The characterization of anisotropic materials and complex systems by ellipsometry has pushed the design of instruments to require the measurement of the full reflection Mueller matrix of the sample with a great precision. Therefore Mueller matrix ellipsometers have emerged over the past twenty years. The values of some coefficients of the matrix can be very small and errors due to noise or systematic errors can induce distored analysis. We present a detailed characterization of the systematic errors for a Mueller Matrix Ellipsometer in the dual-rotating compensator configuration. Starting from a general formalism, we derive explicit first-order expressions for the errors on all the coefficients of the Mueller matrix of the sample. The errors caused by inaccuracy of the azimuthal arrangement of the optical components and residual ellipticity introduced by imperfect optical elements are shown. A new method based on a four-zone averaging measurement is proposed to vanish the systematic errors.
©2008 Optical Society of America
Ellipsometry is a well-established technique for determining optical properties of material samples and complex systems by measuring polarization changes that occur on reflection and refraction. For isotropic samples a significant roughness of the surface induces depolarization and cross-polarization phenomena which affect the ellipsometric measurements. In these cases, the Jones formalism and the Jones matrix associated cannot describe correctly the sample. It is necessary to use the Mueller-Stokes formalism and complete studies require the development of ellipsometers able to measure all the parameters of the Mueller matrix of the sample. In the recent years, Mueller Matrix Ellipsometers (MME) based on coupled photelastic modulator , or on coupled ferroelectric liquid crystal cell  or on dual-rotating compensator  technologies have been developped. In the same time, several works dealing with the relationship between the acquired Mueller matrix and the physical properties of the sample have been made [4, 5]. The values of some measured coefficients of the matrix can be very small and errors due to noise or systematic errors can induce distored analysis.
Identical configurations have been used in polarimetry [6, 7, 8]. Studies to optimize polarimeters to measure the Mueller matrix of a sample in transmission have been performed. For example, Smith  has demonstrated that the optimum retardance value of the retarders of a dual-rotating-retarder (DRR) instrument must be equal to 127 ° compared to the quarter-wave (90°) retarders generally used. An error analysis have been investigated by Goldstein et al where three polarization elements are considered and numerical values are presented when the polarimeter operates with no sample. The authors have shown that the error magnitude from misalignment can be strongly coupled to some off-diagonal elements and can be large.
To our knowledge, in ellipsometry, theorical and experimental investigations of the systematic errors have been performed only for conventional ellipsometry such as rotating compensator  or rotating polarizer [10, 11] ellipsometers. Recently, an experimental study of these errors for a MME has been performed by our laboratory . This paper describes and characterizes the theorical study of the systematic errors of a dual-rotating compensator MME. The first part reminds the principle of measurement of the ellipsometer. The systematic errors caused by optical elements will be detailed individually in the second section. We also give the new approach based on a four-zone averaging measurement method to eliminate these errors.
2. Dual rotating-compensator ellipsometer
This ellipsometer developed in our laboratory is based on the optical PC1SC2A arrangement (Fig. 1) where C1 and C2 are the synchronized rotating compensators (quarter-wave plate). The polarizer and the analyzer are characterized by the azimuthal angles P and A respectively. The Sample is denoted by S.
The state of polarization of a monochromatic light beam is described by its Stokes vector . The light flux passing through the analyzer is equal to the first element of the stokes vector of the detected light. It is described by the equation
where S i is the stokes vector describing the light incident beam. M A, , M, and M P are the Mueller matrices of the analyzer, the second rotating compensator, the sample, the first rotating compensator and the polarizer respectively. The rotation matrix R(Θ) performs the transformation of S from one frame of reference to another, rotated over an angle Θ:
The detected light has the following form :
where C=ωt is related to C 1 and C 2 by the relations C 1=ω 1 t=m 1(C-C S1) and C 2=ω 2 t=m 2(C-C S2) respectively. C 1 and C 2 are the angles of the fast axes of the first and second compensators at time t. The fixed azimuths C S1 and C S2 are imposed by the mechanical assembly of the quarter-wave plates and correspond to the azimuth of the fast axis of the compensators with respect to the plane of incidence at the beginning of the acquisition. The frequency of the two compensators are synchronized and related to the base mechanical frequency ω by the two integers m 1 and m 2. We must choose the ratio m 1:m 2 that provides all Mueller matrix elements of the sample with a good stability in the rotation of the compensators. The ratio 5:k (or k:5) where k is an integer ≠5 or ≠10, are usually used. In these cases, 24 non-zero Fourier coefficients (n=m 1, m 2, 2m 1, 2m 2, m 1±m 2, 2m 1±m 2, m 1±2m 2 and 2m 1±2m 2) allow to deduce all normalized coefficients (Mij=M′ij/M 11; i=1, …, 4; j=1, …, 4) of the matrix. We suggest to the reader to refer to the works of Collins et al for data reduction and details.
3. Systematic-error propagation
All errors independent of their origins, propagate through the Fourier coefficients a 0, a 2n and b 2n. These coefficients are regarding as the elements of a vector a. The systematic errors due to the positioning of the optical elements can be reduced by a suitable calibration procedure, but other errors due to the imperfection of the components affect the measurements. They are evaluated by calculating the Jacobian matrix relating all the coefficients δMij of the Mueller matrix of the sample to the errors δa 0, δa 2n and δb 2n. Taking into account the contributions from all the coefficients a 0, a 2n and b 2n, we find an expression for the complete values of δMij as follows:
where δ a T is the transpose of δ a and is the jacobian matrix of M ij (i.e. the matrix of all first order partial derivatives of M ij). For the Mueller matrix of the sample, the corresponding expression of error matrix is:
The next step is to evaluate the shifts δa 0, δa 2n and δb 2n introduced by the errors. They are evaluated using a Taylor expansion of Eq. (1) . At the first order, the Stokes vector of the detected light is:
where S 0 f is the Stokes vector for the ellipsometer without any imperfect elements. For a given perturbation x, the second element of the second term in Eq. (6) allows to obtain the corresponding Fourier coefficients δa 0, δa 2n and δb 2n to be inserted in Eq. (5). The perturbation of the Stokes vector caused by an error is given by replacing the ideal matrix in the Eq. (1) by the corresponding error matrix δ M (Eq. (5)). For all the optical elements, we will look at the effects individually.
For the sake of clarity in the following expressions of the error matrices in this paper, we introduce the function:
where ρ is the position of an optical element such as the analyzer or the polarizer. We remark that this function changes it sign if , . If a coefficient δMij of the error matrix (Eq. (5)) is proportionnal to f(x,y,ρ), is it possible to eliminate the corresponding error by performing a two-zone measurement given by:
According to the value of ρ, it is possible to eliminate or to vanish the influence of x and/or y in the results. For example, if , , the influence of y is cancelled. On the opposite, if ρ=0, f(x,y,0)=y, the influence of x is cancelled. For the particular angle , x and y have an identical impact on f(x,y,ρ). In this case, .
3.1. Errors due to an azimuthal error
This error perturbation of the Stokes vector caused by the first-order azimuthal error of the elements (i.e the analyzer, the polarizer and the compensators) is given by
where Θ is the azimuthal angle of the element with respect to the plane of incidence.
3.1.1. The analyzer and the polarizer
For the analyzer, an azimuthal error induces an error on the sample matrix as:
where , . δ 1 and δ 2 are the retardances of the first and the second compensator respectively.
The matrix δ M (Eq. (10)) shows that only M 44 is not affected by an error on the position of the analyzer. Moreover, δM 12→δM 14 can be cancelled by a two-zone measurement in A (Eq. (8)) independent of the position of the polarizer. δM 41 can be cancelled by a two-zone measurement in P (Eq. (8)) independent of the position of the analyzer. The other coefficients cannot be vanished with a particular position of the analyzer or the polarizer.
As the same, we can calculate the pertubation on the Mueller matrix for the polarizer:
The Eq. (11) shows that only M 44 is not affected by an error on the position of the polarizer. The coefficients on the first column, δM 21→δM 41, can be cancelled by a two-zone measurement in P independent of the position of the analyzer. δM 14 can be cancelled by a two-zone measurement in A independent of the position of the polarizer. The other coefficients cannot be vanished.
3.1.2. The compensators
For the first compensator (i.e. i=1) the coefficients δMij are shown in Table 1. As can be seen, only the errors on M 21, M 31 and M 41 can be removed by a two-zone measurement in P. The others cannot be suppressed because C S1 and C S2 are imposed by the mechanical assembly. They can not be modified during the measurements and then zone averaging on C S1 and C S2 are impossible. Nevertheless, the mechanical assembly of the compensators can be made with great precision in order to reduce errors of some coefficients like δM 44=0 if C S1=C S2=0. According the sample under test, other coefficients can be eliminated. For example, the variation of δM 21 versus C S1 and C S2 for an isotropic sample 106nm-SiO 2/Si at λ=633nm and 70° of incidence with Ψ=46.14° and Δ=79.61° is presented Fig. 2. δM 21 ranges from -1 up to +1 if P=0° and δ 1=90°. The blue lines represents δM 21=0. So, C S1 and C S2 can be experimentally ajusted in order to obtain δM 21=0.
For the second compensator (i.e. i=2) the coefficients δMij are shown in Table 2 and similar comments may be made as for the first compensator. Only δM 12, δM 13, δM 14 can be removed by a two-zone measurement in A.
3.2. Errors due to an imperfect element
3.2.1. The linear polarizers
Two Glan-Taylor polarizers for the analyzer and polarizer elements are used in our ellipsometer. Depending on the quality of the elements, errors can appear. The error perturbation of the Stokes vector caused by an imperfect linear polarizer at the first-order is:
where γx is the ellipticity of the optical element x.
If the analyzer is imperfect (x=A), we obtain:
with δM 41=-f (M 31-c 1 f (M 22,-M 23,P),M 21+c 1 f (M 32,-M 33,P),A), δM 42=f (M 23-M 32,-M 22-M 33,A) and δM 43=-f (M 22+M 33,M 23 -M 32,A).
The first line of the matrix can be vanish if the second compensator is an ideal quarter wave plate (i.e. cosδ 2=0). The last line δM 41→δM 44, can always be eliminated if a two-zone measurement in A is performed.
The imperfection of the polarizer can be written as:
with δM 14=-f (M 13-c 2 f (M 22,-M 32,A),M 12+c 2 f (M 23,-M 33,A),P). The first column of the matrix will vanish because the first compensator is an ideal quarter wave plate (i.e. cosδ 1=0). The coefficients of the last column can always be equal to zero if a two-zone measurement in P is performed.
3.2.2. The compensators
For a monochromatic ellipsometers, a compensator is generally a quarter wave plate in quartz. In spectroscopic ellipsometry, achromatic retardation plate made with birefringent polymer film stack or composed of two different birefringent crystals (quartz and magnesium fluoride) separated with a thin air-spaced is used. An imperfect compensator induces an error of the ellipticity, γC, and a perturbation matrix can be defined in the first order as:
If an imperfect compensator is used as first compensator, the calculated perturbation matrix is given by:
for the second compensator. We can see on the Eq. 17 that only the two coefficients δM 24 and δM 34 can not be null. The others are eliminated by a two-zone measurement in A and P. For the second compensator (Eq. 18) the same procedure removes all errors except for M 42 and M 43.
3.3. Errors due to imperfect windows and the retardance of the compensator
If imperfect windows are used, they can be modeled as a small-retardation wave plate. Consequently, the entrance or exit windows can be modeled as where the matrix M Ci has the form of a compensator with a small retardation δ Wi. θ Wi is the azimuthal angle of the fast axis of the window i. Inserting the matrix in the Eq. (1) which correspond to the placement of the window in the configuration (i.e. between the polarizer and the sample for the entrance window and between the sample and the analyzer for the exit window), we can evaluate the perturbation of the measured matrix of the sample. The results are not detailed here in the general case. But for particular positions of the optical elements, is it possible to vanished the coefficients δM 14, δM 21 and δM 31 for the entrance window and the coefficients δM 12, δM 13 and δM 41 for the exit window (see Table 3.).
The retardance of the compensators used in a spectroscopic ellipsometer is not strictly equal to 90° in the used spectra. It can be necessary to perform a calibration step to determine the retardance of each compensator individualy for all wavelenghts used in the measurements. These errors are often committed in ellipsometry. In the case of the retardance of a compensator are not well known, errors on the measured Mueller matrix of the sample are:
for the first compensator and
for the second, with , .
If each compensator are an ideal quater wave plate, and the errors on the last column in Eq. (19) and on the last line in Eq. (20) vanish. A two-zone measurement in P for Eq. (19) removes the errors on the first column and a two-zone measurement in A removes the errors on the first line in Eq. (20). For the other coefficients, it is not possible to cancel them by this method.
4. The four-zone averaging measurement
In the previous sections, we show that a two-zone measurement performed as describe in Eq. 8 for the analyzer or for the polarizer eliminates the systematic errors when it was possible independently the position of the other optical element. It is possible to proceed in the same time a two-zone measurement for each element. This method applied to the measurement of the all Mueller matrix parameters is called “four-zone averaging measurement” and is performed as the following expression:
The values of A and P can be choose according to the sample for optimizing precision. In the general case, we suggest to use the four-zone averaging measurement method with . Table 3. shows the systematic errors on the measured Mueller matrix of a sample in the ideal case when C S1=C S2=0°, δ 1=δ 2=90° if the fourzone averaging measurement method is performed.
The systematic errors due to a bad position of the optical elements of the PSG (Polarizer State Generator) (i.e. the polarizer and the first compensator) are eliminated for M 14, M 21→M 41 and M 44. The systematic errors due to a bad position of the elements of the the PSA (Polarizer State Analyzer) (i.e. the analyzer and the second compensator) are eliminated for M 12→M 14, M 41 and M 44. According to the sample under test, some errors could disappear or not. The measured matrix of a sample is not affected by errors due to imperfect analyzer or polarizer. Nevertheless, imperfect compensator can affected the measurements of M 24 and M 34 for the first compensator and measurements of M 42 and M 43 for the second compensator. The case C S1=C S2=0° and δ 1=δ 2=90° coupled with the four-zone averaging measurement method allows to simplify the error matrices of the compensators so the first and last columns, δM 21→δM 41 and δM 14→δM 44 respectively, of Eq. (19) are null. As the same, the first and the last rows of Eq. (20), δM 12→δM 14 and δM 41→δM 44 respectively, are null. Finally, the errors due to the windows are given in this particular configuration. We can see that the entrance window affects the coefficients of the matrix of the sample except M 14, M 21 and M 31. For the exit window, the coefficient M 12, M 13 and M 41 are free of errors by using the four-zone averaging measurement method.
We have characterized the systematic error effects on the measurement of all the Mueller matrix coefficients of a sample for a double rotating compensator ellipsometer. Influence of each
optical element of the MME on the acquired parameters has been described individually. When some errors can not be vanish by our measurement method, only a precisely calibration step can avoid to make errors. We demonstrate that a four-zone averaging measurement is the best way to remove some systematic errors. When the errors cannot be eliminated by our method, it is possible to reduce them by ajusting experimentaly the position of the compensators and/or of the windows according the sample under test. The results are in agreement with the error analysis or a DRR polarimeter presented by Goldstein et al and with the experimental study made recently .
References and links
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