simultaneous characterization of detector and source imperfections in infrared ellipsometry

Simultaneous characterization of detectorand source imperfections in infrared

ellipsometry

Herbert Wormeester,1,* Pepijn R. Kole,1,2 and Bene Poelsema1

1Solid State Physics, MESA+ Institute for Nanotechnology, University of Twente, Enschede, The Netherlands2Current address: Cavendish Laroratory, University of Cambridge, Cambridge CB3 0HE, UK

*Corresponding author: [email protected]

Received 17 November 2008; revised 6 April 2009; accepted 19 April 2009;posted 20 April 2009 (Doc. ID 103892); published 12 May 2009

Optical components required for infrared (IR) ellipsometry have distinctly worse characteristicscompared to those available for the visible spectrum. The calibration of the optical components usedis therefore essential for obtaining reliable results. Here a powerful method is outlined to calibrate si-multaneously the polarization characteristics of a source and detector through the synchronous rotationof two polarizers. The performance of this method is to a large degree independent of the quality of (com-mercially available) polarizers. This renders this method robust and highly suitable for the IR range.Moreover, it is also inherently insensitive toward a nonlinear response of the detector. This enablesus to use this method as the first step in the quantification of component imperfections. © 2009 OpticalSociety of America

OCIS codes: 240.2130, 240.6490, 300.6340.

1. Introduction

Polarization imperfections of optical componentsstrongly affect the measurement in optical character-ization techniques such as ellipsometry. The polari-zation sensitivity of source or detector implies thatthe rotation of a polarizer in the light beam leadsto an additional attenuation of the measured inten-sity that complicates the measurement. Polarizerswith excellent specifications are readily available forthe visible and near visible range. This allows an ac-curate characterization of the polarization sensitiv-ity of especially the light source and detector. Inthe mid-infrared (MIR) range, the quality of commer-cially available polarizers is much lower. Detectorsthat are not sensitive to the polarization of the inci-dent light or sources that emit unpolarized light arealso not readily available. Sufficient intensity canonly be obtained by using polarizing focusing optics

as well as the use of a Fourier transform instrument.This complication of the nonideal character of compo-nents used can be removed with a calibration of thepolarization characteristics of all components used.An extensive analysis of the influence of imperfectpolarizers for infrared ellipsometry has been givenby Wold and Bremer [1]. They used the presence ofhigh-order Fourier components in the signal upon ro-tation of one of the polarizers to determine and cor-rect for polarizer imperfections. Unfortunately, thismethod neglects nonlinear behavior of the detectorand tries to determine all considered imperfectionsin one iterative fit procedure. A separate calibrationof the polarizers helps to overcome the underdeter-mination of fit parameters. den Boer [2] used MgF2polarizers with excellent optical properties down to1500 cm−1 to characterize the polarization behaviorof dedicated MIR optical components. In addition,a Ge Brewster angle polarizer with superior polari-zation properties (attenuation factor γ down to 2×10−4) compared to the commercially available goldwire on KRS-5 polarizers (attenuation of only 1–10%)

0003-6935/09/152853-07$15.00/0© 2009 Optical Society of America

20 May 2009 / Vol. 48, No. 15 / APPLIED OPTICS 2853

[3] was used. However, Brewster angle polarizers areprone to show a large deviation of the beam directionupon rotation of the polarizer, and this effect as wellas the lack of commercial availibility limits the use ofsuch polarizers to characterize the polarization prop-erties of the used components.The imperfections of readily available polarizers

inherently limit the capability to characterize the po-larization imperfections of, especially, source and de-tector. Polarizer imperfections themselves can beevaluated from a comparison of the light transmittedin an aligned position of two polarizers to the lighttransmitted in a cross position. However, these twointensities are in the two limiting ranges (high andlow) of the sensitivity of a detector. Luttmann et al.[3] showed that a mercury cadmium telluride (MCT)detector has a nonlinear response. This complicationhas to be dealt with before polarizer imperfectionscan be determined. The alternative to aMCT, the lesssensitive deuterated triglycine sulfate detector, is inview of the increased measurement time not alwaysan acceptable option. Also the polarization featuresof the detector and the source influence the ratio γof the light intensity measured in a cross positionand an aligned position of two polarizers. Both thesource and the detector polarization and the nonli-nearity of the detector response have to be dealt withbefore the imperfection properties of nonideal polar-izers (γ > 10−5) can be determined. This leads to a vi-cious problem circle that has to be broken by eitherthe use of a very good polarizer (γ < 10�5) that can beregarded as an ideal polarizer or a calibration meth-od for source and detector imperfections that doesnot rely on the quality of the other components used.The actual value of γ required for regarding the im-perfections of a polarizer as negligible depends on theactual required accuracy. In this paper we describe asimultaneous calibration procedure of the polariza-tion sensitivity of the source and detector in an ellip-someter setup. This calibration method is to a largeextent insensitive to the quality of the polarizersused as well as the nonlinearity of the detector.Therefore, it can be used as a first calibration step.This requires that the ellipsometer is positioned ina straight-through configuration, i.e., without a sam-ple present. In a second calibration step the detectornonlinearity can be characterized using the methodoutlined by Luttmann et al. [3]. In a final step thepolarization attenuation of the polarizers used canbe determined.

2. Experimental Setup

A Bruker Vertex 70 spectrometer equipped with aglobar light source and an externally positionedMCT detector were used as a light source and detec-tor in an ellipsometer setup. A tungsten halogenlamp and an InGaAs diode were used to check theperformance in the near infrared (NIR). A deuteratedL-analine-doped triglycine sulfate detector was usedto verify operation in the MIR region. A KBr beamsplitter is used for theMIR range, while a CaF2 beam

splitter is used for the NIR range. The polarizersused in the MIR range were so-called KRS-5 polari-zers that were made of gold grids deposited on thal-lium iodide bromide. They show a polarizationefficiency of about 93%. This was improved by pla-cing two polarizers in tandem [2], which provides apolarization efficiency of 98–99%. The tandem, how-ever, reduces the transmission, which is only 75–80%for one polarizer to about 60% for the tandem. Thetotal of four polarziers used had very similarcharacteristics, as we did not find any influence ofa specific combination on the values characterizingthe imperfections of the components. StandardGlan–Thompson prisms were used in the NIR range.The angle of incidence could be varied, but typicallyan angle of incidence of 70° was used for metallicsamples and 55° for glass samples. A gold-coated el-lipsoidal mirror with a focal length of 150mm wasplaced directly after the exit port of the spectrometerto focus the light beam to a spot size of 2mm dia-meter on the sample. An aperture with a diameterof 8mm was placed in the focused beam to limitthe angular divergence to 3°.

3. Detector and Source Polarization Imperfections

To first order, polarization imperfections of a detectorcan be described by the azimuth D for which it ismost sensitive for incident linearly polarized lightand an attenuation factor αD [2]. The attenuation fac-tor has a similar role to the polarizer attenuationcoefficient. It describes the ratio of the sensitivityfor incident light polarized perpendicular and paral-lel to the azimuth D. However, its role is reversed asa value close to 1 is usually preferred, indicatingsmall polarization sensitivity of the detector. The in-fluence of component imperfections on the measuredsignal intensity can be analyzed following the Muel-ler method as described by Wold and Bremer [1]. Theactual detector is replaced by a perfect, polarization-insensitive detector and a virtual polarizer. The vir-tual polarizer is placed at an angle D with respect tothe reference plane to simulate the azimuth with thehighest transmission direction, and it has an at-tenuation αD. The parameters αD and D of thisvirtual polarizer describe the polarization character-istics of the real detector and all components betweenthe detector and the polarizer, such as mirrors. TheMueller matrices required for the analysis are thestandard matrix for rotation R [4] and the matrixthat describes the attenuation by an imperfect polar-izer, MpolðαÞ:

MpolðαÞ ¼12

26641þ α 1 − α 0 01 − α 1þ α 0 00 0 2

ffiffiffiαp0

0 0 0 2ffiffiffiαp

3775:

Wold and Bremer [1] showed that the determina-tion of the Fourier coefficients from a rotation of

2854 APPLIED OPTICS / Vol. 48, No. 15 / 20 May 2009

analyzer position A at frequencies 2A and 4A allowsthe characterization of the detector imperfections.This analysis is complicated by the polarizer imper-fections. Moreover, they neglected nonlinear beha-vior of the detector, which leads to effects thatcannot be distinguished from source and detectorimperfections in their calibration procedure. Theproblem of the influence of the quality of the polar-izers used in the determination of source or detectorcharacteristics can be overcome by a synchronous ro-tation of two polarizers, one in the analyzer positionand one in the polarizer position, as depicted in Fig. 1.The signal intensity as a function of a synchronous(i.e., phase-fixed) rotation of the two polarizers wouldbe constant in the absence of any polarization sensi-tivity of source and detector. Deviations from a con-stant intensity with a period of half of themechanicalrotation period can be attributed to source and detec-tor polarization sensitivity. This effect also occurswith the rotation of only one polarizer. However,

two polarizers are required for this method to workas outlined below.The setup with both an imperfect source and detec-

tor as well as real polarizers in the polarizer and ana-lyzer position leads to a Stokes vector for the detectorintensity given by

ID ¼ MpolðαDÞ · RðD − AÞ ·MpolðγÞ · RðA − PÞ ·MpolðγÞ· RðP − SÞ ·MpolðαSÞ · IS: ð1Þ

The polarization position of the polarizer is givenby the angle P with respect to the reference frame.The real source consists of the virtual polarizerand an unpolarized source that emits light with in-tensity IS. We use the same value of the extinctioncoefficient γ for both polarizers as we found experi-mentally that these two elements were interchange-able without any effect on the signals. Moreimportant, the actual value is to a large extent ofno consequence in this calibration procedure. The im-perfection of the source is characterized by a polari-zation angle S and an attenuation factor αS. Thedetected intensity upon simultaneous rotation ofthe polarizer and analyzer, (A ¼ P) is given by aconstant intensity and four Fourier coefficients perwavelength

ID0;A¼P ¼ I04·

2666664

fð1þ αDÞð1þ αSÞ þ 12 ð1 − αDÞð1 − αSÞ cosð2ðD − SÞÞg1

þfð1 − αDÞð1þ αSÞ cosð2DÞ þ ð1þ αDÞð1 − αSÞ cosð2SÞg cosð2AÞþfð1 − αDÞð1þ αSÞ sinð2DÞ þ ð1þ αDÞð1 − αSÞ sinð2SÞg sinð2AÞ

þf12 ð1 − αDÞð1 − αSÞ cosð2ðDþ SÞÞg cosð4AÞþf12 ð1 − αDÞð1 − αSÞ sinð2ðDþ SÞÞg sinð4AÞ

3777775: ð2Þ

The attenuation γ of the two imperfect polarizersinfluences only the overall intensity I0. Therefore,the four Fourier coefficients normalized by the posi-tion-independent term are not sensitive to the qual-ity of the polarizers used. The values of these fournormalized coefficients should in principle suffice

Fig. 1. (Color online) Setup for characterizing the detector and source with imperfect polarizers. The two virtual polarizers in the setuprepresent the detector and source polarization dependence and are characterized simultaneously by measuring the intensity as a functionof analyzer angle, where A ¼ P and A ¼ P� 45°.


to characterize the source and detector imperfec-tions. However, no explicit expression for the at-tenuation and polarization azimuth of the detectorand source can be evaluated. A fitting routine forthe determination of the parameters can be used,but Eq. (2) is symmetric in source and detector. Thissymmetry not only removes any direct identificationof either component but unfortunately also results ina very unstable fit situation that will introduce largeerrors for the determined values. This problem canbe solved by also measuring the intensity for syn-chronous rotation of polarizer and analyzer, but withsettings A ¼ Pþ 45° and A ¼ P − 45°. Also in thesetwo situations, the deviation from a constant signalprovides the source and detector imperfections. Thesets of equations obtained for these two cases are

ID0;A¼P−45° ¼I04·

2666664

fð1þ αDÞð1þ αSÞ þ 12 ð1 − αDÞð1 − αSÞ sinð2ðS −DÞÞg1

þfð1 − αDÞð1þ αSÞ cosð2DÞ þ ð1þ αDÞð1 − αSÞ sinð2SÞg cosð2AÞþfð1 − αDÞð1þ αSÞ sinð2DÞ − ð1þ αDÞð1 − αSÞ cosð2SÞg sinð2AÞ

þf12 ð1 − αDÞð1 − αSÞ sinð2ðDþ SÞÞg cosð4AÞ−f12 ð1 − αDÞð1 − αSÞ cosð2ðDþ SÞÞg sinð4AÞ

3777775; ð3Þ

ID0;A¼Pþ45° ¼I04·

2666664

fð1þ αDÞð1þ αSÞ þ 12 ð1� αDÞð1 − αSÞ sinð2ðD − SÞÞg1

þfð1 − αDÞð1þ αSÞ cosð2DÞ − ð1þ αDÞð1 − αSÞ sinð2SÞg cosð2AÞþfð1 − αDÞð1þ αSÞ sinð2DÞ þ ð1þ αDÞð1 − αSÞ cosð2SÞg sinð2AÞ

−f12 ð1 − αDÞð1 − αSÞ sinð2ðDþ SÞÞg cosð4AÞþf12 ð1 − αDÞð1 − αSÞ cosð2ðDþ SÞÞg sinð4AÞ

3777775: ð4Þ

Normalization of theFourier coefficients fromthesemeasurements provides eight additional values perwavelength. However, the two 4A Fourier coefficientsare the same for all the polarizer/analyzer combina-tions. The 12 Fourier coefficients obtained from thethree polarizer/analyzer combinations provide 7 inde-pendent Fourier coefficients. The other 5 are a linearcombination of the latter 7. Any relative rotation ofthe polarizer and analyzer results in an intensity sig-nal that can be described by 4 normalized Fouriercoefficients. These 4 are always a linear combinationof the 7 independent values. Therefore, the use ofmore relative positions of analyzer and polarizer doesnot provide any additional information. Moreover, si-tuations close to A ¼ Pþ 90° should be avoided asthese are low signal situations and will enhancethe noise contribution especially. The 7 independentFourier coefficients areno longer symmetric in the im-perfection parameters of the detector and source. It isthus possible to attribute the imperfection character-istics to either one.A complicating factor arises from the fact that a

nonlinear behavior of the detector will generate botha 2A signal and a 4A signal upon rotation of an ana-

lyzer [3]. The influence of this nonlinearity will belargest if a large dynamic range of the signal hasto be used. However, the synchronous rotation of bothanalyzer and polarizer will usually result in a mar-ginal deviation from a constant level, depending onlyon the detector and source imperfections. In a limitedintensity range the nonlinear behavior is basicallyeliminated, and only the linear response has to be ta-ken into account. Therefore, this measurement al-lows us to disregard nonlinear behavior of thedetector and also diminishes the influence of the im-perfections of the rotating polarizers. The deter-mined detector and source characteristics representthe combined characteristics of all components fromthe source to the polarizer and from the analyzer tothe detector. They include mirrors, windows, beamsplitters, and the source and detector themselves.

The intensity measured in an actual rotating ana-lyzer or rotating polarizer ellipsometer, with lightreflected from a sample characterized by the ellipso-metric quantitiesΨ andΔ, can be evaluated by incor-porating the source and detector characteristics. Thedetected intensity for a rotating analyzer ellips-ometer is measured as a function of the analyzerrotation angle A:

ID0 ¼ I04ð1 − cos 2Ψ cos 2PÞ

·�1þ cos 2P − cos 2Ψ

1 − cos 2Ψ cos2Pcos 2A

þ sin 2Ψ cosΔ sin 2P1 − cos 2Ψ cos2P

sin 2A

�

·��

12þ αS

2

�þ�12� αS

2

�cos 2ðP� SÞ

�

·��

12þ αD

2

�þ�12� αD

2

�cos 2ðD − AÞ

�

ID0 ¼ I04ð1 − cos 2Ψ cos2PÞð1þ α cos 2Aþ β sin 2AÞ

· FcorrectionðαD;D; αS;S;P;AÞ: ð5Þ


The polarizer P is at a constant angle during ameasurement loop. Only the position A is changedfor the determination of the Fourier coefficients αand β. As a result, only the detector imperfectionsare relevant in this case. The measured intensityhas to be corrected for the detector polarization sen-sitivity imperfection before α and β are determined.This is in contrast to the influence of the nonlinearityof the detector and imperfections of the polarizers.These imperfections do not depend on the actual re-lative position of the analyzer A with respect to thedetector D [3] and can therefore be carried out on αand β after correction for the detector polarizationsensitivity. For a rotating polarizer ellipsometer con-figuration, the roles of the polarizer and analyzer areinterchanged, accompanied by a similar change insource and detector influence. In this situation, theactual polarization sensitivity of the detector is ofno consequence.

4. Simulation and Measurement of Detector andSource Imperfection

A simulation of the simultaneous calibration proce-dure for detector and source was performed to verifythe validity range of the method. The input for thesimulation is a value for the source and detector po-larization attenuations, αinputS and αinputD , as well astheir azimuths S andD. With these values a detectorsignal was calculated according to Eq. (5). This pro-vides three series of signals ID0 as a function of a syn-chronous rotation of the polarizer and analyzerangle. The three series differ by the phase differencebetween the polarizer and analyzer, i.e., a phasedifference of 0°, 45°, and −45°. To mimic an actualsignal, a noise level of 1% was added. To study theinfluence of a considerable misalignment of the com-ponents, a 1A component was added with an inten-sity of 5% of the maximum intensity. In the firstsimulation the performance of the method as a func-tion of the strength of the source and detector at-tenuation was studied. In this simulation anextinction of 99% was used for the polarizers. Thisvalue resembles the quality of the polarizers usedin the actual setup. The input for the simulation is

a value for αinputD between 0.2 and 1. The same valueis used for αinputS . The calibration procedure for deter-mination of the detector and source imperfectionswas carried out for each simulated signal based onthe mentioned input parameters. The result of thecalibration procedure are values that characterizethe imperfections of the source and detector, i.e.,αsimD that characterizes the attenuation of the detec-tor following the calibration procedure. The accuracyof the calibration procedure is determined by the re-lative difference between the input and the simu-lated values of the attenuation coefficient:jαinputD − αsimD j=αinputD . This relative difference as a func-tion of the value of αinputD is shown in Fig. 2. A randomposition of the azimuth of the source and detectorpolarization was used. The actual value of this azi-muth had no influence on the simulation result.The relative difference for the source attenuation,jαinputS − αsimS j=αinputS , shows the same result, as ex-pected from symmetry. The simulation result pre-sented in Fig. 2 shows that the calibration methodworks very well for 0:6 ≤ αinputD;S ≤ 1:0. In this rangethe error in the value for the attenuation coefficient

Fig. 2. Simulated accuracy of the calibration method in which acalibration analysis is made on a simulated signal with a knowndetector attenuation αinputD . Displayed is the relative difference be-tween the result from the calibrationmethod on the test signal αsimDas a function of the value αinputD used to construct the test signal.

Fig. 3. (Color online) Relative difference between the calibrationresults αsimD and the input values αinputD as a function of polarizerattenuation γ for both calibration methods: separate (lower curve)and simultaneous (upper curve) calibration. The input values fordetector/source attenuation were chosen to be 0:8 < αinputD;S < 1:0,and their angles D and S were chosen randomly.

Fig. 4. Determined attenuation factors and angles of the detectorand source, illustrating the components’ polarization dependence.


with this calibration method is <1%. The actual at-tenuation factors of detector and source are typicallybetween 0.8 and 1.0 [5], i.e., well within the effectiverange of this calibration method. The lower boundaryof the attenuation for the performance of this methodis directly related to the fact that the deviation froma constant signal is used as the calibrationmonitor. Avariation of the noise and 1A signal showed that thecalibration methods performance was not influencedsignificantly by a normal level of added signal noise.A deviation from the input values as a result of thesimulated misalignment was observed for 1A levelsover 30% of the signal intensity. In reality 1A levelsnever exceeded 5% and are more likely 1–2%.A second simulation was done to verify the

influence of the quality of the polarizers. In this si-mulation a test signal was constructed with an at-tenuation of the source and detector in the rangeof 0:8 < αinputD;S < 1:0. For various values of the polar-izer imperfection γ, the calibration procedure on thetest signals provided αsimD and αsimS . Figure 3 showsthe dependence of the relative difference between thecalibrated value and the input value for the sourceand detector attenuation. Random values for anglesD and S are used. Also shown is the result from amore standard analysis (see, for example, Ref. [3]).In this method the position of the polarizer behindthe source is fixed and the rotation of the analyzeris used to characterize the detector imperfections.The latter method is accurate only for very good po-larizers such as Brewster angle polarizers [3], withattenuation factors of γ ≤ 0:005. Large errors resultif standard wire grid polarizers (γ between 0.005and 0.020, which is the case in our setup, Fig. 5)are used with this method. It is noted that this meth-od is also not robust with regard to detector nonli-nearity. In contrast, the simultaneous calibrationmethod performs very well with these imperfect po-larizers, with an error in αD;S below 0.01.The values characterizing the polarization imper-

fections of the detector and source used, as deter-mined using the approach outlined above, areillustrated in Fig. 4. The values that characterizeboth the detector and the source polarization depen-dence are well within the range in which the applied

calibration method is accurate. The observed polari-zation direction around 90° coincides with the reflec-tion direction from mirrors used to guide the opticalbeam. It is also clear that the imperfections of thedetector are less pronounced in magnitude and showa smaller wavenumber dependence than the source.The source also shows a quite distinctive change inazimuth of the attenuation around 1300 cm−1. Theorigin of this is not clear. Neither the emission spec-trum of the globar nor the transmission spectrumof the beam splitter show any distinctive changearound this wavenumber. The smoother imperfectioncharacteristics of the detector indicate that a rotat-ing analyzer setup is preferred for the combinationof components used.

After characterization of the detector and sourceimperfections, the effect of a nonlinear response ofa detector as well as the actual polarization degreeof the polarizers used can be determined [3].

5. Measurement

After characterizing the detector/source polarizationdependence, the detector nonlinearity and the polar-izer efficiency can be measured and used to calibratethe instrument. The detector and source correction isthe first step in this calibration sequence, and it isapplied to the measured intensity required forfurther calibration as given by Eq. (5). The evalua-tion of the nonlinearity of the detector has been out-lined by Luttmann et al. [3]. The correction of themeasured signal for this effect is applied beforethe attenuation factor γ; see Fig. 5. For this purposethe standard procedure that relates the intensity forparallel positions of the two polarizers to the inten-sity in the cross position is used. These three calibra-tion steps provide a measured signal whose Fouriercoefficients α0 and β0 are given by

α0 ¼ ð1 − γÞ2 cos 2P − ð1þ γÞð1 − γÞ cos 2Ψð1þ γÞ2 − ð1þ γÞð1þ γÞ cos 2Ψ cos 2P

;

β0 ¼ ð1 − γÞ2 sin 2Ψ cosΔ sin 2P

ð1þ γÞ2 − ð1þ γÞð1 − γÞ cos 2Ψ cos 2P: ð6Þ

The influence of the polarizer imperfections γ can-not be dealt with through a direct inversion of Eq. (6).This is in marked contrast to a situation in which onegood (Brewster) and one grid polarizer are used. Inthat case direct inversion is possible. However, themeasured values of α0 and β0, calibrated for sourceand detector polarization sensitivity as well as detec-tor nonlinearity, can be transformed to the ellipso-metric parameters N ¼ cos 2Ψ and C ¼ sin 2Ψ cosΔif angle P is known:

N ¼1−γ1þγ cos 2P −

1þγ1−γ α0

1 − α0 cos 2P ;

C ¼ β0sin 2P

0@�1þγ1−γ

�2− cos22P

1 − α0 cos 2P

1A: ð7Þ

Fig. 5. Measured polarizer attenuation γ values for grid polari-zers in tandem. The values were determined after calibratingand correcting for the detector and source polarization depen-dences and nonlinear behavior of the detector.


The precise value of P is determined by the plane ofincidence of the light beam. The plane of incidencecan be obtained by either an extra calibration orby measuring α0 and β0 at various values of P. A fitwith the position of the plane of incidence as a fitparameter provides the actual values of N and C.To verify the effectiveness of these three calibra-

tion and correction steps, the ellipsometer was testedin a straight-through position. The ellipsometricparameters for this areN ¼ 0 and C ¼ 1, and any de-viation from these values in the rough data is in thecalibration steps assigned to the various componentimperfections. This straight-through measurementis therefore a critical step to determine the accuracyof the ellipsometer after corrections. The measuredN and C values in Fig. 6 show that the instrument’sperformance is limited to the range of 600–7000 cm−1. In this wavenumber range, the deviationof N and C from their expected value is well below0.01. The apparent features in C are associated withH2O absorption regions (1300–1700 cm−1 and 3600–3950 cm−1) as a result of changes in humidity duringthe calibration. The sensitivity of the MCT detectordetermines the lower bound of the measurementrange to 600 cm−1, while the performance of the beamsplitter provides the upper boundary of 7000 cm−1.The deviations in N and C from a straight line,and effectively setting the deviation in N and C to0.01, show the same wavenumber dependence asthe polarizer’s attenuation γ, as seen in Fig. 5. Wetherefore attribute the obtained main deviations toeither a calibration accuracy of the polarizers or achange in polarizer alignment. The accuracy of thepresented method might be limited by the assump-tion that all polarizers have the same polarizationcoefficient.Johs and Herzinger [6] introduced a quantification

of the accuracy of an ellipsometer by the parameter

E ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=3n

PððN −ŃÞ2 þ ðC −ĆÞ2 þ ðS − ŚÞ2Þq

. Thesummation is over all n measured wavenumber va-lues, and the prime values are the expected values.The quantity S is related to the sinðΔÞ value. Thisvalue cannot be measured independently with the el-lipsometer used and is therefore disregarded. For airwe find a value of E ¼ 0:0035 in the wavenumberrange from 630 to 7300 cm−1. This is only about a fac-tor of 10 larger than obtained for a typical visiblelight ellipsometer [6]. The inability to measure theellipsometric parameter S independently also im-plies that we cannot use the degree of polarizationto determine the accuracy of the ellipsometer fornonpolarizing samples, such as gold and fused silica.Independent performance evaluation of the ellip-someter around other N and C values is thereforenot possible.

6. Summary

A method for the simultaneous characterization ofthe polarization sensitivity of a source and detectorwas developed. For this purpose we use two polari-zers that are rotated synchronously with three differ-ent phase differences. The deviation from a constantsignal upon rotation is a measure of the polarizationsensitivity. The use of this deviation from a constantsignal makes this method robust with respect to thenonlinearity of the detector response and the polar-ization quality of the polarizers used. A simulationshows that polarizers with a polarization of only95% suffice, and the method is rather robust with re-spect to measurement noise and misalignment of op-tical components.

References

1. E. Wold and J. Bremer, “Mueller matrix analysis of infraredellipsometry,” Appl. Opt. 33, 5982–5993 (1994).

2. J. H. W. G. den Boer, G. W. M. Kroesen, M. Haverlag, andF. J. de Hoog, “Spectroscopic IR ellipsometry with imperfectcomponents,” Thin Solid Films 234, 323–326 (1993).

3. M. Luttmann, J.-L. Stehle, C. Defranoux, and J.-P. Piel, “Highaccuracy IR ellipsometry working with a Ge Brewster anglereflection polarizer and grid analyzer,” Thin Solid Films313–314, 631–641 (1998).

4. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polar-ized Light (North-Holland, 1979).

5. J. H. W. G. den Boer, “Spectroscopic infrared ellipsometry:components, calibration, and application,” Ph.D. dissertation(University of Eindhoven, 1995).

6. B. Johs and C. M. Herzinger, “Quantifying the accuracy ofellipsometer systems,” Phys. Status Solidi C 5, 1031–1035(2008).

Fig. 6. (Color online) N and C values for air, after correcting fordetector and source polarization dependence, nonlinearity in thedetector and inefficiencies in the polarizers.


simultaneous characterization of detector and source imperfections in infrared ellipsometry

Documents