Systematic errors for a Mueller matrix dual rotating compensator ellipsometer

Laurent Broch; Aotmane En Naciri; Luc Johann

doi:10.1364/OE.16.008814

1. Introduction

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 [1], or on coupled ferroelectric liquid crystal cell [2] or on dual-rotating compensator [3] 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 [7] 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[8] 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 [9] or rotating polarizer [10, 11] ellipsometers. Recently, an experimental study of these errors for a MME has been performed by our laboratory [12]. 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 [13]. 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

S_{f} = [M_{A} . R (A)] . [R^{- 1} (C_{2}) . M_{C_{2}} . R (C_{2})] . M . [R^{- 1} (C_{1}) . M_{C_{1}} . R (C_{1})] . [R^{- 1} (P) . M_{P}] . S_{i},

Fig. 1. Diagram of the Mueller Matrix Ellipsometer in PC₁SC₂A arrangement.

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where S _i is the stokes vector describing the light incident beam. M _A, $M_{C_{2}}$ , M, $M_{C_{1}}$ 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:

R (Θ) = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \cos 2 Θ & \sin 2 Θ & 0 \\ 0 & - \sin 2 Θ & \cos 2 Θ & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) .

The detected light has the following form [3]:

I = I_{0} [a_{0} + \sum_{n} (a_{2 n} \cos 2 n C + b_{2 n} \sin 2 n C)],

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 (M_ij=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[3] 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 δM_ij 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 δM_ij as follows:

δ M_{ij} = δ a^{T} . J_{M_{ij}},

where δ a ^T is the transpose of δ a and $J_{M_{ij}}$ 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:

δ M = {δ a}^{T} . \frac{\partial M}{\partial a} = (\begin{matrix} δ a^{T} . \frac{{\partial M}_{11}}{\partial a} & δ a^{T} . \frac{{\partial M}_{12}}{\partial a} & δ a^{T} . \frac{{\partial M}_{13}}{\partial a} & δ a^{T} . \frac{{\partial M}_{14}}{\partial a} \\ δ a^{T} . \frac{{\partial M}_{21}}{\partial a} & δ a^{T} . \frac{{\partial M}_{22}}{\partial a} & δ a^{T} . \frac{{\partial M}_{23}}{\partial a} & δ a^{T} . \frac{{\partial M}_{24}}{\partial a} \\ δ a^{T} . \frac{{\partial M}_{31}}{\partial a} & δ a^{T} . \frac{{\partial M}_{32}}{\partial a} & δ a^{T} . \frac{{\partial M}_{33}}{\partial a} & δ a^{T} . \frac{{\partial M}_{34}}{\partial a} \\ δ a^{T} . \frac{{\partial M}_{41}}{\partial a} & δ a^{T} . \frac{{\partial M}_{42}}{\partial a} & δ a^{T} . \frac{{\partial M}_{43}}{\partial a} & δ a^{T} . \frac{{\partial M}_{44}}{\partial a} \end{matrix}) .

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) [14]. At the first order, the Stokes vector of the detected light is:

S_{f} = S_{f}^{0} + \sum_{k} \frac{\partial S_{f}}{{\partial x}_{k}} {δ x}_{k} = S_{f}^{0} + \sum_{k} (δ S_{f}) x_{k},

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:

f (x, y, ρ) = x \sin 2 ρ + y \cos 2 ρ,

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 $ρ = ρ + \frac{π}{2}$ , $f (x, y, ρ) = - f (x, y, ρ + \frac{π}{2})$ . If a coefficient δM_ij 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:

{({δ M}_{ij})}_{ρ} + {({δ M}_{ij})}_{ρ + \frac{π}{2}} = 0 .

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 $ρ = \pm \frac{π}{4}$ , $f (x, y, \pm \frac{π}{4}) = \pm x$ , 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 $ρ = \pm \frac{π}{8}$ , x and y have an identical impact on f(x,y,ρ). In this case, $f (x, y \pm \frac{π}{8}) = \frac{\sqrt{2}}{2} (\pm x + y)$ .

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

δ R (Θ) = 2 (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & - \sin 2 Θ & \cos 2 Θ & 0 \\ 0 & - \cos 2 Θ & - \sin 2 Θ & 0 \\ 0 & 0 & 0 & 0 \end{matrix}) δ Θ,

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:

δ M = 2 .

where $c_{1} = \cos^{2} \frac{δ_{1}}{2}$ , $c_{2} = \cos^{2} \frac{δ_{2}}{2}$ . 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:

δ M = 2 (\begin{matrix} 0 & - M_{13} & M_{12} & c_{2} f (M_{24}, {- M}_{34}, A) \\ 2 c_{1} f (- M_{22}, M_{23}, P) & - M_{23} & M_{22} & M_{34} \\ 2 c_{1} f (- M_{32}, M_{33}, P) & - M_{33} & M_{32} & - M_{24} \\ 2 c_{1} f (- M_{42}, M_{43}, P) & - M_{43} & M_{42} & 0 \end{matrix}) δ P .

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

The errors of the azimuth C_i correspond to a bad determination of the angle C_Si, and consecutively, Θ=C_Si in Eq. (9). So, The error matrix (Eq. (5)) of the sample is given by:

δ M = 2 (\begin{matrix} 0 & δ M_{12} & δ M_{13} & δ M_{14} \\ δ M_{21} & δ M_{22} & δ M_{23} & δ M_{24} \\ δ M_{31} & δ M_{32} & δ M_{33} & δ M_{34} \\ δ M_{41} & δ M_{42} & δ M_{43} & δ M_{44} \end{matrix}) {δ C}_{Si}

For the first compensator (i.e. i=1) the coefficients δM_ij 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 δM_ij 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.

Table 1. The calculated elements of the matrix δ M (Eq. 12) for the first compensator. The functions B ₁=c ₂[cos(2A+4C _S2)-cos2A] and B ₂=c ₂[sin(2A+4C _S2)-sin2A] are null if a two-zone measurement in A is performed independently the angle C _S2.

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Table 2. The calculated elements of the matrix δ M (Eq. 12) for the second compensator. The functions B ₃=c ₁[cos(2P+4C _S1)-cos2P] and B ₄=c ₁[sin(2P+4C _S1)-sin2P] are null if a two-zone measurement in P is performed independently the angle C _S1.

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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:

Fig. 2. δM ₂₁ versus the azimuth of the compensators C _S1 and C _S2 for an isotropic sample with Ψ=46.14° and Δ=79.61° and for P=0 and δ ₁=90°. The value of δM ₂₁ ranges from -1 (black area), up to +1 (white area). The blue lines represents δM ₂₁=0

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δ M = 2 (\begin{matrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \end{matrix}) γ_{x},

where γ_x is the ellipticity of the optical element x.

If the analyzer is imperfect (x=A), we obtain:

δ M = 2 (\begin{matrix} 0 & M_{42} \cos δ_{2} & M_{43} \cos δ_{2} & M_{44} \cos δ_{2} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ {δ M}_{41} & {δ M}_{42} & {δ M}_{43} & - f (M_{24}, M_{34}, A) \end{matrix}) {δ γ}_{A},

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:

δ M = 2 (\begin{matrix} 0 & 0 & 0 & δ M_{14} \\ M_{24} \cos δ_{1} & 0 & 0 & - f (M_{23} - M_{32}, M_{22} + M_{33}, P) \\ M_{34} \cos δ_{1} & 0 & 0 & - f (M_{22} + M_{33}, {- M}_{23} + M_{32}, P) \\ M_{44} \cos δ_{1} & 0 & 0 & - f (M_{42}, M_{43}, P) \end{matrix}) {δ γ}_{P}

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, γ_C, and a perturbation matrix can be defined in the first order as:

{δ M}_{C} = 2 (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & {\sin δ}_{C} & 1 - \cos δ_{C} \\ 0 & - \sin δ_{C} & 0 & 0 \\ 0 & 1 - \cos δ_{C} & 0 & 0 \end{matrix}) γ_{C} .

If an imperfect compensator is used as first compensator, the calculated perturbation matrix is given by:

δ M = 2 \sin δ_{1} (\begin{matrix} 0 & 0 & 0 & \frac{c_{2}}{2 c_{1}} f (- M_{24}, M_{34}, A) \\ f (M_{22}, - M_{23}, P) & 0 & 0 & \frac{- 1}{2 c_{1}} M_{34} \\ f (M_{32}, - M_{33}, P) & 0 & 0 & \frac{1}{{2 c}_{1}} M_{24} \\ f (M_{42}, - M_{43}, P) & 0 & 0 & 0 \end{matrix}) δ γ_{C_{1}}

and

δ M = - 2 \sin δ_{2}

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 $M_{Wi} = [R^{- 1} (θ_{w_{i}}) \cdot M_{Ci} \cdot R (θ_{w_{i}})]$ 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:

δ M = \frac{1}{t_{1}} (\begin{matrix} 0 & M_{12} & M_{13} & \frac{t_{1} M_{14}}{\tan δ_{1}} \\ - f (M_{23}, M_{22}, P) & M_{22} & M_{23} & \frac{t_{1} M_{24}}{\tan δ_{1}} \\ - f (M_{33}, M_{32}, P) & M_{32} & M_{33} & \frac{t_{1} M_{34}}{\tan δ_{1}} \\ f (M_{43}, {- M}_{42}, P) & M_{42} & M_{43} & \frac{t_{1} M_{44}}{\tan δ_{1}} \end{matrix}) δ δ_{1},

for the first compensator and

δ M = \frac{1}{t_{2}} (\begin{matrix} 0 & f (M_{32}, {- M}_{22}, A) & f (M_{33}, {- M}_{23}, A) & - f (M_{34}, M_{24}, A) \\ M_{21} & M_{22} & M_{23} & M_{24} \\ M_{31} & M_{32} & M_{33} & M_{34} \\ \frac{t_{2} M_{41}}{\tan δ_{2}} & \frac{t_{2} M_{42}}{\tan δ_{2}} & \frac{t_{2} M_{43}}{\tan δ_{2}} & \frac{t_{2} M_{44}}{\tan δ_{2}} \end{matrix}) δ δ_{2},

for the second, with $t_{1} = \tan \frac{δ_{1}}{2}$ , $t_{2} = \tan \frac{δ_{2}}{2}$ .

If each compensator are an ideal quater wave plate, $\frac{1}{\tan δ_{i}} = 0$ 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:

(M_{ij}) sample = \frac{1}{4} \sum_{A; A + \frac{π}{2}} \sum_{P; P + \frac{π}{2}} (M_{ij}) measured .

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 $[A; P] = {[- \frac{π}{4}; - \frac{π}{4}]; [- \frac{π}{4}; \frac{π}{4}]; [\frac{π}{4}; - \frac{π}{4}]; [\frac{π}{4}; \frac{π}{4}]}$ . 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.

5. Conclusion

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

Table 3. Systematic errors in MME if C _S1=C _S2=0° and δ ₁=δ ₂=90° when the four-zone averaging measurement method is performed.

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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[8] and with the experimental study made recently [12].

References and links

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11. A. En Naciri, L. Broch, L. Johann, and R. Kleim, “Fixed polarizer, rotating-polarizer and fixed analyzer spectroscopic ellipsometer: accurate calibration method, effcet of errors and testing,” Thin Solid Films 406, 103–112 (2002). [CrossRef]

12. G. Piller, L. Broch, and L. Johann, “Experimental study of the systematic errors for a Mueller matrix double rotating compensator ellipsometer,” Physica Status Solidi (c) 5, 1027–1030 (2008). [CrossRef]

13. R. M. A Azzam and N. M. Bashara, ellipsometry and Polarized Light (North-holland, Amsterdam, 1977).

14. J. M. M. De Nijs and A. Van Siflhout, “Systematic and random errors in rotating-analyzer ellipsometry,” J. Opt. Soc. Am. A 5, 773–781 (1988). [CrossRef]

Coefficient	Value
δM ₁₂	2(M ₁₃+M ₂₃ B ₁+M ₃₃ B ₂)cos4C _S1+2(-M ₁₂+M ₂₂ B ₁+M ₃₂ B ₂) sin4C _S1
δM ₁₃	2(-M ₁₂+M ₂₂ B ₁+M ₃₂ B ₂)cos4C _S1+2(-M ₁₃+M ₂₃ B ₁+M ₃₃ B ₂) sin4C _S1
δM ₁₄	c ₂ f (-M ₂₄,M ₃₄,A+2C _S2)cos2C _S1+(-M ₁₄+M ₂₄ B ₁+M ₃₄ B ₂) sin2C _S1
δM ₂₁	2c ₁[f (M ₂₂,-M ₂₃,P+2C _S1)cos4C _S2+f (M ₃₂,-M ₃₃,P+2C _S1) sin4C _S2]
δM ₂₂	2f (M ₃₃,M ₂₃,2C _S2)cos4C _S1-2f (M ₃₂,M ₂₂,2C _S2) sin4C _S1
δM ₂₃	-2f (M ₃₂,M ₂₂,2C _S21)cos4C _S1-2f (M ₃₃,M ₂₃,2C _S2) sin4C _S1
δM ₂₄	-f (M ₂₄,M ₃₄,C _S1-2C _S2)
δM ₃₁	2c ₁[f (-M ₃₂,M ₃₃,P+2C _S1)cos4C _S2+f (M ₂₂,-M ₂₃,P+2C _S1) sin4C _S2]
δM ₃₂	2f (-M ₂₃,M ₃₃,2C _S2)cos4C _S1+2f (M ₂₂,-M ₃₂,2C _S2) sin4C _S1]
δM ₃₃	2f (M ₂₂,-M ₃₂,2C _S2)cos4C _S1+2f (M ₂₃,-M ₃₃,2C _S2) sin4C _S1
δM ₃₄	f (-M ₃₄,M ₂₄,C _S1-2C _S2)
δM ₄₁	2c ₁ f (-M ₄₂,M ₄₃,P+2C _S1-C _S2)
δM ₄₂	2f (-M ₄₂,M ₄₃,2C _S1-C _S2))
δM ₄₃	-2f (M ₄₃,M ₄₂,2C _S1-C _S2)
δM ₄₄	M ₄₄sin(2C _S1-2C _S2)

Coefficient	Value
δM ₁₂	2c ₂[f(M ₂₂,-M ₃₂,A+2C _S2)cos4C _S1+f (M ₂₃,-M ₃₃,A+2C _S2) sin4C _S1]
δM ₁₃	2c ₂[f (M ₂₃,-M ₃₃,A+2C _S2)cos4C _S1+f (-M ₂₂,M ₃₂,A+2C _S2) sin4C _S1]
δM ₁₄	2c ₂ f (M ₂₄,-M ₃₄,A-C _S1-2C S ₂)
δM ₂₁	2(M ₃₁+M ₃₂ B ₃+M ₃₃ B ₄)cos4C _S2+2(-M ₂₁+M ₂₂ B ₃+M ₂₃ B ₄) sin4C _S2
δM ₂₂	2f (M ₃₃,M ₃₂,2C _S2)cos4C _S1-2f (M ₂₃,M ₂₂,2C _S1) sin4C _S2
δM ₂₃	2f (-M ₃₂,M ₃₃,2C _S1)cos4C _S2+2f (M ₂₂,-M ₂₃,2C _S1) sin4C _S2
δM ₂₄	2f (M ₂₄,M ₃₄,C _S1 -2C _S2)
δM ₃₁	2(-M ₂₁+M ₂₂ B ₃+M ₂₃ B ₄)cos4C _S2+2(-M ₃₁+M ₃₂ B ₃+M ₃₃ B ₄) sin4C _S2
δM ₃₂	-2f (M ₂₃,M ₂₂,2C _S1)cos4C _S2-2f (M ₃₃,M ₃₂,2C _S1) sin4C _S2
δM ₃₃	2f (M ₂₂,-M ₂₃,2C _S1)cos4C _S2+2f (M ₃₂,-M ₃₃,2C _S1) sin4C _S2
δM ₃₄	2f (M ₃₄,-M ₂₄,C _S1 -2C _S2)
δM ₄₁	(-M ₄₁+M ₄₂ B ₃+M ₄₃ B ₄) sin2C _S2+c ₁ f (-M ₄₂,M ₄₃,P+2C _S1)cos2C _S2
δM ₄₂	f (M ₄₂,-M ₄₃,2C _S1-C _S2)
δM ₄₃	f (M ₄₃,M ₄₂,2C _S1-C _S2)
δM ₄₄	M ₄₄ sin(2C _S1-2C _S2)

Origin	δ M	Origin	δ M
δA	$2 (\begin{matrix} 0 & 0 & 0 & 0 \\ - M_{31} & - M_{32} & - M_{33} & - M_{34} \\ M_{21} & M_{22} & M_{23} & M_{24} \\ 0 & M_{43} & - M_{42} & 0 \end{matrix})$	γA	$(\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix})$
δP	$2 (\begin{matrix} 0 & - M_{13} & M_{12} & 0 \\ 0 & - M_{23} & M_{22} & M_{34} \\ 0 & - M_{33} & M_{32} & - M_{24} \\ 0 & - M_{43} & M_{42} & 0 \end{matrix})$	γP	$(\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix})$
δC _S1	$4 (\begin{matrix} 0 & M_{13} & - M_{12} & 0 \\ 0 & M_{23} & - M_{22} & \frac{- M_{34}}{2} \\ 0 & M_{33} & - M_{32} & \frac{M_{24}}{2} \\ 0 & M_{43} & - M_{42} & 0 \end{matrix})$	γ C ₁	$\sqrt{2} (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - M_{34} \\ 0 & 0 & 0 & M_{24} \\ 0 & 0 & 0 & 0 \end{matrix})$
δC _S2	$4 (\begin{matrix} 0 & 0 & 0 & 0 \\ M_{31} & M_{32} & M_{33} & M_{34} \\ - M_{21} & - M_{22} & - M_{23} & - M_{24} \\ 0 & \frac{- M_{43}}{2} & \frac{M_{42}}{2} & 0 \end{matrix})$	γC ₂	$\sqrt{2} (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & M_{43} & - M_{42} & 0 \end{matrix})$
δδ ₁	$(\begin{matrix} 0 & M_{12} & M_{13} & 0 \\ 0 & M_{22} & M_{23} & 0 \\ 0 & M_{32} & M_{33} & 0 \\ 0 & M_{42} & M_{43} & 0 \end{matrix})$	δδ ₂	$(\begin{matrix} 0 & 0 & 0 & 0 \\ M_{21} & M_{22} & M_{23} & M_{24} \\ M_{31} & M_{32} & M_{33} & M_{34} \\ 0 & 0 & 0 & 0 \end{matrix})$
δW ₁	$(\begin{matrix} 0 & M_{14} \sin 2 θ_{w_{1}} & {- M}_{14} \cos 2 θ_{w_{1}} & 0 \\ 0 & \frac{M_{24} \sin 2 θ_{w_{1}}}{2} & \frac{M_{24} \sin 2 θ_{w_{1}}}{2} & f (- M_{22}, M_{23}, θ_{w_{1}}) \\ 0 & f (M_{34} + \frac{M_{24}}{2}, - M_{24}, θ_{w_{1}}) & f (\frac{M_{24}}{2}, - M_{34}, θ_{w_{1}}) & f (- M_{32}, M_{33}, θ_{w_{1}}) \\ f (M_{21}, {- M}_{31}, θ_{w_{1}}) & M_{44} \sin 2 θ_{w_{1}} & {- M}_{44} \cos 2 θ_{w_{1}} & f (M_{42}, M_{43}, θ_{w_{1}}) \end{matrix})$
δW ₂	$(\begin{matrix} 0 & 0 & 0 & f (M_{12}, M_{13}, θ_{w_{2}}) \\ {- M}_{41} \sin 2 θ_{w_{2}} & - \frac{M_{42} \sin 2 θ_{w_{2}}}{2} & - \frac{M_{42} \sin 2 θ_{w_{2}}}{2} & {- M}_{44} \sin 2 θ_{w_{2}} \\ M_{41} \cos 2 θ_{w_{2}} & f (- \frac{M_{42}}{2} - M_{43}, M_{42}, θ_{w_{2}}) & f (- \frac{M_{42}}{2}, M_{43}, θ_{w_{2}}) & M_{44} \cos 2 θ_{w_{2}} \\ 0 & f (M_{22}, {- M}_{32}, θ_{w_{2}}) & f (M_{23}, {- M}_{33}, θ_{w_{2}}) & f (M_{24}, {- M}_{34}, θ_{w_{2}}) \end{matrix})$

Coefficient	Value
δM ₁₂	2(M ₁₃+M ₂₃ B ₁+M ₃₃ B ₂)cos4C _S1+2(-M ₁₂+M ₂₂ B ₁+M ₃₂ B ₂) sin4C _S1
δM ₁₃	2(-M ₁₂+M ₂₂ B ₁+M ₃₂ B ₂)cos4C _S1+2(-M ₁₃+M ₂₃ B ₁+M ₃₃ B ₂) sin4C _S1
δM ₁₄	c ₂ f (-M ₂₄,M ₃₄,A+2C _S2)cos2C _S1+(-M ₁₄+M ₂₄ B ₁+M ₃₄ B ₂) sin2C _S1
δM ₂₁	2c ₁[f (M ₂₂,-M ₂₃,P+2C _S1)cos4C _S2+f (M ₃₂,-M ₃₃,P+2C _S1) sin4C _S2]
δM ₂₂	2f (M ₃₃,M ₂₃,2C _S2)cos4C _S1-2f (M ₃₂,M ₂₂,2C _S2) sin4C _S1
δM ₂₃	-2f (M ₃₂,M ₂₂,2C _S21)cos4C _S1-2f (M ₃₃,M ₂₃,2C _S2) sin4C _S1
δM ₂₄	-f (M ₂₄,M ₃₄,C _S1-2C _S2)
δM ₃₁	2c ₁[f (-M ₃₂,M ₃₃,P+2C _S1)cos4C _S2+f (M ₂₂,-M ₂₃,P+2C _S1) sin4C _S2]
δM ₃₂	2f (-M ₂₃,M ₃₃,2C _S2)cos4C _S1+2f (M ₂₂,-M ₃₂,2C _S2) sin4C _S1]
δM ₃₃	2f (M ₂₂,-M ₃₂,2C _S2)cos4C _S1+2f (M ₂₃,-M ₃₃,2C _S2) sin4C _S1
δM ₃₄	f (-M ₃₄,M ₂₄,C _S1-2C _S2)
δM ₄₁	2c ₁ f (-M ₄₂,M ₄₃,P+2C _S1-C _S2)
δM ₄₂	2f (-M ₄₂,M ₄₃,2C _S1-C _S2))
δM ₄₃	-2f (M ₄₃,M ₄₂,2C _S1-C _S2)
δM ₄₄	M ₄₄sin(2C _S1-2C _S2)

Coefficient	Value
δM ₁₂	2c ₂[f(M ₂₂,-M ₃₂,A+2C _S2)cos4C _S1+f (M ₂₃,-M ₃₃,A+2C _S2) sin4C _S1]
δM ₁₃	2c ₂[f (M ₂₃,-M ₃₃,A+2C _S2)cos4C _S1+f (-M ₂₂,M ₃₂,A+2C _S2) sin4C _S1]
δM ₁₄	2c ₂ f (M ₂₄,-M ₃₄,A-C _S1-2C S ₂)
δM ₂₁	2(M ₃₁+M ₃₂ B ₃+M ₃₃ B ₄)cos4C _S2+2(-M ₂₁+M ₂₂ B ₃+M ₂₃ B ₄) sin4C _S2
δM ₂₂	2f (M ₃₃,M ₃₂,2C _S2)cos4C _S1-2f (M ₂₃,M ₂₂,2C _S1) sin4C _S2
δM ₂₃	2f (-M ₃₂,M ₃₃,2C _S1)cos4C _S2+2f (M ₂₂,-M ₂₃,2C _S1) sin4C _S2
δM ₂₄	2f (M ₂₄,M ₃₄,C _S1 -2C _S2)
δM ₃₁	2(-M ₂₁+M ₂₂ B ₃+M ₂₃ B ₄)cos4C _S2+2(-M ₃₁+M ₃₂ B ₃+M ₃₃ B ₄) sin4C _S2
δM ₃₂	-2f (M ₂₃,M ₂₂,2C _S1)cos4C _S2-2f (M ₃₃,M ₃₂,2C _S1) sin4C _S2
δM ₃₃	2f (M ₂₂,-M ₂₃,2C _S1)cos4C _S2+2f (M ₃₂,-M ₃₃,2C _S1) sin4C _S2
δM ₃₄	2f (M ₃₄,-M ₂₄,C _S1 -2C _S2)
δM ₄₁	(-M ₄₁+M ₄₂ B ₃+M ₄₃ B ₄) sin2C _S2+c ₁ f (-M ₄₂,M ₄₃,P+2C _S1)cos2C _S2
δM ₄₂	f (M ₄₂,-M ₄₃,2C _S1-C _S2)
δM ₄₃	f (M ₄₃,M ₄₂,2C _S1-C _S2)
δM ₄₄	M ₄₄ sin(2C _S1-2C _S2)

Systematic errors for a Mueller matrix dual rotating compensator ellipsometer

Abstract

1. Introduction

2. Dual rotating-compensator ellipsometer

3. Systematic-error propagation

3.1. Errors due to an azimuthal error

3.1.1. The analyzer and the polarizer

3.1.2. The compensators

3.2. Errors due to an imperfect element

3.2.1. The linear polarizers

3.2.2. The compensators

3.3. Errors due to imperfect windows and the retardance of the compensator

4. The four-zone averaging measurement

5. Conclusion

References and links

Cited By

Figures (2)

Tables (3)

Equations (23)

Optics Express