## Abstract

We consider the Mueller matrix ellipsometry (MME) measuring the ellipsometric parameters of the isotropic sample and the anisotropic sample under certain conditions in the presence of either Gaussian additive noise or Poisson shot noise. In this case, the ellipsometric parameters only relate to partial elements in Mueller matrix, and we optimize the instrument matrices of polarization state generator (PSG) and analyzer (PSA) to minimize the total measurement variance for these elements, in order to decrease the variance of the estimator of ellipsometric parameters. Compared with the previous instrument matrices, the optimal instrument matrices in this paper can effectively decrease the measurement variance and thus statistically improve the measurement precision of the ellipsometric parameters. In addition, it is found that the optimal instrument matrices for Poisson shot noise are same to those for Gaussian additive noise, and furthermore, the optimal instrument matrices do not depend on the ellipsometric parameters to be measured, which means that the optimal instrument matrices of MME proposed in this paper can be widely applied in various cases.

© 2017 Optical Society of America

## 1. Introduction

Ellipsometry measures the changes of polarization state upon reflection or transmission on the sample of interest, which is a powerful technique for the characterization of thin films and surfaces [1–5]. In particular, Mueller matrix ellipsometry (MME) is a typical ellipsometry, which measures all the 16 elements of Mueller matrix of the sample, and it can provide the ellipsometric parameters related to physical properties of the sample [3–7]. Therefore, MME has many applications in the accurate determination of dielectric function, optical properties and geometric characteristics of materials [8–14].

Compared with conventional optical polarimetry, MME can achieve better measurement sensitivity and precision [6–12]. Although MME requires a high measurement precision (Typical values for the estimated uncertainty in the experimental values of ellipsometers are around order 0.5 and 0.1% [4]), it is always perturbed by the noise (such as sensor noise, quantum fluctuations, etc.), which causes the retrieved Mueller matrix to deviate from the true value. Statistically speaking, the measurement precision is determined by the variance of estimator. Reducing the variance of the estimator is beneficial for a higher measurement precision [15–17]. Up to now, various methods for reducing the variance of the estimator have been proposed [18–27], in which optimizing the instrument matrices is an effective way [25–27]. Furthermore, previous methods of optimizing the instrument matrices aim to minimize the total variance of all the 16 elements in the Mueller matrix [16-17]. However, in most applications of MME, the samples are isotropic or anisotropic under certain conditions, and in this case, some elements in the Mueller matrix equal to zero, which are independent on the ellipsometric parameters. Therefore, the measurement precision only depends on those nonzero elements. In other words, the optimal instrument matrix for all the 16 Mueller elements could no longer be optimal for the measurement of ellipsometric parameters. It is thus interesting to further improve the estimation performance by the optimized couples of illumination/analysis polarization states for nonzero elements in the Mueller matrix [28-29].

In this paper, we address the issue of measuring the ellipsometric parameters for the case of samples presenting zero cross-terms of the Jones matrix, which include anisotropic bulk samples and some anisotropic thin film samples under certain conditions of the alignment. In this case, the Mueller matrix of the sample is a block diagonal, and there are only eight elements related to the ellipsometric parameters. In fact, we can calculate the two ellipsometric parameters by these eight elements or even four elements. Two types of noise statistics that are frequently encountered are considered: Gaussian additive noise, representative of sensor noise, and Poisson shot noise that results from the quantum fluctuations of the useful or ambient light flux. We deduce the closed-form functions for the total variance of these partial Mueller matrix elements related to the ellipsometric parameters, and obtain the optimal instrument matrices of polarization state generator (PSG) and polarization state analyzer (PSA) by minimizing the functions of total variance. In addition, we also make a comparison with two common instrument matrices of PSA/PSG, and the results demonstrate that the optimal instrument matrices proposed in this paper can lead to lower total variance of the Mueller matrix elements related to the ellipsometric parameters.

## 2. Mueller matrix ellipsometry (MME)

Standard ellipsometry measures the ellipsometric parameters$\psi $and$\Delta $of the sample. The parameters$\psi $and $\Delta $ are usually defined from the ratio of the complex Fresnel coefficients (*r _{p}*, parallel and,

*r*, perpendicular to the plane of incidence) as:

_{s}*r*is the surface power reflectance.

According to Eq. (2), one can estimate the ellipsometric parameters$\psi $and$\Delta $by measuring the Mueller matrix *M*. Let us denote

*W*and a PSA with instrument matrix

*A*. The measured intensities of the scattered light are thus given by:where${}^{T}$denotes the transpose of the matrix. The instrument matrices

*W*and

*A*represent 4 Stokes vectors that characterize the different states of the PSG and PSA, respectively. The matrix

*M*is the product of the sample Mueller matrix by the source intensity.

*I*is a 4 × 4 matrix containing the intensities obtained from the 16 measurements using the polarization states defined in the matrices

*W*and

*A*. According to the measured intensities

*I*and the instrument matrices

*W*and

*A*, one can obtain the Mueller matrix of the sample.

In particular, we consider the case of a MME with 4 PSG/PSA states, and furthermore, those states are identical, which means$A=W$ [16,20,22]. In this case, Eq. (4) can be expressed in the form of vector as:

Where$\otimes $ denotes the Kronecker product, ${V}_{I}$ and${V}_{M}$are 16 dimensional vectors obtained by reading the matrices*I*and

*M*in the lexicographic order, respectively.

In addition, since the noise in each intensity measurement is statistically independent from the other, the variations of the measured intensities are independent from each other. So ${V}_{I}$ is a random vector such that each of its element${\left[{V}_{I}\right]}_{i}$ is a random variable. Therefore, the covariance matrix ${\Gamma}_{{V}_{I}}$of the random variables is a diagonal matrix, and each diagonal element of it equals to the variance of each measured intensity.

To estimate the vector of Mueller matrix ${V}_{M}$from the noisy intensity measurements stacked in vector ${V}_{I}$, we obtain the estimator by using some properties of Kronecker product and matrix operation [30]:

It’s clear that $\u3008{\widehat{V}}_{M}\u3009={\left[{A}^{-1}\otimes {A}^{-1}\right]}^{T}\u3008{V}_{I}\u3009{\text{=V}}_{M}$, where$\u3008\cdot \u3009$denotes ensemble averaging, and thus,${\widehat{V}}_{M}$is an unbiased estimator. For an unbiased estimator, the measurement precision is determined by the variance. The variance of ${\widehat{V}}_{M}$can be characterized by its covariance matrix${\Gamma}_{{\widehat{V}}_{M}}$. According to Eq. (6), the covariance matrix has the following expression:

However, it’s noticed in Eq. (2) that for the isotropic reflecting surface and some anisotropic thin film samples under certain conditions, the Mueller matrix is a block diagonal, and there are only eight nonzero elements related to the ellipsometric parameters $\psi $and$\Delta $, which is shown as follows:

*m*,

_{11}*m*,

_{12}*m*,

_{21}*m*,

_{22}*m*,

_{33}*m*,

_{34}*m*,

_{43}*m*) in Mueller matrix. In order to minimize the total variance of these eight elements in Mueller matrix, one should try to find an optimal instrument matrix that follows:

_{44}Furthermore, the ellipsometric parameters $\psi $and$\Delta $can be even calculated only by four elements indicated in the following Muller matrix:

In order to reduce the total variance of these four elements (*m _{11}*,

*m*,

_{12}*m*,

_{33}*m*) in Mueller matrix, according to Eq. (9), it’s thus interesting to find an optimal instrument matrix following:

_{34}#### 2.1 Optimal instrument matrix in the presence of Gaussian additive noise

We first consider the measurement perturbed by Gaussian additive noise. In this case, ${V}_{I}$ is a random vector that each of its elements ${\left[{V}_{I}\right]}_{i}$, $i\in \left[1,16\right]$ is a Gaussian random variable with mean value $\u3008{\left[{V}_{I}\right]}_{i}\u3009$ and variance ${\sigma}^{2}$. The variance of each element of ${V}_{M}$ is given by [16]:

*A*and do not depends on the matrix ${V}_{M}$ to be measured. Thus, the total variances of the eight and four elements in Mueller matrix indicated in Eqs. (8) and (11) can be calculated by Eq. (14), and the corresponding optimal instrument matrices in the presence of Gaussian additive noise are:

#### 2.2 Optimal instrument matrix in the presence of Poisson shot noise

Let us now consider the measurement perturbed by Poisson shot noise. In this case, ${V}_{I}$ is a random vector such that each of its elements ${\left[{V}_{I}\right]}_{i}$ is a Poisson random variable of mean value $\u3008{\left[{V}_{I}\right]}_{i}\u3009$ and variance$\u3008{\left[{V}_{I}\right]}_{i}\u3009$. The covariance matrix ${\Gamma}_{{V}_{I}}$of ${V}_{I}$ is a diagonal matrix, the diagonal elements of ${\Gamma}_{{V}_{I}}$are given by:

However, it should be noted that, different from the case of Gaussian additive noise, the variance${\sigma}_{i}^{2}$ of each element ${\left[{V}_{M}\right]}_{i}$ might vary with the value of ${V}_{M}$in the presence of Poisson noise, which is given by [16]:

Since each element in the first row of the matrix *A* is always equal to 1/2, and thus each element in the first column of the matrix ${\left[A\otimes A\right]}^{T}$equals to 1/4. Consequently, Eq. (17) can be expressed as:

It can be seen in the second term of Eq. (18) that the variance ${\sigma}_{i}^{2}$ could depend on the Mueller matrix ${V}_{M}$ to be measured, which means that the estimation precision depends on both the instrument matrix and the measured Mueller matrix. The issue of optimizing the instrument matrix for partial Mueller elements in this work is quite similar to the case for complete Mueller polarimetry in Ref [16]. In particular, the optimal instrument matrix should be independent of the measured Mueller matrix, which is also similar to the definition for complete Mueller polarimetry in Ref [16]. Therefore, the optimal instrument matrix should make the second term of Eq. (8) equal to zero. In addition, under the premise that the variance does not depend on the measured Mueller matrix, the optimal instrument matrix should also minimize the total variance of the Mueller elements to be estimated, which means that it should lead to a low value of the first term of Eq. (18). It needs to be clarified that the optimal instrument matrix mentioned above is not optimized for every individual sample, but rather optimized “in average” for all possible samples, because the second term of Eq. (18) can be negative for some specific combinations of the instrument matrix *A* and the measured Mueller matrix *M*, and thus the estimation variance can be even lower in these specific cases. This property is also similar to the case of complete Mueller polarimetry, which is discussed in detail in Ref [16]. The optimal instrument matrix fulfilling these properties will be discussed in detail in Section 4.

## 3. Computational Issue for the Optimization

To perform the optimization of instrument matrix that minimizes the total variances of partial elements in Mueller matrix in Eq. (15), one needs to search four optimal states of the PSG and PSA. It is considered in this work that the PSG and PSA have the same instrument matrix, and each state of the PSG or PSA involves two parameters (azimuth and ellipticity). Therefore, optimizing the instrument matrix involves searching the optimal values of eight parameters of the PSG or PSA. We employ the shuffled complex evolution (SCE) method [31]. This algorithm consists in generating different sets of illumination and analysis polarization states, and changing them by using a global evolution framework to finally converge to a parameter set given a well-minimized variance. Concerning our issue, we have verified that SCE method can converges rapidly to the global minimum.

## 4. Optimization Results and Discussions

#### 4.1 Optimization in the presence of Gaussian additive noise

Let us first consider the first optimization problem in Eq. (15), which refers to the minimum total variance of the interested eight elements of Mueller matrix in the presence of Gaussian additive noise. By performing SCE global optimization algorithm, we find the optimal instrument matrix as follows:

The optimal total variance of the eight elements related to the $\psi $and$\Delta $is calculated to be 51.4${\sigma}^{2}$. It needs to be clarified that the variance matrix given by Eq. (20) does not depend on the Mueller matrix to be measured in presence of Gaussian additive noise, which means that the optimal instrument matrix has the same performance for different Mueller matrices.

There are another two representative instrument matrices for MME. The first one is the regular tetrahedron instrument matrix given by:

This instrument matrix can minimize the total variance of all the 16 elements in Mueller matrix [16,19]. Besides, there is another instrument matrix being widely employed in practice due to its simplicity, which is given by [32–34]:

In order to compare the optimal instrument matrix in Eq. (19) with the two representative instrument matrices in Eqs. (21) and (22) in the presence of Gaussian additive noise, the variances of Mueller matrix elements for these three instrument matrices are shown in Table 1.

It can be seen in Table 1 that the optimal instrument matrix ${A}_{8-elems}^{Gau}$ proposed in this work corresponds to the minimum total variance of the eight elements in the box. In particular, compared with the regular tetrahedron instrument matrix${A}_{tetra}$, the total variance can be slightly decreased by 1.2%, while for the instrument matrix${A}_{simp}$, the total variance can be decreased by 67.9%.

According to Eqs. (11) and (12), one can also calculate the ellipsometric parameters$\psi $and$\Delta $only by four Mueller elements. In this case, the optimization problem is given by the second equation in Eq. (15), which refers to minimizing the total variance of these four elements of Mueller matrix. By performing global optimization with the SCE algorithm, we find the optimal instrument matrix as follows:

The total variance of the four elements is about 16.5${\sigma}^{2}$. Comparing with the two representative matrices in Eqs. (21) and (22), the variance of each element in Mueller matrix obtained by using different instrument matrices is also shown in Table 2.

It can be seen in Table 2 that the optimal instrument matrix${A}_{4-elems}^{Gau}$ proposed in this work corresponds to the minimum total variance. In particular, compared with the regular tetrahedron instrument matrix${A}_{tetra}$, which was considered as the best instrument matrix to lead a lowest total variance of all the 16 elements of the Mueller matrix in previous methods, the total variance of these four elements can be decreased by 25%, while with the instrument matrix ${A}_{simp}$, the total variance can be decreased by 79.4%. Actually, it can be seen in Table 2 that the optimal instrument matrix${A}_{4-elems}^{Gau}$reduces the total variance of these four elements at the expense of increasing the variances of other elements (such as *m _{22}*,

*m*,

_{24}*m*).

_{42}#### 4.2 Optimization in the presence of Poisson shot noise

Let us now consider the Poisson shot noise. As with analysis in Section 2, the variance in Eq. (18) depends on both the instrument matrix and the measured Mueller matrix, it is thus interesting to find out the optimal instrument matrix that allows us to optimize the variance whatever the measured Mueller matrix, which means that the optimal matrix *A* can make the second term of Eq. (18) equal to zero:

*A*for the two noise models discussed in this work. Therefore, the optimal matrices in the presence of Poisson noise equal to the ones of Gaussian noise, which are given by

Considering the optimizations of eight and four elements of Mueller matrix in the presence of Poisson shot noise, the corresponding estimation variance of each element in the Mueller matrix are given by:

In particular, the results for Poisson shot noise in Eq. (28) are similar to those for Gaussian additive noise (see Eqs. (20) and (24)). The only difference is that the variance ${\sigma}^{2}$is replaced by the element ${\left[{V}_{M}\right]}_{1}/4$in the case of Poisson noise.

According to the deduction and calculation above, we get the optimal instrument matrix with the definition introduced by Ref [16], which is independent of the measured Mueller matrix and minimize the total variance of the interested partial Muller elements in average for all possible samples. It needs to be clarified that the estimation variance could be lower than that in Eq. (28) for some specific combinations of the instrument matrix *A* and the measured Mueller matrix *M*, because the second term of Eq. (18) can be negative in such case [16]. However, the corresponding instrument matrix *A* can lead to a higher variance for other Mueller matrices, and thus the global performance of such instrument matrix is worse than that of the optimal instrument matrix given by Eq. (27).

Let us now compare the performances of optimization with different instrument matrices in the presence of Poisson noise, which are presented in Eqs. (21), (22) and (27). It has to be noted that, with the instrument matrix${A}_{tetra}$and${A}_{8-elems}^{Poi}$(or${A}_{4-elems}^{Poi}$), the variances of estimators do not depend on the ellipsometric parameters of samples, while the variance with${A}_{simp}$depends on the ellipsometric parameters to be measured, which are consistent with analyses in theory. In order to search the optimal variance for the instrument matrix${A}_{simp}$, by varying $\psi $and$\Delta $ from 0° to 180°, it is possible to generate all the possible Mueller matrices to be measured, the total variances of eight and four elements for the instrument matrix${A}_{simp}$are shown in Fig. 1. We can observe in Fig. 1 that the variance for the instrument matrix${A}_{simp}$ depends on the ellipsometric parameters to be measured. It is noticed that, with.$\left(\psi ,\Delta \right)\text{=}\left({\text{135}}^{\circ},{0}^{\circ}\right)$.(or$\left(\psi ,\Delta \right)\text{=}\left({\text{45}}^{\circ},{180}^{\circ}\right)$), the total variance of eight elements reaches the minimum value of 128 shown in Fig. 1(a), while with$\left(\psi ,\Delta \right)\text{=}\left({\text{135}}^{\circ},{45}^{\circ}\right)$, the total variance of four elements reaches the minimum value of 57.4 shown in Fig. 1(b).

We have checked the validity of these results with Monte Carlo numerical simulations: when a sufficient number of realizations (more than 100,000) are used, one obtains a good agreement with the theoretical values for all the Mueller matrices (which is generate by varying both$\psi $and$\Delta $from 0° to 180°). The ratios of optimization with the optimal instrument matrices for eight and four elements are shown in Table 3. Since the variance for${A}_{simp}$depends on the Mueller matrix to be measured, we give the minimum variance with a symbol “at least” to present the variance for${A}_{simp}$in Table 3.

According to Table 3, compared with the instrument matrix${A}_{simp}$, the optimal instrument matrix found in this work has a reduction of variance by at least 59.8% and 71.3% for eight and four elements of Mueller matrix, respectively. Compared with the regular tetrahedron instrument matrix${A}_{tetra}$, the total variance can be also decreased 25% if only four elements related to the ellipsometric parameters are considered. These results demonstrate the optimal instrument matrices proposed can lead to a lower estimation variance of the partial Mueller elements related to the ellipsometric parameters, and thus the estimation precision is considerably improved, especially with the optimal instrument matrix for four elements.

It needs to be clarified that all the analyses above are based on the ideal optical elements composing PSG and PSA. In practice, since the variance of instrument matrix changes in function of the azimuthal alignment of the optical elements on the PSG and PSA, the generated instrument matrix can be deviate from the optimal one due to the imperfection of optical elements. Therefore, selecting high quality optical elements for the PSG and PSA is beneficial to generate the instrument matrix much close to the optimal one to reduce simply the propagation noise and thus the estimation variance.

## 5. Conclusion

In conclusion, we consider the Mueller matrix ellipsometry for measuring the ellipsometric parameters $\psi $and$\Delta $ in the presence of either Gaussian additive noise or Poisson shot noise. We decrease the variance of the estimators of ellipsometric parameters by optimizing the instrument matrices of PSG and PSA, only considering the interested partial elements of Mueller matrix related to the ellipsometric parameters. Compared with the regular tetrahedron instrument matrix, which is the optimal instrument matrix for the complete 16 Mueller matrix elements, the variance of the estimators of ellipsometric parameters can be decreased. In particular, the reduction of variance is more distinct if only four elements related to the ellipsometric parameters are considered. It is interesting to find that the optimal instrument matrices for Poisson shot noise are same to that for Gaussian additive noise. Furthermore, the optimal instrument matrices do not depend on the Mueller matrix to be measured and thus the ellipsometric parameters to be measured, and with the optimal instrument matrices, the variances of estimator do not depend on the ellipsometric parameters. In addition, the idea of decreasing the variance by optimizing the instrument matrix proposed in this paper can be extended to the more general case where matrices *W* and *A* are different. In such case, optimizing the instrument matrix would involves searching the optimal values of 16 parameters of PSG or PSA while searching for 8 parameters when these two matrices are the same, so the variance of elements in Mueller matrix can be readjusted more flexibly, and thus may further reduce the estimation variance of the particular elements.

These results are particularly important in Mueller polarimetry, because it makes it possible to further increase the precision of ellipsometers only by adjusting the states of the PSG and PSA to realize the optimal instrument matrix proposed in this work on the experimental MME. This work has many perspectives. For example, the idea of this work can be also applied to measure any interested elements of the Mueller matrix, not limited to the block diagonal Mueller elements discussed in this paper (such as the Mueller matrix transformation (MMT) parameters [35], the interested elements of which are ${m}_{11},{m}_{22},{m}_{23},{m}_{32},{m}_{33}$), our method can also be applied to find the corresponding optimal instrument matrix by optimizing the total variance of those interested partial elements in Mueller matrix.

## Funding

National Natural Science Foundation of China (No. 61405140), National Instrumentation Program (No. 2013YQ030915), Natural Science Foundation of Tianjin (No. 15JCQNJC02000), China Postdoctoral Science Foundation (No. 2016M601260).

## Acknowledgments

Haofeng Hu acknowledges the Fondation Franco-Chinoise pour la Science et ses Applications (FFCSA) and the China Scholarship Council (CSC).

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