2 sin j8] x571

x [-i\r|sin0 + \{*e\ + jftcos #)]}

where use was made of the notation Ej = \S^ etc.This is easily seen to equal

fc pAv/i + iW + -^ Re (*** * ) cos plFor two coherent waves with a path difference Az at a given

of space it holds that

S z = qS' l eJ x

, q proportionality


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INTRODUCTION 33

This is the interference term of the superposition considered.

It obviously vanishes for ft = n/2, which results in the following general

statement: no interference can take place between two mutually perpendicularlinear polarizations. As a consequence, any other type of polarization maybe decomposed in an orthogonal system of coordinates into two linearly

polarized oscillations with appropriate phase relationships, which may thenbe treated independently as far as detection of energy is concerned.

Having more oscillations of different polarization states, they aredecomposed in one orthogonal system and then each set of projections is

superposed by the pattern (1-37).

In thin film theory the common orthogonal system is automaticallydetermined by the plane of incidence and the corresponding normal. Wecompute sets of coefficients for the p- and s-polarizations which are eventu-

ally combined into a final result according to the polarization state of the

incident wave.

In most practical situations the incident wave is nonpolarized, which

means that it is a sequence of short eliptically polarized wave packets with

random orientation. The input energy may be regarded as equally distribut-ed between the p- and s-components. Correspondingly, the energy coeffici-

ents of reflection and transmission are

e* - y(e P + e s ) ** = y(* P + *.) (1-40)

When there are more thick substrate plates in tandem, their individualsurfaces having coefficients p' v ,p v , ...

p'% ,pl , ... t p , Tp , ... t' s , %l, ..., and if

the indicence is co-planar, then


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formula of instrumental optics is easily established by emerging in

dielectric medium (y = 0) and writing intensities for the energy flu

^1 + 2 = ^1 +3^2+ 2 7(^1^2)008 T^-dzJ

References 1

[1] Lord Rayleigh, Proc. Roy. Soc. (London), A 93, 565 (1917).[2] P. Rouard, Ann. d. Phys., 7, 291 (1937).

[3] A. VaSicek, /. Opt. Soc. Am., 37, 623 (1947), with later developments.

[4] A. W. Crook, J. Opt. Soc. Am., 38, 954 (1948).[5] W. Geffcken, German Patents BDP 902191 and 904357 (1954).[6] A. G. Vlasov et al, Prosvetlenie optiky, Gostechizdat, 1946.

[7] L. N. Hadley and D. M. Dennison, /. Opt. Soc. Am., 37, 451 (1947) and 38

(1948).

[8] F. Abeles, Ann. d. Phys., 5, 596 (1950).

[9] writing as W. Weinstein, J. Opt. Soc. Am., 37, 576 (1947).[10] special issue of Journ. de Phys. et Rad., 11, (1950).

[11] H. Mayer, Physik dunner Schichten I and //, Stuttgart, 1950 and 1955.

[12] O. S. Heavens, Optical Properties of Thin Solid Films, Butterworth's, London,

[13] A. Vasicek, a) Optics of Thin Films, (in Czech), Prague, 1956,

b) Measurement and Preparation of Thin Films, (in Czech), Prague,

c) Optics of Thin Films, North Holland, Amsterdam, 1960.

[14] G. Ross, Opt. Acta, 16, 95 and 451 (1969).

[15] R. Clark-Jones, /. Opt. Soc. Am., 31, 488 (1941).

[16] D. W. Berreman, /. Opt. Soc. Am., 62, 502 (1972).[17] J. M. Pearson, Thin Solid Films, 6, 349 (1970).[18] H. K. Pulker and K. H. Gunther, Vakuum-Technik, 21, 201 (1972).

[19] W. N. Hansen, /. Opt. Soc. Am., 63, 793 (1973).[20] P. Bousquet and P. Rouard, Appl. Opt., 3, 435 (1964).

[21] C. v. Fragstein and H. Romer, Z. Phys., 151, 54 (1958).[22] seeref. 1. in Ch. 2.

[23] see ref. 7. in Ch. 2.

[24] see ref. 29. in Ch. 3.


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2

Optics of dielectric layers

2.1 Notation and preparatory remarks

From the two possible ways of numbering a system of layers we choosethe one beginning at the left of a drawing (which usually, but not necessarily,

is the input side of the system), see Fig. 2-1.

1 I , ? k I

f>, t h*

^s ^-TT- Q

E 1L

^ ^^

n n 1 Z k

k+1

k+1

Fig. 2-1 Notation in a layer system

starting from the front.

For the left half-space we reserve the subscript 0, for the right k + 1

,

where k is the total number of layers. (There is very small risk of confusing

this k with the propagation constant k = Ink -1 of Ch. 1.) In many situa-tions n will stand for the refractive index of air (= 1), k + 1 for that of


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36 OPTICS OF THIN F

The incident wave gives rise to reflected and refracted waves in

the media except for the output half-space, where only the refracted

exists.

The characteristic parameters of right- and left-going waves in Fi

have the subscripts R and L, respectively.In dealing with the boundary conditions, we provide all the ent

quantities with subscripts identical with the order numbers of the respec

boundaries and with a directional subscript R or L. Thus E 1R is the elcomponent of the right-going electromagnetic wave, measured on the

boundary,


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OPTICS OF DIELECTRIC LAYERS 37

The orientation of the coordinate system with respect to the stack of

layers and to the plane of incidence is shown in Fig. 2-2.

The Oxy plane is usually placed in the first boundary, but any otherboundary would do as well, because we only work with phase differences.

All the cases of oblique incidence can be treated as /- and s-polariza-

tions only. This is no limitation in practice, since any state of polarization

may be split into these two cases in the orthodox way (see 1.7).In every plane electromagnetic wave the vectors {E, H, $} form a right-

handed trihedral. We shall orient these trihedrals in the incident and reflectedwaves in such a way that for normal incidence both the/- case and the s-case

yield the same result as regards the phases of the E-vector: the H-axischanges orientation upon reflection, and the E-axis does not. In the re-

fracted wave the orientation of the trihedral is identical with that of the

incident wave.

Note: Most of the literature on physical optics uses, for/>-polarization

in reflected light, a trihedral whose E- and H-axes are oriented negatively

with respect to the drawing in Fig. 2-2. The convention adopted here

follows a recommendation by Rozenberg [1] and has the advantage of

unifying some signs in Fresnel's coefficients. (See also [2]).

2.2 Boundary conditions in admittance notation

The theory of optical stratifications is based on repeated use of the

boundary conditions for a simple dividing plane between two media, the

coherent coupling between adjacent boundaries being effected by phase

shifts applied to the separated R- and L-waves. A concise form of theboundary conditions, possibly unifying Hasp- and s-polarizations under one

formalism, is therefore desirable. This purpose is met by the concept of

optical admittance, borrowed from similar problems of network theory.

Bearing in mind that only the tangential components enter into the

boundary conditions, we define the characteristic optical admittance of

a medium as the ratio of the tangential components of the magnetic and


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38 OPTICS OF THIN F

Specifying now for the/>- and s-polarizations, projecting accordingly*

into the boundary plane and using (1-2) we have (see Fig. 2-3)

, = * = -


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2.3 Fresnel's formulae

Let us first consider the v-th boundary in isolation, illuminated by anR-wave. We then have E^ L = and a simple algebraic elimination yieldsthe two formulae of the Fresnel type

E v l _ ^v-i 'v E vR _ 2j v _ 1 (2-4ab1'vtL Yv +i + Yv E vR Fv _i + Yv

By symmetry of the incident and reflected rays, the ratio of tangential

components E cL /E dR is the same as that of the actual complex amplitudes

^vl/^vr Th e formula (2-4a) therefore covers the classical Fresnel amplitudereflection formula at oblique incidence both for its p- and s-versions, (r vR ) p ,

0\r) s -

Things are somewhat more complicated in transmitted light: (2-4b)

is identical with Fresnel's ratio -polarization on the

righthand side of (2-4b).

To cover this feature generally for the transition of light between anytwo media of order numbers i and j, we define

(2-5)/cos 6>s/cos 6>j for p

iJ \1 for s

The Fresnel transmission coefficient in particular is then

1Y _'vr c v-i/v-r?

,

v(2-4c)

Writing out the full form of the Fresnel coefficients is left to the reader

as a problem.

Graphs describing Fresnel reflection and refraction at a boundary were

presented in [3] and [4].


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2.4 Input admittance at a boundary

Another useful concept of circuit theory which may be used infilms is the input admittance of a boundary. It is defined as the ratio of

tangential components of the total-field magnetic and electric intensities:

[XIy(v) = _^vS_

LE s

The order number of the boundary is now contained in the ubracket, thus expressing the cumulative character of this kind of admittance.

We also introduce the cumulative reflection coefficient r Rv) coveringinfluence of all the boundaries v, v + 1 . . . k, k + 1 so that

E vL = r^E, vR

Moreover, expressing the magnetic components by the electric

according to (2-2a, b), we may write

y(v) _ yv -iE vR Yy-xrR^Eyn _ 1 r. RE vR + rE vR v_1 1 + #>

from which

Y Y* ' r

v) = r:i ^7 ( f or either of the polarizations)

This formula shows that the cumulative reflection coefficient macomputed by a Fresnel type pattern, inserting the characteristic admittanc

for the input medium and the input admittance of the boundary forsecond medium. A method of recurrent calculation for the Y(v) willshown in 2.7.

Let us simply check for a single boundary: E vs = E v Yw = Yvr Rj = r vR as expected (see Eq. (2-4a)).

Note: The use of input admittance tp compute the transmission factor is lessvenient (see [5]), and is also of limited importance for dielectric systems where= l- e .


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2.5 Matrix treatment of reflection & transmission on a thinfilm system

2.5.1 Admittance matrix

Introducing the admittance matrix by the relations

v'-(w,) v -

-1

- ( _) v * v;1=1 (2 - 9)

the boundary conditions (2-3a, b) may be written as the matrix equation

v-'fc)-

v-() (2

- io)

or alternatively

We shall call the single-column matrices in these relations matricesof the separated field (separated in the sense that one knows the R- andL-components of the total field). Eq. (2-11) describes the transforma-

tion of these matrices upon penetration through a boundary from its rightface to the left one. The transforming matrix is accordingly called the matrixof refraction. Writing out in full one has with respect to (2-4)

(2-12)

(2-13)

Note: We have established Eq. (2-11) in such a form as corresponds to workingthrough the system from the right to the left, i.e. from what is usually the output towardsthe input. However, this does not prevent us from choosing the actual input on any endof the system ( 2 5 4)

W-i/ 1

2

(i + Yjr-.\, i-r,y,-_YV

foip or s

Cv-l/v

a

It is useful to know the determinant of this matrix. Obviously

detW D _ 1/D = YJY,.


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42 OPTICS OF THIN

2.5.2 Phase matrix

Writing Eq. (2-3a, b) for v and v + 1, we have the boundary condi

for two adjacent planes. Both sets of complex amplitudes E are not inde

ent, because the undulations on the two boundaries are coherent a

liable to the obvious phase transformation

-ft -)(&)where


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Let us start with a right-going input wave. We then have Ek+1L =0and a simple computation yields:

r R = |iL = 2I, fR = J%idL = i2l (2 -l 7 a, b)s 1.1 1R

where c 0/k+1 is the reduction for obliquity between the outer media intro-

duced by (2-5). We recall that

XI_ 'cos O J cos &k+1 /^-polarization

^o/k+i-Vi s-polarization

and that f, t are read as f pR , f sR or f pR , f sR , according to the admittances

used.

For the other case of the input wave E 1R = and we have

r L = 3Uk. =-hl t f L = _i^_ = hUS. det S (2-18a, b)E k + 1/R S H ^k+l/L 5l1

Considering S as the product of the W'j and ll's 1 and knowing the values oftheir determinants,

wefind

detS = F k+1 Fo I (2-19)

so that (2- 18b) and (2- 17b) yield the important relation

2 Ik+if- = n k+l cos0 k+ 1 .

Y R n cos 6>2 J k+1 r _ k+l^ ^k+l - (7.7C0k + 1 /0 *R ya R V- LK} )

This set of formulae uses all four elements of the system transfer

matrix S and describes the behaviour of the layer system with respect to theouter fields for both the possible cases of illumination (R, L). In oblique

incidence the basic linear polarizations are respected by taking the Yp

or Ysvalues in the refraction matrices WD/u+1 .

For further work it is of use to introduce the full notations for the

complex transfer coefficients.

We put

? = re\ i=t


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44 OPTICS OF THIN F

These values are extracted from r and i, given by (2-17) and (

by the usual manipulations with complex numbers, e.g.

r = 7(ff *) e 2Ja = f\t*

Following the discussion of Sect. 1.5 we obtain energy transfer

ficients, measured as the ratio of energy fluxes per unit area

ffR.L r R,L TR ~ *R>0

tln ' n k+1

It is sometimes useful to make a correction for the cross-sections

beam of finite aperture before and after refraction (those in reflec

remaining equal), Fig. 2-4. This leads to the definition of a new transmissiocoefficient which will be called normal:

_ cos@ t+1Si =

cos>

cos6> L cos@ k+1

It may be interpreted as measuring the ratio of the normal components

energy fluxes. It is only with the normal coefficient of transmission tha

energy balance p + t = 1 (both for the R- and L-directions) is sati(see problem 2-5).

There is yet another combination of the elements s^ which

a valuable result; we find that

|S| + s 12 s 2i = s z2

-biOtn arta

Mi'L Vl =

riw:rf.-HM : , '


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47/550

46 OPTICS OF THIN F

In particular we have

=mlt - Yq 1 Yg m22 + Yg m12 - Yp ^m^mu + 7o 'fynjz + Y^m^ + x Yp

1 m21

i = / Cos . , (2-27asame denominator

2.5.6 Reduced matrix work R-coefficients only

Working out the product of the 2x2 matrices in (2-16) orrepresents a considerable amounjt of work. However, for various re

one does not usually need the R- and L-coefficients simultaneously. A

turn out later, a number of symmetries exist among the energy coefficient

particularly when the media are dielectric. It is more for the sake of p

changes that one may have to compute the complete set of coefficients.

It is therefore useful to note that the number of operations m

reduced by one half if one sets out to compute the R-coefficients only.

is done byputting

E + 1L =0and carrying out the product (2-16) or

starting with the one-column matrix

/Ek +1>R \

v o ;

on the right of these equations, proceeding leftwards.

It is easily verified that the intermediate results will be one-colu

matrices from which E^ +1R may be factored out, and that the final

must be

()-()*from which we are able to evaluate f R and f R by (2- 17a, b).

In computer practice this is accomplished by starting with the

When one explicitly needs the L-coefficients, the layer systeminto the computer in reverse order.


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OPTICS OF DIELECTRIC LAYERS

2.6 Recursion formulae for multiple reflections

47

2.6.1 Generalized Airy summation

The matrix method may be regarded as a modern approach to thetheory of stratified media, having the advantages of succinctness and ge-nerality both for basic research and for numerical procedures. On the otherhand there have existed methods of computing the transfer coefficients ofa single dielectric layer since as early as 1833 (Airy). These were laterextended to include metallic films and arbitrary numbers of layers and haveserved as the basis for much of the theoretical work in the 1940's and 50's.The idea here is the summation of multiple reflections inside each layer anda recursion pattern for working across the system of layers. This methodforms the basis of the classical book by Vasicek, [6], and more recently ofAnders' book, [7]. It retains its importance for the future owing to the intuitiveapproach it offers for many practical situations. We shall derive the methodn a very short way again using the matrix notation, [8].

Let us divide the layer system according to Fig. 2-5 into two subsets Iand II, separated by a distance layer of phase angle


49/550

48 OPTICS OF THIN

Similarly

_C0v Cv/k+l

_*ir*iir e

SllSll e + S 1221 e 1 ~ r Il/lIR e

It should be emphasized that the r IR , r R etc. concern the subsystems

sidered isolated (in the actual embedding of the adjacent layer v consid

as a half-space). The formulae (2-29ab) thus reduce the computation

starting system to the computation of two simpler systems I & II. Apthe same procedure to I and II, respectively, a further simplification

obtained.

In practice this amounts to starting with a single layer either

left end or at the right end of the system, regarding it as subsystem

respectively. The transfer coefficients of this single layer are again giv

(2-29a, b), where I and II now stand for simple Fresnel boundaries,

We then take one by one further boundaries into consideration, regathem as the corresponding subsystems II or I, respectively. This is the

of the so called recursion method (see below).

The formulae (2-29a, b) may easily be developed in infinite geom

series

r R = r IR + * IR f IL i- IIR e-2j

(l + q + q2 + . . . .),

Ir =


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whole system by a formula resembling that for one layer (the v-th layer

singled out) with composite boundaries I and II. This is a useful way of

looking at multilayer interference in certain problems (see e.g. 3.4.2, 3.5, 5.2).

2.6.2 An algorithm for complete Airy summation

We shall first show how the recursion mentioned in the precedingsection is written out in full for two and three layers and then state (without

actually proving it) a general algorithm for writing the complete formulae

for r and t directly for any number of layers.

Having a two-layer according to Fig. 2-6 and subdividing it accordingly

for recursion from the back (which is simpler than from the front owing to

the identi ty (2-3 lb)), we obtain system I as a Fresnel boundary with thefollowing simplifications:

'il = r 1L = -r 1R = -r IR , t IR t lL - r lR r lh = 1 (2-31a, b)

Inserting this into (2-29a, b) we obtain for the two-layer

r iR + *i R e ' , 'ir'hr^ ,*, ,-> isr =

,

t R = -t \l-5ldi, b)1 + r 1R r I1R e m same denominator

The left boundary in subsystem II also being simple, we may immediatelywrite

r =r 2R + r 3R e

t_ r 2R'3R e (2-33a b),IR

1 + ^R^Re-2 ^ 2 ' * same denominator

Everything now being expressed in terms of the R-coefficients, we canomit this subscript entirely. Inserting (2-33a, b) into (2-32a, b) we have forthe two-layer

r R = l+2

,.+ 3

,.

+r \*+ t (2-34a)

tJih e i(


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50 OPTICS OF THIN

It should be noticed that the scheme presented in Fig. 2-6 calls fo

to be computed first during numerical work, since it is the only quantity

the start expressed entirely in terms of the known Fresnel coefficients.is why the process is called recursion from the back. For this purpos

would have been more convenient to have the films numbered from

back also, as is the case in some books. However, aiming at the ge

algorithm, it remains more advantageous to keep to our present system

This will only cost us some occasional renumbering when switching

a two-layer to a three-layer according to Fig. 2-7 and making a subdiviti

for a back-recursion:

Subsystem I is a simple boundary; subsystem II may be computed(2-34a, b) effecting there the changes

* r v+i. v = .2, 3 v ->

l9l+9ii + r 4e- a 1+ + > + r^r^- 2

r = + r 1 r 2r 4 e

2,( +w> + JV^e -2 * 3 + r a r 3 r 4 e- a,(

+ r 2 r 3 e- 2j * 2 + r 2 r A Q-2 '^ 1+ ^ + r 3 r A e~ 2iV3 + r 1 r a r 3 r 4 e- 2,( *

(

tiff -j(


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(ii) define the phase exponential of the odd type as being derived froma monotonic sequence of integers / t < l 2 < .

.

. / L (L being any odd

number between / and k + 1) and following the pattern

exp {2j[-

l3 + ... - J}(#, being defined by (i))

(iii) define the phase exponential of the even type as being derived froma monotonic sequence of integers // > l 2 . . . > 1$ (S being any evennumber between i and k + 1) and following the pattern

exp {2j[ h . + $ h . - ... -

i + q> 2 + ...


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52 OPTICS OF THIN F

2.6.3 Vector approximation of r

In many practical situations, particularly in the visible region ospectrum, the Fresnel amplitude coefficients r are of the order 0.20

0.30. Their products of increasingly higher order therefore tend to

One is lead to the idea of neglecting some terms in (2-36a, b). More

mation can be extracted from the reflection formula.

As a first approximation we have

rJ^Ti + r 2 e jM + r 3 e -KX1 + X2) + r k+i' -j(M + X2++Xk)

where the / are positive or negative quantities and together with the

nentials form a set of complex vectors in the Gaussian plane (Argand

gram). For brevity we now write x v for 2 v .The formula (2-37) is sometimes called approximation by si

reflections , the underlying illustration being given by Fig. 2-8. Howe

adhering strictly to the idea of simple reflections, one would have to reduc

by a factor < 1R f 1L , r 3 by t 1K t 2ii t 21 t 1L , etc -> in order to account for the Fre

attenuation of the emergingrays. This actually leads to a different appro

mation, which cannot be seen from our general formulae (2-36a,

Intuitively one would expect more exact results here, but the actual

is dubious. We shall adhere, as is customary, to the expansion (2-37).usually referred to as the &-ray approximation, although it is clear that w

layers the number of approximating rays is k + 1.The modulus squared T 2 is the object of interest in most applications.The expansion (2-37) has been of great use in the theory of mult

antireflection films: either one fixes the relative optical thicknesses by

integral ratios and obtains explicit solutions for the r v (and consequentl

for the v ) by annulling T at several points of the spectrum, or the n v aare determined by the technology available and the Argand diagram of

is run across the spectrum, adjusting the relative optical thicknesses in

a way as to keep T 1 small. Both methods will be used in the subsequechapters.


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The.validity of the expansion (2-37) decreases with the value of the

reflection coefficient approximated. About 20 percent may be taken asa generally safe limit, beyond which increasingly erroneous results would be

obtained, including the physically meaningless values T > 1 (for detailssee 2.6.5.)

2.6.4 Vector approximation of r/t

As a second step one may think of the next higher approximation.Since the binary products are contained in the denominator, the second

approximation would not yield a useful simplification. There is a betterway to follow.

Noticing thatj^2-35a) and (2~35b) have the same denominator, one is

naturally lead to form the ratio

rjt = (^/ 2 fk + i)_1

ex P [J(^i +


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can be written computing q, T 2 and B 2 , and possibly also 5, y and /?latter quantities will not be commented on here).

For example, let us take three reflection levels q = 0.33, 0.500.66 respectively, and design tuned (quarter-wave) stacks of the type

l( n i n 2) Tg k even


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in such a way, as to keep q at

= n[2, B 2 and T 2 increase sharplycompared to q, their mutual difference increasing slightly. The ratio between

B 2 , r 2 and q increases with the number of layers and with the value e(7t/2),as illustrated by the following Table 2-1.

Both quantities seem to converge with k to some ceiling values.

The same tendencies can be observed with tuned multilayers having

Table 2-1

Constant reflectivities e(7t/2) produced by increasing numbers of layers of decreasing

refractivity: comparison of approximations by B 2 and T 2 .

k 2 4 6 8 12 16 e

B2

r 2.42

.42

.45

.42

.46

.43

.49

.44

.49

.44

.50

.46.33

B 2

r 2.64

.62

.83

.75

.88

.77

.90

.77

.92

.77

.95

.78.50


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56 OPTICS OF THIN F

pronounced maxima outside of the region q> = rc/2 (see for examplecase (D depicted i Fig. 3-27).

A particular fact of practical importance covered by these rules imultilayer antireflection systems are well approximated by T 2 . We shallpresent a specific figure to this effect because the situation is qualitatively

illustrated by the curves in Fig. 2-9 if we imagine them to be plotted ove

interval n/2 < q> < n.A certain study of refractivity influence can be made by taking

preceding designs and multiplying the same number of high indices

low indices, respectively, by a constant factor. The reader is again refe

to Sect. 3.2.3 for the substantiation of the fact that this scaling does


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58 OPTICS OF THIN F

about the same degree of fit between the three functions q, T 2 ,described in the previous tuned cases. Analogous situations have

observed in irregularly detuned systems as well.

To look more closely at the dependence on configuration, one

devise the following test: a more or less arbitrary configuration of the Ar

diagram is chosen such that the tip of the v-th vector r v passes through

JDf

% f

so

so n*

7/ ' ^00

to \\ f/ /a*10

*s-*Vs180 060

b)


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origin, the remaining vectors r v+l , r v+2 , r k+l determining by their con-figuration F e jy and f 2 . If this Argand diagram is varied so that only x vchanges, then (2-37) changes only in its phase y, T remaining constant. Thisdoes not correspond to an actual crossing of the spectrum (because with Xall the x v must run proportionally to the optical thicknesses Av ), but itdefinitely is a configuration test in the following sense: we compute thetrue reflectivity q and B 2 for the various sets {x l ,x 2 ,.x v ..x k } with x vchanging and observe their possible deviations from the constant T 2 .

Fig. 2-12 shows several situations of this kind. A six-layer was chosenwith a formal sequence of indices 1/2 1 2 121/2 ensuringthat all the vectors r are equal. This facilitates the choice of the basic

configurations.

Figs. 2-12a, b illustrate how q and B 2 change with x 6 or x s within theframework of two basic configurations. The levels of constant F 2 are 0.11and 0.42 respectively. Drastic differences are observed between q and T 2and between B 2 and T 2 .

The third case (Fig. 2- 12c) was devised so as to keep T 2 constantly atzero value. This required changing two of the values x v at a time; x t and x swere chosen in the configuration shown.

Again, one is tempted to say that the ties between T 2 and g or B 2 couldnot be looser.

Each of the three cases was subsequently checked for spectral behaviourby taking a specific value of the variable x v and developing this combination{* , x 2 , . .

.

, x v , . . . , x 6 } across a range of phase angles in the constant tun-ing ratio. Conventional phase diagrams g(


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60 OPTICS OF THIN

as well as cut-on and cut-off edges of various steepness, are approxima

with sufficient accuracy for the overall shape of the curve to be adequa

described. The approximation F 1 - q is quite reliable in the desigmultilayer antireflection films. Broad-band semireflectors of up to 50 p

in q should be amenable to the approach T 2 -> B 2 with the chancesleast a good starting design is obtained for further refinement by

methods. All this holds almost independently for any number of

below 10, say, and for any technologically conceivable indices of refrac

within the range 1-6.

Fig. 2-13 Reflectivity diagrams

corresponding to case (c) of figure 2-1

2.6.6 Internal transfer coefficients

It is sometimes of interest to compute the internal field inside

the layers. Let us again have recourse to Fig. 2-5 and compute the

and left-going waves in the v-th layer, given a right-going wave at the

We shall use the transformation relations for subsystem I, borroE E Th


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Solving these equations for the required ratios, we obtain

vR_

.. EyR_

C

0v (f \

L 0vTj5 T7T7V 22 _ r R5 12îr E 1R | sI

E 'L _ C0v it _ t \Ov u, ,^,V. s ll r R s 2\)

^vL _ E vL(2-39a, b)

' ir E 1R | |

The two brackets may now be manipulated as follows:

(*22 Vll) = jil('lR*IL - ''iRÎL + r Rr II.) =

-'

i t 114-r n* r iL e

>v

\ _' ^r'il

Sj tfiRriL I 1 -t- - - I S t IIR e

where use was made of (2-18a), (2-22) and (2-29a).Further

(s' n r R - s 21 ) = ^(n - r 1R ) = s' n' lKt,Lrme

~ 2

_Zl-r IL r IIR e 2j *

where again (2- 17a) and (2-29a) were invoked.

Inserting these results into (2-39a, b) and interpreting by (2-1 8b)

Aîi I S |-1

as fji.1

, we obtain the two internal transfer coefficients as

1 _ * 'iRÎIR e V r~> ac\ u\Vrr - -; ^liT' *o,rl = rrj (2-40a, b)l--, L r UR e 2j * l-i iL-iiRe 2J ^

where abridged symbols were introduced, reading for example: t 0o RR - transfer from medium o to v, input R-going, result R-going . Similarly*Gv,RL-

Evidently, the formulae may be expanded in infinite geometricseries, again allowing an interpretation in terms of multiple reflections.

It is interesting to note that

*0v,RL = *0v,RR r iiR e v (2-41)

One would perhaps hesitate to set up this relation inside a layer intuitively.We shall often use it in Ch. 4.

The computed transfer coefficients refer to the left boundary of the


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62 OPTICS OF THIN

2.7 A recursion pattern for the input admittance. The SChart

. We shall now deduce a recursion formula for the computationinput admittance of a system of layers, which then determines r R by fo

(2-8). Since it is esentially a recursion from the back, we shall renumbethe layers and boundaries according to the Fig. 2-14.

Yjc,i

\ 1

M

1

1 1

*1 i

nmt

f(iml)

n s n,\ n.

Y >-Yo

Fig. 2-14 Notation in a layer syste

starting from rear.

Using the definition of input admittance (2-6) and the transform

properties of the interference matrix (see 2.5.5), we may immediately

y(v+l) _ Hy+l,S _ m2lE T s + Wt22 vS _E v+i,s mnE vS + m12 HvS

m.+21 ,

W22 y(Y)

i + Hhi. y (v >m,,

where the mik are elements of the v-th interference matrix. Obviously

m21m,

so that

= jy v tan v ,

Y( v+1 )

m,m,

= 1,nt.

m.= -^-tan


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To this end it is first normalized by introducing the reduced input ad-mittance

y v) _ *

so that

r w =1 _ VW 1 _(v)1 y

v(v> i - r

1 + ,W ' * 1 + r (v)

(2-43) then reads

where

y v+1) =y v y v-+1 1 .? (v)

p)_ J>(v) + j tan


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34 OPTICS OF THIN F

(-TT7J+ q

2 =1 + r 2

(r, x partwise const

These are two sets of orthogonal circles, which actually constitute

Smith Chart shown in Fig. 2-15.

The origin of the Opq system lies in the centre of the great circle,

real axis + Op pointing upwards, the imaginary axis + Oq pointingleft. The system Opq is not actually drawn.

The circles form a curvilinear coordinate system whose lines

labelled with convenient positive values of r and with positive and neg

values of x.

At this stage of the presentation we are evidently able to state

following: given a reduced admittance r + jx, we locate the point

in the curvilinear system and interpret its radiusvector p + jq in th

system as the complex refaction coefficient

r = r e 6 , where r 2 = q = p 2 + q 2 , 5 = arctan (q/p)

Practically, r is evaluated as the ratio of the vector OP to thelength of the diagram, which is the distance between the points r =r = 1 on the vertical axis x = 0. This must then be squared to obta

5 could be read on a linear scale (in degrees) on the circumference

the chart starting on the uppermost point and proceeding counter-clockwise.

However, the angles we can actually read on commercially available

are 180 - 5.Let us now suppose we know some y

(v) and wish to compute

(2-45b). We locate the point [r (v) , x (v) ] in the chart, interpret it as r(v)

and switch over from (2-45b) to its equivalent (2-46): the radiusvector

must be turned through the angle -2cp to obtain r(v)

. Reading the

nates of this new radiusvector in the curvilinear system then yields the

ponents of j;(v)

directly.

Renormalization by (2-45a) is a matter for the desk-calculator o


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D K> 20 30 40 50 60 70 80 90 1001 i l I i I i I t I i l i l

swrm CHAR

Fig. 2-15 The Smith Chart


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Table 2-2

Example of a form for computations with the Smith Chart,

unity* = 125 mm

V v J n v + 1 c v X 135 180 225

1.516

0.6593 r il x i 0.6593/0 0.6593/0 0.6593/0

1 2.3 1 Aj 0.1875 0.25 0.3125

r 'll x 'l 1.28/0.38 1.523/0 1.28/0.38-

1.6912 r 2 /x 2 2.164/0.64 2.575/0 2.164/0.64-

2 1.36 0.85 Au 2 0.1594 0.2125 0.2656

r' 2 lx' 2 0.64/0.64- 0.416/0.20- 0.446/0.21

0.5913 r 3l X3 0.378/0.378- 0.246/0.12- 0.264/0.123

3 2.3 1 A 3 0.1875 0.25 0.3125

r 3l x 3 0.585/0.8- 3.33/1.6 0.85/1.35-

2.3 r Ajx A 1.345/1.84- 7.66/3.68 1.95/3.10

4 1.0

u 0.193 0.2421 0.2868

8 -139.0 -174.3 -206.5

~r(mm) 77.8 100.5 94.2

r 0.623 0.804 0.753

e 0.388 0.644 0.567

exact values: d -138.1 -174.0 -206.5

e 0.382 0.656 0.574

To facilitate laying off the angles 2q> in the clock-wise direction,th i f id d ll l b ll d b f


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66 OPTICS OF THIN F

and the angle laid off is 4nu. Thus for example for A = A/4 weu = 0.25 at 2

(meas

in the unit layers) at normal incidence.

In the era of automatic computers it is only a pale reminiscence to

that this graphical procedure reduced the amount of work in compari

with the Airy summation by a factor of 10 and that a team of two

run a 7-layer in one phase angle in 1 5 minutes. However, the Smith

does retain its importance in various odd jobs of filter design, yielding

useful insight through the concept of admittance matching or mismatchin

Its use may be extended to metallic films, as described in [5]. We coourselves to remarking that the chart is already of great help in the

case of a dielectric/metal boundary, yielding immediately the com

Fresnel coefficient of reflection, if we regard the complex refractive ind

the metal normalized with respect to the adjacent dielectric as the

admittance. (See Problem 4-1.)

Note: An affiliated graphical method is based on the useof the exponential

admittance chart, where the logarithms of the complex admittance are plotted. It

scribed in [14] and sometimes referred to as the Kard calculator, [15].

2.8 Notes on numerical computations

A general analysis of the complete formulae for a multilayer iseldom possible. Such cases are confined to some special situations

encountered later in this book. Most of the insight into the behaviour

multilayers had to be computer-aided. Catalogues of useful sys

fitting various purposes and corrected to a lesser or greater degree,

been piling up in research laboratories during the last 15 years or s

proprietory reasons caused only a minor part of these results to be publish

Corresponding to subsequent spectral measurements, the independ

variable in these computations is the wavelength A, or the wave -number


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one of the layers regarded as unit layer. The phase angles in the rest of thelayers for any X are then proportional to the relative optical thicknesses of

these layers with respect to the unit layer. With varying

v which enter into the general formulae

and yields, for example, the function g((p). This is more general than theknowledge of a g(X), since it is only now that we can fix the optical thicknessof the unit layer (and thereby proportionally the thicknesses of the other

layers) and place the function in a specific spectral region. The correspond-ence rule is:

2%choosing nh = A, plot g(


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68 OPTICS OF THIN F

Problems 2-1 Write out in full the Fresnel coefficients by introducing the pr

admittances into (2-4a, c). Compare with formulae in other literatur

taking notice of a possible difference in the sign for r p . Verify that

& = we have r p = r s , while with the other trihedral orientationone has r p = s , although the same electric vector is describe

2-2 What would be the Fresnel Coefficients if the characteristic impedan

Z = Y~ l were used?2-3 Derive the amplitude coefficients for the magnetic vector in terms

either Y or Z.2-4 Verify that:

1 + r v R = f VR c v/v-i

where 9 are the energy coefficients of normal transmission.

2-5 Verify in detail the energy balance q + S = 1 for FresneFs coefficientsin oblique incidence.

2-6 Show that the product of any number of interference matrices re

the property of the single layer matrix that the main diagonal

are real and those in the secondary diagonal are pure imaginary.

2-7 Show that in building up the S-matrix, any partial product

WUW... has the property that the terms of the same diagonalcomplex conjugate.

2-8 Consider the simplifications ensuing from Problems 2-6 and

writing a simple program for dielectric films.

2-9 Using the property proved in Problem 2-7 for the complete S ma

s it = s 2 , s 12 = sti and tllie formulae (2-17, 2-18)show that f R+ 'r'l = 1 fR^R + ^l ^ (principle of reversibility for dielecfilms; see also 6-2.).

2-10 Factoring out the c v _ 1/v . t~1 terms of (2-12) in the product

in (2-16), develop the rule for building up the product

.(>(:> / = 1, 2,


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3

Basic dielectric design units

3.1 The single layer

3.1.1 Amplitude and energy coefficients

The formulae corresponding to Airy summation are mostly known.

With the simplifications already explained in Sect. 2.6.2, with subset II for

a two-layer, we have now for the monolayer

r t + i- 2 e- 2J * . Mie~* n i m= ; t = rr- (3-la, b)l + r^e 2 - l + ri r 2 e- 23

where all the coefficients are understood as R-coefficients of the p- or s-type;

e.g. (r 1R \ and (r 2R ) p , leading to (j-r),,.Sometimes formula (3.1b) is encountered in the literature without the

factor e-J,>

in the numerator. In this case the transmitted amplitude is mea-sured at the input boundary, which is somewhat less logical.

Another set of formulae, which is very convenient for a straightforward

analysis, can be obtained from the formula (2-27) involving the interference

matrix. After some arrangement we have

y t (Fo - Y2 )cosq> + j(Y Y2 - Y^singY^Yo + Y2 ) cos


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Some simple complex algebra applied to the formulae (3- la, b) yfor the energy coefficients

g = r * + r * + 2ri/

2 COS 2


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BASIC DIELECTRIC DESIGN UNITS 71

To investigate q k/2 let us first put Yt = Y or Y2 . Then after somereducing in (3-6b), q k/2 = Qx - (This is an obvious result, particularly fornormal incidence, because the condition n x = n or n 2 means the amalgama-tion of the layer with one of th e embedding media.) Further, for Yt =

= \Iy Y2 , q %i2 = 0. Since V Y Y2 must lie between Y and Y2 , and since Qn/2is a continuous function of the admittances, we may summarize these

results into the following statement:

Yo^Y 1 %Y2 **0^ Qn/2 < Qn (3-7a)or in words: a monotonic sequence of admittances leads to a reduction of the

original bare reflection.

The opposite of condition (3-7a) is that Yx lies outside the intervalbetween Y and Y2 . We then expect

Y ^Y l %Y2 ^ Qlc/2 > Qn (3-7b)

or in words: a symmetrical combination of admittances leads to an increase

of the bare reflection.

These situations are termed as reflection-reducing or reflection-

increasing layers. Another way of putting it is antireflection and semireflection

00

90

80

f% 10.0

60\ / \

70k \ \ I \ \

60

/^\\\ I /~\\\503.0 \ \l // / \\|

W / ~\\\

302.35 \\l

202.0 \v|

10

8


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72 OPTICS OF THIN F

(both ways being later extende4 even to multilayers with a similar

slightly enhanced effect).

Outside of the cardinal points g(


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BASIC DIELECTRIC DESIGN UNITS 73

through equality to its opposite. Now having both inequalities changedin (3-7a, b) we restore the original condition of the admittances being

monotonic or symmetrical (although in a different way than the v , every-thing being reversed). The layer then again behaves as in normal incidence.

Fig. 3-2a, b shows the regions of antireflectio

Knittel Optics Of Thin Films Text

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