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Faraday Modulation Spectroscopy

Theoretical Description and ExperimentalRealization for Detection of Nitric Oxide

Jonas Westberg

Department of PhysicsUmea University, SwedenDoctoral Thesis, 2013

Institutionen for FysikUmea UniversitetSE-901 87 Umeawww.umu.se

ISBN 978-91-7459-616-8

Faraday Modulation SpectroscopyTheoretical Description and ExperimentalRealization for Detection of Nitric Oxide

Jonas Westberg

Doctoral Thesis, 2013Department of PhysicsUmea UniversitySweden

Department of Physics

Umea University

SE-901 87 Umea , Sweden

In collaboration with the Industrial Post Graduate School

c© Jonas Westberg

Front cover by Jonas Westberg

ISBN 978-91-7459-616-8

Electronic version available at: http://umu.diva-portal.org/

Printed by Print & Media, Umea University

Umea, Sweden 2013

”I am no poet,

but if you think for yourselves,

as I proceed,

the facts will form a poem in your minds.”

— Michael Faraday

Abstract

Faraday modulation spectroscopy (FAMOS) is a laser-based spectroscopic

dispersion technique for detection of paramagnetic molecules in gas phase.

This thesis presents both a new theoretical description of FAMOS and

experimental results from the ultra-violet (UV) as well as the mid-infrared

(MIR) regions. The theoretical description, which is given in terms of the

integrated linestrength and Fourier coefficients of modulated dispersion and

absorption lineshape functions, facilitates the description and the use of the

technique considerably. It serves as an extension to the existing FAMOS

model that thereby incorporates also the effects of lineshape asymmetries

primarily originating from polarization imperfections. It is shown how the

Fourier coefficients of modulated Lorentzian lineshape functions, applicable

to the case with fully collisionally broadened transitions, can be expressed in

terms of analytical functions. For the cases where also Doppler broadening

needs to be included, resulting in lineshapes of Voigt type, the lineshape

functions can be swiftly evaluated (orders of magnitude faster than previous

procedures) by a newly developed method for rapid calculation of modulated

Voigt lineshapes (the WWA-method). All this makes real-time curve fitting

to FAMOS spectra feasible. Two experimental configurations for sensitive

detection of nitric oxide (NO) by the FAMOS technique are considered and

their optimum conditions are determined. The two configurations target

transitions originating from the overlapping Q22(21/2) and QR12(21/2)

transitions in the ultra-violet (UV) region (∼227 nm) and the Q3/2(3/2)-

transition in the fundamental rotational-vibrational band in the mid-infrared

(MIR) region (∼5.33 µm). It is shown that the implementations of FAMOS

in the UV- and MIR-region can provide detection limits in the low ppb

range, which opens up the possibility for applications where high detection

sensitivities of NO is required.

v

Sammanfattning

Faraday modulationsspektroskopi (FAMOS) ar en laser-baserad spektroskopisk

detektionsteknik, som bygger pa dispersion, for detektion av paramagnetiska

molekyler i gasfas. I den har avhandlingen presenteras bade experimentella

resultat fran det ultravioletta och det infraroda omradet samt en ny teoretisk

modell av FAMOS, vilken uttrycks i termer av integrerad linjestyrka och

baseras pa Fourier koefficienter av modulerade dispersions- och absorption-

slinjeformer. Dessa koefficienter kan, under tryckbreddade forhallanden, skri-

vas i termer av analytiska uttryck, vilket forenklar den teoretiska beskrivnin-

gen avsevart som harmed ocksa inkluderar ett asymmetriskt bidrag fran

polarisationsimperfektioner. Under andra tryckforhallanden, vid vilka ocksa

Dopplerbreddning maste tas hansyn till, kan dessa koefficienter, som da ar av

Voigt-typ, effektivt beraknas med hjalp av en, av oss, nyligen utvecklad metod

(WWA-metoden), vilken ger upp till 1000 ganger snabbare berakningstider

jamfort med tidigare berakningsrutiner, vilket mojliggor kurvanpassning i

realtid.

Tva experimentella uppstallningar for detektion av kvaveoxid (NO)

med hjalp av Faradaymoduleringstekniken ar beskrivna. Dessa riktar sig

mot kvaveoxidovergangar fran de overlappande Q22(21/2) och QR12(21/2)

overgangarna i det ultravioletta (UV) omradet, samt franQ3/2(3/2) overgangen

i det fundamentala rotationsvibrationsbandet i det infraroda (MIR) omradet.

Experimentella observationer visar att de tva Faradaymoduleringssystemen

tillhandahaller detektionsgranser i det laga ppb omradet, vilket mojliggor

framtida tillampningar av tekniken for kanslig matning av kvaveoxidkoncen-

trationer.

vi

Preface

This doctoral thesis is the result of the experimental and theoretical research

I have been involved in during the last years in the Laser Physics Group at

Umea University. This project was initiated in the fall of 2008, initially with

the intention of investigating the characteristics of a then relatively new type

of laser, the quantum cascade laser (QCL), and its future use in wavelength

modulation spectrometry (WMS). The project, however, gradually progressed

towards an alternative detection technique based on magnetic modulation, the

Faraday modulation spectroscopy (FAMOS/FRS) technique. While awaiting

procurement and construction of suitable instrumentation, we realized that there

were benefits of reconstructing the theoretical base of the FAMOS technique in the

same way as had previously been done for WMS, opening up for rapid calculation of

lineshapes suitable for real-time curve fitting. This proved to be a time-consuming

process running in parallel to the experimental work, which also came to include a

fully-diode-laser-based UV laser system producing laser light at wavelengths close

to 227 nm. Finally, the work resulted in eight articles: five of which concern the

theoretical description of FAMOS and in particular the derivation and application of

Fourier coefficients, two regarding the strong electronic transitions in NO targeted

by the UV-laser, and one concerning a mid-IR QCL FAMOS system used to target

the most sensitive transition in the fundamental rotational-vibrational band of NO,

the Q3/2(3/2) transition.

This thesis is conveniently divided into chapters of which the first three introduce

basic spectroscopic concepts and descriptions of the theoretical principles behind

the Magnetic rotation spectroscopy (MRS) technique for the case of a static external

magnetic field. In Chapter 4 a modulation of the magnetic field is introduced,

which leads to the Faraday modulation spectroscopy technique. Here it is shown

how the FAMOS signals can be expressed in terms of Fourier coefficients and

the optimum pressures and magnetic field strengths are analyzed. Chapter 5

derives analytical expressions for the Fourier coefficients under pressure dominated

conditions and Chapter 6 introduces a new method for rapid calculation of their

Voigt counterparts. The last three chapters comprise the experimental setups,

vii

the results and the conclusions, respectively. Lastly, an appendix that includes

details regarding definitions and concepts that are useful for the reader is given.

All articles included in this thesis are appended and labeled by roman numbering

as Paper I-VIII.

Jonas Westberg

Umea, March 25, 2013

viii

List of publications

This thesis is based on the following publications:

I. Quantitative Description of Faraday Modulation Spectrometry in Termsof the Integrated Linestrength and 1st Fourier Coefficients of the Modu-lated Lineshape Function

Journal of Quantitative Spectroscopy & Radiative Transfer, Vol. 111,pp 2415–2433, 2010.

J. Westberg, L. Lathdavong, C. M. Dion, J. Shao, P. Kluczynski, S.Lundqvist, and O. Axner

II. Faraday Modulation Spectrometry of Nitric Oxide Addressing its Elec-tronic X2Π (ν′′ = 0)−A2σ+ (ν′ = 0) Band: I. Theory

Applied Optics, Vol. 49, No. 29, pp 5597–5613, 2010.

L. Lathdavong, J. Westberg, J. Shao, C. M. Dion,P. Kluczynski, S. Lundqvist, and O. Axner

III. Faraday Modulation Spectrometry of Nitric Oxide Addressing its Elec-tronic X2Π (ν′′ = 0)−A2σ+ (ν′ = 0) Band: II. Experiment

Applied Optics, Vol. 49, No. 29, pp 5614–5625, 2010.

J. Shao, L. Lathdavong, J. Westberg, P. Kluczynski, S. Lundqvist,and O. Axner

IV. Faraday Rotation Spectrometer With Sub-Second Response Time forDetection of Nitric Oxide Using a CW DFB Quantum Cascade Laser at5.33 µm

Applied Physics B: Lasers and Optics, Vol. 103, No. 2, pp 451-459,2011.

P. Kluczynski, S. Lundqvist, J. Westberg, and O. Axner

ix

V. Analytical expression for the nth Fourier coefficient of a modulatedLorentzian dispersion lineshape function

Journal of Quantitative Spectroscopy & Radiative Transfer, Vol. 112,pp 1443-1449, 2011.

J. Westberg, P. Kluczynski, S. Lundqvist, and O. Axner

VI. Fast and non-approximate methodology for calculation of wavelength-modulated Voigt lineshape functions suitable for real-time curve fitting

Journal of Quantitative Spectroscopy & Radiative Transfer, Vol. 113,No. 16, pp 2049–2057, 2012.

J. Westberg, J. Wang, and O. Axner

VII. Lineshape asymmetries in Faraday modulation spectroscopy

Submitted to Applied Physics B, 2013.

J. Westberg and O. Axner

VIII. Methodology for fast curve fitting to modulated Voigt dispersion lineshapefunctions

Submitted to Journal of Quantitative Spectroscopy & Radiative Transfer,2013.

J. Westberg, J. Wang and O. Axner

x

Other publications by the author, not included in the thesis:

IX. Detection of acetylene impurities in ethylene and polyethylene manufac-turing processes using tunable diode laser spectroscopy in the 3µm range

Applied Physics B, Vol. 105, No. 2, pp 427–434, 2011.

P. Kluczynski, M. Jahjah, L. Nhle, O. Axner, S. Belahsene,M. Fischer, J. Koeth, Y. Rouillard, J.Westberg, A. Vicet,and S. Lundqvist

X. Speed-dependent Voigt dispersion line-shape function: applicable to tech-niques measuring dispersion signals

Journal of the Optical Society of America. B, Vol. 29, No. 10, pp2971-2979, 2012.

J. Wang, P. Ehlers, I. Silander, J. Westberg, and O. Axner

xi

Contents

Abstract v

Sammanfattning vi

Preface vii

List of publications ix

1 Introduction 1

2 Absorption spectroscopy 72.1 Beer-Lambert’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Absorption lineshapes . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Natural broadening . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 Doppler broadening . . . . . . . . . . . . . . . . . . . . . . 112.2.3 Collision broadening . . . . . . . . . . . . . . . . . . . . . . 122.2.4 The Voigt profile . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Attenuation and phase shift – Absorption and dispersion lineshapes 142.3.1 Transmission of the electrical field . . . . . . . . . . . . . . 142.3.2 Attenuation and phase shift of the electric field . . . . . . . 15

2.4 Detector signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5 Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.5.1 Thermal noise . . . . . . . . . . . . . . . . . . . . . . . . . 182.5.2 Shot noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.5.3 Flicker noise . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.6 Limit of detection . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.6.1 Direct comparison . . . . . . . . . . . . . . . . . . . . . . . 212.6.2 Allan deviation . . . . . . . . . . . . . . . . . . . . . . . . . 21

xiii

3 Magnetic Rotation Spectroscopy – the static case 253.1 Basic principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 The Zeeman effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3 The transmitted intensity . . . . . . . . . . . . . . . . . . . . . . . 28

3.3.1 Perfect polarization conditions . . . . . . . . . . . . . . . . 283.3.2 The influence of polarization imperfections . . . . . . . . . 303.3.3 Frequency dependent laser intensity . . . . . . . . . . . . . 33

3.4 The phase shift and attenuation due to a transition . . . . . . . . . 343.4.1 Phase shift in terms of integrated linestrength in the absence

of magnetic field . . . . . . . . . . . . . . . . . . . . . . . . 353.4.2 Phase shift in terms of integrated linestrength in the presence

of magnetic field . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Faraday Modulation Spectroscopy – the modulated case 454.1 Introducing the modulation of the magnetic field . . . . . . . . . . 464.2 The signal in terms of Fourier coefficients . . . . . . . . . . . . . . 474.3 Lineshape asymmetries . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.3.1 The influence of polarization imperfections . . . . . . . . . 504.4 In the mid-infrared region – MIR-FAMOS . . . . . . . . . . . . . . 52

4.4.1 Adressing an arbitrary ro-vib transition . . . . . . . . . . . 534.4.2 Addressing a ro-vib Q-transition . . . . . . . . . . . . . . . 54

4.5 In the ultra-violet region – UV-FAMOS . . . . . . . . . . . . . . . 624.5.1 A simple two-transition model . . . . . . . . . . . . . . . . 654.5.2 Magnetic field dependence . . . . . . . . . . . . . . . . . . . 684.5.3 Total pressure dependence . . . . . . . . . . . . . . . . . . . 684.5.4 Optimum conditions . . . . . . . . . . . . . . . . . . . . . . 70

4.6 Comparison of the signals from MIR-FAMOS and UV-FAMOS . . 714.7 Noise analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.7.1 Noise sources . . . . . . . . . . . . . . . . . . . . . . . . . . 734.7.2 Signal-to-noise ratio as a function of uncrossing angle . . . 734.7.3 Noise in measured signals . . . . . . . . . . . . . . . . . . . 76

5 Analytical expressions for Fourier coefficients 775.1 Fourier expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.2 Analytical expression . . . . . . . . . . . . . . . . . . . . . . . . . . 795.3 Derivation of a non-complex analytical expression . . . . . . . . . . 80

6 Fast and non-approximative calculation of modulated Voigt dispersionlineshape functions 856.1 Calculation of the 1st Fourier coefficient . . . . . . . . . . . . . . . 866.2 The Westberg-Wang-Axner method . . . . . . . . . . . . . . . . . . 87

7 Experimental methods and procedures 917.1 Basic configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . 917.2 Experimental setup for MIR-FAMOS . . . . . . . . . . . . . . . . . 92

xiv

7.3 Experimental setup for UV-FAMOS . . . . . . . . . . . . . . . . . 94

8 Results and discussion 978.1 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . 97

8.1.1 MIR-FAMOS . . . . . . . . . . . . . . . . . . . . . . . . . . 988.1.2 UV-FAMOS . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

8.2 Theoretical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1048.2.1 Effects of lineshape asymmetries on measured signals . . . . 1048.2.2 Evaluation of the Westberg-Wang-Axner method . . . . . . 104

9 Conclusions and outlook 1099.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

A Appendix 113A.1 Definition of linestrength . . . . . . . . . . . . . . . . . . . . . . . . 113A.2 The wave vector and integrated linestrength – a different approach 115A.3 Coupling of electronic and rotational motion . . . . . . . . . . . . 117

A.3.1 Hund’s case (a) . . . . . . . . . . . . . . . . . . . . . . . . . 117A.3.2 Hund’s case (b) . . . . . . . . . . . . . . . . . . . . . . . . . 118

Bibliography 121

Acknowledgements 129

List of Figures 131

Nomenclature 135

Summary of the papers 145

xv

Chapter 1

Introduction

”I believe that, in the experiments that I describe in the paper,light has been magnetically affected, i.e. that that which ismagnetic in the forces of matter has been affected, and in turnhas affected that which is truly magnetic in the force of light...”

— Michael Faraday, 1845

In 1845 Michael Faraday made one of the most influential discoveries in classicalphysics of his generation. His discovery led him to believe that electromagnetismand light are closely intertwined, i.e. that optical activity can be induced in matterby a magnetic field. The initial experiment was quite simple [1]. A piece of leadborate glass, about two inches square and half an inch thick with flat and polishededges, was placed between two crossed polarizers, denoted the polarizer and theanalyzer, after which a screen was placed to catch any light from a lamp flametransmitted through the system. This configuration effectively extinguished all lightand no image appeared on the screen, but by placing the piece of glass betweenthe poles of an electromagnet, which created a static magnetic field whenevercurrent was supplied, an image would miraculously appear on the screen andremain as long as the static magnetic field was present [2]. The explanation for thisremarkable result is that Faraday had, by subjecting the beam of unpolarized lightto a polarizer, eliminated all but one direction of oscillation of the light. In theabsence of the external magnetic field this light was completely extinguished afterimpinging upon the perpendicularly oriented analyzer. However, by passing thelight through a dielectric material, such as a piece of glass, subjected to a magneticfield parallel to the direction of propagation of the light, the plane of polarizationwas rotated by an angle proportional to the magnetic field and the interactionlength whereby a fraction of the impinging light was transmitted through the

1

Introduction

analyzer. This rotation was named after its discoverer and the effect is now knownas the Faraday effect.

Figure 1.1. Michael Faraday (1791-1867) giving a Christmas lecture at the Royal Institu-tion in London.

A basic understanding of this phenomenon can be achieved by considering thelinearly polarized light, transmitted through the polarizer, as a combination of twocircularly polarized components: left-handed and right-handed circularly polarizedlight, LHCP and RHCP, respectively, which propagate in the same direction as thelinearly polarized light but with planes of polarization that continuously rotate anti-clockwise and clockwise, respectively, as the light propagates in the medium. Thesetwo components experience different indices of refraction due to the magneticallyaffected dielectric material and thereby propagate with different speeds throughthe medium. This results in a phase shift of the two components, and hence arotation of the plane of polarization. This phenomenon is referred to as magneticcircular birefringence and provides a simple explanation to the image Faraday wasable to observe on his screen.

Faraday’s discovery has led to numerous experimental applications of whichone of the earliest for spectroscopic use was made in 1937 by Carroll [3] whoused a massive water-cooled 34 kg solenoid subjected to about 125 A of current,which produced a magnetic field of roughly 0.2 T to investigate the magneticrotation spectra of diatomic molecules. This early implementation, as well asothers during this era, relied on a broadbanded light source, which severely limitedits spectroscopic applicability. However, the discovery of the laser by Maiman [4]in 1960 initiated a rapid progress of the development of spectroscopic techniques,including such based on Faraday rotation, and with early prominent works by forinstance Kaldor et al. [5] and Liftin et al. [6] the Faraday modulation/rotationspectroscopy (FAMOS/FRS) technique was established as a highly selective and

2

Introduction

sensitive technique for spectroscopic studies of paramagnetic molecules in gasphase. In recent years the technique has seen an increase in activity, primarilydue to the development of narrow-banded, room temperature, and continuouswave light sources in the mid-infrared (MIR) region that target the fundamentalrotational-vibrational (ro-vib) band of various paramagnetic molecules. Theselasers show great potential for spectroscopic applications and as the manufacturingtechniques mature, systems built around these light sources will be increasinglyubiquitous. The benefit of reaching these wavelengths can be seen in Figure 1.2,where the linestrengths for NO are displayed. By targeting the fundamental ro-vibband of NO in the MIR region an increase in sensitivity of 2-3 orders of magnitude,as compared to the overtones in the near-infrared (NIR) region, may be achieved(all other parameters being equal).

1 2 3 4 5 610

−24

10−22

10−20

10−18

10−16

wavelength [µm]Integratedlinestrength

[cm

−1/molecule

cm−2]

∼ 5.3 µmfundamentalro-vib band

∼ 2.65 µm1st overtone

band∼ 1.8 µm2nd overtone

band

Figure 1.2. Integrated linestrengths for nitric oxide from the HITRAN database [7] as afunction of frequency. The linestrengths are roughly two and three orders of magnitudelarger when comparing the fundamental rotational-vibrational band to the 1st and 2nd

overtones, respectively.

The experimental configuration of FAMOS in its most basic form is shown inFigure 1.3 and consists of a laser that emits light that first propagates through aninitial polarizer, which effectively dampens all but one plane of polarization. Thisresults in essentially linearly polarized light, which can be decomposed into the twocircularly polarized components mentioned above. By applying a magnetic field,parallel to the propagation direction of the light, these two helical componentsof the light will, by interaction with the medium, experience different indices ofrefraction due to the Faraday effect, which results in a phase shift between thetwo components and thereby a rotation of the plane of polarization. After theinteraction region the light impinges on an analyzer, which is oriented nearlyperpendicular with respect to the initial polarizer before it is incident on thedetector. By employing a sinusoidal current to the solenoid an alteration of themagnetic field is introduced. This causes the intensity of transmitted light to

3

Introduction

vary accordingly, which produces a modulation of the power impinging on thedetector. A lock-in amplifier is often used to both create the sinusoidal modulationof the magnetic field and to perform the subsequent demodulation, which is usuallyperformed at the modulation frequency.

Figure 1.3. Basic experimental layout of FAMOS.

There are a number of advantages of using the Faraday modulation techniquewhen compared to other laser-based spectroscopic techniques. Firstly, since theFaraday effect only applies to paramagnetic molecules, background signals fromspectrally interfering diatomic compounds can largely be suppressed, thus yieldingan excellent species selectivity. Secondly, by modulating the transition frequencyof the molecular transition instead of modulating the light source implies that thebenefits of the modulation, i.e. decreasing the influence of the frequency dependentflicker noise is kept while the influence of etalons is greatly reduced compared totechniques that rely on light source modulation. This results in a sensitivity thatis 2-3 orders of magnitude greater than that of direct absorption spectrometry(DAS) and often in parity or even superseding the sensitivity of the more commonlyused wavelength modulation spectrometry technique (WMS) [8–12], which, whenperformed by diode lasers, often is referred to as wavelength modulated (wm)tunable diode laser absorption spectrometry technique (TDLAS)[13–16]. TheFAMOS technique has predominantly been used for detection of nitric oxide (NO)[5, 6, 17–28] although also a few other paramagnetic molecules have been detected,e.g. NO2 [29], OH [30] and O2 [31–33]. NO is an environmentally hazardous andstrongly toxic pollutant [34–36] that causes toxic rain, smog and may induce severeadverse health effects in the mammalian body. It is also used as a biomarkerfor diagnosis of various diseases in the respiratory system [37–40] and may thuscontribute to the decision basis before the implementation of medical treatments.All this makes it important to be able to monitor and assess the concentration ofNO, which is typically very low. Moreover, due to the abundant presence of waterand carbon dioxide in most industrial and biological processes, a technique thatis highly insensitive to the influence of these diamagnetic molecular compoundsis needed. FAMOS employing MIR quantum cascade lasers has the potential offulfilling these requirements and have repeatedly proven to yield detection limits forNO in the low ppb range [24, 41]. Due to the larger transition strengths in the UVregion, FAMOS addressing such transitions has the potential to yield even higherdetection sensitivities than those from MIR-FAMOS. However, the instrumentation

4

Introduction

required for such systems is far more complex, which limits the applicability ofFAMOS systems targeting ultra-violet transitions for commercial use.

5

Chapter 2

Absorption spectroscopy

”The number of different optical phenomena has become in ourtime so great that caution must be taken so as to avoid beingdeceived, and also to refer the phenomena to the simple laws.”

— Joseph von Fraunhofer

The simplest realization of laser-based spectroscopy is that given by the directabsorption technique. This technique is appealing primarily due to the fact that ithas the ability to provide reasonably accurate concentration assessments. However,although this technique suffers from a few technical limitations whereby it does notgenerally demonstrate the lowest detectability, it is often used, due to its simplicity,as an illustrative example in order to introduce concepts and features that arecommon to many other types of laser-based spectroscopy techniques includingthose that rely on dispersion to which most of this work is dedicated.

2.1 Beer-Lambert’s law

The basis for direct absorption spectroscopy (DAS) is the Beer-Lambert law whichstates that the frequency dependent intensity, I(ν), (W/m2) of light (preferablymonochromatic) transmitted through a sample of length L containing absorbers,which are assumed to act independently, is related to the incident intensity, I0,through

I(ν) = I0e−α(ν), (2.1)

7


where ν is the frequency (cm−1) of the light and α(ν) is a dimensionless entityoften referred to as the absorbance, which can be given in either of the forms

α(ν) = SNxLχ(ν) = S′crelpLχ(ν), (2.2)

where S and S′ are the integrated (molecular) linestrength (cm−1/molecule·cm−2)and the integrated (gas) linestrength (cm−2/atm) of the transition, respectively, Nxis the density of absorbers (cm−3), L is the interaction length (cm), p is the totalgas pressure (atm), crel is the relative concentration of the absorber (dimensionless),and χ(ν) is the frequency dependent area-normalized lineshape function (1/cm−1).The latter fulfills the normalization requirement∫ ∞

0

χ(ν) dν = 1, (2.3)

where ν is the frequency in cm−1. The integrated gas linestrength, S′, can berelated to the integrated molecular linestrength, S, through

S′ =ntotpS =

1.01325 · 105

kBTS = 2.6868 · 1019T0

TS, (2.4)

where ntot is the total number density of species, kB is the Boltzmann constant(1.38× 10−23 [J/K]), T is the temperature of the gas (K) and T0 is the referencetemperature (K) in this case taken as 273.15 K. The constant in equation (2.4) isknown as the Loschmidt constant and represents the number of molecules of anideal gas at 0 C and 1 atm. It is often convenient to introduce α0, which is thepeak-value (on resonance) of the absorbance, i.e.

α0 = α(0)

= S′crelpLχ0, (2.5)

where χ0 is the peak-value of the area-normalized lineshape function, i.e. χ0 = χ(0).Figure 2.1 shows a schematical layout of the direct absorption technique. Theattenuation of the intensity of the light is illustrated by the horizontal arrows.

8


I0 I

Aα(Δν)

L

detector

detector signal

Figure 2.1. The basic principle of absorption spectroscopy. As light passes through amedium containing an absorber it is attenuated according to Beer-Lambert’s law. Thetransmitted light is then focused on a detector that produces a signal that is proportionalto the incident intensity.

Equation (2.1) suggests that the absorbance can be written in terms of theratio of transmitted and incident intensity, i.e. as

α(ν) = lnI0I(ν)

. (2.6)

For an optically thin medium, defined as α(ν) 1, this can be series expanded,which yields,

I(ν) = I0[1− α(ν)

], (2.7)

which shows that the absorption of the analyte is directly proportional to therelative change of intensity,

α(ν) =I0 − I(ν)

I0=

∆I(ν)

I0. (2.8)

Sometimes it is convenient to also define an entity referred to as the integratedabsorbance, defined as

∫ ∞0

α(ν) dν =

∫ ∞0

ln

[I0I(ν)

]dν ≈

∫ ∞0

∆I(ν)

I0dν. (2.9)

This entity has the appealing property that, under the condition that the mediumis optically thin, it is proportional to the concentration and integrated linestrengthwhile being independent of the lineshape function, which can easily be seen by thefollowing equations

9


∫ ∞0

α(ν) dν = S′crelpL∫ ∞

0

χ(ν) dν

= S′crelpL. (2.10)

This opens up for calibration-free assessments for optically thin samples since therelative concentration can be calculated from the area under the relative absorption,∫ ∞

0

∆I(ν)

I0dν = S′crelpL. (2.11)

2.2 Absorption lineshapes

Earlier in this chapter the lineshape function, χ(ν), was introduced although itsphysical origin was not stated clearly. The reason for introducing this entity isthat the absorption does not occur at only one particular wavelength, or within aninfinitely narrow wavelength region. Rather, the absorption lines have a certainwidth, which implies that the lineshape function describes the relative absorptionaround the center frequency of the transition. The remainder of this chapter isdevoted to various broadening mechanisms and the corresponding models requiredto accurately assess these under various experimental conditions.

2.2.1 Natural broadening

Natural broadening of lineshapes follows from Heisenberg’s uncertainty principle,

∆E∆t ≥ ~, (2.12)

where ∆E is the energy of an excited state, ∆t its lifetime, and ~ Planck’s constantdivided by 2π. This implies that the energy of an excited state can never be fullydetermined because its lifetime is finite. This leads to a Lorentzian lineshape,χL(ν), which in its area-normalized absorption form is given by,

χL(νd) =1

π

δνL/2

(νd)2 + (δνL/2)2, (2.13)

where νd is the detuning frequency (cm−1), given by νd = ν − ν0, where ν0 is thecenter frequency of the transition, δνL is the full-width-half-maximum (FWHM) ofthe profile, which, in turn, is given by

10


δνL =1

2πc

∑i

A2i, (2.14)

where c is the speed of light (cm/s) and A2i is the spontaneous emission rate (Hz)from the upper to a lower state. The summation is taken over all lower states towhich the upper state can decay. In most measurement situations the magnitude ofthe natural broadening is inferior to other more dominant broadening mechanismssuch as Doppler broadening or collision (pressure) broadening, whereby it can beneglected.

2.2.2 Doppler broadening

The thermal motion in random directions of the absorbing molecules in a gaseousmedium leads to a distribution of velocities in a given direction, which can bedescribed by the Maxwell-Boltzmann distribution function, given by

fG(vz) =1√πue−v

2z/u

2

, (2.15)

where u is the most probable velocity of a thermal distribution of molecules, givenby√

2kBT/m, where kB is Boltzmann’s constant (J/K), T the temperature (K),and m the molecular mass (kg) and where vz is the velocity component in thez-direction. Whenever the molecules have a velocity-component in the directionof the propagation of the incoming light (here assumed to be the z-direction)their absorption frequency will be slightly shifted due to the Doppler effect. Suchmolecules will not solely absorb at the frequency of the transition, ν0, but ratherat a shifted frequency ν = ν0(1 + vz/c). This means that light with a certainfrequency can only interact with a specific velocity group of molecules, which leadsto a distribution of absorption frequencies and subsequently an inhomogeneousbroadening of the transition. The lineshape function describing this phenomenon,which is illustrated in Figure 2.2, has a Gausssian form and can be written as

χG(νd) =

√4 ln 2

π

1

δνDexp

[−4 ln 2

ν2d

δν2D

], (2.16)

where δνD is a measure of the full-width-half-maximum (FWHM) of the Gaussianprofile, commonly referred to as the Doppler width, given by

δνD = 2ν0

√2 ln 2kBT

mc2, (2.17)

which also can be written as

11


−0.4 −0.2 0 0.2 0.40

2

4

6

8

10

Detuning frequency [cm−1]

Absorption

lineshap

e[cm]

Figure 2.2. Gaussian absorption lineshape.

δνD = 7.162 · 10−7ν0

√T

M, (2.18)

where M is the molecular weight (g/mol) of the species.

2.2.3 Collision broadening

If the absorbing molecules suffer from frequent collisions with other molecules thelifetimes of the molecular states will be shortened. This occurs predominantlyfor higher pressures where the molecular density is larger and the collision ratesare higher. This is yet another mechanism that will alter the broadening of theabsorption profile. Since it affects all molecules equally the broadening is said to behomogeneous and gives rise to a lineshape of the same form as natural broadening,i.e. a Lorentzian form, which is visualized in Figure 2.3. The lineshape is given byequation (2.13) where δνL in this case represents the collision broadened FWHMof the profile, which can be written as

δνL =∑i

γipi, (2.19)

where γi is the FWHM pressure broadening coefficient for collisions with species iand pi its partial pressure. Usually, the homogeneous broadening is a combinationof different broadening mechanisms including pressure broadening and naturalbroadening, where the coefficient δνL will be a sum of the different broadenings.

2.2.4 The Voigt profile

In general, neither the Doppler profile nor the Lorentzian profile is sufficientlyaccurate to model the lineshape for the pertient measurement conditions. Rather,

12


−0.4 −0.2 0 0.2 0.40

1

2

3

4

5

6


Absorption

lineshape[cm]

Figure 2.3. Lorentzian absorption lineshape.

both the phenomena of homogeneous and inhomogeneous broadening need to betaken into account. Under the assumption that the two broadening mechanismsact independently, the resulting lineshape is the so called Voigt profile, which isillustrated in Figure 2.4 and given by

χV (x, y) =1

δνD

√4 ln 2

πK(x, y), (2.20)

where K(x, y) is given by

K(x, y) =y

π

∫ ∞−∞

e−t2

y2 + (x− t)2dt, (2.21)

where, in turn, the dimensionless parameters x and y are the Doppler-widthnormalized homogeneous detuning frequency and the Doppler-width normalizedhomogeneous linewidth, given by

x =√

4 ln 2νdδνD

, (2.22)

andy =√

ln 2δνLδνD

, (2.23)

respectively. K(x, y) represents the real part of the complex error function, w(z),that can be written as

w(z) = K(x, y) + iL(x, y) =i

π

∫ ∞−∞

e−t2

z − t dt, (2.24)

where z = x+ iy.

13


−0.4 −0.2 0 0.2 0.40

1

2

3

4


Absorption

lineshape[cm]

Figure 2.4. Voigt absorption lineshape.

2.3 Attenuation and phase shift – Absorption and dis-persion lineshapes

So far only the absorbance of the intensity of the light as it propagates throughthe gaseous medium has been described, but the electric field of the light will alsoexperience a phase shift due to the interaction with the molecular species. In directabsorption spectroscopy the phase shift information is lost since only intensitychanges are measured. However, there are techniques that rely on detecting thephase shift. These are often referred to as dispersion techniques since the phaseshift is related to a change in the refractive index of the medium, n.

2.3.1 Transmission of the electrical field

A plane-polarized monochromatic electric field vector propagating in the z-directionin free-space can in general be expressed as

E(z, t) = E0e cos(k0z − ωt), (2.25)

or in complex notation as

E(z, t) =E0

2eeik0z−ωt + c.c., (2.26)

where E0 is the electric field amplitude, e the polarization unit vector, k0 theamplitude of the wave vector in vacuum, ω the angular frequency of the electricfield and t the time. This gives an intensity, defined as the time-average squared ofthe electric field, that can be written

14


I0 = 2cε0E(z, t)E∗(z, t) =1

2cε0E

20 . (2.27)

The electric field transmitted through a sample of length L containing a certainamount of absorbers can, for z > L, be expressed as,

Et(z, t) =E0

2ee−i

[k0(z−L)+k(ν)L−ωt

]+ c.c.

=E0

2ee−i

k0z+[k(ν)−k0]L−ωt

+ c.c., (2.28)

where k(ν) is the amplitude of the complex wave vector in the presence of absorbers.

It is now convenient to express the transmitted electric field in terms of acomplex transmission function, CT (ν), and the incident electric field, E(z, t), as

Et(z, t) = CT (ν)E(z, t), (2.29)

where CT (ν) is given by

CT (ν) = e−i[k(ν)−k0]L = e−δ(ν)−iφ(ν), (2.30)

where we have introduced δ(ν) = −Im[k(ν)

]L and φ(ν) =

Re[k(ν)

]− k0

L,

which represent the attenuation and the phase shift of the electric field due to thepresence of the absorber, respectively. This implies that the transmitted intensitythrough a sample of length L can then be written,

It = 2cε0Et(z, t)E∗t (z, t) = I0e

−2δ(ν), (2.31)

which shows that the information regarding the phase is lost in direct absorptionspectroscopy, and by comparison with equation (2.1), that the absorption of theintensity is equal to twice the attenuation of the electric field, i.e. α(ν) = 2δ(ν).

2.3.2 Attenuation and phase shift of the electric field

The attenuation and phase shift due to interaction with the absorbers can therebybe related to general absorption and dispersion lineshape functions, χabs(ν) andχdisp(ν), respectively, according to

δ(ν) =S′crelpL

2χabs(ν) =

α0

2χabs(ν) (2.32)

and

15


φ(ν) =S′crelpL

2χdisp(ν) =

α0

2χdisp(ν), (2.33)

where, for the case of Voigt lineshape functions, χabs(ν) and χdisp(ν) are written

as χabsV (ν) and χdispV (ν), and are given by

χabsV (x, y) =χ0

π

∞∫−∞

ye−s2

(x− s)2 + y2ds

= χ0Re[w(x+ iy)], (2.34)

and

χdispV (x, y) = −χ0

π

∞∫−∞

(x− s)e−s2

(x− s)2 + y2ds

= −χ0Im[w(x+ iy)], (2.35)

where

χ0 =

√4 ln 2

π1/2δνD, (2.36)

and where w(z) is the complex error function, also referred to as the Faddeevafunction [42], given by equation (2.24), whereas x and y are given by equation (2.22)and equation (2.23), respectively.

To summarize we have defined the absorption and dispersion lineshapes forthree different pressure conditions: (i) for low pressure conditions where the collisionbroadening is negligible and thus the homogeneous linewidth is much smaller thanthe Doppler linewidth (δνL δνD), which leads to lineshape functions of Gaussianform, χG(ν), (ii) for intermediate pressure conditions where the homogeneouslinewidth is of similar magnitude as the Doppler linewidth (δνL ∼ δνD), which leadsto lineshape functions of Voigt form, χV (ν), and (iii) for high pressure conditionswhen the homogeneous linewidth if much larger than the Doppler linewidth (δνL δνD), which leads to lineshape functions of Lorentzian form, χL(ν). In panel (a)and (b) of Figure 2.5 the Gaussian, Voigt and Lorentzian absorption and dispersionlineshape functions are displayed for an electronic transition in NO at 226.57 nm

16


−0.4 −0.2 0 0.2 0.40

2

4

6

8

10

(a)


Absorptionlineshape[cm]

Gaussian

Lorentzian

Voigt

−0.4 −0.2 0 0.2 0.4

−6

−4

−2

0

2

4

6

(b)


Dispersionlineshape[cm]

Gaussian

Lorentzian

Voigt

Figure 2.5. Panel (a) shows the area-normalized Gaussian, Voigt and Lorentzian ab-sorption lineshapes for a transition in NO at 226.57 nm. The Doppler linewidth, δνD, is0.0496 cm−1 and the pressure broadening coefficient, δνL is 0.0573 cm−1 at the pertinentpressure (150 Torr). Panel (b) shows the dispersion lineshape counterparts for the sametransition under the same measurement conditions.

2.4 Detector signal

If the frequency dependence of the transmission in the system and of the powerof the laser is taken into account, the power of the light transmitted, and thusimpinging on the detector, can be written as

PD(ν) = T (ν)PL(ν)e−α(ν), (2.37)

where T (ν) is the transmission through the optical system in the absence ofabsorbers including losses in reflective surfaces, interferences and scattering, andPL(ν) is the power of light from the laser. The signal, SD(ν), can then, under theassumption that the sample is thin, be written as

SD(ν) = ηT (ν)PL(ν)[1− α(ν)

], (2.38)

where η is an instrumentation factor (V/W) comprising the detector sensitivity,input impedance of the current-to-voltage converter and any contribution from thegain of the amplifiers used. It is now possible to divide the signal into two parts:(i) a background signal, which is given by

SBG(ν) = ηP0(ν), (2.39)

where P0(ν) is the power of the light incident on the detector in the absence ofabsorber, given by P0 = T (ν)PL(ν), and (ii) an analytical signal, given by

17


SAS(ν) = −ηP0(ν)α(ν), (2.40)

which can also be given in terms of an area-normalized lineshape function by usingequation (2.2) as

SAS(ν) = −ηP0(ν)α0χ(ν), (2.41)

where χ(ν) is the peak-normalized lineshape function, given by χ(ν)/χ0.

2.5 Noise

There are several types of noise that affect the measured signal in an absorptionspectroscopy setup. In this section the three main noise contributions will be brieflycharacterized; (i) thermal noise, (ii) shot noise, and (iii) flicker noise. The currentsthese noise sources give rise to are denoted ith, ishot, and ifl, respectively. Thesignal-to-noise ratio of the system is defined as the detector current signal from themeasured analyte, iAS , divided by the root-mean-sum of the noise contributions,

SNR =iAS√

i2th + i2shot + i2fl

, (2.42)

where iAS is related to the signal defined in equation (2.41) by SAS = GiAS , whereG (V/A) is the transimpedance gain of the detection system.

2.5.1 Thermal noise

The detection electronics of the measurement setup will always contribute to thetotal noise, regardless of the laser power impinging on the detector. This type ofnoise is thus laser power independent and subsequently dominates for low laserpowers and originates from the random fluctuations of the velocity of the chargecarriers in the resistive material. The current this gives rise to can be modelled as

ith =

√4kBT∆f

R, (2.43)

where ∆f is the electronic bandwidth and R is the input impedance of the detectionelectronics. Since iAS ∝ P0 while ith is independent of P0, when this type of noisedominates the system the SNR will be proportional to the incident power on thedetector.

18


2.5.2 Shot noise

Shot noise arises from the quantum nature of light, i.e. that the distribution ofphotons arriving at the detector surface will have a Poissonian form. This type ofnoise is thus unavoidable and constitutes the fundamental noise limit. The shotnoise current is given by,

ishot =√

2eidc∆f, (2.44)

where e is the electron charge and idc represents the average photo current givenby,

idc = η′P0, (2.45)

where η′ (A/W), given by η′ = η/G, represents the responsivity of the detector.When this type of noise dominates the SNR is proportional to the square root ofthe laser power, SNR ∝ √P0, and can be written,

SNR =

√η′P0χ(ν)α0√

2e∆f, (2.46)

where P0 is the power incident on the detector.

Assuming that the minimum detectable absorbance, (α0)min, corresponds to aSNR of unity implies that it can be written as

(α0)min =1

χ(ν)

√2e∆f

η′P0. (2.47)

In a typical scenario where ∆f = 1 Hz, P0 = 1 mW, η′ = 1 A/W, and χ = 1, theshot-noise limited on-resonance absorption is in the order of ∼ 10−8. However, inpractice these low absorbance values are never reached due to the existence of laserexcess noise that will dominate at the low frequencies where the direct absorptionsignal is detected.

2.5.3 Flicker noise

Flicker noise is the most common type of noise for direct absorption spectrometryand usually sets the limitation on the detectability of the DAS technique. Thedominating origin of this type of noise is usually the light source, and the currentthe flicker noise gives rise to is denoted, ifl. This type of noise can be modelled as

19


ifl = σP (ν)idc, (2.48)

where σP (ν) is the relative standard deviation of the power of the light impingingon the detector, which, in turn, can be modelled as an integral over the detectionbandwidth,

σP (ν) =

∫ f2

f1

σP,f (ν) df, (2.49)

where σP,f is the relative standard deviation of the spectral power of the lightand f1 and f2 are the lower and upper detection bandwidth limits, respectively.Although this can vary substantially between different laser systems, and depend onboth the laser and the driver, it is often assumed that this has a 1/fa dependence,where a > 0 (often above or around 1). For flicker noise limited systems the SNRcan be written as

SNR =η′P0χ(ν)α0

σP (ν)η′P0=

α0

σP (ν). (2.50)

This implies that the minimum detectable absorbance, (α0)min, is given by

(α0)min = σP (ν). (2.51)

This expression shows that under flicker noise limited conditions the minimumdetectable absorbance is solely given by the standard deviation of the power of thelight within the detection bandwidth.

As was mentioned above, the contributions to this type of noise can haveseveral origins of which the most prominent would be power fluctuations from thelaser source. Another contributor might be fluctuations in the transmission of theoptical system, in particular multiple reflections between various optical surfaces, socalled etalons. In order to minimize these effects certain precautions can be made,for instance by employing a modulation technique where the analytical signal isencoded and decoded at a frequency for which the flicker noise is low (preferablyin the kHz or MHz range) or by simply dithering one or several of the surfaces inthe optical system.

20


2.6 Limit of detection

2.6.1 Direct comparison

There are several ways to define the limit of detection in direct absorption spec-troscopy. One is to simply compare the magnitude of the analytical signal froma sample with a known concentration, taken under a given set of conditions, tothe standard deviation of the noise in the zero-gas background signal, taken underotherwise identical conditions. For example, assume that the absorbance from70 Torr of 100 ppm of NO at 5.331 µm for an interaction length of 15 cm correspondsto a measured signal of 1 V. This gives an α0, given by equation (2.5), of

α0 = S′crelpLχ0 = 0.7835 · 100× 10−6 · 70/760 · 15 · 48.6919 = 0.0053. (2.52)

Under the assumption that the estimated standard deviation of the noise, σP , is1 mV this gives a minimum detectable absorption of

(α0)min = α0 ·σP1≈ 5× 10−6. (2.53)

However, this type of approach of calculating the detection limit only provides theminimum detectable quantity for the particular integration time used to obtainthe signal. It does not provide any information for which integration time that willbe optimum for this particular detection system. In order to address this problemthe use of Allan variance plots has become a standard practice in laser absorptionspectroscopy measurements since such plots provide a simple and more accuratemeans of comparing the performance of different detection systems.

2.6.2 Allan deviation

The Allan variance is obtained by measuring a large set of data (e.g. concentrationvalues) obatined at time intervals. Optimally, the time intervals are as short aspossible. It is however required that they exceed the response time of the system,which is limited by the averaging time of the electronics, data acquisition timeand curve fitting time (for real-time curve fitting). In practice, N data points aremeasured with equal time intervals, τ . The data points are then divided into Msubset groups each containing k = N/M data points. This implies that each suchsubset is acquired with an integration time of kτ . The Allan variance for a givenintegration time can then be calculated according to

21


σ2y(k) =

1

2(M − 1)

M−2∑i=0

(yi+1 − yi)2, (2.54)

where yi denotes the mean of each subset and is given by

yi =1

k

k∑j=1

xi·k+j , (2.55)

where x is the measured value. An Allan (deviation) plot consists of the squareroot of the Allan variance as a function of the integration time with both axes givenin logarithmic scales. This type of plot contains information on the dominatingnoise sources for different detection bandwidths. For instance, the Allan deviationof white noise has a τ−1/2 dependence and is equivalent to the ordinary standarddeviation. In Figure 2.6 a simulation of different amounts of noise on a measuredentity (here assumed to be concentration) is shown. The upper panel shows thesimulated concentration values whereas the lower panel displays the correspondingAllan plot.

Optimum conditions are often considered as the minimum of the Allan plot.As long as the response follows the white noise behaviour, the Allan deviationcorresponds to the standard deviation of the signal. Under these conditions, theAllan deviation corresponds to the detection limit for a SNR of unity.

22


0 100 200 300 400 500 600 700 800 900 1000−2

0

2

4

6x 10

−6

Time [s]

Signal[a.u.]

(a)

y = ny = a1 + bt+ ny = a2 + 2bt+ ny = a3 + 2bt+ 5n

10−1

100

101

102

10−9

10−8

10−7

10−6

Integration time [s]

Allandeviation,σ[a.u.]

(b)

Figure 2.6. Panel (a) displays four different simulations of measured concentration valuesacquired continuously at a rate of 10 Hz. The red curve corresponds to measured valuesthat show only random fluctuations (white noise) without any drift. The blue curvecorresponds to measured values that have random fluctuations and a linear drift in time.The black curve has the same noise level but twice the linear drift, while the green curvehas 5 times the noise level compared to the other curves and the same linear drift as theblack curve. Panel (b) shows the Allan deviation plots of the corresponding simulatedmeasurements. By examining these plots conclusions can be drawn regarding the noiselevels and the optimum integration times. For instance the white noise level of thegreen curve is 5 times that of the others, as expected from the input parameters in thesimulation.

23

Chapter 3

Magnetic RotationSpectroscopy – the static case

”Nature is our kindest friend and best critic in experimentalscience if we only allow her intimations to fall unbiassed onour minds. Nothing is so good as an experiment which, whilstit sets an error right, gives us (as a reward for our humility inbeing reproved) an absolute advancement in knowledge.”

— Michael Faraday - Letter to John Tyndall

3.1 Basic principle

Magnetic Rotation Spectroscopy (MRS) [3, 43–45] is a spectroscopic detectiontechnique for detection of paramagnetic species in gas phase that enhances thesensitivity and selectivity of the direct absorption techniques (DAS). The MRStechnique utilizes the Faraday effect, which states that linearly polarized lightpropagating through a dielectric medium will rotate the plane of polarization wheninfluenced by a magnetic field parallel to the propagation direction. A schematicillustration of a typical experimental setup is shown in Figure 3.1.

The laser light first propagates through a polarizer, which defines a polarizationaxes for the linearly polarized light that continues through the interaction regionwhere the paramagnetic species to be detected resides. A surrounding solenoidprovides an external magnetic field that is parallel (depending on the direction of the

25

Magnetic Rotation Spectroscopy – the static case

Figure 3.1. A schematic illustration of the basic experimental layout of MRS. The laserlight is first polarized through a polarizer situated in front of the cell which containsthe paramagnetic species to be detected. A solenoid is placed around the cell in orderto provide a magnetic field parallel to the propagation direction of the laser light. Therotation of the plane of polarization due to the difference in retardation of the circularcomponents of the linearly polarized light is transformed into a change of the transmittedpower by a nearly crossed second polarizer (analyzer) placed after the cell.

current in the solenoid) to the propagation of the light. The linearly polarized lightcan be seen as a combination of left-handed and right-handed circularly polarizedlight, LHCP and RHCP, respectively, which will experience different indices ofrefraction when propagating through the absorbing medium. This introduces asmall phase-shift between them, which is translated into a change in transmittedpower by placing a second polarizer (called the analyzer) after the cell, orientednearly perpendicular with respect to the first polarizer. Since the two polarizersare nearly crossed any small rotation of the plane of polarization caused by theparamagnetic species in the cell will lead to a change of the transmitted powerthrough the analyzer which can be detected by the detector.

3.2 The Zeeman effect

The MRS technique is based upon the Zeeman effect1, which states that in thepresence of an external magnetic field, B, a state with a given total angularmomentum, J , is split into several sub-states due to the lifting of the degeneracyof the atomic or molecular energy levels. The number of sub-states depends onthe magnetic quantum number, M , of the states which is schematically shown inFigure 3.2.

In general, each state will be split into a multitude of sub-states with anenergy separation of MgµBB, where g is the g−factor, B the magnetic field andµB the Bohr magneton. This implies that the frequency (in units of cm−1) ofa magnetically induced M ′′ → M ′ transition between two states i and j can beexpressed as

1When the absorption lines rather than the spectral lines are split the Zeeman effect is sometimesreferred to as the inverse Zeeman effect.

26


Figure 3.2. A schematic illustration of the splitting of a typical rotational-vibrationaltransition in NO, in this case the R(3/2) transition. The figure shows the splitting ofthe lower J ′′ = 3/2 and the upper J ′ = 5/2 levels into several sub-states caused by anexternally applied static magnetic field, B. The number of sub-states is determined by themagnetic quantum number MJ , which ranges from −J to J . Only the allowed transitionsinduced by the LHCP and RHCP light for which ∆MJ = ±1, respectively, are shown bythe vertical arrows in the figure.

νM ′′M ′ = ν0 +(M ′g′ −M ′′g′′

)µBB, (3.1)

where the single and double primed quantities refer to the upper and lower staterespectively, and ν0 is the center frequency in the absence of magnetic field.

The quantum mechanical selection rules state that linearly polarized lightthat propagates along the direction of the external magnetic field can only inducetransitions for which the magnetic quantum number, M , differs by one unit,∆M = M ′ −M ′′ = ±1 (assuming that the states adhere to the same Hund’scoupling case2). The induced transitions are illustrated by the vertical arrows inFigure 3.2, where each group of arrows correspond to transitions induced by theLHCP and RHCP light, respectively. As the frequency of the light is detunedfrom the transition frequency, the two circularly polarized components of the lightwill, in general, experience different indices of refraction and thereby propagate at

2 See Appendix A.3

27


different speeds which introduces a phase shift between them, which, in turn, leadsto a rotation of the plane of polarization of the linearly polarized light.

3.3 The transmitted intensity

There are several different phenomena that affect the light that propagates througha Zeeman split medium; (i) ordinary absorption, which is the absorption of the lightdue to the medium, (ii) magnetic circular birefringence (MCB), which constitutesthe retardation of the different circular components of the light due to interactionswith the medium, and (iii) magnetic circular dichroism (MCD), which occurs whenthe LHCP and RHCP are absorbed differently. A derivation of the transmittedintensity including these effects in the presence of ideal polarizers is given below.In the proceeding section the situation in the presence of polarization imperfections(PI), due to not-fully polarized light transmitted through the system, is given.

3.3.1 Perfect polarization conditions

For simplicity, consider an electric field polarized in the x-direction and propa-gating in the z-direction that is incident on the sample. Using the plane waveapproximation this electric field can be written as

E = E0

(1

0

)ei(kz−ωt), (3.2)

where E0 is the electric field amplitude containing the normalization, k the wavevector, ω the angular frequency of the electric field, and t the time. It is convenientto rewrite the electric field in terms of the unit vectors for LHCP and RHCP light,eL and eR, respectively. This implies that the expression the electric field in thepresence of absorbers can be written as

E =E0√

2eLe

i(kLz−ωt) +E0√

2eRe

i(kRz−ωt), (3.3)

where

eL =1√2

(1

−i

)and eR =

1√2

(1

i

), (3.4)

and kL and kR are the wave vectors for the two circular components of the light inthe presence of absorbers, defined by

28


kL,R = (nL,R + iκL,R)ω

c, (3.5)

where nL,R and κL,R are the refractive index and the absorption coefficient forLHCP and RHCP light, respectively. Following the discussion in Section 2.3 andassuming an interaction length of L it is possible to decompose kL,RL as

kL,RL = k0L+ φL,R + iδL,R, (3.6)

where k0 is the wave vector in the absence of absorbers, given by k0 = ω/c, andφL,R and δL,R are the phase shift and the attenuation experienced by the electricfield, which thus are given by (nL,R − 1)ωL/c and κL,RωL/c, respectively. Thisimplies that the electric field can be written as

E =E0√

2ei(k0L−ωt)

[(1

−i

)e−δL+iφL +

(1

i

)e−δR+iφR

]. (3.7)

Consider now a polarizer that is oriented nearly perpendicular with respect to theincoming polarized light but offset slightly at an uncrossing angle θ. The Jones’matrix for such a polarizer can be written as

P (θ) =

(sin2 θ sin θ cos θ

sin θ cos θ cos2 θ

). (3.8)

This means that the transmitted electric field, which is given by Et = P (θ) ·E, canbe written as

Et =E0

2e−iωteiφR−δR

(sin2 θ + i sin θ cos θ

sin θ cos θ + i cos2 θ

)

+E0

2e−iωteiφL−iδL

(sin2 θ − i sin θ cos θ

sin θ cos θ − i cos2 θ

). (3.9)

The transmitted intensity is proportional to the time-averaged electric field squared,It ∝ 〈|Et|2〉, which implies that it can be expressed as,

It = I0e−2δ[cosh ∆δ − cos(2θ + ∆φ)

], (3.10)

where I0 is the power of the radiation after the polarizer, δ the average attenuation,given by

δ =1

2(δL + δR) =

α0

4

(χabsL + χabsR

), (3.11)

29


where α0 is the absorption strength, given by α0 = α0/χ0 = S′crelpL, and where ∆δand ∆φ are the attenuation and phase shift for LHCP and RHCP light, respectively,given by

∆δ = δL − δR =α0

2

(χabsL − χabsR

), (3.12)

and∆φ = φL − φR =

α0

2

(χdispL − χdispR

), (3.13)

where χabsL/R3 and χdispL/R are the lineshapes for the attenuation and phase shift for

the two circular components of the light, respectively. A phase shift of ∆φ betweenthe LHCP and RHCP components of the light gives rise to a rotation of the planeof polarization of ∆φ/2. Taylor expansion of equation (3.10), assuming θ ∆φ,δ 1 and ∆δ 1, yields, for the transmitted intensity,

It =I02

(1− 2δ)[1− cos(2θ) + ∆φ sin(2θ)

]. (3.14)

Equation (3.14) is equivalent to eq. (1) given by Liftin et al. [6] or eq. (1) givenby Blake et al. [17] albeit with different notation. This expression shows that thetransmitted intensity and thereby the MRS signal, is affected by the species underinvestigation primarily through absorption, given by the 2δ-term, and the frequencydependent phase shift between the two ciruclar components of the light, givenby the ∆φ sin(2θ)-term. The difference in attenuation between the two circularcomponents of the light, ∆δ, does not enter in the lowest order. This implies thatthe signal, under these assumptions, is fully symmetrical when scanned across atransition.

3.3.2 The influence of polarization imperfections

The theoretical description given above indicated that the signal is fully symmetrical.However, this is not always true as asymmetries in the lineshape have been observed[45]. These asymmetries have mainly two origins: (i) polarization imperfectionsdue to not-fully polarized light transmitted through the system and (ii) a frequencydependent laser intensity (FDLI), where the former can stem from the initialellipticity of the light partly transmitted through an imperfect polarizer, stress-induced circular birefringence in the cell windows or imperfections of the analyzer.In order to accurately model the lineshapes in MRS one or both of these effectsmust be considered.

3Theˆsign indicates that the entity is area-normalized [1/cm−1]

30


Polarization imperfections

An analysis of the effects of polarizaton imperfections, which follows that by Brechaand Pedrotti [45], is given below.

Assume that the light impinging upon the analyzer is, in the absence ofabsorbers, essentially linearly polarized in the x-direction but contains a smallamount of ellipticity which stems from either an initial ellipicity of the lighttransmitted through the first polarizer or from stress-induced circular birefringencein the cell windows. The electric field for this light propagating in the z-direction,corresponding to that previously given by equation (3.2), can be written as,

E = E0

(1

iε

)ei(kz−ωt), (3.15)

where ε is an unbalancing term between the LHCP and the RHCP components ofthe light. This expression can, just as before, be conveniently rewritten as a sumof the contributions from the LHCP and RHCP components of light and in termsof their unit vectors, as

E =E0√

2ei(k0L−ωt)

[eL(1− ε)e−δL+iφL + eR(1 + ε)e−δR+iφL

]. (3.16)

The elliptically polarized light is transmitted through the analyzer, which isoffset by an angle θ with respect to the first polarizer. If we assume that theanalyzer has a finite extinction ratio, i.e. that it transmits a fraction of the lightalong the polarization direction orthogonal to its main axis, then we can write theJones’ matrix for the analyzer as,(

α sin2(θ) + β cos2(θ) α sin(θ) cos(θ)− β cos(θ) sin(θ)

α sin(θ) cos(θ)− β cos(θ) sin(θ) α cos2(θ)β sin2(θ)

), (3.17)

where α and β denotes the fractions of incident light that are transmitted along themain and the orthogonal polarization axis of the analyzer, i.e. α ' 1 and β 1.Multiplying the Jones’ matrix in equation (3.17) with the electric field vector inequation (3.16) gives us the electric field transmitted through the analyzer, Et. Thetransmitted intensity is given by the product of this electric field and its complexconjugate, and can be written as

31


It =I02

exp(−2δ)

(α2 + ε2β2)

[cosh(∆δ)− cos(∆φ+ 2θ)

]+ 2(α2 + β2)ε sinh(∆δ)

+ (ε2α2 + β2)

[cosh(∆δ) + cos(∆φ+ 2θ)

], (3.18)

where we have used the definitions from equations (3.11)-(3.13).

This expression, which is the same as that derived by Brecha and Pedrotti [45]although expressed in entities more suitable for FAMOS, includes the effects ofmagnetic circular dichroism (MCD), magnetic circular birefringence (MCB) andpolarization imperfections (PI) without making any assumptions regarding theirrelative sizes. In order to simplify this expression and to compare this with theexpressions that have been used earlier in the field [6, 17] it is appropriate to makea couple of justified approximations.

Small polarization imperfections

First, let us consider a good quality analyzer, i.e. α ' 1 and β2 1. This impliesthat α2 + β2 ' α2 ' 1, ε2α2 ' ε2 and that ε2β2 can be neglected, which gives

It ≈I02e−2δ

[cosh(∆δ)− cos(∆φ+ 2θ) + ξ

], (3.19)

where we have introduced a term ξ that incorporates all effects of polarizationimperfections, which can be written as

ξ = 2ε sinh(∆δ) +(β2 + ε2

)[cosh(∆δ) + cos(∆φ+ 2θ)

]. (3.20)

For small ellipticity of the light (ε 1) and a good analyzer (β 1), ξ can beneglected and equation (3.19) reduces to

It ≈I02e−2δ

[cosh(∆δ)− cos(∆φ+ 2θ)

], (3.21)

which is of the same type as the equations given earlier in the literature, e.g. eq.(1) given by Blake et al. [17].

32


Small attenuations and phase shifts

If we assume that also the phase shifts and the overall absorption are small,equation (3.19) can be Taylor expanded, which gives

It ≈I02

[1− cos(2θ) + sin(2θ)∆φ+ ξ

]. (3.22)

where ξ is now given by

ξ = (β2 + ε2)[1 + cos(2θ)

]+ ε∆δ. (3.23)

Neglecting all the second order effects gives us a final expression for the transmittedintensity,

It ≈I02

[1− cos(2θ) + sin(2θ)∆φ+ ε∆δ

]. (3.24)

This expression provides an extension to the expressions used previously in theliterature and contains the effects of magnetic circular dichroism and polarizationimperfections to the first order.

3.3.3 Frequency dependent laser intensity

As was mentioned above there is another effect that can cause asymmetries in theMRS lineshapes, namely a frequency dependent laser intensity (FDLI). As the laserfrequency is swept across the transition by tuning the injection current to the laser,the intensity of the laser light is varied slightly across the scan which influences theshape of the MRS signal. In order to account for this effect a non-static frequencydependent intensity, I(ν), can be incorporated in the model. For example, this cansimply be achieved by replacing I with I(ν) given by a parabolic expression of theform

I(ν) = I0 + κ1(ν − ν0) + κ2(ν − ν0)2, (3.25)

where I0 is the intensity at ν = ν0 and κ1 and κ2 are the linear and non-linearintensity coefficients, respectively.

33


3.4 The phase shift and attenuation due to a transi-tion

As essentially linearly polarized light propagates through a medium consisting ofparamagnetic molecules in gas phase whose transitions are split by an externalmagnetic field, the splitting gives rise to a phase shift, ∆φ, and attenuationdifference, ∆δ, between the two circularly polarized components of the light, LHCPand RHCP, which can be written as,

∆φ = φL − φR,∆δ = δL − δR, (3.26)

where the subscripts, R and L stand for RHCP and LHCP, respectively. It waspreviously shown that these can be rewritten in terms of the difference betweenthe corresponding wave vectors, ∆k = kL − kR, as

∆φ = Re[∆k]L,

∆δ = − Im[∆k]L, (3.27)

where L is the interaction length. Since all allowed transitions between the differentmagnetic quantum numbers, M , contribute at the same time, which was illustratedby the arrows in Figure 3.2, the total phase shift and attenuation difference,originating from the M ′′ →M ′ transitions, must in general be written as a sum oftheir individual contributions, viz. as

∆φ =∑M ′′M ′

[φM

′′M ′L − φM ′′M ′

R

](3.28)

=∑M ′′M ′

Re[kM

′′M ′L − kM ′′M ′

R

]L

and

∆δ =∑M ′′M ′

[δM

′′M ′L − δM ′′M ′

R

](3.29)

= −∑M ′′M ′

Im[kM

′′M ′L − kM ′′M ′

R

]L.

34


3.4.1 Phase shift and attenuation due to a transition in termsof the integrated linestrength and the dispersion lineshapefunction in the absence of magnetic field

For the FAMOS technique it has become customary [6, 17] to express the contribu-tion to the wave vector, k, experienced by monochromatic light passing a mediumwith a given density of molecules, in terms of the transition dipole moment, as

k(νd) = (Ni −Nj)|〈i|µ |j〉|2

2u~ε0Z[ cuν

(νd + iδνL)], (3.30)

where Ni and Nj are the population densities of the states i and j, respectively,

|〈i|µ |j〉|2 the transition dipole moment squared between states i and j, u the mostprobable molecular speed, ~ Planck’s constant divided by 2π, ε0 the permittivityof free space, Z[...] the plasma dispersion function (Faddeeva function) [42], δνLthe homogeneous broadening (HWHM) of the transition, and νd the frequencydetuning from the peak absorption of the transition given by

νd = ν − ν0, (3.31)

where ν0 is the center frequency of the transition and ν the laser frequency.

The plasma dispersion function can be rewritten in terms of the area-normalizedVoigt lineshape function as

Z[ cuν

(νd + iδνL)]

=πuν

c

[χdispV (νd) + iχabsV (νd)

]. (3.32)

This implies that it is possible to express the dispersion lineshape function andabsorption lineshape function4, χdispV (νd) and χabsV (νd) in terms of the complexerror function5, w, as

χdispV (νd) = −χ0 Imw[ cuν

(νd + iδνL)]

, (3.33)

and

χabsV (νd) = χ0 Rew[ cuν

(νd + iδνL)]

, (3.34)

4under the assumption that the lineshape of the transition can be described by the Voigt profile.5since the complex error function, w(x), and the plasma dispersion function, Z(x), are related

through Z(x) = i√πw(x)

35


respectively, where χ0 is the peak value of the area-normalized absorption Gaussianline-shape function given by

√ln 2/ (

√πδνD), νd is the Doppler-width-normalized

frequency detuning given by

νd =√

ln 2νdδνD

, (3.35)

where δνD is the half-width-half-maximum Doppler width of the transition, δνL isthe corresponding homogeneous broadening parameter (henceforth referred to asthe Voigt parameter) given by

δνL =√

ln 2δνLδνD

, (3.36)

and w(z) is defined as,

w(z) =i

π

∞∫−∞

e−t2

z − t dt. (3.37)

By using this convention it is possible to express the phase shift, φ, and attenuation,δ, of a monochromatic wave interacting with a transition between the states i andj as

φ(νd) =

Re[k(νd)]− k0

L =

π2

hε0cν (Ni −Nj) |〈i|µ |j〉|2 LχdispV (νd) (3.38)

and

δ(νd) = − Im[k(νd)

]L =

π2

hε0cν (Ni −Nj) |〈i|µ |j〉|2 LχabsV (νd), (3.39)

where χdispV (νd) and χabsV (νd) are given by equation (3.33) and (3.34), respectively.Although these expressions are formally correct they are not given in their mostuseful form, since transition strengths are seldom expressed in terms of transitiondipole moments. Instead, it is possible to show6, since δ(νd) can be written asα(νd)/2, that by using the definition of integrated linestrength, which is relatedto the absorption of narrowbanded light due to a transition, α(νd), through thefollowing expression,

α(νd) = SNxLχabsV (νd) = α0χ

absV (νd), (3.40)

6see Appendix A.1 for more information.

36


that the phase shift and attenuation as a function of normalized detuning frequency,φ(νd) and , δ(νd), can be expressed succinctly as

φ(νd) =

Re[k(νd)

]− k0

L =

α0

2χdispV

(νd), (3.41)

and

δ(νd) = − Im[k(νd)

]L =

α0

2χabsV

(νd). (3.42)

The panels (a) of Figure 3.3 and 3.4 show the individual contributions to thephase shift and attenuation difference from a R(3/2) transition without appliedmagnetic field, respectively. Their sums are displayed in the corresponding (b)panels. As is clearly seen the difference between the contributions from LHCP andRHCP light in the absence of magnetic field is zero and thus no MRS signal isproduced neither from magnetic circular birefringence nor from magnetic circulardichroism. Equation (3.41) describes the phase shift of a plane monochromaticwave in the vicinity of a transition and agrees well with expressions by Ma etal. [46] derived for the same entity for the wavelength-modulated noise-immunecavity-enhanced optical heterodyne molecular spectroscopy (wm-NICE-OHMS)technique.

−0.1 −0.05 0 0.05 0.1

(a)


Phase

shift[a.u.]

φM′′M′L

−φM′′M′R

−0.1 −0.05 0 0.05 0.1

(b)


Phase

shift[a.u.]

φL

−φR

φL − φR

Figure 3.3. Individual contributions to the phase shift from a R3/2(3/2) transition withoutapplied magnetic field. Panel (a) displays the phase shifts from the M ′′ →M ′ transitionsfor an unsplit R3/2(3/2) transition, where the phase shifts corresponding to LHCP lightare given in red whereas the ones for RHCP are given in blue. Panel (b) shows the sumover all M ′′ → M ′ transitions for ∆M = ±1, which correspond to LHCP and RHCPlight, respectively. The black curve shows the total phase shift, given by ∆φ = φL − φR,which cancels out in the absence of magnetic field.

37


−0.1 −0.05 0 0.05 0.1

(a)


Attenuation

[a.u.]

δM′′M′

L

−δM′′M′

R

−0.1 −0.05 0 0.05 0.1

(b)


Attenuation

[a.u.]

δL

−δR

δL − δR

Figure 3.4. Individual contributions to the attenuation from a R3/2(3/2) transitionwithout applied magnetic field. Panel (a) displays the attenuations from the M ′′ →M ′

transitions for an unsplit R3/2(3/2) transition, where the phase shifts corresponding toLHCP light are given in red whereas the ones for RHCP are given in blue. Panel (b)shows the sum over all M ′′ →M ′ transitions for ∆M = ±1, which correspond to LHCPand RHCP light, respectively. The black curve shows the total attenuation, given by∆δ = δL − δR, which cancels out in the absence of magnetic field.

3.4.2 Phase shift and attenuation due to a transition in termsof the integrated linestrength and the dispersion lineshapefunction in the presence of magnetic field

If an external magnetic field is applied to a gaseous medium containing paramagneticmolecules the degeneracy of the various states will be lifted and each state will besplit into several sub-states according to their quantum number M . The two circularpolarizations (LHCP and RHCP) applied parallel/anti-parallel to the directionof the magnetic field will experience different phase shifts for each M ′′ → M ′

transition they interact with. It is thus possible to define a specific integratedlinestrength for each of the split M ′′ → M ′ transitions from a lower state i toan upper state j, denoted SM ′′M ′ . It is also convenient to introduce a transitionspecific dimensionless integrated linestrength, SM ′′M ′ , which can be related tothe specific integrated linestrength and the linestrength of the unsplit transitionthrough,

SM ′′M ′ = SM ′′M ′ · S, (3.43)

where an inherent property of this dimensionless entity is that the sum of all splittransitions must equal unity, i.e.,∑

M ′′M ′

SM ′′M ′ = 1. (3.44)

38


The phase shifts for the two helical components of the light, induced by eachM ′′ →M ′ transition from a lower state to an upper state, can thereby be rewrittenas,

φM ′′M ′(ν) =SM ′′M ′NxL

2χdispV

(ν)

=SNxL

2SM ′′M ′ χdispV

(ν)

=α0

2SM ′′M ′ χdispV

(ν). (3.45)

where the shifted normalized detuning frequency, ν, is given by

ν = νd − νa, (3.46)

where νa in turn, is the Doppler-width normalized shift of the transition due to anexternal magnetic field, given by

νa = (M ′g′ −M ′′g′′)µBB√

ln 2/δνD, (3.47)

where g′ and g′′ are the g-factors for the upper and lower state, respectively, µB isthe Bohr magneton and B the amplitude of the magnetic field.

The corresponding expression for the attenuation induced by each M ′′ →M ′

transition can be written as

δM ′′M ′(ν) =SNxL

2SM ′′M ′ χabsV

(ν)

=α0

2SM ′′M ′ χabsV

(ν). (3.48)

Phase shift induced by a general transition In the general case the phase shiftsfor the LHCP and RHCP light can be expressed as a sum over all M ′′ → M ′

transitions,

φL/R(ν) =α0

2

∑M ′′M ′

SM ′′M ′ χdispV (ν). (3.49)

The corresponding expression for the attenuation can thereby be written as

δL/R(ν) =α0

2

∑M ′′M ′

SM ′′M ′ χabsV (ν). (3.50)

39


Phase shift induced by a rotational-vibrational transition The relative inte-grated linestrength, SM ′′M ′ , for a ro-vib transition between two sub-states bothadhering to the same Hund’s coupling case, which implies that they can be ex-pressed in the same basis set, is proportional to a Clebsch-Gordan coefficient, which,in turn, can be expressed in terms of the Wigner 3-j symbol [47]. This occurs inthe fundamental ro-vib band of NO, where the sub-states split according to theirMJ -values. For the experimental configuration of MRS when the magnetic field isparallel to the propagation of the light, the relative integrated linestrength can bewritten as,

SM ′′JM

′J

= 3

(J ′ 1 J ′′

−M ′J ±1 M ′′J

)2

, (3.51)

where a factor of 3 originates from the normalization requirement given in equa-tion (3.44), since an inherent property of the Wigner 3-j symbol is that the sumover all M ′′J →M ′J transitions and a given type of excitation (∆MJ = 0,+1,−1)must equal unity. This implies that the total phase shifts for LHCP and RHCPlight can be expressed as,

φL/R(ν) =α0

2

∑M ′′M ′

3

(J ′ 1 J ′′

−M ′J ±1 M ′′J

)2

χdispV (ν), (3.52)

where ν is given by

ν = νd − (M ′Jg′J −M ′′J g′′J)µBB

√ln 2/δνD. (3.53)

In an analogous manner the attenuations for LHCP and RHCP light can beexpressed as,

δL/R(ν) =α0

2

∑M ′′M ′

3

(J ′ 1 J ′′

−M ′J ±1 M ′′J

)2

χabsV (ν). (3.54)

Panels (a) of Figure 3.5 and 3.6 show the individual contributions to the phaseshift and attenuation difference, under the influence of an external magnetic field,from a R3/2(3/2) transition while the corresponding (b) panels display their sums.As can be seen the magnetic field gives rise to differences between the LHCPand RHCP light, which thus produces a MRS signal both from magnetic circularbirefringence and from magnetic circular dichroism, although at different strengths.

40


−0.1 −0.05 0 0.05 0.1

(a)


Phase

shift[a.u.]

φM′′M′L

−φM′′M′R

−0.1 −0.05 0 0.05 0.1

(b)


Phase

shift[a.u.]

φL

−φR

φL − φR

Figure 3.5. Individual contributions to the phase shift from a R3/2(3/2) transition. Panel(a) displays the phase shifts from the individual M ′′ →M ′ transitions for a magneticallysplit R3/2(3/2) transition, where the phase shifts corresponding to LHCP light are givenin red whereas the ones for RHCP are given in blue. Panel (b) shows the sum overall M ′′ → M ′ transitions for ∆M = ±1, which correspond to LHCP and RHCP light,respectively. The black curve shows the total phase shift, given by ∆φ = φL − φR.

−0.1 −0.05 0 0.05 0.1

(a)


Attenuation[a.u.]

δM′′M′

L

−δM′′M′

R

−0.1 −0.05 0 0.05 0.1

(b)


Attenuation[a.u.]

δL

−δR

δL − δR

Figure 3.6. Individual contributions to the attenuation from a R3/2(3/2) transition. Panel(a) displays the attenuations from the individual M ′′ →M ′ transitions for a magneticallysplit R3/2(3/2) transition, where the phase shifts corresponding to LHCP light are givenin red whereas the ones for RHCP are given in blue. Panel (b) shows the sum overall M ′′ → M ′ transitions for ∆M = ±1, which correspond to LHCP and RHCP light,respectively. The black curve shows the total attenuation, given by ∆δ = δL − δR.

Phase shift induced by a rotational-vibrational Q-transition It is now appro-priate to also consider the particular case of the ro-vib Q-transitions for whichJ ′′ = J ′ = J and g′′J = g′J = gJ . This implies that all transitions for a given helicalcomponent of the light will experience the same frequency detuning shift, givenby ±gJµBB. Furthermore, the sum of relative integrated linestrengths over allM ′′J →MJ transitions for a given type of excitation equals unity. This means thatthe phase shift for LHCP and RHCP light conveniently reduces to,

41


φL/R(ν) =α0

2χdispV (ν), (3.55)

where ν = νd − νa, where, in turn, νa = ∓gJµBB√

ln 2/δνD for the two circularcomponents, respectively. The total phase shift for a Q-transition is given by thedifference in phase shifts for LHCP and RHCP light can be written as

∆φ(ν) =α0

2∆χdispV (ν), (3.56)

where ∆χdispV (ν) is the difference between the dispersion lineshape functions ex-perienced by the LHCP and RHCP components of the light. The correspondingdifference in attenuation between LHCP and RHCP light and can be written as

∆δ(ν) =α0

2∆χabsV (ν). (3.57)

where ∆χabsV (ν) is the difference between the corresponding absorption lineshapefunctions.

Panels (a) of Figure 3.7 and 3.8 show the individual contributions to the phaseshift and attenuation difference from a Q3/2(3/2) transition, under the influenceof an external magnetic field, whereas the corresponding (b) panels display theirsums. The magnetic field causes the same effect as that described by Figure 3.5and 3.6. In this case, however, the individual M ′′ → M ′ transitions all occur atthe same detuning frequency (for a given helical component of the light). Thissignificantly simplifies the expressions, which is further discussed for the case ofFaraday modulation spectroscopy in Section 4.4.2.

42


−0.1 −0.05 0 0.05 0.1

(a)


Phase

shift[a.u.]

φM′′M′L

−φM′′M′R

−0.1 −0.05 0 0.05 0.1

(b)


Phase

shift[a.u.]

φL

−φR

φL − φR

Figure 3.7. Individual contributions to the phase shift from a Q3/2(3/2) transition. Panel(a) displays the phase shifts from the M ′′ → M ′ transitions for a magnetically splitQ3/2(3/2) transition, where the phase shifts corresponding to LHCP light are given in redwhereas the ones for RHCP are given in blue. Panel (b) shows the sum over all M ′′ →M ′

transitions for ∆M = ±1, which correspond to LHCP and RHCP light, respectively. Theblack curve shows the total phase shift, given by ∆φ = φL − φR.

−0.1 −0.05 0 0.05 0.1

(a)


Attenuation[a.u.]

δM′′M′

L

−δM′′M′

R

−0.1 −0.05 0 0.05 0.1

(b)


Attenuation[a.u.]

δL

−δR

δL − δR

Figure 3.8. Individual contributions to the attenuation from a Q3/2(3/2) transition. Panel(a) displays the attenuations from the M ′′ → M ′ transitions for a magnetically splitQ3/2(3/2) transition, where the phase shifts corresponding to LHCP light are given in redwhereas the ones for RHCP are given in blue. Panel (b) shows the sum over all M ′′ →M ′

transitions for ∆M = ±1, which correspond to LHCP and RHCP light, respectively. Theblack curve shows the total attenuation, given by ∆δ = δL − δR.

43

Chapter 4

Faraday ModulationSpectroscopy – the modulatedcase

”I was at first almost frightened when I saw such mathematicalforce made to bear upon the subject, and then wondered to seethat the subject stood it so well.”

— Michael Faraday - Letter to James Clerk Maxwell

Faraday modulation spectroscopy (FAMOS) [20, 21, 26, 48–50], sometimes alsoreferred to as Faraday rotation spectroscopy (FRS) [24, 30], Zeeman modulationspectroscopy (ZMS) [5, 18, 28], modulated laser magnetic resonance spectroscopyor modulated Magnetic rotation spectroscopy (MRS) [6, 17]1, is a spectroscopicdetection technique that further enhances the sensitivity of the unmodulatedmagnetic rotation spectrometry (MRS) technique [3, 44, 45] for detection of param-agnetic species in gas phase. The sensitivity of the FAMOS technique is enhanced,compared to that of MRS, by using a modulation of the magnetic field togetherwith lock-in detection at the modulation frequency, f , (typically f > 1 kHz). Thissuppresses, to a large extent, the influence of etalons and other interferences suchas laser amplitude noise and background signals.

1From here on the abbreviation FAMOS will be used when referring to this technique.

45

Faraday Modulation Spectroscopy – the modulated case

4.1 Introducing the modulation of the magnetic field

The theoretical description given in Chapter 3 only considered the case of a staticmagnetic field, i.e. it does not include the effects of a sinusoidal modulation of themagnetic field, followed by a demodulation of the detector signal at an harmonicof the modulation frequency, on the measured signals. Such a modulation cansignificantly increase the signal-to-noise ratio and has therefore been frequentlyadopted. The modulation procedure implies that the magnetic field, B(t), istime-dependent and can be expressed as,

B(t) = B0 cos(ωt), (4.1)

where ω is the angular modulation frequency. This, in turn, implies that themodulation amplitude, given by equation (3.47), will become time dependent,which subsequently yields a time dependent shifted normalized detuning frequency,introduced in equation (3.46), that can now be expressed as,

ν(t) = νd − νa(t), (4.2)

where νa(t) is given by

νa(t) = (M ′g′ −M ′′g′′)µBB0 cos(ωt)√

ln 2/δνD. (4.3)

It can be intuitively understood that this will lead to a corresponding modulationof the phase and the attenuation, i.e. φ(t) and δ(t), and thereby a rotation ofthe plane of polarization ∆φ(t)/2, which leads to a modulation of the transmittedintensity, It(t). An expression for this entity was given in equation (3.24) andis restated here, with the time-independent terms that do not contribute to theFAMOS signal omitted, as

It(t) =I0 sin(2θ)

2

[∆φ(t) + ε∆δ(t)

], (4.4)

where ∆φ(t) and ∆δ(t) take slightly different forms depending on which type oftransition that is targeted and ε is the uncrossing angle normalized unbalancingterm between LCHP and RHCP light, given by ε/ sin(2θ).

46


4.2 The signal in terms of Fourier coefficients of mod-ulated lineshape functions

As was mentioned above, the FAMOS signal is extracted by the use of a lock-inamplifier set for detection at the modulation frequency2. This implies that theFAMOS signal, SF , can be written as

SF =η

τ

τ∫0

It(t) cos(ωt) dt =

=η sin(2θ)I0

2

1

τ

τ∫0

∆φ(t) cos(ωt) dt+ ε · 1

τ

τ∫0

∆δ(t) cos(ωt) dt

, (4.5)

where η is an instrumentation factor and τ is the integration time. By examiningequation (4.6) it can be concluded that the FAMOS signal can be expressed interms of integrals of the form

2

τ

τ∫0

χdispV (ν) cos(ωt) dt (4.6)

and

2

τ

τ∫0

χabsV (ν) cos(ωt) dt, (4.7)

where χdispV and χabsV are the normalized modulated Voigt lineshape functions, given

by the ratios of the area-normalized modulated lineshape functions, χdispV and χabsV ,

and the peak-value of the Gaussian lineshape function, χ0. Since all χdispV and χabsVare periodic, it is possible to identify the expressions above, in line with what hasbeen done for the wavelength modulation technique (WMS) [11, 12, 51], i.e. asthe even components of the 1st Fourier coefficient of the Doppler-peak-normalizedmodulated dispersion and absorption lineshape functions, respectively. Denotingthese integrals χdisp,evenV,1 and χabs,evenV,1 , i.e.

2In general the demodulation can occur at any harmonic of the modulation frequency, but sincethe first harmonic gives a maximum at resonance this is usually preferred and therefore onlyconsidered here.

47


χdisp,evenV,1 =2

τ

τ∫0

χdispV (ν) cos(ωt) dt (4.8)

and

χabs,evenV,1 =2

τ

τ∫0

χabsV (ν) cos(ωt) dt, (4.9)

makes it possible to express the FAMOS signal as

SF = S0χF , (4.10)

where the entity S0 is introduced as the FAMOS signal strength given by

S0 = ηα0χ0

8I0 sin(2θ), (4.11)

and where χF is the Doppler-peak-normalized FAMOS lineshape function, which,in general, is given by

χF =∑M ′′M ′

[SM ′′M ′∆χdisp,evenV,1 + SM ′′M ′ ε∆χabs,evenV,1

], (4.12)

where ∆χdisp,evenV,1 and ∆χabs,evenV,1 denote the difference in 1st Fourier coefficients ofthe modulated Voigt absorption and dispersion lineshape functions between LHCPand RHCP light, respectively.

Rather than using the number density, Nx, it is often convenient to expressthe FAMOS signal in terms of the partial pressure, px, or the concentration (molefraction), cx, of the constituent in question. This is achieved by expressing theFAMOS signal strength in terms of the gas integrated linestrength, S′, whichenables us to express the FAMOS signal strength as,

S0 = ηSNxLχ0

8sin(2θ)I0 (4.13)

= ηS′crelpLχ0

8sin(2θ)I0 (4.14)

= Satm0 crelp

p0, (4.15)

(4.16)

48


Detuning frequency, νd

χdisp,even

L,1

(a)


χabs,even

L,1

(b)

Figure 4.1. Panel (a) shows the 1st Fourier coeffients of modulated dispersion lineshapefunctions for LHCP (red) and RHCP (blue) as a function of normalized detuning frequency,νd for different normalized modulation amplitudes, νa. The different curves correspondto νa of 1.27, 2.54, 3.81, 5.08, and 6.35. Panel (b) displays the 1st Fourier coeffients ofmodulated absorption lineshape functions for LHCP (red) and RHCP (blue) as a functionof normalized detuning frequency, νd for the same modulation amplitudes.

where p is the total pressure of the sample, p0 is a reference pressure (here taken as1 atm) and Satm0 the FAMOS signal strength under atmospheric pressure conditions,given by

Satm0 = ηI0S′p0Lχ0

8sin(2θ). (4.17)

The equations stated above show that it is possible to express the FAMOSsignal as a weighted sum of the difference between the 1st Fourier coefficients ofthe Doppler-peak-normalized dispersion and absorption lineshape functions for theparticular transition addressed. Furthermore, they also show that the FAMOSlineshape is a function of only two normalized variables, namely, the Doppler-width normalized modulation amplitude, νa, and the Doppler-width normalizeddetuning frequency, νd. The 1st Fourier coefficients of modulated dispersionand absorption lineshape functions are visualized in Figure 4.1 for a variety ofmodulation amplitudes. The left axes correspond to the 1st Fourier coefficient ofthe Doppler-peak-normalized modulated lineshape function (henceforth referred toas either the 1st Fourier coefficient of the modulated lineshape function or simplythe 1st Fourier coefficient), i.e. χdisp,even1 or χabs,even1 as a function of the Doppler-width normalized detuning, νd, for various Doppler-width normalized modulationamplitudes (henceforth referred to as normalized modulation amplitudes).

49


4.3 Lineshape asymmetries

In Sections 3.3.2 and 3.3.3 a theoretical model including the effects that giverise to asymmetric lineshapes for the static case of MRS was described. It is nowappropriate to extend this description to include also the modulated case of FAMOSand the Fourier coefficient description introduced in the previous section. This givesa theoretical model for the FAMOS signals that include lineshape asymmetries andcan be evaluated swiftly through the Fourier coefficients of modulated dispersionand absorption lineshape functions, which opens up the possibility of real-timecurve fitting to FAMOS signals in the same way that has been done for WMS[11, 12, 51].

As was discussed for the MRS case there are mainly two phenomena thatgive rise to asymmetries in the lineshapes; (i) polarization imperfections and (i)a frequency dependent laser intensity. The latter can be handled by the sameapproach as was introduced in Section 3.3.3, whereas the former needs to beadjusted to account for the modulation procedure.

4.3.1 The influence of polarization imperfections

Just as in the case for MRS the part of the electric field propagating in thez-direction that originates from the absorber can, in general, be written as,

E =E0√

2ei(k0L−ωt)

[eL(1− ε)e−δL+iφL + eR(1 + ε)e−δR+iφR

], (4.18)

which upon transmission through the imperfect analyzer, with a Jones’ matrixaccording to equation (3.17), results in the following expression for the transmittedintensity, which has become time-dependent due to the modulation

It(t) =I02

exp[−2δ(t)]

(α2 + ε2β2

cosh[∆δ(t)]− cos[∆φ(t) + 2θ]

+ 2α2 + β2ε sinh[∆δ(t)]

+ ε2α2 + β2

cosh[∆δ(t)] + cos[∆φ(t) + 2θ]

), (4.19)

where the differential phase shift, the differential attenuation, and the averageattenuation are now, on account of the modulation, time dependent. The corre-sponding time-averaged entities, here deonted ∆φ, ∆δ, and δ are now given bythe nth Fourier coefficients of the modulated dispersion and absorption lineshapefunctions according to,

50


∆φ = φL − φR =α0

2

(χdisp,even,LV,n − χdisp,even,RV,n

)=α0

2∆χdisp,evenV,n , (4.20)

∆δ = δL − δR =α0

2

(χabs,even,LV,n − χabs,even,RV,n

)=α0

2∆χabs,evenV,n , (4.21)

and

δ =1

2

(δL + δR

)=α0

4

(χabs,even,LV,n + χabs,even,RV,n

). (4.22)

We can now write the 1f FAMOS signal for small absorbances (SA-FAMOS),corresponding to the SA-MRS expression in equation (3.22), where terms that donot contribute to the modulated signals have been neglected, as

SF ≈ ηα0I0 sin(2θ)

4

∑M ′′M ′

[SM ′′M ′∆χdisp,evenV,1 + SM ′′M ′ ε∆χabs,evenV,1

], (4.23)

where η is a lock-in instrumentation factor and ∆χdisp,evenV,1 and ∆χabs,evenV,1 are the

1st Fourier coefficients of the modulated lineshapes for the differential phase shiftand the differential attenuation, respectively. The dispersion term gives rise tosignals that are fully symmetrical and are identical to the ones derived in Chapter 3and agree well with those derived by Liftin et al. [6] and Blake et al. [17] for theunmodulated case. The absorption term provides an asymmetric contributionto the FAMOS signals, given by a product of the normalized unbalancing termε and the 1st Fourier coefficient of the modulated lineshapes for the differentialattenuation. As can be seen from the expression such an asymmetry results onlyas a consequence of polarization imperfections (ε 6= 0) that originate either froman initial ellipticity of the light passing through an imperfect polarizer or fromstress-induced birefringence in the cell windows [45] that are transmitted throughthe system due to imperfections of the analyzer. Figure 4.2 illustrates the influenceof different ε on the FAMOS lineshape.

51



FAMOSsign

al

(a) ǫ = -0

ǫ = -0.002

ǫ = -0.004

ǫ = -0.006

ǫ = -0.008


FAMOSsign

al

(b) ǫ = -0

ǫ = -0.002

ǫ = -0.004

ǫ = -0.006

ǫ = -0.008

Figure 4.2. FAMOS lineshapes for different normalized unbalancing terms ε. Thelineshapes are calculated from equation (4.23)

4.4 In the mid-infrared region – MIR-FAMOS

1840 1850 1860 1870 1880 1890 1900 1910

P Q R

Wavenumber, [cm−1]

Transm

ission

[a.u.]

Figure 4.3. Transmission spectra for NO in the fundamental ro-vib band includingtransitions in both the 2Π1/2 and 2Π3/2 sub-systems. The integrated linestrengths,pressure broadening coefficients, and the wavenumbers of the transitions were taken fromthe HITRAN 2008 database [7]. The simulation represent pressure broadened conditions.

As was discussed in the introduction and visualized in Figure 1.2 the largestlinestrengths for rotational-vibrational transitions are found in the fundamentalro-vib band. The transmission spectra of NO for parts of the P, R and Q-branchis shown in Figure 4.3. The most prominent peaks are located in the P- andR-branch. However, the largest linestrengths do not always provide the largestFAMOS signals. In Figure 4.4 a FAMOS spectra from NO in the fundamental

52


ro-vib band is shown. By inspection it can be concluded that the most sensitivetransitions are situated in the Q-branch. In the following section the theoreticalmodel required to produce such a spectra will be described.

1840 1850 1860 1870 1880 1890 1900 1910

P Q R

Wavenumber, [cm−1]

FAMOSsign

al[a.u.]

1875 1875.2 1875.4 1875.6 1875.8

Q-branch

Figure 4.4. FAMOS spectra from transitions in the fundamental ro-vib band of NO in-cluding transitions in both the 2Π1/2 and 2Π3/2 sub-systems. The integrated linestrengths,pressure broadening coefficients and the wavenumbers of the transitions were taken fromthe HITRAN 2008 database [7], whereas the gJ factors were taken from Herrmann et al.[18].

4.4.1 Adressing an arbitrary ro-vib transition

In the mid-infrared (MIR) region the rotational-vibrational transitions in NO alladhere to Hund’s coupling case (a) which means that the magnetic field splits thestates into several sub-states according to their magnetic quantum number, MJ ,which ranges from −J to J . This implies that the phase shifts and attenuations,first introduced in equations (3.52) and (3.54), can be rewritten in terms of the1st Fourier coefficients of modulated dispersion and absorption lineshape functionsaccording to equations (4.8) and (4.9), respectively. This gives us the followingexpression for the 1st Fourier coefficients for a general ro-vib transition in FAMOS,

φL/R(νd, νa) =

=α0χ0

2

∑M ′′M ′

3

(J ′ 1 J ′′

−M ′J ±1 M ′′J

)2

χdisp,evenV,1 (νd, νa) (4.24)

and

53


δL/R(νd, νa) =

=α0χ0

2

∑M ′′M ′

3

(J ′ 1 J ′′

−M ′J ±1 M ′′J

)2

χabs,evenV,1 (νd, νa), (4.25)

where νd and νa are given by

νd = (ν − ν0)√

ln 2/δνD (4.26)

and

νa = (M ′Jg′J −M ′′J g′′J)µBB0

√ln 2/δνD, (4.27)

respectively.

4.4.2 Addressing a ro-vib Q-transition

Unlike the cases for P- and R-transitions, for which transitions occur betweenstates with dissimilar rotation quantum numbers, i.e. ∆J ± 1, respectively, theQ-transitions occur between states that have the same rotational quantum number,i.e. J ′′ = J ′ = J and subsequently g′′J = g′J = gJ . This implies that all transitionsaddressed by a certain helical component of the light (LHCP or RHCP light) willappear at the same frequency detuning, given by ±gJµBB. Also, the sum overall M ′′J →M ′J transitions for a given type of circular component of light, LHCPor RCHP, equals unity. This implies that the total phase shifts and attenuationsof the two circular components of the light can be written in terms of 1st Fouriercoefficients, as

φL/R =α0χ0

2χdisp,evenV,1 (4.28)

and

δL/R =α0χ0

2χabs,evenV,1 , (4.29)

with LHCP (L) corresponding to transitions for which M ′ −M ′′ = 1 and RHCP(R) to M ′ −M ′′ = −1.

Let us now introduce ∆φ(νd, νa) and ∆δ(νd, νa) as the Fourier coefficients ofthe total phase shift and the total attenuation difference between the two helicalcomponents of the light, just as was done for MRS in Chapter 3, as

54


∆φ(νd, νa) =α0χ0

2

[χdisp,evenV,1 (νd, ν

La )− χdisp,evenV,1 (νd, ν

Ra )]

(4.30)

and

∆δ(νd, νa) =α0χ0

2

[χabs,evenV,1 (νd, ν

La )− χabs,evenV,1 (νd, ν

Ra )], (4.31)

where νLa and νRa is the magnitude of the normalized modulation amplitude forLHCP and RHCP light, given by

νLa = gJµBB0

√ln 2/δνD (4.32)

and

νRa = −gJµBB0

√ln 2/δνD, (4.33)

respectively.

For a Q-transition further simplifications can be made by examining thesymmetries of the modulated lineshape functions. Only the differential phase shiftwill be considered here as the same applies also to the attenuation difference.

Since the normalized modulation amplitude of RHCP light is equal to thenegative of that of the LHCP light, it can be concluded that the 1st Fouriercoefficient for RHCP light for a Q-transition can be written as the negative of the1st Fourier coefficient for LHCP light with negative detuning, i.e.,

χdisp,evenV,1 (νd, νLa ) = −χdisp,evenV,1 (−νd,−νRa ). (4.34)

Furthermore, since the 1st Fourier coefficient is an even function with respect tothe detuning frequency, i.e.,

χdisp,evenV,1 (νd, νL/Ra ) = χdisp,evenV,1 (−νd, νL/Ra ), (4.35)

it can be concluded that for a Q-transition,

χdisp,evenV,1 (νd, νLa ) = −χdisp,evenV,1 (νd, ν

Ra ). (4.36)

This implies that the FAMOS expression for the total phase shift for a Q-transitioncan be written concisely as

∆φ(νd, νa) =SNxLχ0

22χdisp,evenV,1 (νd, νa), (4.37)

55


where νd and νa are, again, the normalized detuning frequency and the magnitudeof the normalized modulation amplitude given by

νd =√

ln 2(ν − ν0)

δνD(4.38)

and

νa = gJµBB0

√ln 2/δνD, (4.39)

respectively.

By using the 1st Fourier coefficients it is straightforward to assess the propertiesof a Q-transition in terms of a signal-strength- and concentration-normalizedFAMOS signal (henceforth referred to as normalized FAMOS signal), SF , given by

SF = SF /Satm0 cx =

2δνLδνatmL

[χdisp,evenV,1 + εχabs,evenV,1

], (4.40)

where δνatmL is the Voigt parameter under atmospheric pressure conditions given

by γp0

√ln 2/δνD and Satm0 is the FAMOS signal strength under the same pressure

conditions. In Figure 4.5 the FAMOS spectra from NO in the Q3/2-branch is shown.The difference in signal strength stems from different integrated linestrengths andgJ -values. It is clearly seen that the Q3/2(3/2) transition is the most suitable forFAMOS. Table 4.1 shows the integrated linestrengths of the most magneticallysensitive Q3/2-transitions taken from the HITRAN database [7].

1875.2 1875.3 1875.4 1875.5 1875.6 1875.7 1875.8 1875.9

Wavenumber [cm−1]

FAMOSsign

al[a.u.]

Q3/2(3/2)

Q3/2(5/2)

Q3/2(7/2)Q3/2(9/2)Q3/2(11/2)

Figure 4.5. FAMOS spectra from nitric oxide in the Q3/2-branch showing only the mostmagnetically sensitive transitions.

If we assume that NO exists in such low concentrations that the collisionbroadening is dominated by collisions with air, the collision broadening parameter,γ, can be extracted directly from the HITRAN database [7] which, for the Q3/2(3/2)transition in NO at 5.331 µm, is 0.0659 cm−1/atm. Moreover, by assuming a

56


temperature of 296 K for which the Doppler width is 0.002 11 cm−1, the Voigtparameter takes a value of 26. This is used in the right axes of Figure 4.6, wherethe normalized FAMOS signal for the Q3/2(3/2) transition in the fundamentalrotational-vibrational band of NO is calculated for the case without polarizationimperfections, i.e. ε = 0, using SF =

(δνL/13

)· χdisp,evenV,1 .

Table 4.1. Integrated linestrength values and wavenumbers for the Q transi-tions in the 2Π3/2 sub-system from the HITRAN 2008 database [7].

Transition Wavenumber Wavelength Integrated linestrength[cm−1] [µm] [cm−1/molecule cm−2]

Q3/2(3/2) 1875.81 5.33102 3.1582× 10−20

Q3/2(5/2) 1875.72 5.33128 1.9471× 10−20

Q3/2(7/2) 1875.60 5.33164 1.3601× 10−20

Q3/2(9/2) 1875.43 5.33210 1.0036× 10−20

Q3/2(11/2) 1875.24 5.33266 7.6016× 10−21

Q3/2(13/2) 1875.00 5.33333 5.8355× 10−21

Q3/2(15/2) 1874.73 5.33410 4.4979× 10−21

The six panels A-F in Figure 4.6 represent Voigt parameters of 0.1, 0.5, 2, 5 and20, respectively. Panel A represents a case close to the Doppler limit whereas panelsB, C, and D represent cases where the collision broadening is slightly smaller, similarto, and slightly larger than the Doppler-width, respectively. The panels E and Ffinally represent cases where the collision broadening dominates. The eleven curvesin each panel correspond to normalized modulation amplitudes, νa, of 0.1, 0.2, 0.4,0.5, 0.67, 1, 1.5, 2, 2.5, 5 and 10, respectively. The various panels all show that the1st Fourier coefficients are fully symmetric with respect to detuning frequency andthat their maximum values appear on resonance (νd = 0). It is also interesting tonote that when the modulation amplitude is increased to an overmodulated statethe shape of the curves has changed markedly from the previously “2f -like” shaperecognized from WMS to a shape characterized by a plateau in the center. However,reaching these modulation amplitudes require significantly larger magnetic fieldsthan are normally applied in FAMOS and does not yield any advantages when itcomes to signal strength wherefore such conditions are of less importance from anexperimental point of view.

The on-resonance value of the normalized signal

Panel A in Figure 4.7 shows how the modulation amplitude influences the on-resonance (maximum) value of the normalized FAMOS signal from a Q-transition

57


Figure 4.6. Left axes: the 1st Fourier coefficient of the Doppler-peak-normalized magnetic-field-modulated dispersion lineshape function, i.e. χdisp,even

1 ; right axes: the signal-strength- and concentration-normalized FAMOS signal, SF = SF /S

atm0 cx, both as a

function of the Doppler-width-normalized detuning, νd. The six panels A - F representVoigt parameters of 0.1, 0.5, 1, 2, 5, and 20, respectively. The eleven curves in each panelcorrespond to modulation amplitudes, νa, of 0.1, 0.2, 0.4, 0.5, 0.67, 1, 1.5, 2, 2.5, 5, and10, respectively. The curves representing modulation amplitudes of 0.4,1 and 2.5 aredrawn as dashed, dotted and dashed-dotted, respectively.

for various Voigt parameters ranging from 0.1 to 25 (0.1, 0.25, 0.5, 1, 2, 5, 10, 15, 20

58


and 25, respectively). The figure also shows that the on-resonance value increaseslinearly with modulation amplitude for small modulation amplitudes (magneticfield strengths) until the slope decreases and the on-resonance value reaches amaximum, after which it starts to decrease. Hence, for each Voigt parameter(pressure) there is a modulation amplitude for which the FAMOS signal takes amaximum value. The upper axis in panel A in Figure 4.7 represents the magneticfield corresponding to the modulation amplitude for the Q3/2(3/2) transition inthe fundamental rotational-vibrational band of NO. For this particular transitionthe various curves represent pressures of ∼3, 7, 15, 29, 58, 146, 290, 440, 585 and730 Torr, respectively. This shows, for example, that the optimum magnetic fieldamplitude for a sample at 58 Torr (the dashed curve) is slightly above 200 G or ∼141 Grms.

Panel B in Figure 4.7 illustrates the dependence of the Voigt parameter onthe maximum value of the normalized FAMOS signal for various modulationamplitudes ranging from 0.1 to 20 (0.1, 0.2, 0.4, 0.67, 1, 1.5, 2, 2.5, 5, 10, 15 and 20,respectively). The curves display the same behavior as in panel A, meaning that foreach modulation amplitude there exists a Voigt parameter for which the normalizedFAMOS signal takes a maximum value. The upper axis represent the pressureconditions for the Q3/2(3/2) transition in the fundamental rotational-vibrationalband of NO. For this particular transition the 13 curves represent magnetic fieldamplitudes of ∼7, 14, 28, 35, 47, 70, 105, 140, 175, 350, 700, 1050 and 1400 G,respectively.

59


Figure 4.7. Panel A: The 1st Fourier coefficient of the normalized modulated dispersionlineshape function as a function of normalized modulation amplitude. The ten curvesrepresent Voigt parameters (δνDL ) of 0.1, 0.25, 0.5, 1, 2, 5, 10, 15, 20, and 25, respectively.The upper axis represents the magnetic fields that correspond to the various modulationamplitudes for the Q3/2(3/2) transition in the fundamental rotational-vibrational band ofNO. Panel B: The 1st Fourier coefficient of the normalized modulated dispersion lineshapefunction as a function of Voigt parameter. The 13 curves represent normalized modulationamplitudes (νa) of 0.1, 0.2, 0.4, 0.5, 0.67, 1, 1.5, 2, 2.5, 5, 10, 15, and 20, respectively. Theupper axis represents the pressure that corresponds to the various modulation amplitudesfor the Q3/2(3/2) transition in the fundamental rotational-vibrational band of NO.

The optimum conditions for the signal

The data shown in Figure 4.7 can be summarized in a plot as given in Figure 4.8where panel A shows the modulation amplitude that maximizes the on-resonancevalue of the normalized FAMOS signal for a given Voigt parameter, henceforthreferred to as the optimum modulation amplitude. In a similar manner, Panel Bshows the Voigt parameter that maximizes the on-resonance value of the normalizedFAMOS signal for a given modulation amplitude, henceforth referred to as optimumVoigt parameter. The upper and rightmost axes in both panels represent thecorresponding values for the Q3/2(3/2) transition in the fundamental rotational-vibrational band of NO. For instance, these plots show that the optimum magneticfield under low pressure conditions for this particular transition is around 105 Grms

and increases roughly with a slope of 3 Grms/Torr.

Figure 4.8 correspond well with previously reported data. For example it wasshown by Lewicki et al. [24] that for a pressure of 40 Torr the optimum magneticfield was 110 G (rms), which corresponds to a magnetic field amplitude of 156 G.

60


The simulations in Figure 4.8 give for the same set of parameters an optimummagnetic field of 158 G.

Figure 4.8. Panel A: The optimum modulation amplitude as a function of Voigt parameter.The upper and the right axes represent the corresponding pressure and magnetic field forthe Q3/2(3/2) transition in the fundamental rotational-vibrational band of NO, respec-tively. Panel B: The optimum Voigt parameter as a function of normalized modulationamplitude. The upper and the right axes represent the corresponding magnetic field andpressure for the Q3/2(3/2) transition in the fundamental rotational-vibrational band ofNO, respectively.

The maximum signal under optimum conditions

Panel A in Figure 4.9 displays the maximum normalized FAMOS signal on resonancethat can be obtained for a given Voigt parameter (lower axis) and for a givenpressure for the case when the Q3/2(3/2) transition in NO is addressed. Thisfigure corresponds well with predictions previously reported by Blake et al. [17]where the maximum signal increases rapidly for small Voigt parameters after whichit levels off towards a constant value. This shows that it is possible to reducethe sample-pressure down to a Voigt parameter of ∼ 2 with only a small loss(∼ 10%) in signal sensitivity. Similarly, panel B displays the maximum normalizedFAMOS signal on resonance that can be obtained for a given modulation amplitude(lower axis) and for a given magnetic field when the Q3/2(3/2) transition in thefundamental rotational-vibrational band of NO is addressed. As is shown, thereis little need to reach modulation amplitudes larger than ∼2 (representing 140 Gfor the rotational-vibrational Q3/2(3/2)-transition) at which already ∼ 90% of themaximum signal has been reached.

61


Figure 4.9. Panel A: The maximum normalized FAMOS signal on resonance that canbe obtained for a given Voigt parameter (lower axis) and for a given pressure for theQ3/2(3/2) transition in the fundamental rotational-vibrational band of NO (upper axis).Panel B: The maximum normalized FAMOS signal on resonance that can be obtained fora given modulation amplitude (lower axis) and for a given magnetic field for the Q3/2(3/2)transition in the fundamental rotational-vibrational band of NO (upper axis).

4.5 In the ultra-violet region – UV-FAMOS

Paper II and III focus on FAMOS when electronic transitions in the ultra-violet (UV)region in NO are addressed. The main advantage when addressing these transitionscomes from their considerably larger integrated linestrengths as compared to those inthe MIR region. Figure 4.10 displays the linestrengths for different wavelengths andshows that UV-FAMOS has the potential of reaching significantly lower detectionlimits due to the increase in integrated linestrengths of the targeted transitions.

There are several properties of the electronic transitions that differentiate themfrom the rotational-vibrational transitions. Most of these originate from the factthat the upper and lower state do not adhere to the same Hund’s coupling case andtherefore cannot be expressed in the same basis set. For instance, as is discussedin Paper II, when addressing the electronic X2Π(ν′′ = 0) - A2Σ+(ν′ = 0) band inNO the upper electronic state is a Σ+-state which lacks orbital angular momentum(Λ = 0). This implies that it follows Hund’s coupling case (b) as opposed to thelower level which adheres to Hund’s couling case (a). The rotational states in theupper level, whose energy levels are not given by the quantum number J but ratherby the N -quantum number, split according to their MS-value instead of theirMJ -value, which was the case for the rotational-vibrational transitions discussedpreviously. This has three major consequences for FAMOS addressing these states:

62


0 1 2 3 4 5 610

−24

10−22

10−20

10−18

10−16

wavelength [µm]Integratedlinestrength

[cm

−1/molecule

cm−2]

∼ 5.3 µmfundamentalro-vib band

∼ 2.65 µm1st overtone

band∼ 1.8 µm2nd overtone

band

∼ 227 nmelectronicband

Figure 4.10. Integrated linestrengths for the electronic band in NO around 227 nm.The linestrengths in this region are 2-3 orders of magnitude larger than those in thefundamental ro-vib band around 5.33 µm.

(i) The X2Π(ν′′ = 0

)−A2Σ+

(ν′ = 0

)band contains 12 types of branches,

not three: The energies of the rotational states in the upper electronic state arenot given by the total angular momentum, J , but rather by the quantum numberN . This means that compared to FAMOS targeting ro-vib transitions there aremore branches that can be addressed; not only the P, Q, and R branches but alsosatellite branches for which ∆N 6= ∆J [52]. In addition, since the lower state issplit into two by spin-splitting (the 2Π1/2 and 2Π1/2 substates) there are 6 typesof branches for each spin-split state, i.e. in total 12 branches [52, 53]. However,due to the weak ρ-type doubling present in NO, some of these branches overlap.Effectively, therefore the 12 branches appear frequency-wise as 8 branches. Overlapoccurs, for instance, in the case with the Q22(J ′′) and the QR12(J ′′) branches thatare addressed in Paper III and schematically illustrated in Figure 4.11.

Transitions within the Q22(J ′′) and the QR12(J ′′) bands are induced fromcommon lower states of type 2Π3/2(J ′′) to two types of upper states, 2Σ

(F ′1, N

′ =

N ′′)

and 2Σ(F ′2, N

′ = N ′′), respectively. Hence, an excitation of a transition in

the Q22(J ′′) branch implies that two transitions, both a Q22(J ′′) and a QR12(J ′′)transition, to two upper states with the same N but different J , are simultaneouslyinduced [54, 55].

(ii) Each 2Σ state is split by a magnetic field into two states, not 2J + 1states: In the upper A2Σ+

(ν′ = 0

)state it is the electronic spin, Sel (and not

the total angular momentum, J , as is the case for the lower X2Π(ν′′ = 0

)state),

that orients itself with respect to the external magnetic field. This gives rise toa splitting of the upper level into sub-states according to the magnetic quantum

63


Figure 4.11. Schematic illustration of the energy level structure for a Q22(J ′′) and aQR12(J ′′) transition in NO. The ground state is split into two (2Π1/2 and 2Π3/2) due tospin-splitting. In addition, each state is split according to its rotational quantum number,J , while also splitting into two states of opposite parity because of Λ-doubling. Theupper 2Σ state is split according to the rotation of the nuclei, N . Each rotational state isthen split into two because of the coupling between the electronic spin and the rotationof the nuclei (ρ-type doubling). The Q22(J ′′) and the QR12(J ′′) transitions correspondto transitions originating from a given 2Π3/2 ground state, but they couple to differentexcited states. Since the ρ-type doubling is very small or non-existent in NO, thesetransitions are often considered fully overlapping.

number of the electron, MS , not according to the magnetic quantum number of thetotal angular momentum, MJ , which is the case for ro-vib transitions. Therefore,each sub-state in the upper electronic state will experience a frequency shift givenby

∆ν = MSgSµBB0, (4.41)

where gS is the g-factor of the free electron, gS = 2.0023192. For a molecule witha single valence electron, such as NO, MS can only take two values, ±1/2. Thisimplies that the splitting of each rotational state in the upper electronic state willbe split solely into two sub-states whose relative splitting is gSµBB0. This is in

64


direct contrast to the ro-vib transitions discussed earlier for which the magneticfield splits each state into 2J + 1 equidistant sub-states whose energy separationdepends on J .

(iii) The relative transition strength between magnetically split sub-stateswithin the X2Π

(ν′′ = 0

)− A2Σ+

(ν′ = 0

)band cannot be simply expressed

in terms of the Wigner 3-j symbol. Unlike the case for a general ro-vib transitionwhere the relative transition strength can be expressed in terms of the Wigner 3-jsymbol given by equation (3.51), the relative transition strength of an electronictransistion needs to be expressed in dissimilar basis sets. This is caused by the factthat the lower and upper states belong to different coupling cases which complicatesthe calculations of the relative transition strength significantly, but at the sametime this opens up the possibility of an approximative theoretical model, which isdescribed in detail in Paper II and summarized here in Section 4.5.1.

4.5.1 A simple two-transition model

For most electronic transitions in NO, the total splitting of the upper level issignificantly larger than for the lower. For instance, the previously mentionedoverlapping Q22(J ′′) and QR12(J ′′) branches have magnetically split upper levelswhose total splitting is 2µBB0, which can be compared to the total splitting of thelower level given by 2JgJµBB0. For this particular pair of branches addressed bythe experimental investigation in Paper III the gJ -factor has a value of 2.47× 10−3.This implies that the lower level has a total splitting of ∼ 0.05µBB0, which is onlya factor of 1/40 of that of the upper state. The difference in splitting opens up fora convenient approximation, namely to neglect the splitting of the lower state withrespect to that of the upper. This can be schematically illustrated as is done inFigure 4.12.

Since each of the sub-states can be addressed by both LHCP and RHCP light,this actually implies that the system can be regarded as having four transitions.However, as will be shown below, it is possible interpret the system as having onlytwo transitions, making the model even simpler, and applicable to all transitionsfor which the splitting of one of the levels can be neglected in favour of the other.

In order to make this approximation, let us first conclude that in the absenceof magnetic field the total linestrengths for LHCP and RHCP are each equal to thetotal transition linestrength, previously given by Si,j , henceforth referred to as SΠ,Σ

because of the lower state 2Πi(J′′, N ′′) and the upper state 2Σ(F ′i , J

′, N ′,MS =±1/2). This implies that if we denote the relative linestrength for the unsplittransition for LHCP and RHCP light by, SL and SR, respectively, they are bothunity, as is illustrated in the left hand side of Figure 4.12. By introducing an

65


(a) (b)

+1/2

-1/2

+1/2

-1/2

-1/2

Figure 4.12. A simplified two-transition model of the energy level structure associatedwith an electronic transition in NO. The two panels, (a) and (b), correspond to cases forwhich B = 0 and B 6= 0, respectively. It is assumed that the upper level is magneticallysplit into two states, denoted by 2Σ(1/2) and 2Σ(−1/2). The splitting of the lower level isnegligible in comparison to that of the upper and can be neglected. Both LHCP (L) andRHCP (R) can induce transitions to both upper states, although with dissimilar relativelinestrengths.

external magnetic field, and under the approximation of the two-transition model,the upper state will be split, and four transitions appear. For simplicity, let usdenote the linestrengths of the transitions according to their upper states, as is donein Figure 4.12. This means, that the transition linestrength induced by LHCP andRHCP light from the lower 2Πi(J

′′, N ′′) state to the upper 2Σ(F ′i , J′, N ′,MS =

±1/2) state is denoted by SL/RMS

, which implies that it is possible to write the FAMOS

lineshape function for an electronic transition, χEV,1, according to equation (4.12),for ε = 0, as

χEV,1(νd, νa±MS

) = ∆SnetMS· χdisp,evenV,1 (νd, ν

aMS

)

−∆Snet−MS· χdisp,evenV,1 (νd, ν

a−MS

), (4.42)

where the Fourier coefficients are given by equation (4.6) and the normalizedmodulation amplitudes are given by νa±MS

= ±MSgSµBB0

√ln 2/δνD. We have

also introduced the equally strong ∆SnetMSand ∆Snet−MS

referred to as net relative

linestrengths given by ∆Snet±MS= SL±MS

− SR±MS. Since the net relative linestrength

are equally strong it is possible to write the FAMOS lineshape function as

66


χEV,1(νd, νa±MS

) = ∆Snet[χdisp,evenV,1 (νd, ν

aMS

)− χdisp,evenV,1 (νd, νa−MS

)]

= ∆Snet2χdisp,evenV,1 (νd, νa), (4.43)

where νa = gSµBB0

√ln 2/δνD and where symmetry properties of the Fourier

coefficients similar to the case for the rotational-vibrational Q-transitions havebeen utilized3. This means that the FAMOS signal strength can be written in asimilar manner to that of the ro-vib Q-transition, i.e. as

SEF (νd, νa) = S0χEV,1(νd, νa)

= S0∆Snet2χdisp,evenV,1 (νd, νa). (4.44)

This shows that the FAMOS signal strength addressing an electronic transitionin NO, S0∆Snet, is the product of the FAMOS signal strength according toequation (4.13), S0, and the net relative linestrength, ∆Snet. As is shown in PaperIII, this implies that the FAMOS signal strength addressing an electronic transitionin NO can be expressed as

SEF (νd, νa) = Satm0 crelp

p02χdisp,evenV,1 (νd, νa), (4.45)

where crel is the concentration (mole fraction or relative partial pressure) of NOin the gas, p is the total pressure, p0 is the reference pressure of 1 atm, and Satm0

is the FAMOS signal strength addressing an electronic transition in NO from asample with pure NO under atmospheric pressure conditions, which thus containthe net relative linestrength.

Since the FAMOS signal strength for electronic transitions, given in equa-tion (4.44), contains factors such as the integrated linestrength, the interactionlength and the peak value of the Gaussian lineshape function, it is possible to definea linestrength-, interaction length-, concentration- and Doppler peak-normalizedFAMOS signal (henceforth, in short, referred to as the ”normalized FAMOS signal”)denoted by SEF and defined as

SEF = SEF /(Satm0 crel)

=p

p02χdisp,evenV,1

(νd, νa). (4.46)

By the use of this expression it is possible to assess some of the properties ofthe FAMOS technique addressing electronic transitions in NO, in particular themagnetic field dependence and the pressure dependence. It is important to notethat in the following simulations the gS-factor was taken as 2, which is in line

3See Paper III for more information.

67


with measurements made by Takazawa et al. [55]. Also, the pressure broadeningcoefficient for NO in N2 was taken as that measured by Shao et al. [56] by directabsorption spectrometry for the particular transitions discussed in Section 4.5.1,i.e. as 0.582 cm−1/atm. In accordance with the same reference, a pressure shift of−0.174 cm−1/atm has been introduced in the simulations.

4.5.2 Magnetic field dependence

Figure 4.13 displays by the 10 curves the peak-value of the normalized FAMOSsignal, SEF , for an electronic transition in NO as a function of magnetic fieldamplitude, for pressures ranging from 1 to 760 Torr. The figure shows that foreach pressure there is a magnetic field amplitude that maximizes the normalizedFAMOS signal and that the optimum magnetic fields range from 1700 G andupwards depending on pressure.

0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0

0 . 0 0

0 . 0 5

0 . 1 0

0 . 1 5

Norm

alize

d FAM

OS si

gnal

M a g n e t i c f i e l d ( G a u s s )

Figure 4.13. Peak value of the normalized FAMOS signal, SEF , as a function of magnetic

field amplitude for a variety of total pressures, ranging from 1 to 760 Torr. The ten curvesrepresent, from the bottom to the top (at the highest magnetic field amplitudes), totalpressures of 1, 10, 25, 50, 100, 200, 300, 400, 550, and 760 Torr, respectively.

4.5.3 Total pressure dependence

Figure 4.14 illustrates the dependence of pressure on the normalized FAMOS signal,SEF , addressing electronic transitions in NO. The four panels A-D represent, by thesets of curves in each panel, a wide range of total pressures, p, ranging from 5 to760 Torr, for four different magnetic field amplitudes: 100, 750, 1250 and 2500 G,respectively. The figure shows that the pressure has a significant effect on theFAMOS signals, both regarding shape and magnitude. It is also interesting tonote that, just as in the case of ordinary absorption signals, for low pressures, theFAMOS signals increase with pressure. However, as the pressure is increased the

68


Figure 4.14. Normalized FAMOS signal, SEF , from a sample with a given concentration

of NO in N2 for various total pressures (ranging from 5 to 760 Torr) as a function offrequency detuning, νd , for four different magnetic field amplitudes: 100, 750, 1250 and2500 G, respectively. The 12 curves in each panel correspond to total pressures of 5, 10,20, 35, 50, 75, 100, 150, 250, 350, 500 and 760 Torr, respectively. Note the different y-axesin the panels.

signals eventually reaches as maximum and start to decrease. This means that foreach pressure there is a magnetic field amplitude that maximizes the normalizedFAMOS signal.

Figure 4.15 shows, by the 14 curves, the peak-value of the normalized FAMOSsignal, SEF , as a function of total pressure for a wide range of magnetic fields,ranging from 50 to 5000 G. The figure illustrates clearly what was concludedabove, i.e. that by controlling the pressure of the sample, the magnitude of theFAMOS signal can be maximized. The reason for this is that by increasing thepressure for a given magnetic field amplitude the FAMOS signal eventually becomesunder-modulated and starts to decrease. It is also worth to note that the peaks inFigure 4.15 are rather broad, making the requirements on a specific total pressurenot crucial.

69


0 2 0 0 4 0 0 6 0 0 8 0 0

0 . 0 0

0 . 0 5

0 . 1 0

0 . 1 5

Norm

alize

d FAM

OS si

gnal

P r e s s u r e ( T o r r )

Figure 4.15. Peak value of the normalized FAMOS signal, SEF , as a function of total

pressure for a variety of magnetic field amplitudes, ranging from 50 to 5000 G. The 14curves represent, from the bottom to the top (at the highest pressure), magnetic fieldamplitudes of 50, 100, 200, 300, 500, 750, 1000, 1500, 2000, 2500, 3000, 3500, 4000, and5000 G, respectively.

4.5.4 Optimum conditions

It could be seen from Figure 4.13 that for each magnetic field amplitude thereis a certain total pressure which maximizes the FAMOS signal; likewise, forFigure 4.15 it can be concluded that for each total pressure there is a certainmagnetic field amplitude that maximizes the signal. These two figures serve asthe basis for identification of the optimum conditions for the technique and aretherefore summarized by Figure 4.16, which, by the panel A, shows the magneticfield amplitude that maximizes the FAMOS signal for a given pressure, henceforthreferred to as “optimum magnetic field”, whereas the panel B shows the totalpressure that maximizes the FAMOS signal for a given magnetic field amplitude,henceforth referred to as “optimum total pressure”.

Figure 4.17A shows the maximum normalized FAMOS signal as a functionof magnetic field, i.e. using the optimum total pressure given by Figure 4.16A.Figure 4.17A shows the same general behavior as that of a Q-transition displayedin Figure 4.8 and follows the predictions made by Blake et al. [17]. This figure alsoshows that there is little need to utilize magnetic field amplitudes above ∼1800 Gfor which the FAMOS signal takes a large fraction (> 80%) of its maximum value.

Similarly, Figure 4.17B shows the maximum normalized FAMOS signal asa function of total pressure, i.e. using the optimum magnetic field given byFigure 4.16B. The figure also shows that there is little need for pressures above∼150 Torr for which the signal takes roughly 80% of its maximum value.

70


Figure 4.16. Panel A: The magnetic field amplitude that provides the maximum FAMOSsignal (pressure-shifted on-resonance peak-value value) from a gaseous sample with agiven concentration of NO for a given total pressure, referred to as the “optimum magneticfield”. Panel B: The total pressure that provides the maximum FAMOS signal for a givenmagnetic field amplitude, denoted the “optimum total pressure”.

Figure 4.17. Panel A: The maximum FAMOS signal as a function of magnetic fieldamplitude (thus for the optimum pressure). Panel B: The maximum FAMOS signal(pressure-shifted on-resonance peak-value value) as a function of pressure (thus for theoptimum magnetic field).

4.6 Comparison of the signals from MIR-FAMOS andUV-FAMOS

A comparison between the simulations for a rotational-vibrational Q-transition andan electronic transition given in this chapter shows that even though there are somesimilarities, such as general FAMOS signal shape and dependences on magneticfield and pressure, there are also some differences. For instance, the magneticfield amplitudes and total pressures that are considered to be optimum, differs forthe two realizations of the FAMOS technique. This can be seen when comparing

71


Figure 4.7A-B and Figure 4.13 together with Figure 4.15. For the rotational-vibrational Q-transitions in NO, typical optimum magnetic field amplitudes requiredto produce 80% of the maximum signal are found around 100 G while the optimummagnetic fields for an electronic transition are more than one order of magnitudelarger, viz. ∼1800 G. This comes primarily from the larger Doppler width for theUV-region but also, to a certain extent, from dissimilar g-values for the transitionsaddressed. A more detailed study of this phenomenon, which is given in Paper II,shows that magnetic field amplitudes for a Q-transition and an electronic transition,BQ and BE , respectively, that gives the same normalized modulation amplitudeare related to each other according to

BEBQ

=2gJgS

δνEDδνQD

≈ gJνEνQ≈ 18, (4.47)

where νE and νQ denote the center frequencies for the electronic transition and thero-vib Q-transition, respectively. This shows that in order to reach the optimummagnetic field for an electronic transition in NO, a magnetic field amplitude roughly18 times larger than that of a Q-transition in NO is required.

72


4.7 Noise analysis

In order to understand and improve the limiting factors of the Faraday modulationsetup it is important to analyze the various noise contributions in the system.Detailed signal-to-noise analysis are given in refs. [6, 17, 24, 30, 45, 57, 58] and inthe following section a short summary of these descriptions is given.

4.7.1 Noise sources

As for most other laser-based spectroscopic detection techniques and as was dis-cussed in Chapter 2, there are three main noise sources in FAMOS:

1. Thermal noise – this is noise coming from the photo detector and itsamplifiers. This noise is independent of laser power and can be expressed bya constant a.

2. Shot noise – this noise source is angle-dependent fundamental quantum noise(QN) which is proportional to the square-root of the transmitted laser powerand can be represented by b

√Pt, where b is a proportionality constant.

3. Flicker noise – this noise source is proportional to the transmitted powerand can be expressed as cPt, where c is a proportionality constant.

4.7.2 Signal-to-noise ratio as a function of uncrossing angle

It is convenient to define the signal-to-noise ratio as the ratio of the on-resonancesignal to the on-resonance noise as was done by McCarthy and Field [58]. Thissimplifies the SNR calculations considerably since the contribution from magneticcircular dichroism, ∆δ, is zero on-resonance. This means that the power transmittedthrough the analyzer, given by equation (4.19), can be written as

Pt = P0

[sin2

(∆φ

2+ θ

)+ (β2 + ε2) cos2

(∆φ

2+ θ

)], (4.48)

where the overall absorption is assumed to be small, α ' 1 and α2 ε2β2.By taking the square root of the Euclidean sum of the three power equivalentcontributions of noise currents we can express the noise in a measurement of ∆φ as

(∆φ)noise =

∣∣∣∣ (Pt)noise∂(Pt)/∂(∆φ)

∣∣∣∣ =

√(Pt)2

th + (Pt)2shot + (Pt)2

fl∣∣P0

2

sin(∆φ+ 2θ)[1− (β2 + ε2)]

∣∣ , (4.49)

73


where

(Pt)th = a

(Pt)shot = b√Pt

(Pt)fl = cPt. (4.50)

This yields a SNR of

SNR =∆φ

(∆φ)noise=

∆φ(1− β2 − ε2) sin(∆φ+ 2θ)√((Pt)thP0/2

)2

+

((Pt)shot

P0/2

)2

+

((Pt)fl

P0/2

)2. (4.51)

Inserting the expressions for the noise contributions in equation (4.50) into equa-tion (4.51) gives

SNR =∆φ

(∆φ)noise=

∆φ(1− β2 − ε2) sin(∆φ+ 2θ)√a2

(P0/2)2 + b2Pt

(P0/2)2 +c2P 2

t

(P0/2)2

, (4.52)

where Pt is given by equation (4.48). For small phase shifts the SNR can be Taylorexpanded yielding

SNR =∆φ

(∆φ)noise=

∆φ(1− β2 − ε2) sin(2θ)√a2

(P0/2)2 + b2Pt

(P0/2)2 +c2P 2

t

(P0/2)2

, (4.53)

which is of the same form as the expressions given in the literature [24, 30, 45].The SNR as a function of uncrossing angle θ for a system dominated by thermalnoise is given in Figure 4.18. In Figure 4.19 the SNR as a function of uncrossingangle, θ is shown under shot noise limited conditions, whereas Figure 4.20 displaysthe SNR as a function of uncrossing angle, θ, under flicker noise limited conditions.Figure 4.20 shows a clear maximum of the SNR for a given uncrossing angle as hasbeen observed in the literature [24, 41].

74


0 2 4 6

(c)

Analyzer uncrossing angle, θ

SNR

0 2 4 6

(a)


Signal

0 2 4 6

(b)

Analyzer uncrossing angle, θNoise

Figure 4.18. Signal, noise and SNR for a system dominated by thermal noise. Panel (a)displays the signal as a function of analyzer uncrossing angle, θ. Panel (b) shows the noiseas a function of the same angle, θ. Finally, panel (c) displays the SNR as a function ofthe uncrossing angle, θ.

0 2 4 6

(c)


SNR

0 2 4 6

(a)


Signal

0 2 4 6

(b)


Noise

Figure 4.19. Signal, noise and SNR for a system dominated by shot noise. Panel (a)displays the signal as a function of analyzer uncrossing angle, θ. Panel (b) shows the noiseas a function of the same angle, θ. Finally, panel (c) displays the SNR as a function ofthe uncrossing angle, θ.

0 2 4 6

(c)


SNR

0 2 4 6

(a)


Signal

0 2 4 6

(b)


Noise

Figure 4.20. Signal, noise and SNR for a system dominated by flicker noise. Panel (a)displays the signal as a function of analyzer uncrossing angle, θ. Panel (b) shows the noiseas a function of the same angle, θ. Finally, panel (c) displays the SNR as a function ofthe uncrossing angle, θ.

75


4.7.3 Noise in measured signals

In order to evaluate the noise characteristics of the MIR-FAMOS system describedin Section 7.2, the measured noise, here defined as the standard deviation of theFAMOS signal on-resonance, is fitted to equation (4.49), such a fit is shown inFigure 4.21. The data is from Paper IV and the noise analysis indicates that thedominating sources of noise at the optimum analyzer uncrossing angle are lasersource noise and detector noise (which also comprises other intensity independentnoise sources, e.g. noise from electronics, lock-in amplifier and data acquisition).This analysis shows that the optimum analyzer uncrossing angle for this particularsetup is 0.5 to 0.6.

0 0.2 0.4 0.6 0.8 1

10−6

10−4

10−2

(a)


Noise

(a.u)

Total noise - fitTotal noise - measurementDetector noiseShot noiseLaser noise

0 0.2 0.4 0.6 0.8 10

50

100

150

200

250

300

(b)


SNR

SNR - measuredSNR - fit

Figure 4.21. Panel (a) displays the total noise and the three different noise contributions asa function of analyzer uncrossing angle, θ. The noise contributions are fitted according toequation (4.49). Panel (b) shows the SNR as a function of uncrossing angle, θ. The circlescorrespond to measured data and the solid line is a numerical fit based on equation (4.52).

76

Chapter 5

Analytical expressions for theFourier coefficients of amodulated dispersion lineshapefunction

”Mathematical analysis is as extensive as nature itself; it de-fines all perceptible relations, measures times, spaces, forces,temperatures; this difficult science is formed slowly, but it pre-serves every principle which it has once acquired; it grows andstrengthens itself incessantly in the midst of the many vari-ations and errors of the human mind. It’s chief attribute isclearness; it has no marks to express confused notations. Itbrings together phenomena the most diverse, and discovers thehidden analogies which unite them.”

— Joseph Fourier

In Chapter 4 it was shown how to express the FAMOS signal in terms of the 1stFourier coefficient of a modulated dispersion lineshape function. In this chapterthe benefit of this description is realized by deriving a non-complex fully analyticalexpression for these Fourier coefficients, which is orders of magnitude faster toexecute than previous descriptions which involve numerical calculations of nestedintegrals. This allows for a rapid real-time fitting procedure for modulation tech-niques that rely on dispersion, e.g. Faraday modulation/rotation spectroscopy and

77

Analytical expressions for Fourier coefficients

wavelength-modulated noise-immune cavity-enhanced optical heterodyne molecularspectroscopy.

5.1 Fourier expansion of an arbitrary modulated line-shape function

Let us begin by expanding an arbitrary modulated lineshape function, f [νd(t)],into a Fourier series, where νd(t) is the time-dependent detuning frequency fromthe center of the transition, given by

νd(t) = νd + νa cos(ωt), (5.1)

where νa is the modulation amplitude1 and ω the modulation frequency. TheFourier expansion can be achieved by the use of Fourier transforms as follows:

By definition, the Fourier transform of f(νd) is given by,

F (τ) =1√2π

∫ +∞

−∞f (νd) e

−iνdτ dνd, (5.2)

where τ is the conjugate variable to νd. Since the Fourier transform of f(νd + a)is F (τ)eiaτ , this implies that the Fourier transform of the arbitrary modulatedlineshape function, f [νd(t)] can be written,

F (τ, t) = F (τ)eiνaτ cos(ωt). (5.3)

It is now approriate to make use of the Jacobi-Angerer formula, which states that

eiz cos Θ =

∞∑n=0

(2− δn0)inJn(z) cos(nΘ), (5.4)

where δn0 is Kroencker’s delta-function and Jn(z) is the nth Bessel function. Thisimplies that equation (5.3) can be expressed as,

F (τ, t) =

∞∑n=0

(2− δn0)inF (τ)Jn(νaτ) cos (nωt) . (5.5)

By using the inverse Fourier transform,

1νa → −νa if the transition rather than the laser frequency is modulated

78


f(νd) =1√2π

∫ +∞

−∞F (τ) eiνdτ dτ, (5.6)

it is possible to write the arbitrary modulated lineshape function, f [νd(t)], as,

f [νd(t)] =∞∑n=0

χn(νd, νa) cos (nωt) , (5.7)

where χn(νd, νa) is the nth (in-phase) Fourier coefficient of the time-dependentarbitrary modulated lineshape function f [νd(t)], given by

χn(νd, νa) = (2− δn0)1√2π

∫ +∞

−∞inF (τ)Jn(νaτ)eiνdτ dτ. (5.8)

5.2 Analytical expression for a modulated Lorentziandispersion lineshape function

In the derivation above an arbitrary modulated lineshape function was used. Let usnow instead focus on an area-normalized modulated Lorentzian dispersion lineshapefunction, given by

f(ν, δνL) =ν

1 + ν2, (5.9)

where ν is the normalized detuning frequency given by ν/δνL, where δνL is thehalf-width-half-maximum (HWHM) of the Lorentzian absorption profile. TheFourier transform of equation (5.9) is

F (τ) = i

√π

2e−|τ |, (5.10)

where τ is a normalized time given by τδνL. By expressing the modulation ofthe normalized detuning frequency as ν(t) = νd + νa cos(ωt), where νa is thenormalized modulation amplitude, and inserting the expression for the Fouriertransformed modulated Lorentzian dispersion lineshape function into equation (5.8)the following expression for the Fourier coefficients is obtained

χdisp,evenL,n (νd, νa) =1

2

(2− δn0

)∫ ∞−∞

in+1e−|τ |Jn(νaτ)eiνdτ dτ . (5.11)

79


This integral can, in turn, by following the procedure outlined by Arndt [59] bewritten,

χdisp,evenL,n (νd, νa) =1

2

(2− δn0

)in+1

[(1− iνd)2 + ν2

a

] 12 − (1− iνd)

nνna[(1− iνd)2 + ν2

a

])

12

+ c.c.

= Re

[(2− δn0

)in+1

[(1− iνd)2 + ν2

a

] 12 − (1− iνd)

nνna[(1− iνd)2 + ν2

a

])

12

],

(5.12)

where c.c. stands for the complex conjugate.

5.3 Derivation of a non-complex analytical expressionfor the nth Fourier coefficient

As is described in detail in Paper V it is possible to convert this expression intoa non-complex expression by the use of binomial expansions. This leads to thefollowing expression for the nth Fourier coefficient of a modulated Lorentziandispersion lineshape function,

χdisp,evenL,n (νd, νa) =Anνna

[Bn +

CnS+ +DnS−√2R

], (5.13)

where S+ and S− are given by

S+ =√R+M

S− =√R−M, (5.14)

where, in turn,

R =√M2 + 4ν2

d , (5.15)

and

M = 1 + ν2a − ν2

d . (5.16)

The An, Bn, Cn, and Dn coefficients are given by

An = 2− δn0, (5.17)

80


Bn(νd, νa) =

amax∑a=0

a∑b=0

a−b∑r=0

r∑s=0

cmax∑c=cmin

[(−1)a−b−r+c−1

×(

n

n− 2a− 1

)(a

b

)(a− br

)(r

s

)(n− 2a− 1

2c− n− b− 1

)2bν2s

a ν2(a−b−r+c)−n−1d

],

(5.18)

Cn(νd, νa) =

dmax∑d=0

d∑e=0

d−e∑r=0

r∑s=0

fmax∑f=fmin

[(−1)d−e−r+f−1

×(

n

n− 2d

)(d

e

)(d− er

)(r

s

)(n− d

2f − n− e− 1

)2eν2s

a ν2(d−e−r+f)−n−1d

],

(5.19)

and

Dn(νd, νa) =− sign(νd)

dmax∑d=0

d∑e=0

d−e∑r=0

r∑s=0

gmax∑g=gmin

[(−1)d−e−r+g

×(

n

n− 2d

)(d

e

)(d− er

)(r

s

)(n− d

2g − n− e

)2eν2s

a ν2(d−e−r+g)−nd

],

(5.20)

where the summation indices are given in Table 5.1.

For convenience of the reader the expressions for the Bn, Cn and Dn coefficientsfor the first six Fourier components are explicitly given in the Tables 5.2-5.4.

Figure 5.1 shows the first six Fourier coefficients of modulated Lorentziandispersion lineshapes as functions of normalized detuning. Panel (a) shows thefirst three odd coefficients whereas panel (b) shows the first three even ones, allevaluated at their optimum modulation amplitude, i.e. the one that maximizes thepeak-value of each coefficient.

In conclusion, the first Fourier coefficient, χdisp,evenL,1 (νd, νa), which is the entitythat is most often used in FAMOS, can be succinctly written as

81


Table 5.1. Expressions for the summation indicesgiven in equations equation (5.18)-equation (5.20).

Summation index Expressiona

amax Int−(n−1

2

)cmin Int+

(n+b−1

2

)cmax n− a+ Int−

(b−2

2

)dmax Int−

(n2

)fmin Int+

(n+e+1

2

)fmax n− d+ Int−

(e+1

2

)gmin Int+

(n+e

2

)gmax n− d+ Int−

(e2

)

a Mathematically the Int+(x) and Int−(x)functions are defined as sign(x)ceil(|x|) andsign(x)floor(|x|), respectively, where sign(x)represents the sign of the argument, x, and ceil(x)and floor(x) extract the nearest upper and lowerinteger of the argument, x, respectively.

Table 5.2. Expressions for the first six Bn coefficients.

Summation index Expression

B0 0

B1 −1

B2 2νd

B3 4 + ν2a − 4ν2

d

B4 −24νd − 4ν2a νd + 8ν3

d

B5 −16− 12ν2a − ν4

a + 96ν2d + 12ν2

a ν2d − 16ν4

d

82


Table 5.3. Expressions for the first six Cn coefficients.


C0 0

C1 1

C2 −4νd

C3 −4− 3ν2a + 12ν2

d

C4 32νd + 16ν2a νd − 32ν3

d

C5 16 + 20ν2a + 5ν4

a − 160ν2d − 60ν2

a ν2d + 80ν4

d

Table 5.4. Expressions for the first six Dn coefficients, where sign(x) rep-resent the sign of argument x, i.e. +1 if x > 0, 0 if x = 0 and -1 if x < 0.


D0 −sign(νd)

D1 sign(νd)νd

D2 −sign(νd)(−2− ν2a + 2ν2

d)

D3 −sign(νd)(12νd + 3ν2a νd − 4ν3

d)

D4 −sign(νd)(8 + 8ν2a + ν4

a − 48ν2d − 8ν2

a ν2d + 8ν4

d)

D5 −sign(νd)(−80νd − 60ν2a νd − 5ν4

a νd

+160ν3d + 20ν2

a ν3d − 16ν5

d)

−10 −5 0 5 10

−0.2

0

0.2

0.4

0.6

(a)


χL,n

Odd coefficients

χL,1χL,3χL,5

−10 −5 0 5 10

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

(b)


χL,n

Even coefficients

χL,0χL,2χL,4

Figure 5.1. The first six Fourier coefficients of a modulated Lorentzian dispersion lineshapefunction as a function of normalized detuning frequency, νd. Each of the Fourier coefficientshave been evaluated at their optimum normalized modulation amplitude which is 1.27,3.15 and 5.10 for the first three odd coefficients [panel (a)] and 0, 2.84 and 4.44 for thefirst three even coefficients [panel (b)], respectively.

83


χdisp,evenL,1 (νd, νa) = − 2

νa

[1− S+ + |νd|S−√

2R

]=

= − 2

νa

1−

√2

2√

(1 + ν2a − ν2

d)2 + 4ν2d

×[√√

(1 + ν2a − ν2

d)2 + 4ν2d + 1 + ν2

a − ν2d

+ |νd|√√

(1 + ν2a − ν2

d)2 + 4ν2d − 1− ν2

a + ν2d

]. (5.21)

where we have used the fact that sign(νd)νd = |νd|. Figure 5.2 schematically showshow the first Fourier coefficient as a function of normalized detuning behaves whensubjected to various normalized modulation amplitudes.


χdisp,even

L,1

Figure 5.2. Schematic illustration of the effects of normalized modulation amplitudeon the first Fourier coefficient of a modulated Lorentzian dispersion lineshape function,χdisp,evenL,1 (νd, νa), as a function of normalized detuning frequency, νd. The Fourier

coefficient has been evaluated for normalized modulation amplitudes ranging from 1.27 to6.35 with increments of 1.27.

84

Chapter 6

Fast and non-approximativecalculation of modulated Voigtdispersion lineshape functions– the Westberg-Wang-Axnermethod

”Mathematics, rightly viewed, possesses not only truth, butsupreme beauty a beauty cold and austere, like that of sculpture,without appeal to any part of our weaker nature, without thegorgeous trappings of painting or music, yet sublimely pure, andcapable of a stern perfection such as only the greatest art canshow. The true spirit of delight, the exaltation, the sense ofbeing more than Man, which is the touchstone of the highestexcellence, is to be found in mathematics as surely as poetry.”

— Bertrand Russell

In Chapter 5 it was shown how the 1st Fourier coefficient for a modulated Lorentziandispersion lineshape can be expressed as an analytical function of two variables; thenormalized detuning frequency and the normalized modulation amplitude. Thissignificantly decreases the computational time and thereby makes real-time curvefitting feasible. However, there are measurement conditions when the Lorentzianlineshape does not yield sufficiently accurate results since it neglects the Dopplerbroadening that contributes significantly for lower pressure conditions. In fact,

85

Fast and non-approximative calculation of modulated Voigt dispersion lineshape functions

when considering the optimum pressure conditions for FAMOS it is more accurateto include also the Doppler broadening. This is normally done by the use ofthe Voigt lineshape, which is defined as the convolution of the aforementionedlineshapes (the Lorentzian and the Doppler lineshapes). By doing this, however,the ability to express the 1st Fourier coefficient as an analytical function is lost andthe computation of the lineshape needs to be done numerically which increasesthe computational time about a thousand fold. In this chapter a new method ofcalculating the 1st Fourier coefficient of a modulated Voigt dispersion lineshapefunction will be presented. The method, which is referred to as the Westberg-Wang-Axner method (WWA-method), does not contain any approximation andis therefore accurate down to the precision of the software used (in our case10−15), while it increases the computational speed significantly compared to theconventional numerical calculations by about three orders of magnitude (103). Inorder to fully appreciate this method the conventional approach using numericalcalculations of Voigt integrals is first presented.

6.1 Calculation of the 1st Fourier coefficient of a mod-ulated Voigt dispersion lineshape function – Theconventional approach

As was mentioned above, when neither Doppler broadening nor pressure broadeningis the dominating broadening mechanism the lineshapes are usually modelled by aVoigt function. In the absence of modulation such a function takes the form

χV (νd) =y

π3/2δνL

∫ ∞−∞

(x+ s)e−s2

(x+ s)2 + y2ds, (6.1)

where νd is the width-normalized detuning frequency, in this case normalized bythe Lorentzian width, i.e. νd = (ν − ν0)/δνL, x and y are a normalized detuningand the Voigt parameter, respectively, given by

x =ν0 − νcδν′D

= −yνd (6.2)

and

y =δνLδν′D

, (6.3)

where δν′D is given by δνD/√

ln 2 where δνD, in turn, is the HWHM of the Doppler

broadening, given by ν0

√2 ln 2 kT/mc2 and s is a dimensionless integration pa-

86


rameter. It is common to express the Voigt function as the imaginary part of thecomplex error function, w[...], as,

χV (νd) =1

π1/2δν′DIm[w(x+ iy)]. (6.4)

By introducing a modulation, the x-parameter defined above becomes,

x(t) = −y[νd + νa cos(2πft)], (6.5)

which implies that the nth Fourier coefficient of a modulated Voigt dispersionlineshape function can be written,

χdisp,evenV,n (νd, νa) =y

π3/2δνL

×2

τ

τ∫0

∞∫−∞

−y[νd + νa cos(2πft)] + se−s2 ds

−y[νd + νa cos(2πft)] + s2 + y2× cos(2πnft) dt. (6.6)

As was mentioned above there is no known analytical solution to this equationwhich means that it has to be estimated by numerical calculations. Since ameasurement scan might contain thousands of discrete points and several iterationsmust be performed in order to find a suitable fit, these calculations are often verytime-consuming, which prevent real-time curve fitting. In the next section a newmethodology addressing this issue is presented. The methodology thus providesfast and non-approximate calculations of the nth Fourier coefficients of modulatedVoigt dispersion lineshape functions suitable for real-time curve fitting.

6.2 Calculation of the 1st Fourier coefficient of a mod-ulated Voigt dispersion lineshape function – TheWestberg-Wang-Axner method

Let us start by recalling the definition of a modulated Voigt dispersion lineshapefunction from equation (6.7),

χdisp,evenV,n (νd, νa) =y

π3/2δνL

×2

τ

τ∫0

∞∫−∞

−y[νd + νa cos(2πft)] + se−s2 ds

−y[νd + νa cos(2πft)] + s2 + y2× cos(2πnft) dt. (6.7)

87


If we interchange the integration order and make the variable substitution s→ −ys′,this expression can be written as

χdisp,evenV,n (νd, νa) = − y

π3/2δνL

×∞∫−∞

2

τ

τ∫0

[(νd − s′) + νa cos(2πft)] cos(2πnft) dt

[(νd − s′) + νa cos(2πft)]2 + 1× e−y2s′2 ds′. (6.8)

By examining this expression closely it can be seen that the inner integral canbe identified as the nth Fourier coefficient of a modulated Lorentzian dispersionlineshape function defined in Chapter 5, although here with a shifted detuningfrequency, νd → νd − s′. This means that it is possible to write the nth Fouriercoefficient of a modulated Voigt dispersion lineshape function as

χdisp,evenV,n (νd, νa) =y√π

∞∫−∞

χdisp,evenL,n (νd − s′, νa)e−y2s′2 ds′. (6.9)

This equation simply denotes a convolution between a distribution function andthe nth Fourier coefficient of a modulated Lorentzian dispersion lineshape function,for which analytical expressions exist, which can be written succinctly as

χdisp,evenV,n (νd) = χdisp,evenL,n ⊗ fv(νd), (6.10)

where fv(νd) is an area-normalized distribution function given by

fv(νd) =y

π1/2e−y

2ν2d (6.11)

and ⊗ denotes a convolution. Equation (6.10) is referred to as the WWA-method.

This implies that the 1f FAMOS signal from a rotational-vibrational Q-transition can, for the case with no polarization imperfections, be written as,

SF (νd) = ηα0χ0I0 sin(2θ)

4χdisp,evenV,1 (νd), (6.12)

where α0 is the absorption strength, given by α0 = SNxL, η is an instrumentationfactor, S is the integrated linestrength of the transition, Nx is the density ofmolecules, L is the interaction length, χ0 is the peak-value of the area-normalizedGaussian lineshape function given by

√ln 2/ (

√πδνD), θ is the polarizer uncrossing

angle and I0 is the intensity of the light impinging on the detector. As wasillustrated in Chapter 4, for cases when lineshape assymetries need to be taken into

88


account this formulation can be extended to also involve the effects of polarizationimperfections1. The 1f FAMOS signal can then be expressed as

SF (νd, νa) = ηα0χ0I0 sin(2θ)

4

×[χdisp,evenV,1 (νd, νa) + εχabs,evenV,1 (νd, νa)

], (6.13)

where ε denotes the uncrossing angle normalized unbalancing between the LHCPand RHCP components of the light and χabs,evenV,1 (νd, νa) is the modulated Voigtabsorption lineshape function defined by Westberg et al. [50].

1For more information see Paper VII

89

Chapter 7

Experimental methods andprocedures

”ALL THIS IS A DREAM. Still examine it by a few experi-ments. Nothing is too wonderful to be true, if it be consistentwith the laws of nature; and in such things as these experimentis the best test of such consistency.”

— Michael Faraday - Laboratory journal

7.1 Basic configuration

A schematical experimental layout of a typical Faraday modulation spectroscopysetup is shown in Figure 7.1. This layout includes the necessary components foran experimental realization of the Faraday modulation spectroscopy technique.However, numerous variations and additions to this setup are possible, e.g. differentlaser sources, differential detector setups, reference cells, mirrors, lenses etc. Inthis thesis two different experimental setups will be discussed: (i) a setup basedon a distributed feedback (DFB) quantum cascade laser (QCL) from Maxiontechnologies producing light at ∼5.33 µm to target the most sensitive transitionof the fundamental rotational-vibrational band of NO, namely the Q3/2(3/2)-transition, and (ii) a setup based on a UV-laser from Toptica (TA-FHG pro) thatproduces milli-watts of power at ∼226.6 nm for detection of NO addressing thestrong electronic X2Π(ν′′ = 0)−A2Σ+(ν′ = 0) transitions.

91

Experimental methods and procedures

Figure 7.1. Schematical experimental layout of FAMOS.

7.2 Experimental setup for MIR-FAMOS

In Figure 7.2 the experimental setup used for detection of NO in the MIR region isshown. It is based around a room temperature continous-wave (cw) distributedfeedback (DFB) quantum cascade laser (QCL) from Maxion Technologies produc-ing around 70 mW of power at ∼5.331 µm (1875.81 cm−1). The temperature iscontrolled by two separate Peltier elements which allow a temperature tuning inthe range between 15 C and 35 C while keeping the temperature stabilizationwithin 0.1 C. In order to reach the desired Q3/2(3/2) transition at 5.331 µm theQCL was operated at 32 C with a bias current of 708 mA. The threshold currentat 32 C was 460 mA and the side mode suppression was better than 25 dB. Thecollimation of the laser beam was performed by a chalcogenide aspheric lens fromLightpath Technologies with an effective focal length of 1.8 mm.

The laser beam from the QC-laser was split into two by a CaF2 wedge. Thisprocedure enabled a small fraction of reflected light to pass through a reference cellcontaining 1000 ppm of NO at 58 mbar, before it was focused on a Peltier cooledMCT detector. The signal from the detector was then amplified by a DC-coupledtransimpedance amplifier before it was sent to the DAQ board connected to a laptopfor data analysis. The direct absorption signal from the reference cell was used toaccurately determine the laser frequency during the signal processing. The mainpart of the beam continued through the CaF2 wedge and two wire grid polarizersin series before entering the 25 cm long absorption cell with an outer diameterof 12 mm. The cell was made from borofloat glass and the windows consisted of2 mm thick CaF2 plane parallel discs. Inside the shielded aluminum box a 15 cmlong solenoid made from 765 turns of 1 mm thick copper wire with 0.3 mm thick

92


Figure 7.2. Experimental setup of the FAMOS-system adressing the Q3/2(3/2) transition.

isolation was placed. The solenoid was coupled in series with a capacitor stack tocreate a resonance circuit that was driven by a commercially available powerfulaudio amplifier (Cervin Vega). The resonant circuit was constructed to producea resonance at a frequency of 7.4 kHz which was found to be a good compromisebetween the performance of the instrument (detection limit and response time)and the losses due to heat dissipation from the magnetic coil.

The dimensions of the shield box were carefully chosen to minimize the effectsof the magnetic field outside of the aluminum shield box while still allowing for anefficient field generation inside the box. The solenoid was able to produce roughly∼200 G at a modulation frequency of 7.4 kHz under normal operating conditionswhile keeping the temperature of the coil sufficiently low for hours of operationtime. The maximum allowed magnetic field around 640 G, although at this fieldstrength only a few minutes of measurements could be performed before the shieldbox became overheated. The glass cell was equipped with a thermistor in order toconstantly monitor the temperature inside the solenoid to prevent damage to theglass cell the solenoid itself.

Once the light had passed through the absorption cell it encountered a goldwire-grid analyzer (Infraspecs). The laser beam was then focused using an off-axisparabola and an immersion lens situated on a Peltier cooled MCT detector. Thedetector signal was sent to a lock-in amplifier configured for 1f detection. Afterthe demodulation process the signal was sent to the DAQ board connected to alaptop computer for data storage and analysis.

93


Figure 7.3. Basic experimental layout of FAMOS addressing a UV-transition.

7.3 Experimental setup for UV-FAMOS

The experimental setup involving the fully-diode-laser-based UV laser system(Toptica, TA-FHG Pro) producing several mW of power at ∼226.6 nm is shownin Figure 7.3 and consisted of an external-cavity diode laser (ECDL) and twoconsecutive frequency doubling stages in order to produce wavelengths around thedesired 226.6 nm transition. For each series of experiments the center wavelengthwas tuned to 906.30 nm with the aid of a wavelength meter. This ensured thatafter the two frequency doubling stages the resulting wavelength would target thetransitions around 226.6 nm.

The laser light, after leaving the cavity, was sent through two nearly crossedRochon prism polarizers (CVI Melles Griot, RHCP-5.0-MF) with extinction ratiosof about 10−5, placed on each side of a 10 cm long, wedge-window fused-silica cellwhich, in turn, was placed inside a magnetic coil. The two polarizers were placednearly perpendicular to each other with an uncrossing angle of about 12.

The magnetic coil was made from 530 turns of copper wire, wound around aplastic cylinder with an inner diameter of 76 mm. The coil was coupled in serieswith a capacitor (0.31 µF) in order to create a resonant circuit with a resonancefrequency of ∼2.5 kHz. The lock-in amplifier shown in the setup served two purposes(i) to generate the sinusoidal signal which was sent to the commercially availablepower amplifier (Apart Audio PubDriver 2000) whose output power was used to

94


drive the resonant circuit in order to create the alternating magnetic field used tomodulated the NO molecules, (ii) to demodulate the detector signals at 1f .

The scanning procedure was handled using a function generator to provide thepiezo-electric crystal transducer in the ECDL with a slow ramp signal, usually ofsinusoidal form since such provided less strain on the locking at the turning points.The scanning range was around 38 GHz (1.3 cm) which is slightly more than 10times the Doppler-width of the NO transition. A LabView program collected dataat 6 kHz and NO with a concentration of 100 ppm in N2 was used. All experimentswere performed in room temperature, assumed to be 23 C.

95

Chapter 8

Results and discussion

”The secret is comprised in three words Work, finish, publish.”— Michael Faraday

8.1 Experimental results

As was discussed in the previous chapters, the FAMOS technique is a sensitive andselective modulated spectroscopic technique used for detection of paramagneticgaseous compounds. The modulation decreases the influence of flicker noise fromthe laser. Moreover, the modulation of the molecular transitions rather than thelaser light decreases also the influence of etalons as well as unwanted spectroscopicdisturbances from interfering diamagnetic species. This makes the technique highlyapplicable for detection of NO and in recent years low detection limits have beendemonstrated when targeting the fundamental rotational-vibrational transitionaround 5.33 µm [17, 20, 21, 24, 26]. In the following sections the results from thetwo experimental setups will be presented, i.e. (i) a QCL-based FAMOS setuptargeting the fundamental rotational-vibrational Q-transition in NO at 5.33 µm and(ii) a UV-laser based FAMOS system addressing the strong overlapping electronictransitions, Q22(21/2) and QR12(21/2), in NO at 226.6 nm.

Additionally, the theoretical expressions from the appended papers will beevaluated, in particular the results from the evaluation of the effects of lineshapeasymmetries on curve fitting and the results from the implementation of theWWA-method.

97


8.1.1 MIR-FAMOS

Figure 8.1A displays a FAMOS spectrum as a function of frequency (in unitsof cm−1) around the three most commonly targeted and magnetically sensitiveQ3/2(7/2), Q3/2(5/2), and Q3/2(3/2) ro-vib transitions in NO. The spectrum isobtained by averaging 200 scans, at a scan rate of 20 Hz. The data acquisitionrate was in this case 40 kHz and a lock-in time constant of 100 µs was applied.For frequency calibration purposes a one inch (2.54 cm) Fabry-Perot etalon madefrom germanium was temporarily inserted in the beam path. As can be seen inFigure 8.1A there is a slight asymmetry of the curves most likely originating fromelliptical polarization transmitted through the imperfect analyzer1. Figure 8.1Bdisplays the corresponding absorption spectrum made from simulations using theHITRAN 2008 database [7]. Finally, Figure 8.1C shows a simulation using theexpressions derived in Chapter 4, where the asymmetry effects have been neglected.As can be seen the agreement between simulations and experiment is good.

In order to maximize the signal-to-noise ratio (SNR), both the uncrossing angle,θ, and the external magnetic field amplitude need to be optimized. An expression forthe SNR as a function of uncrossing angle, θ, is derived in Section 4.7. Figure 8.2(a)shows the SNR as a function of uncrossing angle together with a fit based on theexpressions derived in Section 4.7. The figure clearly displays a maximum nearθ = 0.6. By performing these types of measurements it is straightforward to setthe uncrossing angle to obtain maximum SNR. Likewise, the magnetic field canbe optimized by slowly changing the modulation amplitude while observing thepeak-value of the FAMOS signal from the Q3/2(3/2) transition. As the field isincreased the slope seen in Figure 8.2(b) decreases and the signal eventually reachesa maximum after which it starts to decrease due to over-modulation. The optimummagnetic field, in this particular case, was measured to ∼150 G, but in practicesince the peak is relatively broad, any value between 120 G and 180 G will providesimilarly large signals.

In order to quantify the performance of the instrumentation a 16 hours longsystem stability test on zero gas was performed and the results are displayed inthe form of an Allan plot shown in Figure 8.3. As can be seen in the figure, thesystem stability is excellent showing white noise limited performance up to a fewthousand seconds. From Figure 8.3 the detection limit (1σ) at 1 s integration timewas found to be 4.5 ppb for an optical path length of 15 cm using scan mode. Also,an estimated Allan variance plot was performed in order to estimate the potentialbenefits of the locked-line mode approach. This yielded a detection limit (1σ) at 1 sintegration time of 1.8 ppb for the same optical path length. Since the system showsexcellent stability it is possible to perform sub-ppb detection of NO by choosing alonger integration time. For instance, for integration times above 20 s the detection

1See Section 4.3

98


Figure 8.1. Panel A: experimental FAMOS data showing the three most magneticallysensitive transitions in the fundamental ro-vib band in NO. Panel B: simulation of thetransmission using constants from the HITRAN database for the same frequency range.Panel C: theoretical simulations for the same frequency range following the model outlinedin Chapter 4.

0 0.2 0.4 0.6 0.8 1

0

200

400

600

800

1000

1200

(a)

Uncrossing angle, θ []

SNR

0 50 100 150 200 250

0

1

2

3(b)

Magnetic field [G]

Signal[a.u.]

Figure 8.2. Panel (a): FAMOS signal as a function of offset angle, θ, for the experimentalconfiguration described in Section 7.2. The markers represent individual data pointswhereas the solid curve corresponds to a fit based on the expressions derived in Section 4.7.Panel (b): the optimum magnetic field amplitude for the same experimental setup.

99


limit is in the ppt (parts-per-trillion) region, e.g. for 100 and 1000 s of integrationthe Allan plot displays detection limits of 440 ppt and 140 ppt, respectively.

Figure 8.3. Allan variance plot to determine the detection limit for scan mode (red)together with an estimated Allan variance for locked-line mode (blue).

100


8.1.2 UV-FAMOS

FAMOS spectra as a function of detuning frequency (in cm−1) targeting theoverlapping Q22(21/2) and QR12(21/2) lines in the electronic Π − Σ band inNO under different pressure conditions is shown in Figure 8.4. The six panels(a) − (f) correspond to six different pressure conditions (4.2, 20, 40, 60, 90 and150 Torr, respectively) where the gray curves (whose width is exaggerated for visualpurposes) represent the measured data from an average of 10 scans. The redand black curves represent fits based on the two-transition model (described inSection 4.5.1) disregarding and including asymmetry effects, respectively. Theintensity is assumed to be linear over the scan.

The panels show a very good agreement between the various spectra and thefits, especially when including the asymmetry effects described in Section 4.3. Theresiduals of the fits show less than 0.1 % deviation over the entire scan and for allpressures investigated. It is interesting to note that the center part occasionallyshows a systematic error which may originate from either non-fully overlap betweenthe two Q22(21/2) and QR12(21/2) transitions, approximations in the two-transitionmodel, or the neglect of Dicke narrowing and speed-dependent effects which maybe significant under these pressure conditions [60–66].

Figure 8.5A shows FAMOS spectra from 100 ppm of NO taken under varioustotal pressures for a fixed magnetic field amplitude. The total pressures rangefrom 0.5 to 250 Torr, thus representing partial pressures of 50 µTorr to 25 mTorr.Figure 8.5B displays simulations according to the two-transition model for thesame range of total pressures for a magnetic field of 130 G. Although not beingidentical, the similarities of the two panels are apparent.

Figure 8.6A displays the peak value of the FAMOS signal as a function oftotal pressure, evaluated from the measurements presented in Figure 8.5A, whereasFigure 8.6B shows the peak values of the FAMOS signal as a function of totalpressure obtained from simulations such as those presented in Figure 8.5B. Althoughthe two curves differ somewhat in slope and shape, the general trend is the same,with a FAMOS signal that increases with total pressure up to around 130 Torrafter which it starts to decrease. The difference is mainly attributed to drifts andfluctuations of the UV laser light power. The agreement between simulations andexperiments verify the validity of the two-level model for the transitions addressed.

101


−0.5 0 0.5

0.3

0.4

0.5 (a)

Farad

aysign

al(a.u.)

−0.5 0 0.5

−0.1

0

0.1


Residual

(%)

−0.5 0 0.5

0.4

0.6

0.8

1

(b)

Farad

aysign

al(a.u.)

−0.5 0 0.5

−0.1

0

0.1


Residual

(%)

−0.5 0 0.5

0.6

0.8

1

1.2

1.4 (c)

Farad

aysign

al(a.u.)

−0.5 0 0.5

−0.1

0

0.1


Residual

(%)

−0.5 0 0.5

0.6

0.8

1

1.2

1.4

1.6

1.8 (d)Farad

aysign

al(a.u.)

−0.5 0 0.5

−0.1

0

0.1


Residual

(%)

−0.5 0 0.5

1

1.5

2

2.5 (e)

Farad

aysign

al(a.u.)

−0.5 0 0.5

−0.1

0

0.1


Residual

(%)

−0.5 0 0.5

1.5

2

2.5

3

3.5 (f )

Farad

aysign

al(a.u.)

−0.5 0 0.5

−0.1

0

0.1


Residual

(%)

Figure 8.4. Panel (a)− (f) display by the gray curves in the upper windows the measuredFAMOS spectra from 4.2, 20, 40, 60, 90 and 150 Torr of the 100 ppm NO addressing theoverlapping Q22(21/2) and QR12(21/2) lines in the electronic Π− Σ band, respectively.The red and black curves represent fits based on the two-transition model, where thered curve corresponds to a fit that neglect asymmetry effects whereas the black curvecorresponds to a fit which includes these effects. The lower parts of each panel display thecorresponding residual normalized with respect to the peak-to-peak value of the FAMOSsignal.

102


Figure 8.5. Panel A: Measured FAMOS spectra of the gaseous mixture of 100 ppm NOin N2 for various total pressures. The 22 curves represent total pressures of 0, 0.5, 1,2, 4.2, 8, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 118, 128, 150, 200, and 250 Torr,respectively. Each spectrum corresponds to an average of ten measurements, correctedfor their background. Panel B: Simulations of the FAMOS signal calculated from thetwo-tranistion model for a magnetic field of 130 G and the same set of pressures as in panelA. The x-axes: detuning measured from the unshifted center frequency of the transition(i.e. under Doppler limited conditions). The y-axes: FAMOS signal (arbitrary scale).

Figure 8.6. Peak values of the FAMOS signal as a function of total pressure from a gasmixture of 100 ppm NO in N2. Panel A shows the experimental data whereas Panel Bshows the simulations built on the two-transition model. The data, which is taken fromFigure 8.5, corresponds to a magnetic field amplitude of 130 G.

103


8.2 Theoretical results

8.2.1 Effects of lineshape asymmetries on measured signals

As was discussed in Section 4.3, there are mainly two phenomena that affect thesymmetry properties of the lineshapes in Faraday modulation signals: (i) frequencydependent laser intensity and (ii) polarization imperfections. In Figure 8.7 and8.8 the residual from different curve fits together with measured data is shown.The residuals in Figure 8.7 originate from FAMOS addressing the Q22(21/2) andQR12(21/2) lines in the electronic Π− Σ band in NO at a pressure of 4.2 Torr andare based on the expressions given in Paper VII, where the effects of a frequencydependent laser intensity have been neglected since the laser was equipped with anintensity stabilization feedback. The residuals in Figure 8.8 (b)− (e) originate fromcurve fits to data from FAMOS addressing the Q3/2(3/2)-transition, for 100 ppmof NO at a pressure of 40 Torr. The curve fits are based on the same expressions asthose for Figure 8.7 both with and without the influence of a frequency dependentlaser intensity.

8.2.2 Evaluation of the Westberg-Wang-Axner method

The main advantage of the WWA-method is the reduction in calculation time(without introducing approximations) as compared to the conventional approachof solving the necessary integrals numerically. This is especially important whenperforming real-time curve fits to measured data, since such a procedure putshigh demands on the efficiency of the curve fitting algorithms. For example,let us assume that we use a laser scanning rate of 20 Hz and that each curvefit needs approximately 10 iterations for sufficient convergence. This gives us1/(20 · 10) · 1000 = 5 ms for each iteration (if the data acquisition is assumed tobe instantaneous). The conventional procedure simply cannot be applied undersuch conditions since the available time is greatly exceeded (by several orders ofmagnitude) by the calculation time of the curve fits. This problem is remedied bythe WWA-method, which significantly reduces the calculation time and therebyallows for real-time curve fitting. The WWA-method applied to a modulatedVoigt absorption lineshape function is shown in Figure 8.9. The figure showsthat the WWA-method, while being accurate, producing residuals solely givenby the accuracy of the software used [see panels (c) and (d)], is also considerablyfaster than any previously used methodologies. The case with a modulated Voigtdispersion lineshape function is illustrated in Figure 2 in Paper VIII, where theaccuracy of two different evaluation algorithms are compared.

104


−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2

0

0.05

0.1(a)

Measured data


Faradaysignal(a.u.)

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2−10

0

10

(b)

C-FAMOS

Residual(%

)

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2−10

0

10

(d)

SPI-FAMOS

Residual(%

)

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2−10

0

10

(e)

SA-FAMOS

Residual(%

)

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2−10

0

10

(c)

F-FAMOS

Residual(%

)

Figure 8.7. Panel (a) shows a FAMOS signal from the overlapping Q22(21/2) andQR12(21/2) lines in the electronic Π − Σ band in NO at 100 ppm NO at a pressureof 4.2 Torr. Panel (b), (c), (d), and (e) display fits to the measured data based onthe expressions for conventional FAMOS [modulated version of the expression given inequation (3.21)], full FAMOS [modulated version of the expression given in equation (4.19)],small polarization imperfections FAMOS [modulated version of the expression given inequation (3.19)] and small attenuation FAMOS [equation (4.23)], respectively.

105


−0.03 −0.02 −0.01 0 0.01 0.02 0.03

−1

0

1

2

3

4

(a)

Measured data


Faraday

signal(a.u.)

−0.03 −0.02 −0.01 0 0.01 0.02 0.03−2

0

2

(b)

C-FAMOS w. or w/o. FDLI

Residual(%

)

−0.03 −0.02 −0.01 0 0.01 0.02 0.03−2

0

2

(d)

SPI-FAMOS w. or w/o. FDLI

Residual(%

)

−0.03 −0.02 −0.01 0 0.01 0.02 0.03−2

0

2

(e)

SA-FAMOS w. or w/o. FDLI

Residual(%

)

−0.03 −0.02 −0.01 0 0.01 0.02 0.03−2

0

2

(c)

F-FAMOS w. or w/o. FDLI

Residual(%

)

Figure 8.8. Panel (a) shows a FAMOS signal originating from the most magneticallysensitive transition in the fundamental rotational-vibrational band of NO, the Q3/2(3/2)-transition, for 100 ppm of NO at a pressure of 40 Torr. Panel (b), (c), (d), and (e) displayfits to the measured data based on the expressions for conventional FAMOS [modulatedversion of the expression given in equation (3.21)], full FAMOS [modulated versionof the expression given in equation (4.19)], small polarization imperfections FAMOS[modulated version of the expression given in equation (3.19)] and small attenuationFAMOS [equation (4.23)], respectively.

106


−20 −15 −10 −5 0 5 10 15 20−0.05

0

0.05

0.1

νd

2ndFourier

coeffi

cient

calculation time: 0.92 s (i) Conventional

(ii) De Tommasi et al.

(iii) Convolution (conv)

(iv) Convolution (FFT)

−20 −15 −10 −5 0 5 10 15 20−2

0

2x 10

−3

νd

Differen

ce calculation time: 0.18 s

−20 −15 −10 −5 0 5 10 15 20

−0.5

0

0.5

1x 10

−14

νd

Differen

ce

calculation time: 0.013 s

−20 −15 −10 −5 0 5 10 15 20

−0.5

0

0.5

1x 10

−14

νd

Differen

ce calculation time: 0.00093 s

a

b

c

d

Figure 8.9. The accuracy and calculation time for different methods of calculatingthe modulated Voigt absorption lineshape function. Panel a displays the four differentevaluation methods discussed in Paper VI and the calculation time for the conventionalapproach which is also used as a reference in the panels b-d. Panel b shows the residualbetween the reference lineshape and the evaluation method given by Tommasi et al. [67].Panel c displays the residual for the WWA-method calculated using Matlab’s built-infunction conv. Finally, panel d displays the residual for the WWA-method calculatedusing a simple FFT-based convolution algorithm.

107

Chapter 9

Conclusions and outlook

”If you would cause your view ... to be acknowledged by scien-tific men; you would do a great service to science. If you wouldeven get them to say yes or no to your conclusions it wouldhelp to clear the future progress. I believe some hesitate becausethey do not like their thoughts disturbed.”

— Michael Faraday

9.1 Conclusions

FAMOS is a spectroscopic technique based on a sinusoidal magnetic field modulationof the magnetic rotation spectroscopy (MRS) technique. FAMOS has demonstratedexcellent detection limits for NO down to low (or even sub-) ppb-levels, whichgreatly supersede those obtained with the direct absorption spectroscopy (DAS)technique and can be advantageously compared to those of the commonly usedwavelength modulation spectroscopy (WMS) technique. Although the experimentalimprovements of the Faraday modulation technique has been extensive during thelast decades the theoretical description has not seen the same level of development.Therefore this thesis has focused, to a significant part, on the development ofa theoretical model for FAMOS, which is simplified significantly by expressingthe FAMOS 1f -signal in terms of the integrated linestrength (conveniently foundin the HITRAN database) and the difference between 1st Fourier coefficientsof modulated lineshape functions for LHCP and RHCP light. It is also shownhow the lineshape asymmetry effects can be accounted for by incorporating intothe theoretical model the Fourier coefficients for modulated absorption lineshape

109


functions. For the case of the rotational-vibrational Q-transitions in NO, whichinclude the most magnetically sensitive transition Q(3/2)3/2 in the MIR region,it has been shown (Paper I) that the FAMOS signal is proportional to solely oneFourier coefficient for dispersion, and if polarization imperfections are included,also a Fourier coefficient for absorption, which results in a compact and simpleexpression for the 1f FAMOS signal. This work also includes the derivation of anew method for rapid calculation of modulated Voigt lineshape functions that canbe readily applied to both dispersion and absorption techniques.

The improvements mentioned above have a number of significant advantages.First, the use of the tabulated integrated linestrength means that no cumbersometransition dipole moments need to be considered. Second, the time-consumingintegrals needed for the numerical calculations of the demodulated lock-in signal canlargely be avoided since they are an inherent property of the 1st Fourier coefficientsof modulated lineshape functions. This often reduces the calculation times ofthe curve fits by several orders of magnitude even when compared to numericalapproximations of the Voigt profile using Fast Fourier Transforms (FFTs). This,in the end, means that real-time curve fitting is feasible also for the techniquesthat rely on dispersion rather than absorption and by applying the WWA-methoddescribed in Paper VI the theoretical model is extended to also include the caseswhere the Voigt lineshape profile is preferred over the otherwise commonly usedLorentzian. Lastly, the effects of lineshape asymmetries are incorporated into thetheoretical description making accurate curve fits possible even when low qualitypolarizers are used or when the laser intensity varies over the frequency scan.

For the case of UV-FAMOS, which addresses the even stronger electronictransitions, a novel two-transition model has been introduced. It is based ona simple but effective approximation that allows a reduction in the number oftransitions needed for an accurate description of the 1f UV-FAMOS signal. Themodel has been verified experimentally in Paper III, which is the first demonstrationof the Faraday modulation technique targeting the strong electronic transitionsin NO. This particular wavelength region shows great potential for obtaininglow detection limits through the FAMOS technique, mostly since the transitionstrengths often exceed those of the fundamental rotational-vibrational band byroughly two orders of magnitude. However, practical limitations in the experimentalsetup implied that only under-modulated measurement conditions have so far beentargeted, which severely restricted the detection limit. Despite this the two-transition model showed good agreement with the measured spectra and predictedplausible optimum measurement conditions.

In summary, this thesis provides a step towards a complete theoretical descrip-tion of the Faraday modulation spectroscopy technique. The expressions derivedherein can swiftly be applied during the data analyzing stages of the measurementprocess in order to facilitate real-time curve-fitting to asymmetric as well as sym-metric FAMOS signals in any wavelength region, which is of great importance when

110


translating the measured signals into their corresponding concentration values. Themodel has been experimentally verified through measurements performed by twodifferent configurations that target two completely different wavelength regions.

9.2 Outlook

Although much work has been done there are still areas of interest that might beexplored. The inherent properties of this technique makes it highly applicable incases that require low ppb detection of a gaseous compound in the presence oflarge quantities of H2O and CO2. One such area might be breath analysis, wherethe measurement conditions are rather controlled and the portability demands onthe instrumentation are quite low, but where the presence of interfering gaseouscompounds is relatively high. There have been successful attempts of measuringexhaled NO in human breath with the FAMOS technique but much work stillremains before a reliable and commercially available instrumentation can be built.

In addition, the heat production and power consumption of the magnetic coilsare major issues when considering applying this technique outside of the laboratoryenvironment. As an alternative, the development of a combination of wavelengthmodulation and magnetic rotation spectroscopy based on permanent static magnetsmight be a viable path to building instrumentation based on this technique forfield use [33]. There have already been a considerable amount of work put into thisparticular experimental realization [33, 44, 45] and this will undoubtedly amountin very interesting future implementations of the magnetic rotation techniques.

111

Appendix A

Appendix

”The experimental researches of Faraday are so voluminous,their descriptions are so detailed, and their wealth of illustrationis so great, as to render it a heavy labour to master them. Themultiplication of proofs, necessary and interesting when the newtruths had to be established, are however less needful now whenthese truths have become household words in science.”

— John Tyndall

A.1 Definition of linestrength

Radiative transfer theory defines the integrated linestrength (also referred to asspectral line intensity) for the two states of a rotational-vibrational system as [68]

Si,j =hνi,jc

NiN

(1− gi

gj

NiNj

)Bi,j , (A.1)

where h is Planck’s constant, νi,j is the transition frequency, c is the speed oflight, Bi,j is the Einstein coefficient for induced absorption, gi and gj are thestatistical weights, Ni and Nj are the populations of the lower and upper states,respectively, and N is the molecular number density. The Einstein coefficientfor induced absorption can, in turn, be related to the weighted transitions dipolemoment squared, Ri,j , given in SI-units, as

113

Appendix

Bi,j =1

4πε0

8π3

3h2Ri,j . (A.2)

The population partition is given by Boltzmann statistics which allows us to write

NjNi

=gjgi

exp (−c2νi,j/T ) (A.3)

and

NiN

=gi exp (−c2Ei/T )

Q(T ), (A.4)

where Ei is the lower state energy, c2 the second radiation constant, given byc2 = hc/k, T the temperature in Kelvin and Q(T ) the total internal partition sumgiven by

Q(T ) =∑i

gi exp (−c2Ei/T ) . (A.5)

By substituting equation (A.2), (A.3) and (A.4) into equation (A.1) neglecting anyisotopics effects gives us an expression for the linestrength (in SI-units)

Si,j(T ) =1

4πε0

8π3

3hcνi,j

gi exp (−c2Ei/T )

Q(T )[1− exp (−c2νi,j/T )]Ri,j , (A.6)

where ε0 is the permittivity of free space, and Ri,j the weighted transition dipolemoment squared, defined as

Ri,j =1

gi|Ri,j |2 , (A.7)

where |Ri,j |2 represents the transition dipole moment squared between the twostates, i, and j. The integrated linestrength can be corrected for temperatureby using the integrated linestrength from HITRAN [7] at a certain referencetemperature, Tref , and using the following expression

114

Appendix

Si,j(T ) = Si,j(Tref )Q(Tref )

Q(T )

exp (−c2Ei/T )

exp (−c2Ei/Tref )

[1− exp (−c2νi,j/T )]

[1− exp (−c2νi,j/Tref )]. (A.8)

A.2 The wave vector and integrated linestrength – adifferent approach

It has become customary for the FAMOS technique to express the real part of thewave vector associated with a transition between a state i and a state j as wasindicated by Liftin et al. [6], namely as

k(ν) = (Ni −Nj)|〈i|µ |j〉|2

2u~ε0ReZ

[ cuν

(ν + iδνL)], (A.9)

where δνL is the homogeneous broadening (HWHM) of the transition, Ni and

Nj the population density of state i and j, respectively, |〈i|µ |j〉|2 the transitiondipole moment squared between states i and j, ε0 the permittivity of free space, ~Planck’s constant divided by 2π, u the most probable molecular speed, Z[...] theplasma dispersion function and ν the shifted frequency detuning from the peakabsorption of the transition given by

ν = νd ∓ gµBB, (A.10)

where νd is the unshifted detuning frequency, g the g-factor, µB the Bohr magnetonand B the magnetic field.

In this section it will be shown that the simplified expressions based on theintegrated linestrength derived in this thesis are fully compatible with equation (A.9)previously used in this field. We start by restating equation (3.49) given inSection 3.4.2, which gives us the total phase shift of the two helical components ofthe light

φL/R(ν) =Si,jNxL

2χdispV (ν). (A.11)

The definition of integrated linestrength, Si,j , in SI-units given in equation (A.6)implies that we can write

115

Appendix

φL/R(ν) =π2νNxL

3hcε0

gie−c2Ei/T

Q(T )

[1− e−c2ν/T

]Ri,j · χdispV (ν). (A.12)

and by using Ri,j = (1/gi) |Ri,j |2 and φ(ν) = k(ν)L we obtain

kL/R(ν) =π2νN

3hcε0

e−c2Ei/T

Q(T )

[1− e−c2ν/T

]|Ri,j |2 χdispV (ν). (A.13)

The definition of the dispersion lineshape function, previously given in equa-tion (3.33), gives

χdispV (ν) = −χ01√π

ReZ[ cuν

(ν + iδνL)]

, (A.14)

where χ0 is the peak value of the area-normalized Gaussian lineshape functiongiven by c/(

√πuν), which leads us to the following expression

kL/R(ν) =πNx

3uhε0

e−c2Ei/T

Q(T )

[1− e−c2ν/T

]|Ri,j |2 Re

Z[ cuν

(ν + iδνL)]

.

(A.15)

Furthermore, by using equation (A.3) and equation (A.4) it can be seen that thefollowing relation holds

e−c2Ei/T

Q(T )

[1− e−c2ν/T

]=gjNi − giNj

gigj

1

N=

(Nigi− Njgj

)1

N, (A.16)

which leads us to a fully simplified expression, to be compared with equation (A.9),

kL/R(ν) =

(Nigi− Njgj

) |〈i|µ |j〉|22u~ε0

ReZ[ cuν

(ν + iδνL)]

, (A.17)

where we have identified |〈i|µ |j〉|2 as the orientationally averaged total transition

dipole moment squared, i.e. as |Ri,j |2 /3, and the degeneracy explicitly has beentaken into account as was done by Blake et al. [17]. Hence, this shows thatequation (3.49) given in Section 3.4.2 above is fully compatible with the commonly

116

Appendix

used expression for the real part of the wave vector given by Liftin et al. [6]in equation (A.9). Since the integrated linestrength for virtually all transitionsnowadays can be found tabulated, this opens up for possibilities to appropriatelyassess the strength of a given transition also in FAMOS.

A.3 Coupling of electronic and rotational motion: Shortdescription of Hund’s cases (a) and (b)

There are various ways in which the different angular momenta can be coupled indiatomic molecules. This gives rise to the so called Hund’s coupling cases [69, 70]named after F. Hund [53]. It is important to remember that the Hund’s casesrepresent idealized situations and are used as an aid to understand the pattern ofrotational levels and the resulting spectra. The following angular momenta areinvolved:

• L – the electronic orbital angular momentum

• S – the electronic spin angular momentum

• J – the total angular momentum

• N – the total angular momentum excluding electron spin

• R – the rotational angular momentum of the nuclei

For a complete description also the nuclear spin angular momentum, I, mustbe taken into account. However, this has a negligible effect on molecular energylevels wherefore it for simplicity often is ignored. Here, only the cases relevant forthis work, (a) and (b), are described.

A.3.1 Hund’s case (a)

In Hund’s case (a), illustrated by the vector diagram in Figure A.1, the electrostaticinteraction is much larger than the spin-orbit interaction which in turn is muchlarger than the spin-rotation interaction. The axial components of L and S arewell defined, and are denoted Λ and Σ, respectively, the sum, Λ + Σ, is denoted Ω.The rotational angular momentum of the nuclei, R is coupled to an axial vectorΩ which forms the resulting total angular momentum, J. The wave functions forcase (a) can be written in ket notation as ‖ζ,Λ;S,Σ; J,Ω,MJ〉, where ζ denotesall quantum numbers that are not expressed explicitly, such as electronic andvibrational quantum numbers.

117

Appendix

For this work the effects of an externally applied magnetic field is of greatinterest. In this case a transition adhering to Hund’s case (a) will be split accordingto its MJ magnetic quantum number. A magnetic field will therefore split a statewith a given angular momentum, J , into 2J + 1 states with energies given byMJgJµBB, where MJ takes half-integer values from −J to J , gJ is the g-factor ofstate J , µB Bohr’s magneton and B the magnetic field amplitude. As J increases,case (a) becomes less appropriate and other cases need to be used.

L

S

R

J

Λ Σ

Ω

Figure A.1. Vector coupling diagram for Hund’s case (a).

A.3.2 Hund’s case (b)

In Hund’s case (b) the spin-orbit interaction is so small that it can be neglected.This means that Λ = 0 while S 6= 0, which in turn implies that Ω is not defined.This takes place predominantly in states that lack orbital angular momentum,e.g. Σ-states, which effectively means that the spin interaction with the rotationalmotion of the nuclei is now the most important interaction. In this case a transitionwill be split, by the presence of an external magnetic field, according to its MS

quantum number, which, for a state with a single valence electron, as is the casefor NO, is the magnetic quantum number of the spin of the electron, MS = ±1/2.This means that the energy separation between the substates in this case will begiven by MSgSµBB, where gS is the g-factor of the free electron (2.0023192), µBBohr’s magneton and B the magnetic field amplitude.

118

L

S

RJ

Λ

N

Figure A.2. Vector coupling diagram for Hund’s case (b).

119

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127

Acknowledgements

Of the many people who deserve thanks, some are particularly prominent: Mysupervisor Ove Axner, always available for theoretical discussions even though Iknow he’s always very busy. My deputy supervisors; Pawel Kluczynski and StefanLundqvist, they have been very helpful in providing solutions to practical problemsregarding the experimental setup (lasers, lenses, mirrors, polarizers, electronicsetc.) which was a tremendous help during the first two years. They also deservespecial thanks for providing me with the opportunity to work in Gothenburg andfor showing me how things work outside the lab (in the industry). I hope thecollaboration will continue in the future in some form. My gratitude also goes tomy co-workers in the Laser Physics Group at Umea University, not only for yourhelp to solve work related problems but also for the non-work related discussionsand for your company during travels to conferences. I would also like to thank theUmea University Industrial Post Graduate School and all participants therein forgiving me the opportunity to share interesting courses, seminars and workshopswith you. I would also like to thank my family, since I believe that it is customaryto do so, although their contributions to the actual thesis has been negligible.

Colophon

This thesis was written in LATEX using a modified version of Andy Buckley’s[71] hepthesis class from ctan.org.

129

List of Figures

1.1 Michael Faraday giving a lecture. . . . . . . . . . . . . . . . . . . . 2

1.2 Integrated linestrengths for nitric oxide. . . . . . . . . . . . . . . . 3

1.3 Basic experimental layout of FAMOS. . . . . . . . . . . . . . . . . 4

2.1 The basic principle of absorption spectroscopy. . . . . . . . . . . . 9

2.2 Gaussian absorption lineshape. . . . . . . . . . . . . . . . . . . . . 12

2.3 Lorentzian absorption lineshape. . . . . . . . . . . . . . . . . . . . 13

2.4 Voigt absorption lineshape. . . . . . . . . . . . . . . . . . . . . . . 14

2.5 Absorption and dipersion lineshapes for a transition in NO at 226.57nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.6 Allan deviation plots for different drifts and noise levels. . . . . . . 23

3.1 Basic experimental layout of MRS. . . . . . . . . . . . . . . . . . . 26

3.2 A schematic illustration of the splitting of a R(3/2) transition in NO. 27

3.3 Individual contributions to the phase shifts from a R3/2(3/2) transi-tion for B = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4 Individual contributions to the attenuations from a R3/2(3/2) tran-sition for B = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.5 Individual contributions to the phase shifts from a R3/2(3/2) transition. 41

3.6 Individual contributions to the attenuations from a R3/2(3/2) tran-sition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

131

3.7 Individual contributions to the phase shifts from a Q3/2(3/2) transition. 43

3.8 Individual contributions to the attenuations from a Q3/2(3/2) tran-sition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.1 1st Fourier coeffients of modulated lineshape functions. . . . . . . . 49

4.2 Dependence of ε on the lineshape . . . . . . . . . . . . . . . . . . . 52

4.3 Transmission spectra in the fundamental ro-vib band of NO. . . . 52

4.4 FAMOS spectra from transitions in the fundamental ro-vib band ofNO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.5 FAMOS spectra from nitric oxide in the Q3/2-branch. . . . . . . . 56

4.6 1st Fourier coefficient as a function of normalized detuning. . . . . 58

4.7 1st Fourier coefficient as a function of normalized modulation ampli-tude and Voigt parameter. . . . . . . . . . . . . . . . . . . . . . . . 60

4.8 The optimum modulation amplitude and the optimum Voigt parameter. 61

4.10 Integrated linestrengths for the electronic band in NO . . . . . . . 63

4.11 Schematic illustration of the energy level structure for a Q22(J ′′)and a QR12(J ′′) transition in NO. . . . . . . . . . . . . . . . . . . 64

4.12 A simplified two-transition model of the energy level structure asso-ciated with an electronic transition in NO. . . . . . . . . . . . . . . 66

4.13 Peak value of the normalized FAMOS signal as a function of magneticfield amplitude for ten total pressures. . . . . . . . . . . . . . . . . 68

4.14 Normalized FAMOS signal as a function of frequency detuning forvarious total pressure for four different magnetic field ampliitudes. 69

4.15 Peak value of the normalized FAMOS signal, SEF , as a function oftotal pressure for a variety of magnetic field amplitudes. . . . . . . 70

4.16 Optimum magnetic field and optimum total pressure. . . . . . . . . 71

4.17 The maximum FAMOS signal as a function of pressure and magneticfield. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.18 Noise, Signal and SNR for a system dominated by thermal noise. . 75

4.19 Noise, Signal and SNR for a system dominated by shot noise. . . . 75

132

4.20 Noise, Signal and SNR for a system dominated by flicker noise. . . 75

4.21 Different noise contributions and SNR. . . . . . . . . . . . . . . . . 76

5.1 χdisp,evenL,n (νd, νa). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.2 χdisp,evenL,1 (νd, νa). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

7.1 Basic experimental layout of FAMOS. . . . . . . . . . . . . . . . . 92

7.2 Experimental setup of the FAMOS-system adressing the Q3/2(3/2)transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

7.3 Basic experimental layout of FAMOS addressing a UV-transition. . 94

8.1 Experimental data and simulations. . . . . . . . . . . . . . . . . . . 99

8.2 Optimum angle and optimum magnetic field amplitude. . . . . . . 99

8.3 Allan variance plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

8.4 Measured FAMOS spectra together with fits and residuals. . . . . 102

8.5 Measured FAMOS spectra together with calculated FAMOS spectra. 103

8.6 Peak values of the FAMOS signal as a function of total pressure. . 103

8.7 FAMOS signal and fit from UV-FAMOS. . . . . . . . . . . . . . . . 105

8.8 FAMOS signal and fit from MIR-FAMOS. . . . . . . . . . . . . . . 106

8.9 Evaluation of the accuracy of the WWA-method. . . . . . . . . . . 107

A.1 Hund’s case (a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

A.2 Hund’s case (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

133

Nomenclature

Abbreviations

atm . . . . . . . . . . . . . . atmosphere

c.c. . . . . . . . . . . . . . . . complex conjugate

cw . . . . . . . . . . . . . . . continuous-wave

DAQ . . . . . . . . . . . . . data acquisition

DAS . . . . . . . . . . . . . . direct absorption spectroscopy

DFB . . . . . . . . . . . . . distributed-feedback

ECDL . . . . . . . . . . . . external cavity diode laser

FAMOS . . . . . . . . . . Faraday modulation spectroscopy

FDLI . . . . . . . . . . . . . frequency dependent laser intensity

FRS . . . . . . . . . . . . . . Faraday rotation spectroscopy

FWHM . . . . . . . . . . . full-width-half-maximum

HITRAN . . . . . . . . . high-resolution transmission molecular absorption

HWHM . . . . . . . . . . half-width-half-maximum

Im . . . . . . . . . . . . . . . imaginary part

LHCP . . . . . . . . . . . . left-handed circularly polarized light

MCB . . . . . . . . . . . . . magnetic circular birefringence

MCD . . . . . . . . . . . . . magnetic circular dichroism

MCT . . . . . . . . . . . . . Mercury Cadmium Telluride

MIR . . . . . . . . . . . . . . mid-infrared

NIR . . . . . . . . . . . . . . near-infrared

NO . . . . . . . . . . . . . . . nitric oxide

PI . . . . . . . . . . . . . . . . polarization imperfections

135

QCL . . . . . . . . . . . . . quantum cascade laser

QN . . . . . . . . . . . . . . . fundamental quantum noise

Re . . . . . . . . . . . . . . . real part

RHCP . . . . . . . . . . . . right-handed circularly polarized light

SA-FAMOS . . . . . . FAMOS signal assumning small absorbances

SA-MRS . . . . . . . . . MRS signal assumning small absorbances

SNR . . . . . . . . . . . . . . signal-to-noise ratio

UV . . . . . . . . . . . . . . . ultra-violet

UV-FAMOS . . . . . . Faraday modulation spectroscopy in the ultra-violet region

wm-NICE-OHMS . wavelength-modulated noise-immune cavity-enhanced opticalheterodyne molecular spectroscopy

WMS . . . . . . . . . . . . wavelength modulation spectroscopy

ZMS . . . . . . . . . . . . . Zeeman modulation spectroscopy

Constants

µB . . . . . . . . . . . . . . . Bohr’s magneton [cm−1/G], i.e. 4.67× 10−5 c/m/G

c . . . . . . . . . . . . . . . . . speed of light [cm/s], i.e. 2.997 924 58× 1010 cm s−1

gS . . . . . . . . . . . . . . . . g-factor of the free electron, gS = 2.0023192

kB . . . . . . . . . . . . . . . Boltzmann constant (1.380 65× 10−23 [J/K])

Definitions

1f signal . . . . . . . . . signal demodulated at the modulation frequency

x . . . . . . . . . . . . . . . . . the entity, x, is dimension less

⊗ . . . . . . . . . . . . . . . . denotes a convolution

ρ-type doubling . . . splitting of rotational state due to coupling between the elec-tronic spin and the rotation of the nuclei

Σ . . . . . . . . . . . . . . . . . molecular electronic state

2Π . . . . . . . . . . . . . . . molecular electronic state

Jn . . . . . . . . . . . . . . . . nth Bessel function

N . . . . . . . . . . . . . . . . quantum number related to the rotation of the molecule as awhole

Qx(y)-transition . . Q-transition for which the total angular momentum quantumnumber of the lower state J ′′ = y and the projection of thetotal angular momentum along the internuclear axis is x

sign(x) . . . . . . . . . . . sign of argument x

exp . . . . . . . . . . . . . . . exponential function

136

P-transition . . . . . . transition for which ∆J = −1

Q-transition . . . . . . transition for which ∆J = 0

R-transition . . . . . . transition for which ∆J = 1

Variables & Functions

(α0)min . . . . . . . . . . minimum detectable absorbance

(∆φ)noise . . . . . . . . . noise in the measurement of the phase shift

(Pt)fl . . . . . . . . . . . . laser noise in the measurement of transmitted power

(Pt)shot . . . . . . . . . . shot noise in the measurement of transmitted power

(Pt)th . . . . . . . . . . . . detection system noise in the measurement of transmittedpower

α . . . . . . . . . . . . . . . . . fraction of light transmitted through the main axis of thepolarizer

α(ν) . . . . . . . . . . . . . . absorption of narrowbanded light due to a transition

α0 . . . . . . . . . . . . . . . . absorbance peak-value (on resonance)

χF . . . . . . . . . . . . . . . normalized FAMOS lineshape function

ε . . . . . . . . . . . . . . . . . uncrossingle angle-normalized unbalancing term defined asε/ sin(2θ)

χdisp,even1 . . . . . . . . . 1st (even) Fourier coefficient of a modulated dispersion line-shape function

χdisp . . . . . . . . . . . . . the product of the area-normalized modulated dispersion line-shape function, χdisp and the peak-value of the Gaussianlineshape function, χ0

χ1 . . . . . . . . . . . . . . . . Doppler-peak-normalized FAMOS lineshape function

χdisp,even1 . . . . . . . . . 1st even Fourier coefficient of the normalized modulated dis-persion lineshape function

χdisp,evenL,n . . . . . . . . . normalized nth (in-phase) Fourier coefficient of a modulatedLorentzian dispersion lineshape function

χabs,evenL/R . . . . . . . . . . even Fourier coefficient of the normalized modulated absorp-

tion lineshape function for LHCP and RHCP light

χdisp,evenL/R . . . . . . . . . even Fourier coefficient of the normalized modulated dispersion

lineshape function for LHCP and RHCP light

χEV,1 . . . . . . . . . . . . . . normalized FAMOS lineshape function for an electronic tran-sition

δ . . . . . . . . . . . . . . . . . average attenuation

ν . . . . . . . . . . . . . . . . . shifted Doppler-width normalized frequency detuning

137

νa±MS. . . . . . . . . . . . . normalized modulation amplitude for ±MS

νL/Ra . . . . . . . . . . . . . magnitude of the normalized modulation amplitude for LHCP

and RHCP light

νd . . . . . . . . . . . . . . . . Doppler-width normalized frequency detuning

νa . . . . . . . . . . . . . . . . normalized modulation amplitude

τ . . . . . . . . . . . . . . . . . normalized time given by τδνL

SEF . . . . . . . . . . . . . . . linestrength-, interaction length-, concentration- and Dopplerpeak-normalized FAMOS signal for electronic transitions

SF . . . . . . . . . . . . . . . signal-strength- and concentration-normalized FAMOS signal

SM ′′M ′ . . . . . . . . . . . transition-specific relative integrated linestrength

β . . . . . . . . . . . . . . . . . fraction of light transmitted through the axis orthogonal tothe main axis of the polarizer

χ . . . . . . . . . . . . . . . . . area-normalized lineshape function [1/cm−1]

χabs . . . . . . . . . . . . . . absorption lineshape function

χdisp,evenV,n . . . . . . . . . nth Fourier coefficient of a modulated Voigt dispersion line-shape

χdisp . . . . . . . . . . . . . dispersion lineshape function

χG . . . . . . . . . . . . . . . Gaussian lineshape function

χL . . . . . . . . . . . . . . . Lorentzian lineshape function

χn . . . . . . . . . . . . . . . nth (in-phase) Fourier coefficient of an arbitrary modulatedlineshape function

χV . . . . . . . . . . . . . . . Voigt lineshape function

∆Snet . . . . . . . . . . . . short notation for the equally strong net relative linestrengths

∆Snet±MS. . . . . . . . . . net relative linestrength for ±MS

∆E . . . . . . . . . . . . . . energy of an excited state

∆f . . . . . . . . . . . . . . . bandwidth

∆I . . . . . . . . . . . . . . . change in intensity

∆k . . . . . . . . . . . . . . . difference between wave vectors for LHCP and RHCP light

∆M . . . . . . . . . . . . . . the difference in magnetic quantum number between the upperand the lower state

∆t . . . . . . . . . . . . . . . lifetime an excited state

∆χabs,even1 . . . . . . . . difference in 1st even Fourier coefficient of the normalizedmodulated absorption lineshape function between LHCP andRHCP light

138

∆χdisp,even1 . . . . . . . difference in 1st even Fourier coefficient of the normalizedmodulated dispersion lineshape function between LHCP andRHCP light

δνDL . . . . . . . . . . . . . . Doppler-width normalized Voigt parameter, also referred toas homogeneous broadening

δνatmL . . . . . . . . . . . . the Doppler-width normalized Voigt parameter under atmo-spheric pressure conditions

∆δ . . . . . . . . . . . . . . . difference between the attenuations for LHCP and RHCP light

δνD . . . . . . . . . . . . . . half-width-half maximum (HWHM) Doppler width of thetransition [cm−1]

δνED . . . . . . . . . . . . . . Doppler-width (HWHM) for an electronic transition [cm−1]

δνQD . . . . . . . . . . . . . . Doppler-width (HWHM) for a Q-transition [cm−1]

δνL . . . . . . . . . . . . . . . full-width-half-maximum (FWHM) of the Lorentzian lineshape

δνL . . . . . . . . . . . . . . . half-width-half maximum (HWHM) Voigt parameter, alsoreferred to as homogeneous broadening [cm−1]

∆φ . . . . . . . . . . . . . . . difference in frequency dependent phase shift between LHCPand RHCP light

∆φ(t)/2 . . . . . . . . . . rotation of the plane of polarization at time t

δM′′M ′

L/R . . . . . . . . . . . attenuation from a M ′′ →M ′ transition for LHCP and RHCPlight, respectively

δL/R . . . . . . . . . . . . . attenuation for LHCP and RHCP light, respectively

δM ′′M ′ . . . . . . . . . . . attenuation induced by a M ′′ →M ′ transition

δmn . . . . . . . . . . . . . . Kroencker’s delta-function, which is 0 for m 6= n and 1 form = n

η . . . . . . . . . . . . . . . . . instrumentation factor comprising detector sensitivity, inputimpedance of current-to-voltage converter and any contribu-tion from the gain of amplifiers

η′ . . . . . . . . . . . . . . . . responsivity of the detector

γ . . . . . . . . . . . . . . . . . the collision broadening parameter [cm−1/atm]

γi . . . . . . . . . . . . . . . . FWHM pressure broadening coefficient for collisions withspecies i

χabsL/R . . . . . . . . . . . . . attenuation lineshape for LHCP and RHCP light, respectively

χdispL/R . . . . . . . . . . . . . phase shift lineshape for LHCP and RHCP light, respectively

χabs . . . . . . . . . . . . . . area-normalized absorption lineshape function [1/cm−1]

χdisp . . . . . . . . . . . . . area-normalized dispersion lineshape function [1/cm−1]

139

χ0 . . . . . . . . . . . . . . . . peak value of the area-normalized absorption Gaussian line-shape function [1/cm−1]

~ . . . . . . . . . . . . . . . . . Planck’s constant divided by 2π

κ1 . . . . . . . . . . . . . . . . linear intensity coefficient

κ2 . . . . . . . . . . . . . . . . non-linear intensity coefficient

κL/R . . . . . . . . . . . . . absorption coefficient for LHCP and RHCP light, respectively

Λ . . . . . . . . . . . . . . . . . axial component of L

e . . . . . . . . . . . . . . . . . polarization unit vector

eL . . . . . . . . . . . . . . . . unit vector for LHCP light

eR . . . . . . . . . . . . . . . . unit vector for RHCP light

Et . . . . . . . . . . . . . . . . transmitted electric field vector

E . . . . . . . . . . . . . . . . electric field vector

E∗ . . . . . . . . . . . . . . . complex conjugate of the electric field vector

I . . . . . . . . . . . . . . . . . nuclear spin angular momentum

J . . . . . . . . . . . . . . . . . total angular momentum

L . . . . . . . . . . . . . . . . . electronic orbital angular momentum

N . . . . . . . . . . . . . . . . total angular momentum excluding electron spin

R . . . . . . . . . . . . . . . . rotational angular momentum of the nuclei

S . . . . . . . . . . . . . . . . . electronic spin angular momentum

ν . . . . . . . . . . . . . . . . . frequency of the light [cm−1]

ν′ . . . . . . . . . . . . . . . . upper vibrational state

ν′′ . . . . . . . . . . . . . . . . lower vibrational state

ν0 . . . . . . . . . . . . . . . . center frequency of a transition [cm−1]

νd . . . . . . . . . . . . . . . . detuning from the center frequency given by ν − ν0 [cm−1]

νE . . . . . . . . . . . . . . . frequency of an electronic transition [cm−1]

νQ . . . . . . . . . . . . . . . frequency of a Q-transition [cm−1]

νi,j . . . . . . . . . . . . . . . transition frequency between state i and j

νM ′′M ′ . . . . . . . . . . . frequency of light originating from a M ′′M ′ transition betweentwo states [cm−1]

Ω . . . . . . . . . . . . . . . . . the sum of the axial components of L and S, Ω = Λ + Σ

ω . . . . . . . . . . . . . . . . . angular frequency

ω . . . . . . . . . . . . . . . . . angular modulation frequency [rad/s]

φM′′M ′

L/R . . . . . . . . . . . phase shift from a M ′′ →M ′ transition for LHCP and RHCPlight, respectively

140

φL/R . . . . . . . . . . . . . phase shift for LHCP and RHCP light, respectively

φM ′′M ′ . . . . . . . . . . . phase shift induced by a M ′′ →M ′ transition

Σ . . . . . . . . . . . . . . . . . axial component of S

σ . . . . . . . . . . . . . . . . . Allan deviation

σ2 . . . . . . . . . . . . . . . . Allan variance

σP . . . . . . . . . . . . . . . relative standard deviation of the power of the light impingingon the detector

σP,f . . . . . . . . . . . . . . relative standard deviation of the spectral power of the light

τ . . . . . . . . . . . . . . . . . integration variable corresponding to time [s]

θ . . . . . . . . . . . . . . . . . angle between the polarization axes of the polarizer and ana-lyzer, also known as uncrossing angle [rad]

α0 . . . . . . . . . . . . . . . . absorption strength, given by S′crelpL

ε . . . . . . . . . . . . . . . . . unbalancing term between the amplitudes of the LCHP andthe RHCP light

ε0 . . . . . . . . . . . . . . . . permittivity of free space

ξ . . . . . . . . . . . . . . . . . polarization imperfections term

ζ . . . . . . . . . . . . . . . . . denotes all quantum numbers that are not expressed explicitly

An . . . . . . . . . . . . . . . coefficient for the expression of the nth Fourier coefficient

A2i . . . . . . . . . . . . . . . spontaneous emission rate from the upper state to the lowerstate

B . . . . . . . . . . . . . . . . magnetic field [G]

B(t) . . . . . . . . . . . . . . magnetic field at time t [G]

B0 . . . . . . . . . . . . . . . amplitude of the time-dependent magnetic field [G]

BE . . . . . . . . . . . . . . . magnetic field amplitude for an electronic transition [G]

Bn . . . . . . . . . . . . . . . coefficient for the expression of the nth Fourier coefficient

BQ . . . . . . . . . . . . . . . magnetic field amplitude for a Q-transition [G]

Bi,j . . . . . . . . . . . . . . Einstein coefficient for induced absorption

c . . . . . . . . . . . . . . . . . speed of light, 29 979 245 800 cm s−1

c2 . . . . . . . . . . . . . . . . second radiation constant, given by c2 = hc/k

Cn . . . . . . . . . . . . . . . coefficient for the expression of the nth Fourier coefficient

CT . . . . . . . . . . . . . . . complex transmission function

crel . . . . . . . . . . . . . . . relative concentraion of the absorber

Dn . . . . . . . . . . . . . . . coefficient for expression of the nth Fourier coefficient

Dn . . . . . . . . . . . . . . . coefficient for the expression of the nth Fourier coefficient

141

e . . . . . . . . . . . . . . . . . electron charge

E0 . . . . . . . . . . . . . . . electric field amplitude containing the normalization

Ei . . . . . . . . . . . . . . . . lower state energy

f1 . . . . . . . . . . . . . . . . lower detection bandwidth limit

f2 . . . . . . . . . . . . . . . . upper detection bandwidth limit

fG . . . . . . . . . . . . . . . . Maxwell-Boltzmann velocity distribution

fv . . . . . . . . . . . . . . . . area-normalized distribution function

G . . . . . . . . . . . . . . . . transimpedance gain of the detection system

g . . . . . . . . . . . . . . . . . g-factor or dimensionless magnetic moment

g′ . . . . . . . . . . . . . . . . g-factor of the upper state

g′′ . . . . . . . . . . . . . . . . g-factor of the lower state

gJ . . . . . . . . . . . . . . . . g-factor of state J

g′J . . . . . . . . . . . . . . . . g-factor of the upper state J ′

g′′J . . . . . . . . . . . . . . . . g-factor of the lower state J ′′

gi . . . . . . . . . . . . . . . . statistical weight for the lower state

gj . . . . . . . . . . . . . . . . statistical weight for the upper state

h . . . . . . . . . . . . . . . . . Planck’s constant

I . . . . . . . . . . . . . . . . . intensity

i . . . . . . . . . . . . . . . . . lower state

I0 . . . . . . . . . . . . . . . . power of radiation after the polarizer [W/m2]

IL . . . . . . . . . . . . . . . . intensity from the laser

It . . . . . . . . . . . . . . . . transmitted intensity

It(t) . . . . . . . . . . . . . . transmitted intensity at time t [W/m2]

iAS . . . . . . . . . . . . . . . detector current signal from the measured analyte

idc . . . . . . . . . . . . . . . average photo current

ifl . . . . . . . . . . . . . . . . current from flicker noise

ishot . . . . . . . . . . . . . . current from shot noise

ith . . . . . . . . . . . . . . . current from thermal noise

j . . . . . . . . . . . . . . . . . upper state

J ′ . . . . . . . . . . . . . . . . total angular momentum quantum number of the upper state

J ′′ . . . . . . . . . . . . . . . . total angular momentum quantum number of the lower state

k . . . . . . . . . . . . . . . . . wave vector

K(x, y) . . . . . . . . . . . real part of the complex error function

142

kM′′M ′

L/R . . . . . . . . . . . wave vector from a M ′′ →M ′ transition for LHCP and RHCPlight, respectively

k0 . . . . . . . . . . . . . . . . amplitude of the wave vector in vacuum

kL/R . . . . . . . . . . . . . wave vector for LHCP and RHCP light, respectively

L . . . . . . . . . . . . . . . . . interaction length [cm]

L(x, y) . . . . . . . . . . . imaginary part of the complex error function

M . . . . . . . . . . . . . . . . molecular weight [g/mol]

M . . . . . . . . . . . . . . . . magnetic quantum number for the projection of angular mo-mentum in the direction of the magnetic field

m . . . . . . . . . . . . . . . . molecular mass [kg]

M ′ . . . . . . . . . . . . . . . magnetic quantum number of the upper state

M ′′ . . . . . . . . . . . . . . . magnetic quantum number of the lower state

M ′J . . . . . . . . . . . . . . . magnetic MJ quantum number of the upper state

M ′′J . . . . . . . . . . . . . . . magnetic MJ quantum number of the lower state

MS . . . . . . . . . . . . . . . secondary spin quantum number

n . . . . . . . . . . . . . . . . . refractive index

Ni . . . . . . . . . . . . . . . . population for the lower state

Nj . . . . . . . . . . . . . . . population for the upper state

Nx . . . . . . . . . . . . . . . density of molecule x [molecules/m3]

nL/R . . . . . . . . . . . . . refractive index for LHCP and RHCP light, respectively

ntot . . . . . . . . . . . . . . total number density of species

p . . . . . . . . . . . . . . . . . total gas pressure [atm]

P (θ) . . . . . . . . . . . . . Jones’ matrix for uncrossing angle θ

P0 . . . . . . . . . . . . . . . . power incident on the detector

p0 . . . . . . . . . . . . . . . . reference pressure of one atmosphere [atm]

pi . . . . . . . . . . . . . . . . partial pressure of species i

PL . . . . . . . . . . . . . . . power of light from the laser

px . . . . . . . . . . . . . . . . partial pressure from species x

ptot . . . . . . . . . . . . . . . total pressure of the sample

Q(T ) . . . . . . . . . . . . . total internal partition sum

R . . . . . . . . . . . . . . . . input impedance of the detection electronics

S . . . . . . . . . . . . . . . . . integrated (molecular) linestrength [cm−1/molecule·cm−2]

S′ . . . . . . . . . . . . . . . . integrated (gas) linestrength [cm−2/atm]

143

SEF . . . . . . . . . . . . . . . UV-FAMOS signal

S0 . . . . . . . . . . . . . . . . FAMOS signal strength

Satm0 . . . . . . . . . . . . . FAMOS signal strength under atmospheric pressure conditions

Satm0 . . . . . . . . . . . . . FAMOS signal strength under atmospheric pressure conditions

SD . . . . . . . . . . . . . . . signal measured by the detector

SF . . . . . . . . . . . . . . . FAMOS signal

SΠ,Σ . . . . . . . . . . . . . total integrated linestrength for transitions from a lower stateΠ to an upper state Σ

SL/RΠ,MJ ,Σ,MS

. . . . . . . transition linestrength for a transition induced by LHCP (L)

and RHCP (R) light from a lower state 2Πi(MJ ) to an upperstate 2Σ(MS)

SAS . . . . . . . . . . . . . . analytical signal

SBG . . . . . . . . . . . . . . background signal

Si,j . . . . . . . . . . . . . . . integrated linestrength (also referred to as spectral line inten-sity) for the two states of a rotational-vibrational system, iand j

SM ′′M ′ . . . . . . . . . . . transition-specific integrated linestrength [cm−1/(molecule·cm2]

T . . . . . . . . . . . . . . . . temperature of the gas [K]

t . . . . . . . . . . . . . . . . . time

T (ν) . . . . . . . . . . . . . transmission through the optical system in the absence ofabsorbers

T0 . . . . . . . . . . . . . . . . reference temperature, usually 273.15 K

u . . . . . . . . . . . . . . . . . most probable velocity of a thermal (Maxwellian) velocitydistribution given by

√2kBT/m [cm/s]

w[...] . . . . . . . . . . . . . complex error function, also referred to as Faddeeva’s function

Z[...] . . . . . . . . . . . . . plasma dispersion function

144

Summary of the papers

Paper I

Quantitative Description of Faraday Modulation Spectrometry in Terms of theIntegrated Linestrength and 1st Fourier Coefficients of the Modulated Line-shape Function

Journal of Quantitative Spectroscopy & Radiative Transfer Vol. 111, pp2415–2433, 2010

J. Westberg, L. Lathdavong, C. M. Dion, J. Shao, P. Kluczynski, S. Lundqvistand O. Axner

In this paper a quantitative description of the strength and shape of FaradayModulation Spectrometry (FAMOS) signals is given. Firstly, it is demonstratedhow the signal can be expressed in terms of the integrated linestrength, Si,j , whichcan be found in the HITRAN-database. Secondly, it is shown that the FAMOSsignals can be expressed concisely using the 1st Fourier coefficients of modulateddispersion lineshape functions. The Q3/2(3/2) transition of NO is carefully studiedand simulations revealing its pressure- and magnetic field dependences for variousconditions are performed.

My contribution to this paper was predominantly involvement in the develop-ment of the theoretical description and I made all the simulations.

145

Paper II

Faraday Modulation Spectrometry of Nitric Oxide Addressing its ElectronicX2Π (ν′′ = 0) −A2σ+ (ν′ = 0) Band: I. Theory

Applied Optics Vol. 49, No. 29, pp 5597–5613, 2010

L. Lathdavong, J. Westberg, J. Shao, C. M. Dion, P. Kluczynski, S. Lundqvistand O. Axner

This paper introduces a new description of the transitions in the strong electronicX2Π (ν′′ = 0)−A2σ+ (ν′ = 0) band of NO. The model is built upon an approxi-mation regarding the splitting of the lower state, which effectively yields a modelwith only two transitions, the two-transition model. Simulations are performedusing this model in order to determine the pressure and magnetic field dependenceof the FAMOS signals.

My involvement in this paper was predominantly contributions to the theoreticaldescription and I preformed a significant part of the simulations.

Paper III

Faraday Modulation Spectrometry of Nitric Oxide Addressing its ElectronicX2Π (ν′′ = 0) −A2σ+ (ν′ = 0) Band: II. Experiment

Applied Optics Vol. 49, No. 29, pp 5614–5625, 2010

J. Shao, L. Lathdavong, J. Westberg, P. Kluczynski, S. Lundqvist and O. Axner

In this paper the first experimental results of FAMOS addressing the strongelectronic X2Π (ν′′ = 0)− A2σ+ (ν′ = 0) band of NO are presented. The resultsagree well with under-modulated simulations from the new two-transition modelpresented in the accompanying paper. It is also shown that NO could be detecteddown to partial pressures of 13 µTorr which demonstrates the feasibility of thetechnique.

My contribution to this paper was predominantly in the curve-fitting anddata-processing part but my contributions are also included in the theory part.

146

Paper IV

Faraday Rotation Spectrometer With Sub-Second Response Time for Detec-tion of Nitric Oxide Using a CW DFB Quantum Cascade Laser at 5.33 µm

Applied Physics B: Lasers and Optics, Vol. 103, No. 2, pp 451-459, 2011

P. Kluczynski, S. Lundqvist, J. Westberg and O. Axner

In this paper a Faraday modulation spectrometer for sensitive and fast detection ofnitric oxide at the fundamental rotational-vibrational band at 5.33 µm is presented.It uses a rapid modulation of the magnetic field at 7.4 kHz made possible by acustom made solenoid/cell capable of reaching magnetic fields of over 600 Gauss.The detection limit from an absorption cell measuring 15 cm was assessed at 4.5 ppbfor a response time of 1 s.

My contribution to this paper was predominantly in the theory and simulationpart but my involvement also include experimental development and data collection.

Paper V

Analytical expression for the nth Fourier coefficient of a modulated Lorentziandispersion lineshape function

Journal of Quantitative Spectroscopy & Radiative Transfer Vol. 112, pp1443-1449, 2011

J. Westberg, P. Kluczynski, S. Lundqvist, and O. Axner

In this paper analytical expressions for the nth Fourier coefficients of modulatedLorentzian dispersion lineshape functions are derived and their applicability tomodulated techniques that rely on dispersion is discussed. The expressions areeasier to implement and orders of magnitude faster to execute than conventionalapproaches involving numerical integrals.

My contribution to this paper includes the theoretical derivations, performingthe simulations and writing the manuscript.

147

Paper VI

Fast and non-approximate methodology for calculation of wavelength-modu-lated Voigt lineshape functions suitable for real-time curve fitting

Journal of Quantitative Spectroscopy & Radiative Transfer Vol. 113, No. 16, pp2049–2057, 2012


In this paper a new non-approximative method for calculating the modulatedVoigt absorption lineshape function is presented. The new method is easy toimplement and orders of magnitude faster to execute than previous methods basedon time-consuming numerical integration of the modulation procedure.

My contribution to this paper includes the theoretical derivations, performingthe simulations and writing the manuscript.

Paper VII

Lineshape asymmetries in Faraday modulation spectroscopy

Submitted to Applied Physics B, 2013

J. Westberg and O. Axner

In this paper a the origins of lineshape asymmetries in Faraday modulation spec-troscopy are investigated. An extension, which includes asymmetry effects, to theconventional theoretical description is given and the resulting expressions are usedfor curve fitting to FAMOS data obtained in the mid-IR and UV region. Theresiduals from the curve fits show that the extension provides consistently moreaccurate results and therefore it is concluded that the lineshape asymmetries inFAMOS need to be accounted for in order to reduce inaccuracies.

My contribution to this paper includes deriving the expressions, obtaining theexperimental data, performing the simulations and writing the manuscript.

148

Paper VIII

Methodology for fast curve fitting to modulated Voigt dispersion lineshapefunctions

Submitted to Journal of Quantitative Spectroscopy & Radiative Transfer, 2013


In this paper the Westberg-Wang-Axner method introduced in Paper VI is ex-tended to also include signals measured from techniques that rely on dispersion.The resulting expressions are applied to experimentally obtained data therebydemonstrating the applicability of this method.

My contribution to this paper includes deriving the expressions, obtaining theexperimental data, performing the simulations and writing the manuscript.

149

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