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Journal of Quantitative Spectroscopy &Radiative Transfer

Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 244–250

0022-40http://d

n CorrE-m

journal homepage: www.elsevier.com/locate/jqsrt

Methodology for fast curve fitting to modulated Voigtdispersion lineshape functions

Jonas Westberg, Junyang Wang, Ove Axner n

Department of Physics, Umeå University, SE-901 87 Umeå, Sweden

a r t i c l e i n f o

Article history:Received 12 April 2013Received in revised form14 August 2013Accepted 18 August 2013Available online 24 August 2013

Keywords:Faraday rotationFaraday modulationSpectroscopyQuantum cascade laser

73/$ - see front matter & 2013 Elsevier Ltd.x.doi.org/10.1016/j.jqsrt.2013.08.008

esponding author. Tel.: þ46 90 786 6754.ail address: ove.axner@physics.umu.se (O. Ax

a b s t r a c t

Faraday rotation spectroscopy (FAMOS) as well as other modulated techniques that rely ondispersion produce lock-in signals that are proportional to various Fourier coefficients ofmodulated dispersion lineshape functions of the molecular transition targeted. In order toenable real-time curve fitting to such signals a fast methodology for calculating theFourier coefficients of modulated lineshape functions is needed. Although there exist ananalytical expression for such Fourier coefficients of modulated Lorentzian absorption anddispersion lineshape functions, there is no corresponding expression for a modulatedVoigt dispersion function. The conventional computational route of such Fourier coeffi-cients has therefore so far either consisted of using various approximations to themodulated Voigt lineshape function or solving time-consuming integrals, which hasprecluded accurate real-time curve fitting. Here we present a new methodology tocalculate Fourier coefficients of modulated Voigt dispersion lineshape functions that issignificantly faster (several orders of magnitude) and more accurate than previousapproximative calculation procedures, which allows for real-time curve fitting to FAMOSsignals also in the Voigt regime.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Dispersion lineshape functions are the basis for spectro-scopic techniques that measure the interference betweentwo (or several) light fields. Examples of such techniquesare Faraday modulation/rotation spectroscopy (FAMOS/FRS)[1–4], frequency modulation spectroscopy (FMS), andthereby noise-immune cavity-enhanced optical heterodynemolecular spectroscopy (NICE-OHMS) [5–7], and chirpedlaser dispersion spectroscopy (CLaDS) [8–11]. They are oftenused to assess the concentration of molecular species in gasphase, most frequently in trace concentrations. In order toperform such assessments with high accuracy the laserlight is often scanned across the transition after which amodel expression of the expected signal is fitted to the

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ner).

measured data. The concentration is then given by the value(s) of one (or several) parameter(s) in the fitted function.

To improve on the signal-to-noise ratio in the measure-ments and to increase the detection sensitivity, some spectro-scopic detection techniques incorporate a modulation ofeither the transition frequency, e.g. as FAMOS, or thewavelength of the laser light, as wavelength-modulatedtunable diode-laser absorption spectrometry (TDLAS). It haspreviously been shown that the signal from modulatedabsorption techniques can be expressed in terms of one orseveral Fourier coefficients of the modulated lineshapefunction [12,13]. As has recently been demonstrated, thesame is valid for techniques based on dispersion [14,15].This has both facilitated the theoretical description ofmodulated techniques and, for the cases when a givenFourier coefficient can be expressed as an analytical func-tion in closed form, speeded up the fitting processes.

However, a limitation of this approach has been thatit is only possible to find analytical expressions for the

J. Westberg et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 244–250 245

Fourier coefficients of modulated lineshape functions(absorption or dispersion) when the samples are fullypressure broadened, i.e. when the lineshape has a Lorent-zian form [14]. For measurement conditions that do notfulfill this assumption, e.g. when the transition needs to bedescribed by a Voigt lineshape, the modulated lineshapefunction has been calculated by numerical means, e.g. interms of a modulated plasma dispersion function or amodulated error function. Since this is utterly time con-suming [16], it has, up to recently, precluded real-timecurve fitting to signals from transitions with Voigt line-shapes assessed by modulated dispersion techniques.A remedy to this for techniques that rely on absorptionwas recently given by Westberg et al. by the developmentof the WWA-methodology (Westberg–Wang–Axner),which reduces the computation time of various Fouriercoefficients of a modulated Voigt absorption lineshapefunction by three orders of magnitude, from the secondto the millisecond range, with no loss of accuracy [17]. Nosuch remedy has yet been demonstrated for techniquesthat rely on dispersion. Therefore, we demonstrate in thiswork how the WWA methodology can be extended tomodulated Voigt dispersion lineshape functions just aseasily, which makes it applicable also to the modulateddetection techniques that rely on dispersion addressingVoigt lineshape functions.

Although the new WWA-methodology is applicable toall modulated techniques that detect dispersive signals itis, in this work, demonstrated on measurements of NOperformed by the FAMOS technique. FAMOS is a techniquethat is based upon the fact that in the presence of amagnetic field a transition in a paramagnetic molecule willrotate the plane of polarization of linearly polarized light.The reason for this is that linearly polarized light can beseen as a superposition of two circularly polarized com-ponents of light with oppositely helicity, left-handed andright-handed circularly polarized light (LHCP and RHCP),respectively. When the two circularly polarized compo-nents of the light propagate through a gas containingparamagnetic species exposed to an external magneticfield, they will experience dissimilar indices of refraction,and therefore propagate at different speed through thesample, which, in turn, results in a tilting of the polarizationplane of the linearly polarized light. By placing the samplebetween two polarizers and modulating the magnetic fieldit is possible to detect this tilting as a modulation of theintensity of the transmitted light. The signal is proportionalto the difference between two or several modulated disper-sive lineshape functions [14].

The FAMOS signal is, in this work, modeled by atheoretical description developed by Westberg et al. [14]that is based on the integrated linestrength of the transi-tion addressed and the first (even) Fourier coefficient of amodulated dispersive line shape function but also takesthe influence of polarization imperfections that give rise toasymmetric lineshape functions into account [18]. Thisdescription requires calculations of the 1st Fourier coeffi-cients of both modulated dispersion and absorption Voigtlineshape functions, which is exceedingly time consumingby standard numerical calculation procedures. It is shownthat the new methodology presented here can model the

FAMOS signal more appropriately than previously usedapproaches, and that it is several (up to four) orders ofmagnitude faster than the ordinary numerical solutionprocedure.

2. Theory

2.1. Calculation of the 1st Fourier coefficient of a modulatedLorentzian dispersion lineshape function

When pressure broadening is the dominating broad-ening mechanism the lineshape function takes a Lorent-zian form. The Lorentzian dispersion lineshape functionthat corresponds to an area normalized Lorentzian absorp-tion lineshape function can be written as

χdispL ðνÞ ¼� 1πδνL

ν�ν0

ðν�ν0Þ2þδν2L; ð1Þ

where ν is the frequency of the laser light, ν0 is the centerfrequency of the transition, and δνL is the half-width-halfmaximum (HWHM) of the Lorentzian profile. For the casewhen modulation is applied, either to the laser frequencyor the transition, the corresponding modulated lineshapefunction can be written as

χdispL ðνd; νaÞ ¼� 1πδνL

νdþνa cos ð2πftÞ½νdþνa cos ð2πftÞ�2þ1

ð2Þ

where νd is the width-normalized detuning, given byðνc�ν0Þ=δνL, where νc is the center frequency of the laser,νa is the width-normalized modulation amplitude, givenby νa=δνL, where νa is the modulation amplitude, definedas positive or negative for the case when the laserfrequency or the transition is modulated, respectively,f is the modulation frequency, and t is the time.

It has previously been shown that the nf-signal from amodulated technique (irrespectively of whether it detectsabsorption [12,13] or dispersion [15]) can be written interms of the nth (even) Fourier coefficient of the modu-lated lineshape function. In the case of a modulatedLorentzian dispersion lineshape function, this Fouriercoefficient can be written as

χdisp;evenL;n ðνd; νaÞ ¼ � 1πδνL

2τ

Z 1

�1

νdþνa cos ð2πftÞ½νdþνa cos ð2πftÞ�2þ1

� cos ð2πnftÞ dt; ð3Þ

where τ is given by 1/f (or an integer thereof).Moreover, it has been shown by Westberg et al. [14] that

this Fourier coefficient can be expressed expediently as

χdisp;evenL;n ðνd; νaÞ ¼ � 1πδνL

An

νnaBnþ

CnSþ þDnS�ffiffiffi2

pR

� �; ð4Þ

where Sþ and S� are given byffiffiffiffiffiffiffiffiffiffiffiffiRþM

pand

ffiffiffiffiffiffiffiffiffiffiffiR�M

p,

respectively, where, in turn, R¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2þ4ν2d

qand M¼ 1þ

ν2a�ν2d . The coefficient An is given by Eq. (16) in Ref. [14]whereas the coefficients Bn, Cn, and Dn are given inTables 2–4 in the same reference.

For the case with 1f-detection, which is the coefficientthat most often is used, as for example in the FAMOS

J. Westberg et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 244–250246

technique, χdisp;evenL;1 ðνd; νaÞ can be written explicitly as

χdisp;evenL;1 ðνd; νaÞ ¼� 1πδνL

2νa

1�Sþ þjνdjS�ffiffiffi2

pR

� �

¼� 1πδνL

2νa

1�ffiffiffi2

p

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þν2a�ν2dÞ2þ4ν2d

q8><>:

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þν2a�ν2dÞ2þ4ν2d

qþð1þν2a�ν2dÞ

r"

þjνdjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þν2a�ν2dÞ2þ4ν2d

q�ð1þν2a�ν2dÞ

r #9>=>;:

ð5Þ

2.2. Calculation of the 1st Fourier coefficient of a modulatedVoigt dispersion lineshape function – the conventionalapproach

When pressure broadening is not the sole dominatingbroadening mechanism the lineshape is affected also bythe Doppler broadening. The dispersion lineshape is thenoften modeled by a Voigt dispersion lineshape function.In the absence of modulation such a function can bewritten as [14,19]

χdispV ðνdÞ ¼1

π1=2δν′DIm½wðxþ iyÞ�

¼ yπ3=2δνL

Z 1

�1

ðxþsÞe�s2

ðxþsÞ2þy2ds; ð6Þ

where w(xþ iy) is the complex error function and x and yare the normalized detuning and the Voigt parameter,respectively, given by

x¼ ν0�νcδν′D

¼�yνd ð7Þ

and

y¼ δνLδν′D

; ð8Þ

where δν′D is given by δνD=ffiffiffiffiffiffiffiffiffiln 2

pwhere δνD, in turn, is the

HWHM of the Doppler profile, given by ν0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ln 2kT=mc2

pand s is a dimensionless integration parameter.

When a modulation is introduced, the normalizeddetuning becomes time-dependent, and can be written as

xðtÞ ¼�y½νdþνa cos ð2πftÞ�: ð9ÞInserting Eq. (9) into Eq. (6) gives us an expression for thenth Fourier coefficient of a modulated Voigt dispersionlineshape function that can be written either as

χdisp;evenV ;n ðνd; νaÞ ¼1

π1=2δν0D� 2

τ

Z τ

0Im½wðxðtÞþ iyÞ� � cos ð2πnftÞ dt;

ð10Þor alternatively as

χdisp;evenV ;n ðνd; νaÞ ¼y

π3=2δνL� 2

τ

Z τ

0

Z 1

�1

f�y½νdþνa cos ð2πftÞ�þsge�s2 ds

f�y½νdþνa cos ð2πftÞ�þsg2þy2

� cos ð2πnftÞ dt: ð11Þ

Unfortunately there is no known analytical solution to thisequation, which means that it has to be calculated bynumerical procedures. As was alluded to above, this isutterly time-consuming. For each detuning, an integral(over time) comprising a product of another integral (overthe integration parameters) or the Faddeeva function and atrigonometric function needs to be numerically calculated,which often requires � 102 calculations for each integral,in total � 104 calculations. Since a scan over an absorbingprofile might involve � 102 discrete points, a significantnumber of calculations, often in the order of 106, is neededfor a full scan. Moreover, if the expression is to be fitted toexperimental data, and each fit requires � 101 iterationsteps, the number of calculations for the fitting to one singlescan can then require 107 discrete calculations. This makesfitting of modulated Voigt dispersion lineshape functionsextremely time-consuming, especially when high accuracyis needed and has so far precluded real-time curve fitting tomodulated Voigt dispersion lineshape functions. Eq. (11)will henceforth be referred to as the conventional metho-dology for calculation of the nth Fourier coefficient ofa modulated Voigt dispersion lineshape function (belowshortly referred to as the conventional methodology).

However, as was mentioned above, a new methodology[17] developed for wavelength modulation spectroscopyaddressed recently this issue for absorption spectroscopyand presented a possibility of using the Fourier coefficientapproach swiftly also in the Voigt regime, reducing com-putational time with around three orders of magnitude.Applying this methodology to the case with a dispersionline shape function provides, in a similar manner, aconvenient means to evaluate the nth Fourier coefficientof a modulated Voigt dispersion line shape function.

2.3. Calculation of the nth Fourier coefficient of a modulatedVoigt dispersion lineshape function – the dispersive WWAmethodology

The nth Fourier coefficient of the modulated Voigtdispersion lineshape function given above can be swiftlyevaluated by the WWA procedure as follows. First, let usinterchange the integration order of the expression givenin Eq. (11) and make the variable substitution s-ys′. Thisimplies that χdisp;evenV ;n ðνd; νaÞ can be written as

χdisp;evenV ;n ðνd; νaÞ ¼y

π1=2�

Z 1

�1

�1πδνL

2τ

�Z τ

0

½ðνd�s′Þþνa cos ð2πftÞ� cos ð2πnftÞ dt½ðνd�s′Þþνa cos ð2πftÞ�2þ1

�e�y2s′2 ds′: ð12ÞBy examining this expression closely and by a comparisonwith Eq. (3) it can be seen that the inner integral can beidentified as the nth even Fourier coefficient of a modu-lated Lorentzian dispersion lineshape function, althoughwith a shifted detuning frequency, νd-νd�s′. This meansthat it is possible to write the nth even Fourier coefficientof a modulated Voigt dispersion lineshape function as

χdisp;evenV ;n ðνd; νaÞ ¼y

π1=2

Z 1

�1χdisp;evenL;n ðνd�s′; νaÞe�y2s′2 ds′: ð13Þ

J. Westberg et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 244–250 247

This equation simply denotes a convolution between thenth Fourier coefficient of a modulated Lorentzian disper-sion lineshape function and an area-normalized distribu-tion function, which formally can be written succinctly as

χdisp;evenV ;n ðνdÞ ¼ χdisp;evenL;n � f vðνdÞ; ð14Þ

where the former is given by Eq. (4), f vðνdÞ is an area-normalized frequency (velocity) distribution functiongiven by

f vðνdÞ ¼y

π1=2e�y2ν2d ; ð15Þ

and � denotes a convolution. Eq. (14) will henceforth bereferred to as the dispersive WWA methodology.

2.4. The 1f FAMOS signal from a rotational–vibrationalQ-transition

For the case with a frequency independent laser inten-sity and neglecting the effects of magnetic circular dichro-ism, it is possible to express the 1f FAMOS signal from arotational-vibrational (ro-vib) Q-transition with Voigt formas [2,18]

S¼ ηSNxLI0 sin ð2θÞ

4χdisp;evenV ;1 ; ð16Þ

where η is an instrumentation factor, S is the integratedlinestrength of the transition, Nx is the density of mole-cules, L is the interaction length, I0 is the power of theradiation after the polarizer, here assumed to be frequencyindependent, θ is the uncrossing angle of the polarizer andanalyzer, and χdisp;evenV ;1 is the 1st Fourier coefficient ofa modulated Voigt lineshape function, given by eitherEq. (10) or (14).

For cases when lineshape asymmetries need to betaken into account the formulation given above can beextended to incorporate also the effects of polarizationimperfections [18]. Under the assumption that theseeffects are small it has been shown that the FAMOS signalcan be expressed as

S� ηSNxLI0 sin ð2θÞ

4½χdisp;evenV ;1 þεχabs;evenV ;1 �; ð17Þ

where ε is the uncrossing angle-normalized unbalancingterm between the LHCP and RHCP components of thelight, given by ε= sin ð2θÞ, where ε is an unbalancing term,defined in Ref. [18], and χabs;evenV ;1 is the 1st Fourier coeffi-cients of the modulated Voigt absorption lineshape [17].

3. Experimental methods and procedures

The experimental setup, which is schematically shownin Fig. 1, is based around a room temperature continuous-wave (cw) distributed feedback (DFB) quantum cascadelaser (QCL) from Maxion Technologies producing around70 mW of power at � 5:33 μm. The temperature is regu-lated by two separate, individually controlled, Peltierelements which allow a temperature tuning in the 15–35 1Crange while keeping the temperature stabilization within0.1 1C. In order to reach the desired Q3=2ð3=2Þ transition inNO at 5:331 μm (1875.81 cm–1) the QCL is operated at 32 1Cwith a bias current of 708 mA. The laser beam is collimated

by a chalcogenide aspheric lens from Lightpath Technologieswith an effective focal length of 1.8 mm.

The collimated laser beam passes two wire grid polar-izers in series before entering the 25 cm glass cell with anouter diameter of 12 mm. The cell is made from borofloatglass with windows comprising 2 mm thick CaF2 planeparallel discs and is situated inside a shielded aluminiumbox which contains a 15 cm long solenoid made from 765turns of 1 mm thick copper wire with 0.3 mm thickisolation. The dimensions of the shield box are carefullychosen to minimize the disturbances caused by the mag-netic field on the electronics outside the box. Due toefficient air-cooling the solenoid is able to produce roughly�200 G at a modulation frequency of 7.4 kHz whilekeeping the temperature of the coil sufficiently low forhours of operation time. The maximum allowed magneticfield is estimated to 640 G, although at this magnetic fieldmagnitude only a few minutes of operation is possiblebefore the shield box becomes overheated. The glass cell isequipped with a thermistor in order to constantly monitorthe temperature inside the solenoid to prevent damage tothe glass cell and the solenoid itself.

The solenoid is coupled in series with a capacitor stackto create a resonance circuit that is driven by a powerfulcommercially available audio amplifier (Apart Audio Pub-drive 2000). The resonant circuit is constructed to producea resonance at a frequency of 7.4 kHz which is found to bea good compromise between the performance of theinstrument (detection limit and response time) and thelosses due to heat dissipation from the magnetic coil.Upon exiting the glass cell the laser beam passes a thirdpolarizer (the analyzer) of gold wire-grid type (Infraspecs),nearly perpendicularly oriented with respect to the pre-vious two polarizers. The laser beam is finally focusedupon a 2-staged Peltier-cooled MCT detector (VIGO) usingan off-axis parabolic mirror. The detector signal is thenpassed on to a lock-in amplifier configured for 1f detection.After the demodulation process the signal is sent to theDAQ board connected to the desktop computer runningMatlab for data storage and analysis.

4. Results

The quality and speed of fitting Eq. (17) to FAMOS datafrom 100 ppm of NO at a pressure of 40 Torr measured at amagnetic field strength of � 150 G were evaluated bythree different methodologies; the conventional metho-dology based both on a fully numerical solution of Eq. (11)and an approximate but up-speeded numerical calculationprocedure based on calculating the Faddeeva function inEq. (10) by an FFT approach [20] and the dispersive WWAmethodology based on Eq. (14). Fig. 2 shows by the upperpanel the set of data to which fits were performed. All fitsused the same set of free parameters: δνL, δνD, ν0, ϵ, and S0,where the latter represents the combination of entities infront of the brackets in Eq. (17). In addition, a second orderpolynomial was used to account for background signalsencountered in the measurements. The panels ðbÞ and ðcÞillustrate the residuals from fits to the data by the threedifferent methodologies: panel ðbÞ, the completely over-lapping two conventional methodologies, and ðcÞ, the

polarizer analyzer

cell/solenoid

laser

detectorgas inlens

asphericlens

power amplifierfunction generator

rampmodulationsync

lock-in amplifier

1f

MatLab

laser current controller

laser temperature controller 1

laser temperature controller 2

Fig. 1. Schematical figure of the basic experimental setup of the Faraday modulation system. The laser current source is an ILX Lightwave LDX-3232 drivercapable of 2000 mA output with a ripple below 20 μm. The modulation signal from the lock-in amplifier (Stanford Research Systems, SR830) is amplified bya commercially available audio amplifier before reaching the solenoid. All data is stored and analyzed by the DAQ-toolbox in Matlab.

−0.03 −0.02 −0.01 0 0.01 0.02 0.03

Measured data

−0.03 −0.02 −0.01 0 0.01 0.02 0.03−2

0

2

−0.03 −0.02 −0.01 0 0.01 0.02 0.03−2

0

2

Fig. 2. Panel ðaÞ: A FAMOS spectrum from the 5:331 μm transition in NO.The concentration of NO (in N2) was 100 ppm and the pressure was40 Torr. Panel ðbÞ: The residual from a fit to the data based on the WWAmethodology. Panel ðcÞ: The corresponding fully overlapping residualsfrom the numerical solutions of Eq. (17).

1 2 3 4 5 6 7 8 9 1010−4

10−3

10−2

10−1

100

101

102

103

Fig. 3. Calculation time of the fitting methodologies, by the use of Eq. (17), asa function of the number of iterations. The upper curve (green in color): theconventional methodology based on a fully numerical solution of the Fouriercoefficients for the modulated Voigt lineshape functions, which for dispersionis given by Eq. (11); the middle curve (blue in color): the approximate but up-speeded numerical calculation procedure of the conventional methodology;the lower curve (red in color): theWWAmethodology, based on convolutionsof the 1st Fourier coefficients of the modulated dispersion and absorptionLorentzian lineshape functions and a Maxwellian frequency (velocity) dis-tribution, which for dispersion is given by Eq. (14). The WWA methodologyperforms almost two orders of magnitude better than the up-speededconventional methodology and more than four orders of magnitude fasterthan the fully numerical conventional methodology. (For interpretation of thereferences to color in this figure legend, the reader is referred to the webversion of this article.)

J. Westberg et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 244–250248

WWA methodology. All fits were implemented in Matlabby the fminsearch function, which uses the Nelder–Mead simplex algorithm described in Lagarias et al. [21].As can be seen in the figure the residuals are virtuallyindistinguishable as is expected since the fits for allmethodologies are based on the same modulated Voigtlineshape function. The remaining residuals are assumedto originate from effects not accounted for in theVoigt model, predominantly Dicke narrowing [22,23] and

Speed dependent effects [24–26]. Moreover, the variousmethodologies produce almost identical sets of fittingparameters, differing typically by no more than a percent,and less the more stringent fitting tolerance, eventually

J. Westberg et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 244–250 249

caused by the accuracy of the numerical algorithms used.This demonstrates that the different methodologies repro-duce, within the numerical accuracy of the calculations,the measured FAMOS signal equally well. However, theytake considerably different amounts of time.

In order to evaluate the computational time for thedifferent curve fitting routines the number of data points ineach scanwas set to 100 while the number of iterations wasvaried from 1 to 10 (irrespectively of the quality of the fit).The individual fitting procedures were repeated severaltimes to obtain an average calculation-time for each proce-dure and a special “warm-up” code was implemented toensure as equal conditions for the three methodologies aspossible. Fig. 3 shows the computational time for the threemethodologies as a function of the number of iterationsfor the case with the time integral in the conventionalapproaches decomposed into 102 timesteps, which is alower limit for adequate accuracy. The figure shows that,for the conditions given, the WWA methodology consis-tently executes about two orders of magnitude faster thanthe up-speeded conventional approach and more than fourorders of magnitude faster than the fully numerical con-ventional methodology.

5. Summary and conclusions

In this article a new methodology of calculating modu-lated Voigt dispersion lineshapes has been described,given by Eq. (14), and verified by the use of a newlyderived curve fitting expression for Faraday modulationthat also takes polarizations imperfections into account,Eq. (17), applied to data from 100 ppm of NO detected at5:33 μm at a pressure of 40 Torr. The methodology, whichis referred to as the dispersive Westberg–Wang–Axner(WWA) methodology, is based on an analytical expressionfor the 1st Fourier coefficient of the modulated Lorentziandispersion lineshape function [15] that is expeditiouslyconvoluted with an area-normalized frequency (velocity)distribution function. This gives, in a swift manner, anaccurate calculation of the 1st Fourier coefficient of themodulated Voigt dispersion lineshape function that can beused for real-time curve fitting to data from modulateddispersion techniques such as FAMOS or NICE-OHMS.When polarization imperfections are to be included, thecorresponding Fourier coefficient of the modulated Voigtabsorption lineshape function is needed and it can then beswiftly calculated by the previously developed absorptionWWA methodology [17]. The advantage of the WWAmethodologies becomes apparent when examining Fig. 3,which displays the calculation times for the three fittingmethodologies as a function of the number of iterations.As can be seen the calculation time, for the conditionsgiven, is significantly reduced (from hundreds of secondsto the milli-second range) as compared to the conven-tional methods (and more if the resolution of the scanneeds to be more than 100 data points). This allows foraccurate real-time curve fitting to FAMOS signals, both inthe absence and presence of polarization imperfections, inthe Voigt regime. In its present form, the WWA methodol-ogy cannot directly be expanded to account for lineshapesbeyond the Voigt profile.

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