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 parametersψandΔof the sample. The parametersψand Δ are usually defined from the ratio of the complex Fresnel coefficients (rp, parallel and, rs, perpendicular to the plane of incidence) as:

ρ=rprs=tan ψeiΔ,
where tanψ=|rp/rs| is the amplitude ratio upon reflection, andΔ=δpδs is the difference in phase shift [1–5], which are related to the Mueller matrix of the sample. Actually, 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 of the plane of incidence versus the sample anisotropy axis of symmetry. In these cases, the Mueller matrix of the sample is given by [1–5]:
M=r[1cos2ψ00cos2ψ10000sin2ψcosΔsin2ψsinΔ00sin2ψsinΔsin2ψcosΔ],
where r is the surface power reflectance.

According to Eq. (2), one can estimate the ellipsometric parametersψandΔby measuring the Mueller matrix M. Let us denote

M=[m11m12m13m14m21m22m23m24m31m32m33m34m41m42m43m44]
the 4 × 4 Mueller matrix of the sample. Typical Mueller matrix ellipsometry (MME) measures ellipsometric parameters consists of a light source, a PSG with instrument matrix W and a PSA with instrument matrix A. The measured intensities of the scattered light are thus given by:
I=ATMW,
whereTdenotes 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 meansA=W [16,20,22]. In this case, Eq. (4) can be expressed in the form of vector as:

VI=[AA]TVM,
Where denotes the Kronecker product, VI andVMare 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 VI is a random vector such that each of its element[VI]i is a random variable. Therefore, the covariance matrix ΓVIof 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 VMfrom the noisy intensity measurements stacked in vector VI, we obtain the estimator by using some properties of Kronecker product and matrix operation [30]:

V^M={[AA]T}1VI=[(AT)1(AT)1]VI=[A1A1]TVI.

It’s clear that V^M=[A1A1]TVI=VM, wheredenotes ensemble averaging, and thus,V^Mis an unbiased estimator. For an unbiased estimator, the measurement precision is determined by the variance. The variance of V^Mcan be characterized by its covariance matrixΓV^M. According to Eq. (6), the covariance matrix has the following expression:

ΓV^M=[A1A1]TΓVI[A1A1],
where ΓVI is the covariance matrix of VI. The total variance of the estimator of all the 16 elements in Mueller matrix equals to the trace of ΓV^M (which is expressed astrace{ΓV^M}). Previous works of optimizing the instrument matrix focus on the reducing of trace{ΓV^M}in order to decrease the variance of estimator [15–18].

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 ψandΔ, which is shown as follows:

M=[m11m1200m21m220000m33m3400m43m44].
The zero elements in the Mueller matrix of Eq. (8) are independent of the ellipsometric parametersψandΔ, and thus ψandΔcan be calculated only from these nonzero elements in Mueller matrix. Let us express the variances for these nonzero elements in the Mueller matrix as:
Var[M]=[σ12σ22σ52σ62σ112σ122σ152σ162],
where the symbol “” refers to the matrix elements irrelevant to corresponding discussion, and σi2 corresponds to the variance for those eight elements (m11, m12, m21, m22, m33, m34, m43, m44) 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:

A8-elems=argminAiΩ1σi2,Ω1={1,2,5,6,11,12,15,16}.

Furthermore, the ellipsometric parameters ψandΔcan be even calculated only by four elements indicated in the following Muller matrix:

M=[m11m12m33m34].
In this case, the ellipsometric parameters ψand Δcan be calculated by [24]:

ψ=12cos1[m12m11],Δ=tan1[m34m33].

In order to reduce the total variance of these four elements (m11, m12, m33, m34) in Mueller matrix, according to Eq. (9), it’s thus interesting to find an optimal instrument matrix following:

A4-elems=argminAiΩ2σi2,Ω2={1,2,11,12}.

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, VI is a random vector that each of its elements [VI]i, i[1,16] is a Gaussian random variable with mean value [VI]i and variance σ2. The variance of each element of VM is given by [16]:

σi2=σ2[[AAT]1[AAT]1]ii,i[1,16].
It’s noticed that the variances σi2 in Eq. (14) only depend on the instrument matrix A and do not depends on the matrix VM 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:
A8elemsGau=argminAiΩ1σ2[[AAT]1[AAT]1]ii,A4elemsGau=argminAiΩ2σ2[[AAT]1[AAT]1]ii,
where A8elemsGau and A4elemsGau refer to the optimal instrument matrices, with which it can get the minimum total variance of eight and four elements of Mueller matrix in the presence of Gaussian additive noise, respectively.

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, VI is a random vector such that each of its elements [VI]i is a Poisson random variable of mean value [VI]i and variance[VI]i. The covariance matrix ΓVIof VI is a diagonal matrix, the diagonal elements of ΓVIare given by:

[ΓVI]ii=[VI]i=k=116[AA]ikT[VM]k,i[1,16].

However, it should be noted that, different from the case of Gaussian additive noise, the varianceσi2 of each element [VM]i might vary with the value of VMin the presence of Poisson noise, which is given by [16]:

σi2=k=116[VM]k[n=116([A1A1]ni)2[AA]nkT].

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 [AA]Tequals to 1/4. Consequently, Eq. (17) can be expressed as:

σi2=[VM]14n=116([A1A1]ni)2+k=216[VM]k[n=116([A1A1]ni)2[AA]nkT].

It can be seen in the second term of Eq. (18) that the variance σi2 could depend on the Mueller matrix VM 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:

A8elemsGau=12[11110.550-0.5500.550-0.550-0.6530.5210.653-0.521-0.521-0.6530.5210.653].
The variance of each element in Mueller matrix is given by:

Var[M]=σ2[1.003.313.3110.942.872.879.489.482.879.482.879.488.228.228.228.22].

The optimal total variance of the eight elements related to the ψandΔis calculated to be 51.4σ2. 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:

Atetra=12[11111/31/31/31/31/31/31/31/31/31/31/31/3].

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]:

Asimp=12[1111110000100001].

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.

Tables Icon

Table 1. Variance of each element in Mueller matrix for different instrument matrices in the presence of Gaussian additive noise. The total variance of the eight Mueller elements and the ratio of optimization are also presented.

It can be seen in Table 1 that the optimal instrument matrix A8elemsGau 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 matrixAtetra, the total variance can be slightly decreased by 1.2%, while for the instrument matrixAsimp, the total variance can be decreased by 67.9%.

According to Eqs. (11) and (12), one can also calculate the ellipsometric parametersψandΔonly 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:

A4elemsGau=12[11110.4430.443-0.443-0.4430.732-0.7320.732-0.7320.518-0.518-0.5180.518]
as well as the matrix A4elemsGau*obtained from A4elemsGauby reversing the signs of all the elements of the last three rows. The corresponding variances of each Mueller element are given by:

Var[M]=σ2[1.005.105.1025.991.873.739.5119.031.879.513.7319.033.486.966.9613.93].

The total variance of the four elements is about 16.5σ2. 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.

Tables Icon

Table 2. Variance of each element in Mueller matrix for different instrument matrices in the presence of Gaussian additive noise. The total variance of the four Mueller elements and the ratio of optimization are also presented.

It can be seen in Table 2 that the optimal instrument matrixA4elemsGau proposed in this work corresponds to the minimum total variance. In particular, compared with the regular tetrahedron instrument matrixAtetra, 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 Asimp, the total variance can be decreased by 79.4%. Actually, it can be seen in Table 2 that the optimal instrument matrixA4elemsGaureduces the total variance of these four elements at the expense of increasing the variances of other elements (such as m22, m24, m42).

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:

k=216[VM]k[n=116([A1A1]ni)2[AA]nkT]=0.
In this case, the estimation variance does not depend on the Mueller matrix to be measured. In addition, under this premise, the optimal instrument matrix is found to minimize the first term of Eq. (18). Consequently, the optimal matrix for Poisson shot noise should also satisfy:
A8elemsPoi=argminAiΩ1[VM]14n=116([A1A1]ni)2,A4elemsPoi=argminAiΩ2[VM]14n=116([A1A1]ni)2,
where [VM]1 refers to the first element of Mueller matrixVM, which depends on the intensity and surface power reflectance. According to some properties of Kronecker product and matrix operation [30], we find that iΩ1n=116([A1A1]ni)2 equals toiΩ1[[AAT]1[AAT]1]ii, which means that Eq. (26) have almost the same expressions with the optimization problems for Gaussian additive noise in Eq. (15). The only difference is that the variance σ2 is replaced by the element[VM]1/4, which also represents a variance in the presence of Poisson noise. It is noticed that both [VM]1/4 and σ2 are constant factors, which do not affect the matrix 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

A8elemsPoi=A8elemsGau=12[11110.550-0.5500.550-0.550-0.6530.5210.653-0.521-0.521-0.6530.5210.653],A4elemsPoi=A4elemsGau=12[11110.4430.443-0.443-0.4430.732-0.7320.732-0.7320.518-0.518-0.5180.518].

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:

Eightelements:Var[M]=[VM]14[1.003.313.3110.942.872.879.489.482.879.482.879.488.228.228.228.22],Fourelements:Var[M]=[VM]14[1.005.105.1025.991.873.739.5119.031.879.513.7319.033.486.966.9613.93].

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 σ2is replaced by the element [VM]1/4in 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 matrixAtetraandA8elemsPoi(orA4elemsPoi), the variances of estimators do not depend on the ellipsometric parameters of samples, while the variance withAsimpdepends on the ellipsometric parameters to be measured, which are consistent with analyses in theory. In order to search the optimal variance for the instrument matrixAsimp, by varying ψandΔ 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 matrixAsimpare shown in Fig. 1. We can observe in Fig. 1 that the variance for the instrument matrixAsimp depends on the ellipsometric parameters to be measured. It is noticed that, with.(ψ,Δ)=(135,0).(or(ψ,Δ)=(45,180)), the total variance of eight elements reaches the minimum value of 128 shown in Fig. 1(a), while with(ψ,Δ)=(135,45), the total variance of four elements reaches the minimum value of 57.4 shown in Fig. 1(b).

 

Fig. 1 With the instrument matrix Asimp, evolution of the total variance of the interested partial Mueller elements (8 elements or 4 elements) in function of parameters ψand Δ.

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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 bothψandΔfrom 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 forAsimpdepends on the Mueller matrix to be measured, we give the minimum variance with a symbol “at least” to present the variance forAsimpin Table 3.

Tables Icon

Table 3. The total variances of the eight and four Mueller elements and the ratio of optimization for different instrument matrices in the presence of Poisson shot noise.

According to Table 3, compared with the instrument matrixAsimp, 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 matrixAtetra, 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 ψandΔ 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 m11,m22,m23,m32,m33), 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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23. A. De Martino, Y. K. Kim, E. Garcia-Caurel, B. Laude, and B. Drévillon, “Optimized Mueller polarimeter with liquid crystals,” Opt. Lett. 28(8), 616–618 (2003). [CrossRef]   [PubMed]  

24. S. Krishnan and P. C. Nordine, “Mueller-matrix ellipsometry using the division-of-amplitude photopolarimeter: a study of depolarization effects,” Appl. Opt. 33(19), 4184–4192 (1994). [CrossRef]   [PubMed]  

25. S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng. 41(5), 965–972 (2002). [CrossRef]  

26. H. Hu, R. Ossikovski, and F. Goudail, “Performance of Maximum Likelihood estimation of Mueller matrices taking into account physical realizability and Gaussian or Poisson noise statistics,” Opt. Express 21(4), 5117–5129 (2013). [CrossRef]   [PubMed]  

27. J. S. Tyo, Z. Wang, S. J. Johnson, and B. G. Hoover, “Design and optimization of partial Mueller matrix polarimeters,” Appl. Opt. 49(12), 2326–2333 (2010). [CrossRef]   [PubMed]  

28. G. Anna, H. Sauer, F. Goudail, and D. Dolfi, “Fully tunable active polarization imager for contrast enhancement and partial polarimetry,” Appl. Opt. 51(21), 5302–5309 (2012). [CrossRef]   [PubMed]  

29. S. Savenkov, R. Muttiah, E. Oberemok, and A. Klimov, “Incomplete active polarimetry: measurement of the block-diagonal scattering matrix,” J. Quant. Spectrosc. Radiat. Transf. 112(11), 1796–1802 (2011). [CrossRef]  

30. F. Hiai and D. Petz, Introduction to matrix analysis and applications (Springer Science & Business Media, 2014).

31. Q. Y. Duan, V. K. Gupta, and S. Sorooshian, “Shuffled complex evolution approach for effective and efficient global minimization,” J. Optim. Theory Appl. 76(3), 501–521 (1993). [CrossRef]  

32. G. R. Boyer, B. F. Lamouroux, and B. S. Prade, “Automatic measurement of the Stokes vector of light,” Appl. Opt. 18(8), 1217–1219 (1979). [CrossRef]   [PubMed]  

33. S. X. Wang and A. M. Weiner, “Fast wavelength-parallel polarimeter for broadband optical networks,” Opt. Lett. 29(9), 923–925 (2004). [CrossRef]   [PubMed]  

34. H. Hu, G. Anna, and F. Goudail, “On the performance of the physicality-constrained maximum-likelihood estimation of Stokes vector,” Appl. Opt. 52(27), 6636–6644 (2013). [CrossRef]   [PubMed]  

35. E. Du, H. He, N. Zeng, M. Sun, Y. Guo, J. Wu, S. Liu, and H. Ma, “Mueller matrix polarimetry for differentiating characteristic features of cancerous tissues,” J. Biomed. Opt. 19(7), 076013 (2014). [CrossRef]   [PubMed]  

References

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  1. R. M. Azzam and N. M. Bashara, Ellipsometry and polarized light (Elsevier Science Publishing, 1987).
  2. D. Goldstein, Polarized Light (Dekker, 2003).
  3. H. Tompkins and E. Irene, Handbook of Ellipsometry (William Andrew, 2005).
  4. E. Garcia-Caurel, A. De Martino, J. P. Gaston, and L. Yan, “Application of spectroscopic ellipsometry and Mueller ellipsometry to optical characterization,” Appl. Spectrosc. 67(1), 1–21 (2013).
    [Crossref] [PubMed]
  5. K. Hinrichs and K. J. Eichhorn, eds., Ellipsometry of Functional Organic Surfaces and Films (Springer, 2014).
  6. W. Li, C. Zhang, H. Jiang, X. Chen, and S. Liu, “Depolarization artifacts in dual rotating-compensator Mueller matrix ellipsometry,” J. Opt. 18(5), 055701 (2016).
    [Crossref]
  7. X. Chen, H. Jiang, C. Zhang, and S. Liu, “Towards understanding the detection of profile asymmetry from Mueller matrix differential decomposition,” J. Appl. Phys. 118(22), 225308 (2015).
    [Crossref]
  8. M. Sun, H. He, N. Zeng, E. Du, Y. Guo, S. Liu, J. Wu, Y. He, and H. Ma, “Characterizing the microstructures of biological tissues using Mueller matrix and transformed polarization parameters,” Biomed. Opt. Express 5(12), 4223–4234 (2014).
    [Crossref] [PubMed]
  9. T. Novikova, A. De Martino, P. Bulkin, Q. Nguyen, B. Drévillon, V. Popov, and A. Chumakov, “Metrology of replicated diffractive optics with Mueller polarimetry in conical diffraction,” Opt. Express 15(5), 2033–2046 (2007).
    [Crossref] [PubMed]
  10. Y. Guo, N. Zeng, H. He, T. Yun, E. Du, R. Liao, Y. He, and H. Ma, “A study on forward scattering Mueller matrix decomposition in anisotropic medium,” Opt. Express 21(15), 18361–18370 (2013).
    [Crossref] [PubMed]
  11. D. A. Ramsey and K. C. Luderna, “The influences of roughness on film thickness measurements by Mueller matrix ellipsometry,” Rev. Sci. Instrum. 65(9), 2874–2881 (1994).
    [Crossref]
  12. H. Gu, X. Chen, H. Jiang, C. Zhang, W. Li, and S. Liu, “Accurate alignment of optical axes of a biplate using a spectroscopic Mueller matrix ellipsometer,” Appl. Opt. 55(15), 3935–3941 (2016).
    [Crossref] [PubMed]
  13. S. N. Savenkov, R. S. Muttiah, K. E. Yushtin, and S. A. Volchkov, “Mueller-matrix model of an inhomogeneous, linear, birefringent medium: single scattering case,” J. Quant. Spectrosc. Radiat. Transf. 106(1), 475–486 (2007).
    [Crossref]
  14. R. M. A. Azzam, “Mueller-matrix ellipsometry: a review in Polarization: Measurement, Analysis, and Remote Sensing,” Proc. SPIE 3121, 396–399 (1997).
    [Crossref]
  15. X. Li, T. Liu, B. Huang, Z. Song, and H. Hu, “Optimal distribution of integration time for intensity measurements in Stokes polarimetry,” Opt. Express 23(21), 27690–27699 (2015).
    [Crossref] [PubMed]
  16. G. Anna and F. Goudail, “Optimal Mueller matrix estimation in the presence of Poisson shot noise,” Opt. Express 20(19), 21331–21340 (2012).
    [Crossref] [PubMed]
  17. K. M. Twietmeyer and R. A. Chipman, “Optimization of Mueller matrix polarimeters in the presence of error sources,” Opt. Express 16(15), 11589–11603 (2008).
    [Crossref] [PubMed]
  18. X. Li, H. Hu, T. Liu, B. Huang, and Z. Song, “Optimal distribution of integration time for intensity measurements in degree of linear polarization polarimetry,” Opt. Express 24(7), 7191–7200 (2016).
    [Crossref] [PubMed]
  19. W. Du, S. Liu, C. Zhang, and X. Chen, “Optimal configurations for the dual rotating-compensator Mueller matrix ellipsometer,” Proc. SPIE 8759, 875925 (2013).
    [Crossref]
  20. H. Hu, E. Garcia-Caurel, G. Anna, and F. Goudail, “Maximum likelihood method for calibration of Mueller polarimeters in reflection configuration,” Appl. Opt. 52(25), 6350–6358 (2013).
    [Crossref] [PubMed]
  21. F. Liu, C. J. Lee, J. Chen, E. Louis, P. J. M. van der Slot, K. J. Boller, and F. Bijkerk, “Ellipsometry with randomly varying polarization states,” Opt. Express 20(2), 870–878 (2012).
    [Crossref] [PubMed]
  22. H. Hu, E. Garcia-Caurel, G. Anna, and F. Goudail, “Simplified calibration procedure for Mueller polarimeter in transmission configuration,” Opt. Lett. 39(3), 418–421 (2014).
    [Crossref] [PubMed]
  23. A. De Martino, Y. K. Kim, E. Garcia-Caurel, B. Laude, and B. Drévillon, “Optimized Mueller polarimeter with liquid crystals,” Opt. Lett. 28(8), 616–618 (2003).
    [Crossref] [PubMed]
  24. S. Krishnan and P. C. Nordine, “Mueller-matrix ellipsometry using the division-of-amplitude photopolarimeter: a study of depolarization effects,” Appl. Opt. 33(19), 4184–4192 (1994).
    [Crossref] [PubMed]
  25. S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng. 41(5), 965–972 (2002).
    [Crossref]
  26. H. Hu, R. Ossikovski, and F. Goudail, “Performance of Maximum Likelihood estimation of Mueller matrices taking into account physical realizability and Gaussian or Poisson noise statistics,” Opt. Express 21(4), 5117–5129 (2013).
    [Crossref] [PubMed]
  27. J. S. Tyo, Z. Wang, S. J. Johnson, and B. G. Hoover, “Design and optimization of partial Mueller matrix polarimeters,” Appl. Opt. 49(12), 2326–2333 (2010).
    [Crossref] [PubMed]
  28. G. Anna, H. Sauer, F. Goudail, and D. Dolfi, “Fully tunable active polarization imager for contrast enhancement and partial polarimetry,” Appl. Opt. 51(21), 5302–5309 (2012).
    [Crossref] [PubMed]
  29. S. Savenkov, R. Muttiah, E. Oberemok, and A. Klimov, “Incomplete active polarimetry: measurement of the block-diagonal scattering matrix,” J. Quant. Spectrosc. Radiat. Transf. 112(11), 1796–1802 (2011).
    [Crossref]
  30. F. Hiai and D. Petz, Introduction to matrix analysis and applications (Springer Science & Business Media, 2014).
  31. Q. Y. Duan, V. K. Gupta, and S. Sorooshian, “Shuffled complex evolution approach for effective and efficient global minimization,” J. Optim. Theory Appl. 76(3), 501–521 (1993).
    [Crossref]
  32. G. R. Boyer, B. F. Lamouroux, and B. S. Prade, “Automatic measurement of the Stokes vector of light,” Appl. Opt. 18(8), 1217–1219 (1979).
    [Crossref] [PubMed]
  33. S. X. Wang and A. M. Weiner, “Fast wavelength-parallel polarimeter for broadband optical networks,” Opt. Lett. 29(9), 923–925 (2004).
    [Crossref] [PubMed]
  34. H. Hu, G. Anna, and F. Goudail, “On the performance of the physicality-constrained maximum-likelihood estimation of Stokes vector,” Appl. Opt. 52(27), 6636–6644 (2013).
    [Crossref] [PubMed]
  35. E. Du, H. He, N. Zeng, M. Sun, Y. Guo, J. Wu, S. Liu, and H. Ma, “Mueller matrix polarimetry for differentiating characteristic features of cancerous tissues,” J. Biomed. Opt. 19(7), 076013 (2014).
    [Crossref] [PubMed]

2016 (3)

2015 (2)

X. Li, T. Liu, B. Huang, Z. Song, and H. Hu, “Optimal distribution of integration time for intensity measurements in Stokes polarimetry,” Opt. Express 23(21), 27690–27699 (2015).
[Crossref] [PubMed]

X. Chen, H. Jiang, C. Zhang, and S. Liu, “Towards understanding the detection of profile asymmetry from Mueller matrix differential decomposition,” J. Appl. Phys. 118(22), 225308 (2015).
[Crossref]

2014 (3)

2013 (6)

2012 (3)

2011 (1)

S. Savenkov, R. Muttiah, E. Oberemok, and A. Klimov, “Incomplete active polarimetry: measurement of the block-diagonal scattering matrix,” J. Quant. Spectrosc. Radiat. Transf. 112(11), 1796–1802 (2011).
[Crossref]

2010 (1)

2008 (1)

2007 (2)

S. N. Savenkov, R. S. Muttiah, K. E. Yushtin, and S. A. Volchkov, “Mueller-matrix model of an inhomogeneous, linear, birefringent medium: single scattering case,” J. Quant. Spectrosc. Radiat. Transf. 106(1), 475–486 (2007).
[Crossref]

T. Novikova, A. De Martino, P. Bulkin, Q. Nguyen, B. Drévillon, V. Popov, and A. Chumakov, “Metrology of replicated diffractive optics with Mueller polarimetry in conical diffraction,” Opt. Express 15(5), 2033–2046 (2007).
[Crossref] [PubMed]

2004 (1)

2003 (1)

2002 (1)

S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng. 41(5), 965–972 (2002).
[Crossref]

1997 (1)

R. M. A. Azzam, “Mueller-matrix ellipsometry: a review in Polarization: Measurement, Analysis, and Remote Sensing,” Proc. SPIE 3121, 396–399 (1997).
[Crossref]

1994 (2)

D. A. Ramsey and K. C. Luderna, “The influences of roughness on film thickness measurements by Mueller matrix ellipsometry,” Rev. Sci. Instrum. 65(9), 2874–2881 (1994).
[Crossref]

S. Krishnan and P. C. Nordine, “Mueller-matrix ellipsometry using the division-of-amplitude photopolarimeter: a study of depolarization effects,” Appl. Opt. 33(19), 4184–4192 (1994).
[Crossref] [PubMed]

1993 (1)

Q. Y. Duan, V. K. Gupta, and S. Sorooshian, “Shuffled complex evolution approach for effective and efficient global minimization,” J. Optim. Theory Appl. 76(3), 501–521 (1993).
[Crossref]

1979 (1)

Anna, G.

Azzam, R. M. A.

R. M. A. Azzam, “Mueller-matrix ellipsometry: a review in Polarization: Measurement, Analysis, and Remote Sensing,” Proc. SPIE 3121, 396–399 (1997).
[Crossref]

Bijkerk, F.

Boller, K. J.

Boyer, G. R.

Bulkin, P.

Chen, J.

Chen, X.

H. Gu, X. Chen, H. Jiang, C. Zhang, W. Li, and S. Liu, “Accurate alignment of optical axes of a biplate using a spectroscopic Mueller matrix ellipsometer,” Appl. Opt. 55(15), 3935–3941 (2016).
[Crossref] [PubMed]

W. Li, C. Zhang, H. Jiang, X. Chen, and S. Liu, “Depolarization artifacts in dual rotating-compensator Mueller matrix ellipsometry,” J. Opt. 18(5), 055701 (2016).
[Crossref]

X. Chen, H. Jiang, C. Zhang, and S. Liu, “Towards understanding the detection of profile asymmetry from Mueller matrix differential decomposition,” J. Appl. Phys. 118(22), 225308 (2015).
[Crossref]

W. Du, S. Liu, C. Zhang, and X. Chen, “Optimal configurations for the dual rotating-compensator Mueller matrix ellipsometer,” Proc. SPIE 8759, 875925 (2013).
[Crossref]

Chipman, R. A.

Chumakov, A.

De Martino, A.

Dolfi, D.

Drévillon, B.

Du, E.

Du, W.

W. Du, S. Liu, C. Zhang, and X. Chen, “Optimal configurations for the dual rotating-compensator Mueller matrix ellipsometer,” Proc. SPIE 8759, 875925 (2013).
[Crossref]

Duan, Q. Y.

Q. Y. Duan, V. K. Gupta, and S. Sorooshian, “Shuffled complex evolution approach for effective and efficient global minimization,” J. Optim. Theory Appl. 76(3), 501–521 (1993).
[Crossref]

Garcia-Caurel, E.

Gaston, J. P.

Goudail, F.

Gu, H.

Guo, Y.

Gupta, V. K.

Q. Y. Duan, V. K. Gupta, and S. Sorooshian, “Shuffled complex evolution approach for effective and efficient global minimization,” J. Optim. Theory Appl. 76(3), 501–521 (1993).
[Crossref]

He, H.

He, Y.

Hoover, B. G.

Hu, H.

Huang, B.

Jiang, H.

H. Gu, X. Chen, H. Jiang, C. Zhang, W. Li, and S. Liu, “Accurate alignment of optical axes of a biplate using a spectroscopic Mueller matrix ellipsometer,” Appl. Opt. 55(15), 3935–3941 (2016).
[Crossref] [PubMed]

W. Li, C. Zhang, H. Jiang, X. Chen, and S. Liu, “Depolarization artifacts in dual rotating-compensator Mueller matrix ellipsometry,” J. Opt. 18(5), 055701 (2016).
[Crossref]

X. Chen, H. Jiang, C. Zhang, and S. Liu, “Towards understanding the detection of profile asymmetry from Mueller matrix differential decomposition,” J. Appl. Phys. 118(22), 225308 (2015).
[Crossref]

Johnson, S. J.

Kim, Y. K.

Klimov, A.

S. Savenkov, R. Muttiah, E. Oberemok, and A. Klimov, “Incomplete active polarimetry: measurement of the block-diagonal scattering matrix,” J. Quant. Spectrosc. Radiat. Transf. 112(11), 1796–1802 (2011).
[Crossref]

Krishnan, S.

Lamouroux, B. F.

Laude, B.

Lee, C. J.

Li, W.

W. Li, C. Zhang, H. Jiang, X. Chen, and S. Liu, “Depolarization artifacts in dual rotating-compensator Mueller matrix ellipsometry,” J. Opt. 18(5), 055701 (2016).
[Crossref]

H. Gu, X. Chen, H. Jiang, C. Zhang, W. Li, and S. Liu, “Accurate alignment of optical axes of a biplate using a spectroscopic Mueller matrix ellipsometer,” Appl. Opt. 55(15), 3935–3941 (2016).
[Crossref] [PubMed]

Li, X.

Liao, R.

Liu, F.

Liu, S.

H. Gu, X. Chen, H. Jiang, C. Zhang, W. Li, and S. Liu, “Accurate alignment of optical axes of a biplate using a spectroscopic Mueller matrix ellipsometer,” Appl. Opt. 55(15), 3935–3941 (2016).
[Crossref] [PubMed]

W. Li, C. Zhang, H. Jiang, X. Chen, and S. Liu, “Depolarization artifacts in dual rotating-compensator Mueller matrix ellipsometry,” J. Opt. 18(5), 055701 (2016).
[Crossref]

X. Chen, H. Jiang, C. Zhang, and S. Liu, “Towards understanding the detection of profile asymmetry from Mueller matrix differential decomposition,” J. Appl. Phys. 118(22), 225308 (2015).
[Crossref]

M. Sun, H. He, N. Zeng, E. Du, Y. Guo, S. Liu, J. Wu, Y. He, and H. Ma, “Characterizing the microstructures of biological tissues using Mueller matrix and transformed polarization parameters,” Biomed. Opt. Express 5(12), 4223–4234 (2014).
[Crossref] [PubMed]

E. Du, H. He, N. Zeng, M. Sun, Y. Guo, J. Wu, S. Liu, and H. Ma, “Mueller matrix polarimetry for differentiating characteristic features of cancerous tissues,” J. Biomed. Opt. 19(7), 076013 (2014).
[Crossref] [PubMed]

W. Du, S. Liu, C. Zhang, and X. Chen, “Optimal configurations for the dual rotating-compensator Mueller matrix ellipsometer,” Proc. SPIE 8759, 875925 (2013).
[Crossref]

Liu, T.

Louis, E.

Luderna, K. C.

D. A. Ramsey and K. C. Luderna, “The influences of roughness on film thickness measurements by Mueller matrix ellipsometry,” Rev. Sci. Instrum. 65(9), 2874–2881 (1994).
[Crossref]

Ma, H.

Muttiah, R.

S. Savenkov, R. Muttiah, E. Oberemok, and A. Klimov, “Incomplete active polarimetry: measurement of the block-diagonal scattering matrix,” J. Quant. Spectrosc. Radiat. Transf. 112(11), 1796–1802 (2011).
[Crossref]

Muttiah, R. S.

S. N. Savenkov, R. S. Muttiah, K. E. Yushtin, and S. A. Volchkov, “Mueller-matrix model of an inhomogeneous, linear, birefringent medium: single scattering case,” J. Quant. Spectrosc. Radiat. Transf. 106(1), 475–486 (2007).
[Crossref]

Nguyen, Q.

Nordine, P. C.

Novikova, T.

Oberemok, E.

S. Savenkov, R. Muttiah, E. Oberemok, and A. Klimov, “Incomplete active polarimetry: measurement of the block-diagonal scattering matrix,” J. Quant. Spectrosc. Radiat. Transf. 112(11), 1796–1802 (2011).
[Crossref]

Ossikovski, R.

Popov, V.

Prade, B. S.

Ramsey, D. A.

D. A. Ramsey and K. C. Luderna, “The influences of roughness on film thickness measurements by Mueller matrix ellipsometry,” Rev. Sci. Instrum. 65(9), 2874–2881 (1994).
[Crossref]

Sauer, H.

Savenkov, S.

S. Savenkov, R. Muttiah, E. Oberemok, and A. Klimov, “Incomplete active polarimetry: measurement of the block-diagonal scattering matrix,” J. Quant. Spectrosc. Radiat. Transf. 112(11), 1796–1802 (2011).
[Crossref]

Savenkov, S. N.

S. N. Savenkov, R. S. Muttiah, K. E. Yushtin, and S. A. Volchkov, “Mueller-matrix model of an inhomogeneous, linear, birefringent medium: single scattering case,” J. Quant. Spectrosc. Radiat. Transf. 106(1), 475–486 (2007).
[Crossref]

S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng. 41(5), 965–972 (2002).
[Crossref]

Song, Z.

Sorooshian, S.

Q. Y. Duan, V. K. Gupta, and S. Sorooshian, “Shuffled complex evolution approach for effective and efficient global minimization,” J. Optim. Theory Appl. 76(3), 501–521 (1993).
[Crossref]

Sun, M.

E. Du, H. He, N. Zeng, M. Sun, Y. Guo, J. Wu, S. Liu, and H. Ma, “Mueller matrix polarimetry for differentiating characteristic features of cancerous tissues,” J. Biomed. Opt. 19(7), 076013 (2014).
[Crossref] [PubMed]

M. Sun, H. He, N. Zeng, E. Du, Y. Guo, S. Liu, J. Wu, Y. He, and H. Ma, “Characterizing the microstructures of biological tissues using Mueller matrix and transformed polarization parameters,” Biomed. Opt. Express 5(12), 4223–4234 (2014).
[Crossref] [PubMed]

Twietmeyer, K. M.

Tyo, J. S.

van der Slot, P. J. M.

Volchkov, S. A.

S. N. Savenkov, R. S. Muttiah, K. E. Yushtin, and S. A. Volchkov, “Mueller-matrix model of an inhomogeneous, linear, birefringent medium: single scattering case,” J. Quant. Spectrosc. Radiat. Transf. 106(1), 475–486 (2007).
[Crossref]

Wang, S. X.

Wang, Z.

Weiner, A. M.

Wu, J.

E. Du, H. He, N. Zeng, M. Sun, Y. Guo, J. Wu, S. Liu, and H. Ma, “Mueller matrix polarimetry for differentiating characteristic features of cancerous tissues,” J. Biomed. Opt. 19(7), 076013 (2014).
[Crossref] [PubMed]

M. Sun, H. He, N. Zeng, E. Du, Y. Guo, S. Liu, J. Wu, Y. He, and H. Ma, “Characterizing the microstructures of biological tissues using Mueller matrix and transformed polarization parameters,” Biomed. Opt. Express 5(12), 4223–4234 (2014).
[Crossref] [PubMed]

Yan, L.

Yun, T.

Yushtin, K. E.

S. N. Savenkov, R. S. Muttiah, K. E. Yushtin, and S. A. Volchkov, “Mueller-matrix model of an inhomogeneous, linear, birefringent medium: single scattering case,” J. Quant. Spectrosc. Radiat. Transf. 106(1), 475–486 (2007).
[Crossref]

Zeng, N.

Zhang, C.

W. Li, C. Zhang, H. Jiang, X. Chen, and S. Liu, “Depolarization artifacts in dual rotating-compensator Mueller matrix ellipsometry,” J. Opt. 18(5), 055701 (2016).
[Crossref]

H. Gu, X. Chen, H. Jiang, C. Zhang, W. Li, and S. Liu, “Accurate alignment of optical axes of a biplate using a spectroscopic Mueller matrix ellipsometer,” Appl. Opt. 55(15), 3935–3941 (2016).
[Crossref] [PubMed]

X. Chen, H. Jiang, C. Zhang, and S. Liu, “Towards understanding the detection of profile asymmetry from Mueller matrix differential decomposition,” J. Appl. Phys. 118(22), 225308 (2015).
[Crossref]

W. Du, S. Liu, C. Zhang, and X. Chen, “Optimal configurations for the dual rotating-compensator Mueller matrix ellipsometer,” Proc. SPIE 8759, 875925 (2013).
[Crossref]

Appl. Opt. (7)

Appl. Spectrosc. (1)

Biomed. Opt. Express (1)

J. Appl. Phys. (1)

X. Chen, H. Jiang, C. Zhang, and S. Liu, “Towards understanding the detection of profile asymmetry from Mueller matrix differential decomposition,” J. Appl. Phys. 118(22), 225308 (2015).
[Crossref]

J. Biomed. Opt. (1)

E. Du, H. He, N. Zeng, M. Sun, Y. Guo, J. Wu, S. Liu, and H. Ma, “Mueller matrix polarimetry for differentiating characteristic features of cancerous tissues,” J. Biomed. Opt. 19(7), 076013 (2014).
[Crossref] [PubMed]

J. Opt. (1)

W. Li, C. Zhang, H. Jiang, X. Chen, and S. Liu, “Depolarization artifacts in dual rotating-compensator Mueller matrix ellipsometry,” J. Opt. 18(5), 055701 (2016).
[Crossref]

J. Optim. Theory Appl. (1)

Q. Y. Duan, V. K. Gupta, and S. Sorooshian, “Shuffled complex evolution approach for effective and efficient global minimization,” J. Optim. Theory Appl. 76(3), 501–521 (1993).
[Crossref]

J. Quant. Spectrosc. Radiat. Transf. (2)

S. Savenkov, R. Muttiah, E. Oberemok, and A. Klimov, “Incomplete active polarimetry: measurement of the block-diagonal scattering matrix,” J. Quant. Spectrosc. Radiat. Transf. 112(11), 1796–1802 (2011).
[Crossref]

S. N. Savenkov, R. S. Muttiah, K. E. Yushtin, and S. A. Volchkov, “Mueller-matrix model of an inhomogeneous, linear, birefringent medium: single scattering case,” J. Quant. Spectrosc. Radiat. Transf. 106(1), 475–486 (2007).
[Crossref]

Opt. Eng. (1)

S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng. 41(5), 965–972 (2002).
[Crossref]

Opt. Express (8)

H. Hu, R. Ossikovski, and F. Goudail, “Performance of Maximum Likelihood estimation of Mueller matrices taking into account physical realizability and Gaussian or Poisson noise statistics,” Opt. Express 21(4), 5117–5129 (2013).
[Crossref] [PubMed]

F. Liu, C. J. Lee, J. Chen, E. Louis, P. J. M. van der Slot, K. J. Boller, and F. Bijkerk, “Ellipsometry with randomly varying polarization states,” Opt. Express 20(2), 870–878 (2012).
[Crossref] [PubMed]

X. Li, T. Liu, B. Huang, Z. Song, and H. Hu, “Optimal distribution of integration time for intensity measurements in Stokes polarimetry,” Opt. Express 23(21), 27690–27699 (2015).
[Crossref] [PubMed]

G. Anna and F. Goudail, “Optimal Mueller matrix estimation in the presence of Poisson shot noise,” Opt. Express 20(19), 21331–21340 (2012).
[Crossref] [PubMed]

K. M. Twietmeyer and R. A. Chipman, “Optimization of Mueller matrix polarimeters in the presence of error sources,” Opt. Express 16(15), 11589–11603 (2008).
[Crossref] [PubMed]

X. Li, H. Hu, T. Liu, B. Huang, and Z. Song, “Optimal distribution of integration time for intensity measurements in degree of linear polarization polarimetry,” Opt. Express 24(7), 7191–7200 (2016).
[Crossref] [PubMed]

T. Novikova, A. De Martino, P. Bulkin, Q. Nguyen, B. Drévillon, V. Popov, and A. Chumakov, “Metrology of replicated diffractive optics with Mueller polarimetry in conical diffraction,” Opt. Express 15(5), 2033–2046 (2007).
[Crossref] [PubMed]

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[Crossref] [PubMed]

Opt. Lett. (3)

Proc. SPIE (2)

W. Du, S. Liu, C. Zhang, and X. Chen, “Optimal configurations for the dual rotating-compensator Mueller matrix ellipsometer,” Proc. SPIE 8759, 875925 (2013).
[Crossref]

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[Crossref]

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[Crossref]

Other (5)

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Figures (1)

Fig. 1
Fig. 1 With the instrument matrix A s i m p , evolution of the total variance of the interested partial Mueller elements (8 elements or 4 elements) in function of parameters ψ and Δ .

Tables (3)

Tables Icon

Table 1 Variance of each element in Mueller matrix for different instrument matrices in the presence of Gaussian additive noise. The total variance of the eight Mueller elements and the ratio of optimization are also presented.

Tables Icon

Table 2 Variance of each element in Mueller matrix for different instrument matrices in the presence of Gaussian additive noise. The total variance of the four Mueller elements and the ratio of optimization are also presented.

Tables Icon

Table 3 The total variances of the eight and four Mueller elements and the ratio of optimization for different instrument matrices in the presence of Poisson shot noise.

Equations (28)

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ρ = r p r s = tan   ψ e i Δ ,
M = r [ 1 cos 2 ψ 0 0 cos 2 ψ 1 0 0 0 0 sin 2 ψ cos Δ sin 2 ψ sin Δ 0 0 sin 2 ψ sin Δ sin 2 ψ cos Δ ] ,
M = [ m 11 m 12 m 13 m 14 m 21 m 22 m 23 m 24 m 31 m 32 m 33 m 34 m 41 m 42 m 43 m 44 ]
I = A T M W ,
V I = [ A A ] T V M ,
V ^ M = { [ A A ] T } 1 V I = [ ( A T ) 1 ( A T ) 1 ] V I = [ A 1 A 1 ] T V I .
Γ V ^ M = [ A 1 A 1 ] T Γ V I [ A 1 A 1 ] ,
M = [ m 11 m 12 0 0 m 21 m 22 0 0 0 0 m 33 m 34 0 0 m 43 m 44 ] .
Var [ M ] = [ σ 1 2 σ 2 2 σ 5 2 σ 6 2 σ 11 2 σ 12 2 σ 15 2 σ 16 2 ] ,
A 8 - e l e m s = arg min A i Ω 1 σ i 2 , Ω 1 = { 1 , 2 , 5 , 6 , 11 , 12 , 15 , 16 } .
M = [ m 11 m 12 m 33 m 34 ] .
ψ = 1 2 cos 1 [ m 12 m 11 ] , Δ = tan 1 [ m 34 m 33 ] .
A 4 - e l e m s = arg min A i Ω 2 σ i 2 , Ω 2 = { 1 , 2 , 11 , 12 } .
σ i 2 = σ 2 [ [ A A T ] 1 [ A A T ] 1 ] i i , i [ 1 , 16 ] .
A 8 e l e m s G a u = arg min A i Ω 1 σ 2 [ [ A A T ] 1 [ A A T ] 1 ] i i , A 4 e l e m s G a u = arg min A i Ω 2 σ 2 [ [ A A T ] 1 [ A A T ] 1 ] i i ,
[ Γ V I ] i i = [ V I ] i = k = 1 16 [ A A ] i k T [ V M ] k , i [ 1 , 16 ] .
σ i 2 = k = 1 16 [ V M ] k [ n = 1 16 ( [ A 1 A 1 ] n i ) 2 [ A A ] n k T ] .
σ i 2 = [ V M ] 1 4 n = 1 16 ( [ A 1 A 1 ] n i ) 2 + k = 2 16 [ V M ] k [ n = 1 16 ( [ A 1 A 1 ] n i ) 2 [ A A ] n k T ] .
A 8 e l e m s G a u = 1 2 [ 1 1 1 1 0 .550 -0 .550 0 .550 -0 .550 -0 .653 0 .521 0 .653 -0 .521 -0 .521 -0 .653 0 .521 0 .653 ] .
Var [ M ] = σ 2 [ 1.00 3.31 3.31 10.94 2.87 2.87 9.48 9.48 2.87 9.48 2.87 9.48 8.22 8.22 8.22 8.22 ] .
A t e t r a = 1 2 [ 1 1 1 1 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 ] .
A s i m p = 1 2 [ 1 1 1 1 1 1 0 0 0 0 1 0 0 0 0 1 ] .
A 4 e l e m s G a u = 1 2 [ 1 1 1 1 0 .443 0 .443 -0 .443 -0 .443 0 .732 -0 .732 0 .732 -0 .732 0 .518 -0 .518 -0 .518 0 .518 ]
Var [ M ] = σ 2 [ 1.00 5 .10 5 .10 25 .99 1 .87 3 .73 9 .51 19 .03 1 .87 9 .51 3 .73 19 .03 3 .48 6 .96 6 .96 13 .93 ] .
k = 2 16 [ V M ] k [ n = 1 16 ( [ A 1 A 1 ] n i ) 2 [ A A ] n k T ] = 0.
A 8 e l e m s P o i = arg min A i Ω 1 [ V M ] 1 4 n = 1 16 ( [ A 1 A 1 ] n i ) 2 , A 4 e l e m s P o i = arg min A i Ω 2 [ V M ] 1 4 n = 1 16 ( [ A 1 A 1 ] n i ) 2 ,
A 8 e l e m s P o i = A 8 e l e m s G a u = 1 2 [ 1 1 1 1 0 .550 -0 .550 0 .550 -0 .550 -0 .653 0 .521 0 .653 -0 .521 -0 .521 -0 .653 0 .521 0 .653 ] , A 4 e l e m s P o i = A 4 e l e m s G a u = 1 2 [ 1 1 1 1 0 .443 0 .443 -0 .443 -0 .443 0 .732 -0 .732 0 .732 -0 .732 0 .518 -0 .518 -0 .518 0 .518 ] .
Eight elements : Var [ M ] = [ V M ] 1 4 [ 1.00 3.31 3.31 10.94 2.87 2.87 9.48 9.48 2.87 9.48 2.87 9.48 8.22 8.22 8.22 8.22 ] , Four elements : Var [ M ] = [ V M ] 1 4 [ 1.00 5 .10 5 .10 25 .99 1 .87 3 .73 9 .51 19 .03 1 .87 9 .51 3 .73 19 .03 3 .48 6 .96 6 .96 13 .93 ] .

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