Unevenly spaced continuous measurement approach for dual rotating&#x2013;retarder Mueller matrix ellipsometry

Kai Meng; Bo Jiang; Christos D. Samolis; Mohamad Alrished; Kamal Youcef-Toumi

doi:10.1364/OE.27.014736

1. Introduction

The Dual Rotating-Retarder Mueller Matrix Ellipsometer (DRR-MME), one of the widely used MME ushered by Azzam [1] and Hauge [2], has been recognized as a powerful instrument to measure geometric dimensions as well as physical properties of various nanoscale thin-film structures [3–6]. Due to its prominent merits including high speed, non-invasiveness and ease of operation, it is an attractive alternative for inline metrology and process control [7–10].

A typical DRR-MME, as shown in Fig. 1, consists of a polarizer, the first retarder (retarder 1), a sample, a second retarder (retarder 2) and an analyzer. The polarizer and first retarder sequence form a Polarization State Generator (PSG), whereas the second retarder and analyzer constitute a Polarization State Analyzer (PSA). The retarders are rotated to get modulated light intensity curves, from which the sample Mueller matrix (SMM) can be extracted.

Fig. 1 Schematic of a typical DRR-MME.

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Data reduction technique plays an important role in extracting the full 16 elements of the SMM from the detected intensity curve in an efficient way. Two data reduction techniques, the Fourier analysis approach and the measurement matrix approach, are commonly suggested [11,12]. The fundamental idea behind both approaches is to formulate a determined or an overdetermined system of equations to solve the 16 unknowns of the SMM. The former has been widely accepted in the DRR-MME. However, it requires more sampling points to obtain the Fourier coefficients, which is then used to solve for the SMM at a given rotation speed ratio, compared to the approach based on measurement matrix algebra. The latter can directly identify the minimum number of equations by using 16 sampling points, which makes it suitable for high-speed and large range metrology applications (e.g. imaging ellipsometry, where each pixel corresponds to a SMM calculation). There are also two ways to implement the measurement matrix approach. One approach is identifying four polarization states in both the PSG and PSA to reach the well-conditioned measurement matrix. The primary limitation of this method lies in the discontinuous retarder control. For each PSG state, four discrete measurements need to be performed via fixing the first retarder while rotating the second one. The other approach is to perform a continuous measurement, in which the two retarders can rotate continuously by setting a specific ratio of angle increments and 16 measurements are made by evenly spaced sampling with respect to the retarder azimuthal angle positions [13]. However, this method can probably miss the optimal sampling points due to its fixed sampling step.

Therefore, the purpose of this work is to propose an unevenly spaced sampling strategy for data reduction to achieve a globally optimal measurement matrix in continuous measurements. To identify the uneven sampling points that can have the best system robustness to both instrument errors and detection noise, there is a strong need to build an appropriate optimization model. Numerous studies have contributed to the optimization of MME and a similar instrument, namely the Stokes vector polarimeter, by minimizing the condition number or Equally Weighted Variance (EWV) [14–28]. Several studies considered expected systematic error by using covariance-based model [29–31]. Modulation efficiency criteria to assess the noise propagated into each element of the measured Stokes vector or SMM was also employed to optimize the Signal-to-Noise Ratio (SNR) [32–35]. Nevertheless, most of the previous research focused on the discrete measurement scenarios rather than continuous rotating measurement. Smith first proposed the evenly spaced sampling strategy for continuous measurement and provided the optimal ratio of retarder rotation increment and the recommended retardance by using CN optimization [13]. Twietmeyer and Chipman further utilized the criteria of CN and the singular value decomposition to identify the optimal retarder rotation increment in an even sampling measurement [36]. However, no existing work is found to provide optimization models and approach as well as specific schemes for unevenly sampled continuous measurement scenarios.

In this paper, we develop multi-objective robust optimization models to achieve optimal system robustness. A combined global search algorithm is also designed based on the Multi-Objective Genetic Algorithm (MOGA) to identify the best sampling points. Optimal unevenly spaced measurement schemes are provided for different design scenarios. Significant improvements of error robustness are found by comparing unevenly spaced measurements with evenly spaced measurements. Simulations and experiments demonstrate that our proposed model and approach can be used as a powerful tool to assist designers in achieving a more flexible optimized design of DRR-MME.

The remainder of this paper is organized as follows. We first describe our formulation in Section 2. Then, Section 3 formulates our optimization models for different scenarios. A MOGA-based search approach is presented in Section 4. In Section 5, we conduct a series of simulations and experiments to validate the proposed approach. Finally, major contributions and future work are summarized in Section 6.

2. Problem description and mathematical formulation

In this section, we review the formulation of the measurement matrix approach and then propose the unevenly spaced measurement problem. The nomenclature used to describe the problem is listed in Table 1.

Table 1. Nomenclature

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At one measurement, the output light intensity is formulated as Eq. (1):

I_{i} = V_{o} A_{i} M_{0} G_{i} S_{i n} = P_{i} M

where

M \in R^{16 \times 1}

is obtained by flattening the sample Mueller matrix

M_{0}

in column,

P_{i}

is the Kronecker product, denoted as

P_{i} = {(G_{i} S_{i n})}^{T} \otimes (V_{o} A_{i})

.

Suppose there are n measurements that are implemented and there are no errors in the instrument system and detection process, the ideal intensity column vector $I_{0}$ is easily given as $I_{0} = P_{0} M$ , and thus the SMM can be obtained by the matrix calculation:

M = P_{0}^{- 1} I_{0}

It should be noted that at least 16 independent measurements are needed to calculate the SMM since there are 16 unknowns in the Mueller matrix. If n >16, then it is an overdetermined system and pseudo inversion can be used $P_{0}^{+} = {(P_{0}^{T} P_{0}^{})}^{- 1} P_{0}^{T}$ . In this work, the unified notation $P_{0}^{- 1}$ is used as we mainly focus on the scenario n = 16.

Theoretically, unbiased SMMs can be obtained by estimated $\hat{P}$ and real measured intensity $I_{r}$ if the calibration is perfect ( $\hat{P} = P_{r}$ ) and the estimated $Δ I_{r}$ is unbiased. However, it is impossible to know the exact state of the instrument under imperfect calibration due to the residual error ( $δ_{P} = P_{r} - \hat{P} = P_{r} - P_{0} - Δ \hat{P}$ ) caused by random noise and imperfect components. Therefore, assuming that the unbiased estimation of detection noise ( $Δ {\hat{I}}_{r} = Δ I_{r}$ ) could be obtained, the real estimation of the SMM is calculated by:

\hat{M} = {\hat{P}}^{- 1} I_{r} = {\hat{P}}^{- 1} (P_{r} M + Δ {\hat{I}}_{r}) = {\hat{P}}^{- 1} [(\hat{P} + δ_{P}) M + Δ {\hat{I}}_{r}]

Furthermore, we can derive the final measurement error of the SMM:

Δ M = {\hat{P}}^{- 1} δ_{P} M + {\hat{P}}^{- 1} Δ {\hat{I}}_{r}

Equation (4) is a generalized expression of the measurement error of the sample Mueller matrix. The first term represents the system calibration error while the second term denotes the random detection errors. In addition, it is straightforward to find that: If $δ_{P} = 0$ , then Eq. (4) represents perfect calibration scenarios. If $δ_{P} = Δ P_{0} = P_{r} - P_{0} \neq 0$ , that represents the scenario where no calibration is performed. If $δ_{P} \in (0, Δ P_{0})$ , imperfect calibration should be considered.

Therefore, the purpose of this study is to identify the global optimal sampling points of retarder azimuth angles ( $x_{1}, x_{2} \dots x_{n}$ , $n \geq 16$ ) within one sampling period to minimize the measurement errors (Eq. (4)) while considering both calibration errors and detection noise at the design phase. Here, the sampling period is equal or less than the period of output intensity changes due to the DRR motion. One complete modulation period can be examined to include all the potential combinations of two-retarder positions. Also, note that the two retarders are continuously rotating with a specified speed ratio $ω$ . The azimuth angle of retarder 2 can be calculated from that of retarder 1. Thus, only the sampling points of retarder 1 are considered as variables in this study. In addition, a flexible sampling period can be determined by the speed ratio. Considering the availability of design sources, we aim to develop our model and approach to be flexible for any rotating control ratio and any design scenarios.

3. Optimization model

In this study, the ultimate goal of our optimization model is to minimize the measurement error in the sample Mueller matrix. According to Eq. (4), the final measurement error includes two parts: instrumental errors ${\hat{P}}^{- 1} δ_{P} M$ and light intensity detection errors ${\hat{P}}^{- 1} Δ {\hat{I}}_{r}$ . In this work, we separately examine the effects of these two error sources.

First, we assume that there are no system errors under perfect calibration so that only detection errors are considered. Second, we take calibration errors into account to improve system robustness to both instrument errors and detection errors. In this case, the calibration is imperfect. However, during the design phase, it is impractical to know either the exact estimated perturbation $Δ \hat{P}$ or the specific estimated measurement matrix $\hat{P}$ because no real calibration information can be known before putting the specific design into practice. Instead, we consider the extreme no-calibration scenario, to optimize for the worst case.

3.1. Optimization under perfect calibration

In the scenario of perfect calibration, the optimization objective is to minimize the measurement error by improving the SNR. We can use the vector/matrix norm to describe the magnitude of total error ( $Δ M$ in Eq. (4), a column vector with 16 elements) and derive an upper bound on the relative error:

\frac{| | Δ M | |}{| | M | |} \leq κ (\hat{P}) (\frac{| | δ_{P} | |}{| | P_{0} | |} + \frac{| | Δ {\hat{I}}_{r} | |}{| | I_{0} | |})

The proof of Eq. (5) is given in Appendix A. where

κ (\hat{P})

is the condition number of the estimated measurement matrix, which has the definition:

κ (\hat{P}) = | | \hat{P} | |_{2} | | {\hat{P}}^{- 1} | |_{2}

Here, we choose the L₂-norm as the measure, which is defined as the ratio of the maximum to the minimum of matrix singular values $| | X | |_{2} = μ_{\max} / μ_{\min}$ .

It can be easily seen from Eq. (5) that the upper bound of relative error depends on the condition number of the estimated measurement matrix, relative instrument errors and relative detection errors. If prior information about the errors is unknown, one can minimize the condition number to improve the SNR. The condition number optimization has been widely used for the optimized design of MME. In practice, minimizing Eq. (6) not only reduces the error propagation but also equalizes the errors over the 16 elements.

During the design phase, instead, the ideal measurement matrix $P_{0}$ is commonly used in the optimization process. Considering the azimuth angles of the first retarder as unknown variables, $P_{0}$ is a function of angle sampling points: $x_{1}, x_{2} \dots x_{n}$ , $n \geq 16$ , thus, the first optimization criterion is given by:

O_{1} = \min_{} κ (P_{0} (x_{1}, x_{2} \dots x_{n}))

However, as discussed by some previous studies, the condition number can only imply the global error performance and is essentially insensitive to any system-level perturbation that can scale all the singular values [17]. Accordingly, there is a need to assess the error in each element of the sample Mueller matrix. Supposing that $P_{0}^{- 1} = Q \in R^{16 \times n}$ and all the sources of detection noise are independently and identically distributed with variance $σ_{I k}^{2}$ , the variance of measurement error in the i^th matrix element can be estimated according to the standard error propagation equation:

Δ M_{i}^{2} = {(\sum_{k} Q_{i k})}^{2} σ_{I k}^{2}

Therefore, to minimize the square of the Mueller matrix estimation error, $Δ M_{i}^{2}$ given by Eq. (8), under different errors for all the matrix elements, we formulate the following objective function:

O_{2} = \min \sqrt{\sum_{i} {(\sum_{k} Q_{i k} (x_{1}, x_{2} \dots x_{n}))}^{2}}

3.2. Optimization under no calibration

In the scenario of no calibration, no estimation can be obtained on the measurement matrix error, which means $Δ \hat{P} = 0$ and $\hat{P} = P_{0}$ . Further, Eq. (4) can be rewritten as Eq. (10) for the no-calibration scenario:

Δ M = Q Δ P_{0} M + Q Δ {\hat{I}}_{r}

To estimate $Δ P_{0}$ in the design phase, Taylor-series expansion is suggested when the perturbation is small and prior error information is available. In this work, we only consider the azimuthal angle errors of the two retarders:

Δ P_{0,}_{i j} \approx {\frac{\partial P_{0, i j}}{\partial θ_{1}} |}_{θ_{1, i}} Δ θ_{1, i} + {\frac{\partial P_{0, i j}}{\partial θ_{2}} |}_{θ_{2, i}} Δ θ_{2, i} = f_{i j} Δ θ_{1, i} + g_{i j} Δ θ_{2, i}

where

Δ P_{0,}_{i j}

is the error estimation for the (i, j)^th element of measurement matrix,

θ_{1}

and

θ_{2}

are respectively the azimuthal angle of retarder 1 and 2, and

Δ θ_{1, i}

and

Δ θ_{2, i}

are their errors in the i^th measurement (i^th sampling point). To simplify the formulations,

f_{i j}

and

g_{i j}

are used here and below to represent the first-order derivative of

P_{0, i j}

with respect to

θ_{1}

and

θ_{2}

.

Here, assuming that $Δ {\hat{I}}_{r} = 0$ and the angle errors of the retarders in different measurements are independent random variables that follow an identical probabilistic distribution: $\forall k \neq t, E (Δ θ_{k} \cdot Δ θ_{t}) = 0$ and $E (Δ θ_{k} \cdot Δ θ_{k}) = σ^{2}$ , the final expectation of the i^th measurement is given in Eq. (12) (proof is given in Appendix B), which agrees with the results of covariance analysis in [29].

E ({(Δ M_{i})}^{2}) = σ^{2} \sum_{k} Q_{i, k}^{2} [{(\sum_{j} M_{j} f_{k j})}^{2} + {(\sum_{j} M_{j} g_{k j})}^{2}]

To assess the total measurement error, we choose the F-norm measure:

E (| | Δ M | |_{_{F}}^{2}) = \sum_{i} σ^{2} \sum_{k} Q_{i, k}^{2} [{(\sum_{j} M_{j} f_{k j})}^{2} + {(\sum_{j} M_{j} g_{k j})}^{2}]

It can be seen from the Eq. (13) above that measurement error is related to the real values of SMM elements. Here, we consider the extreme case of all $M_{j} > 0$ and minimize $E (| | Δ M | |_{_{F}}^{2})$ by using the following criterion:

O_{3} = \min \sum_{i} \sum_{k} Q_{i, k}^{2} [(\sum_{j} f_{k j}^{2}) + (\sum_{j} g_{_{k j}}^{2})]

Finally, incorporating the detection noise optimization:

O_{4} = O_{2} = \min \sqrt{\sum_{i} {(\sum_{k} Q_{i k})}^{2}}

3.3. Multi-objective optimization

Based on the optimization criteria above, we can obtain the multi-objective optimization model:

M o d e l 1 : \min {O_{1} = κ (P_{0}), O_{2} = {(\sum_{i} {(\sum_{k} Q_{i k})}^{2})}^{1 / 2}}

M o d e l 2 : \min {O_{3} = \sum_{i} \sum_{k} Q_{i, k}^{2} (\sum_{j} f_{k j}^{2}) + (\sum_{j} g_{k j}^{2})), O_{4} = {(\sum_{i} {(\sum_{k} Q_{i k})}^{2})}^{1 / 2}}

s . t . \det | P_{0} | \neq 0

0 \leq x_{1} < x_{2} \dots < x_{n} \leq T

where T is the maximum azimuth angle within one sampling period, Model 1 applies to the perfect calibration scenario and Model 2 applies to the no-calibration scenario, constraint (18) ensures that all the measurement matrices should be full rank while constraint (19) means there are no repetitive sampling points in one sampling period.

In order to obtain a near-global optimum of our problem as well as to examine the performance of DRR-MME on different criteria, we search for the Pareto optimal solutions, also referred to as the Pareto front, instead of converting the multi-objective into single objective by weighted sum aggregation. Assuming two feasible solutions $ξ_{i}$ and $ξ_{j}$ , if, $O_{t} (ξ_{i}) \leq O_{t} (ξ_{j})$ for any criterion t and there exists at least one criterion that makes $O_{u} (ξ_{i}) < O_{u} (ξ_{j})$ , $ξ_{i}$ is said to dominate $ξ_{j}$ . The Pareto front can be defined as a solution set in the solution space. Any solution in this set cannot dominate any other solution in this set. Moreover, no other solution in the solution space can dominate any solution in this set.

To identify the best solution from all the alternatives in the Pareto front, we introduce another criterion to further assess the stability of the solution performance. The stability evaluation function is defined by the approximate value of the gradient of the objective function, as shown in Eq. (20). Subsequently, a suggested practical solution S* can be elected by Eq. (21).

S (O_{t}) = \nabla O_{t} (x_{1}, x_{2} \dots x_{n}) \approx \sum_{k = 1}^{n} [O_{t} (x_{k} + Δ x_{k}) - O_{t} (x_{k})]

S^{*} = \underset{i}{\arg \min} {η_{i} = \sum_{t} ω_{t} (O_{t} + S (O_{t}))}

where

η_{i}

is the assessment index for the individual solution

ξ_{i}

in the Pareto front,

ω_{t}

is subjective weight for different criteria,

ω_{t} > 0

and

\sum_{t} ω_{t} = 1

, encapsulating the fact that no preference is placed among different criteria.

4. Optimization approach

In this work, a multi-objective genetic algorithm is developed to search for the approximate optimal solution. The proposed model is a typical non-linear multi-variable continuous optimization problem. Since the objective function involves high-dimension matrix inversion and singular value decomposition, a traditional optimization approach is not straightforward and easy to implement. A meta-heuristic global optimization method has been proven to be effective in solving such problems and has recently been used in the optimized design of MMEs [21,37–39].

To improve the performance of global search and stable convergence, we design a rough-fine combined search strategy. In the rough search stage, we consider the problem as an integer combinatorial optimization problem and design a genetic algorithm to solve it. Integer coding is used to represent each retarder azimuth angle. Heuristic initialization is also suggested to speed up the search process. Then, the output Pareto front of rough search is incorporated into the fine local search as part of the initial input. MATLAB’s function “gamultiobj” with real number coding is utilized to implement a fine search. The specific procedures of our proposed approach are described in Table 2.

Table 2. Combined search strategy based on multi-objective genetic algorithm

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5. Simulation and experiment

In order to validate our proposed model and approach as well as identify the near optimal uneven sampling scheme for typical design scenarios, simulations and experiments are conducted in this section. We also compare the performance and the robustness of our uneven measurements with even measurements.

5.1. Test design

Aiming at finding the best 16 sampling points within one sampling period, two rotating frequency ratios of retarders, ω₁ = 5:3 and ω₂ = 34:26, along with two choices of retardance R₁ = 90° and R₂ = 127°, are examined in our experiments, where ω₁ and R₁ (one quarter wave plate) are widely used in the automated DRR-MME [5,6], while ω₂ and R₂ are respectively the ratio and retardance recommended by the existing research [13]. In practice, 132°-retarder is also suggested [27]. Here, we arbitrarily choose the former one to generate different scenarios for our tests.

As for the selection of polarizer/analyzer angles, P₁/A₁ = 0°/0° and P₂/A₂ = 90°/90° are considered. Theoretically, these two scenarios should have identical performances. Here, we compare their optimal results to validate the consistency and robustness of our algorithm. Furthermore, the condition number O₁ is used as an optimization metric to search for the best polarizer/analyzer angles for the evenly spaced sampling scheme. Evenly sampled spaces of retarder 1 are respectively chosen to be 30° and 34° for ω₁ and ω₂. Some results of logarithmic values of O₁ are shown in Fig. 2, where bright yellow lines indicate that the measurement matrix is singular or ill-posed when choosing these value combinations for polarizer/analyzer.

Fig. 2 Condition number (log(O₁)) distribution under different polarizer/analyzer combinations.

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The results show that optimal areas of the P/A pair are relatively insensitive to the retardance changes but highly sensitive to the speed ratio. Therefore, we arbitrarily choose the optimal solution of R₁ = 90° for our subsequent tests. Here, P₃/A₃ = 39°/141° for ω₁ scenarios and P₃/A₃ = 178°/171° for ω₂ scenarios. Then, our test-bed is designed as listed in Table 3, including 12 combination scenarios S1-S12.

Table 3. Test-bed design

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A sample set used in the simulations is shown in Table 4, including the free-space air (identity matrix) and three anisotropic Mueller matrices.

Table 4. Sample set

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5.2. Simulation

The proposed combined search algorithm is applied for both the perfect calibration scenario (Model 1) and no-calibration scenario (Model 2). Here, the sampling number is set at n = 16, the sampling length is set as 180ω/(ω-1), the population size Npop is set as 20n, the maximum iteration number for rough search is set at MaxGen = 200n, the crossover rate r_c and mutation rate r_m are empirically set to be 0.8 and 0.2. Running extensive experiments demonstrates that our algorithm performs more stably and effectively than direct real-number search. Typical Pareto fronts of rough search and fine search are illustrated in Fig. 3, which shows good spread of Pareto optimal solutions.

Fig. 3 Pareto fronts of Model1-S1 obtained by (a) rough search and (b) fine local search

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Fig. 4 Optimal retarder positions for a perfect calibration scenario (Model 1 S1-S3, free space)

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Fig. 5 Measurement errors of each element in the SMM under perfect calibration scenarios

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The results of Model 1 and Model 2 are listed in Tables 9 and 10 in Appendices C and D, respectively. There are several observations from these results:

(1) The optimal sampling points are always not exactly the evenly spaced points, which proves the necessity of an uneven sampling strategy. However, these optimal positions are still relatively uniformly distributed over the intensity modulation period. Figure 4 visualizes the 16 optimal positions of retarder 1 for model 1 S1-S3. It can be found that S1 and S2 have almost identical results, which indicates a good stability of our algorithm. A similar observation can also be seen in all the other scenarios for both Model 1 and Model 2. Therefore, in the following analysis, Scenarios S2, S5, S8 and S11 are no longer considered.
(2) The optimal angles of Polarizer/Analyzer for evenly spaced sampling also works well for unevenly spaced sampling. Although they are no longer strictly superior to other P/A combinations (e.g. S9), further search experiments show that, compared to the combinations considered in our test-bed, no significant performance improvement can be found by changing the P/A;
(3) Rotation increment ratio ω and retardance R indeed have significant effects on the error performance, which agrees with those fundamental findings from previous studies on polarimeters. Table 5
Table 5. Comparison on detection noise propagation
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summarizes the reduction rate r1-r4 of objective value O1-O4 for Model 1 and Model 2 by comparing (S1, S4), (S7, S10), (S1, S7) and (S4, S10) to examine the impacts of retardance and speed ratio changes. One can see that ω optimization results in a larger error reduction than retardance optimization.

To further validate unevenly spaced sampling measurement schemes, we compare the obtained optimal solutions to evenly spaced sampling schemes as follows.

(1) Perfect calibration scenario: random intensity noise is assumed to follow the normal distribution N (0, σ²). Here, we set the standard deviation to be 2% of the peak value of detected intensity, σ = 0.01. Retarder angle increment ratios are supposed to be 25:15 and 34:26 for evenly spaced sampling. The comparison results are presented in Table 6
Table 6. Comparison on detection noise propagation
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, where 1000 random cases are conducted, O1min and O2min are the minimum values of objective O1 and O2 found in the Pareto fronts. RMS1 and RMS2 are separately the Root-Mean-Squares (RMS) of the SMM total error in unevenly and evenly spaced measurement, calculated by $R M S = \sqrt{\sum_{i = 1}^{N} | | Δ M | |_{F}^{2} / N}$ , where N = 1000, $| | Δ M | |_{F}^{2} = \sum_{j} Δ M_{j}^{2}$ . r is the improvement rate of RMS1 compared to RMS2. Apparently, unevenly spaced measurement schemes can significantly reduce the specimen-independent measurement error caused by intensity noise. We further examine the mean squared error of each element in the SMM as shown in Fig. 5, where notations 1-16 denote 16 elements in the flattened SMM. It can be seen that unevenly spaced measurement schemes are superior in error immunity to evenly spaced measurement schemes over all the 16 elements.

(2) No-calibration scenario: Here, we only focus on the instrumental errors and set the detection noise to be 0. The retarder position error (°) of each sampling point is assumed to follow the normal distribution N (0, 0.1²). Table 7
Table 7. Comparisons on instrumental errors propagation of different samples
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gives the RMS comparison of 1000 random measurements on all the four samples, where RMS has the same definition as above.

All these numerical experiments demonstrate that the unevenly spaced measurement schemes can achieve better robustness to not only detection noise but also instrumental errors than evenly spaced sampling. The fundamental mechanisms behind its superiority are both mathematical and physical. For the detection noise, matrix algebra used to calculate the SMM (Eq. (2)) makes it possible to identify a measurement matrix that can mathematically reduce the noise impact. Apparently, such a matrix is not always the evenly spaced measurement matrix. On the other side, for the retarder position error, the perturbations taking place at different positions have different impacts on the modulated signal as the detected light intensity is a function of the rotating angle. The evenly spaced measurement can easily be trapped into the error-sensitive positions, whereas the uneven measurement tries to avoid such positions. Therefore, the optimal measurement matrix is a competitive or tradeoff solution of these two mechanisms.

Finally, we conduct extended optimizations on overdetermined unevenly sampled measurement systems for S12, in which more than 16 points are chosen to form the measurement matrix. Here, we consider the overdetermined system with sampling number n = 30. Tolerance analysis is conducted on both intensity detection errors and retarder position errors as illustrated in Fig. 6. It is straightforward that overdetermined sampling can improve the robustness. However, we need to make a tradeoff between the data processing efficiency and acceptable measurement accuracy. Tolerance analysis can better assist in such design tasks.

Fig. 6 Tolerance comparison of 16 and 30 measurements with optimal uneven sampling (sample 4)

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5.3. Experimental results

A fundamental DRR-MME is set up to further demonstrate our approach. As shown in Fig. 7, the setup consists of a mixed-laser source (402.6nm, 530.5nm and 630.6nm), a PSG/PSA (Edmund Optics, α-BBO Rochon Polarizer; Thorlabs, rotating retarder components, AQWP05M-600 and KPRM1E), a spectrometer (Ocean Optics, maya2000-pro) and several collimating lenses.

Fig. 7 Experimental setup

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We conduct the experiments with free space (Sample A) and an imperfect waveplate (Sample B). The P/A angles are set at 0°/0°. The worst cases under no calibration are considered. The results are given in Table 8, where the retardance of the achromatic retarder is provided by the vendor. The SMM error is the F-norm of the error matrix. We compare our approach (eM1) to the even measurement approach (eM2). The experimental results prove that the unevenly spaced sampling performs better in error reduction under the no-calibration scenarios. In practice, Eigenvalue Calibration Method (ECM) and noise filtering can be used to further reduce the errors and to ensure the precision/accuracy of the results [4].

Table 8. Experimental results

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In this experiment, a spectrometer is used to collect the modulated signals. Then, only the optimal sampling points are picked up for subsequent data processing and all the other points are dropped off to save the storage burden. The optimal points can be identified by running our algorithm for any given design scenario. It should be noted that our approach can be incorporated into the instrument design for automatic measurement scenarios. One can develop a customized sampling software with flexible sampling interval for any given PSA. The optimal points obtained by our approach can be automatically collected or flexibly adjusted to enable a customized sampling configuration.

6. Conclusion

In this paper, we consider the robust optimization problem of the unevenly spaced sampling configuration for DRR-MME. This is a new attempt to implement Mueller matrix ellipsometry with minimum data requirements by combining error modeling and smart computing techniques. The proposed approach can assist the designer in better understanding the error propagation performance of the system under different scenarios.

The major contributions of our work are summarized as follows: 1) An unevenly spaced sampling strategy is proposed to extract the unknown SMM while maintaining the ease of continuous retarder control. Compared to the discrete measurement and evenly spaced sampling technique, our proposed scheme cannot only reduce the error propagation but also save on the data processing consumption. 2) We build multi-objective optimization models to assess the robustness of different unevenly spaced measurement schemes under different design availability. Both system errors and detection noise are considered. Moreover, our proposed model can be easily extended into the optimization of a variable retarder MME by redefining the physical meanings of the sampling points and the variables of error sources in Eq. (11). 3) A combined global search algorithm is designed as an efficient and simple tool to achieve the near optimal design. 4) We also examine a series of design scenarios to validate our approach and to provide practical insights to the DRR-MME design. The results of our simulations and experiments demonstrate that the optimal configurations are usually not the evenly spaced sampling schemes.

There are several interesting directions for the future research. First, one can model the uncertainty of calibration to incorporate imperfect error estimation into our optimization models. Second, we plan to extend our models to globally optimize the combinations of different retarders while considering unequal retardances for the PSG and the PSA. In addition, more quantitative error analysis and experiments need to be conduct to further improve our approach by comparing it with the Fourier analysis methods.

Appendices

Appendix A: proof of Eq. (5)

According to Eq. (4) $Δ M = {\hat{P}}^{- 1} δ_{P} M + {\hat{P}}^{- 1} Δ {\hat{I}}_{r}$ , the norm of the error matrix is:

| | Δ M | | \leq | | {\hat{P}}^{- 1} δ_{P} M + {\hat{P}}^{- 1} Δ {\hat{I}}_{r} | | \leq | | {\hat{P}}^{- 1} | | | | δ_{P} | | | | M | | + | | {\hat{P}}^{- 1} | | | | Δ {\hat{I}}_{r} | |

Considering

I_{0} = P_{0} M

,

| | I_{0} | | \leq | | P_{0} | | | | M | |

, thus

\frac{1}{| | M | |} \leq \frac{| | P_{0} | |}{| | I_{0} | |}

. Then we can obtain:

\frac{| | Δ M | |}{| | M | |} \leq | | {\hat{P}}^{- 1} | | | | δ_{P} | | | | M | | \cdot \frac{1}{| | M | |} + | | {\hat{P}}^{- 1} | | | | Δ {\hat{I}}_{r} | | \frac{| | P_{0} | |}{| | I_{0} | |}

Given the definition of the condition number

κ (\hat{P}) = | | \hat{P} | | | | {\hat{P}}^{- 1} | |

, the relative error can be written as:

\frac{| | Δ M | |}{| | M | |} \leq κ (\hat{P}) (\frac{| | δ_{P} | |}{| | \hat{P} | |} + \frac{| | Δ {\hat{I}}_{r} | |}{| | I_{0} | |} \frac{| | P_{0} | |}{| | \hat{P} | |}) \approx κ (\hat{P}) (\frac{| | δ_{P} | |}{| | \hat{P} | |} + \frac{| | Δ {\hat{I}}_{r} | |}{| | I_{0} | |})

The approximate equation is strictly valid when

\frac{| | P_{0} | |}{| | \hat{P} | |} < 1

and also holds when

Δ \hat{P}

is very small to make

\frac{| | P_{0} | |}{| | \hat{P} | |} \approx 1

Appendix B: proof of Eq. (12)

Supposing $Δ \hat{I} = 0$ , the measurement error of the i^th matrix element can be calculated from:

Δ M_{i} = \sum_{j} \sum_{k} Q_{i k} (f_{k j} Δ θ_{1, k} + g_{k j} Δ θ_{2, k}) M_{j} = Δ M_{i}^{f} + Δ M_{i}^{g}

It shows that the errors caused by rotating retarder 1 and retarder 2 have a similar form of contribution to the final error. We further expanding $Δ M_{i}^{f}$ to be the sum of the retarder angle error terms in all the measurements.

Δ M_{i}^{f} = Q_{i 1} \sum_{j} M_{j} f_{1 j} Δ θ_{1, 1} + Q_{i 2} \sum_{j} M_{j} f_{2, j} Δ θ_{1, 2} + \dots Q_{i, 16} \sum_{j} M_{j} f_{16, j} Δ θ_{1, 16}

Then, assuming that angle errors of retarders in different measurements are independent random variables and identically distributed: $\forall k \neq t, E (Δ θ_{1, k} \cdot Δ θ_{1, t}) = 0$ and $E (Δ θ_{1, k} \cdot Δ θ_{1, k}) = σ_{1}^{2}$ , then the expectation of the variance of $Δ M_{i}^{f}$ can be calculated as follows:

E ({(Δ M_{i}^{f})}^{2}) = σ_{1}^{2} \sum_{k} Q_{i, k}^{2} {(\sum_{j} M_{j} f_{k j})}^{2}

Similarly, we can obtain the variance of $Δ M_{i}^{g}$ based upon the same assumption on the error covariance $σ_{2}^{2}$ of retarder 2, and thus, $E ({(Δ M_{i}^{g})}^{2}) = σ_{2}^{2} \sum_{k} Q_{i, k}^{2} {(\sum_{j} M_{j} g_{k j})}^{2}$ .Further assuming that $σ_{1}^{2} = σ_{2}^{2} = σ^{2}$ , the final error expectation is given by:

E ({(Δ M_{i})}^{2}) = σ^{2} \sum_{k} Q_{i, k}^{2} [{(\sum_{j} M_{j} f_{k j})}^{2} + {(\sum_{j} M_{j} g_{k j})}^{2}]

Appendix C

Table 9. Results of Model 1 tests obtained by combined search strategy

View Table | View all tables in this article

Appendix D

Table 10. Results of Model 2 tests obtained by combined search strategy

View Table | View all tables in this article

Funding

Wuxi Friedrich Measurement and Control Instruments Co. Ltd, China.

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Notation	Description
_$ω$	Rotating speed ratio of retarder 1 with respect to retarder 2
_{$S_{i n}$}	Stokes vector of the incident light, here considered to be [1 0 0 0]^T
_{$G_{i}$}	Mueller matrix of the polarization state generator in the ith measurement, $G_{i} \in R^{4 \times 4}$
_{$A_{i}$}	Mueller matrix of the polarization state analyzer in the ith measurement, $A_{i} \in R^{4 \times 4}$
_{$V_{o}$}	A row vector for extracting the first component of the output Stokes vector, [1 0 0 0]
$M_{0}$	Real Mueller matrix of the sample, $M_{0} \in R^{4 \times 4}$
_{$I_{i}$}	Output light intensity in the ith measurement
$P_{i}$	Measurement matrix of the ith measurement, $P_{i} \in R^{1 \times 16}$
_{$I_{0}$}	Ideal output light intensity with no system errors in n measurements, $I_{0} = {I_{0,}_{i}}_{n \times 1}$
$P_{0}$	Ideal measurement matrix for n measurements, $P_{0} = {P_{0,}_{i}}_{n \times 1} \in R^{n \times 16}$
$I_{r}$	Real measured light intensity, $I_{r} \in R^{n \times 1}$
$P_{r}$	Real measurement matrix, $P_{r} \in R^{n \times 16}$
$Δ I_{r}$	Real error of the output light intensity, $Δ I_{r} \in R^{n \times 1}$
$Δ P_{0}$	Real error of the measurement matrix, $Δ P_{0} \in R^{n \times 16}$
$\hat{P}$	Estimated real measurement matrix, $\hat{P} \in R^{n \times 16}$
$Δ \hat{P}$	Estimated measurement matrix error, $Δ \hat{P} \in R^{n \times 16}$
_{$Δ {\hat{I}}_{r}$}	Estimated error of the real measured light intensity, $Δ {\hat{I}}_{r} \in R^{n \times 1}$
_{$δ_{P}$}	Residual error of measurement matrix estimation, $δ_{P} \in R^{n \times 16}$
_$\hat{M}$	Estimated sample Mueller matrix using measurement data, $\hat{M} \in R^{16 \times 1}$
_{$Δ M$}	Estimation error of the sample Mueller matrix, $Δ M \in R^{16 \times 1}$

Steps	Description
1	Initialization for integer optimization
1.1	Parameter setting: population size Npop, maximum iteration number MaxGen1, crossover rate r_c, Mutation rate r_m, sampling length T, sampling number n;
1.2	Generate random initial population
	$P o p (0) = [p o p_{1}, p o p_{2}, p o p_{3}] = {ξ_{i} = {[x_{1}, \dots x_{k} \dots x_{n}]}_{i} \| x_{k} \in Ν, x_{1} < x_{2} < \dots x_{n}, i = 1, 2 \dots N_{p o p}}$
	where $x_{k} \neq x_{j} i f j \neq k$ , and: · $\forall ξ \in p o p_{1}$ , $x_{k} \in [(k - 1) T / n, k T / n)$ ; · $\forall ξ \in p o p_{2}$ , $i f^{} k \leq 0.5 n$ , then $x_{k} \in [0, 0.5 T)$ , else $x_{k} \in [0.5 T, T)$ ; · $\forall ξ \in p o p_{3}$ , $x_{k} \in [0, T]$ .
1.3	Calculate the fitness for each individual in Pop(0) using objective functions (Eqs. (16-17))
1.4	Perform non-dominated sorting on Pop and get the initial Pareto Optimal Front POF(0);
2	Evolutionary operation while ( $g e n \leq M a x G e n 1$ ), do:
2.1	Generate crossover population Popc: while size(popc)<0.5 r_c Npop, do:
	· Randomly select two individuals $ξ_{1}$ , $ξ_{2}$ from Pop (gen-1) by tournament mechanism, · $γ = r a n d (1)$ , $ξ_{1}^{'} = ξ_{1} \cdot γ + ξ_{2} (1 - γ)$ , $ξ_{2}^{'} = ξ_{2} \cdot γ + ξ_{1} (1 - γ)$ , · Calculate the fitness for $ξ_{1}^{'}$ and $ξ_{2}^{'}$ ;
2.2	Generate mutation population Popm:
	while size (popc)< r_m Npop, do:
	· Randomly select one individual $ξ$ from Pop(gen-1) by tournament mechanism,
	· Randomly select $x_{k}$ in $ξ$ and $y_{l} \in C_{U} {x_{1} \dots x_{n}}, U = {0, 1, \dots T}$ , set $x_{k} = y_{l}$ , · Calculate the fitness for $ξ_{}^{'}$ ;
3	Re-generate population
3.1	PopTotal = [ Pop(gen-1), Popc, Popm];
3.2	Perform non-dominated sorting on Pop and get the Pareto Optimal Front POF(gen);
3.3	Sort PopTotal in ascending order, and Pop(gen) = PopTotal(1: Npop)
4	Fine local search
4.1	Initialization for real number optimization:
	PopNew(0) = [ POF(MaxGen), PopGap], where Size(PopGap) = Npop-Size(POF(MaxGen1))
	$P o p G a p = {ξ = [x_{1}, \dots x_{k} \dots x_{n}] \| x_{1} < x_{2} < \dots x_{n}, x_{k} \in [0, T], x_{k} \in R^{+}}$
4.2	while ( $g e n \leq M a x G e n 2$ ), do: multi-objective genetic evolution in real number coding mechanism
5	End and output the final POF(MaxGen2)

No.	1	2	3	4
Sample	Free-space	Quarter waveplate ( $θ = 30^{o}$ )^a	Dielectric S ^b	Trapezoidal grating ^c ( $φ = 30^{o}$ )
Mueller matrix	$[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]$	$[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 0.250 & 0.433 & - 0.866 \\ 0 & 0.433 & 0.750 & 0.500 \\ 0 & 0.866 & - 0.500 & 0.000 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.0077 & 0.0247 & 0.0108 \\ 0.0006 & 1.0023 & 0.0207 & 0.0196 \\ - 0.0259 & 0.0259 & - 1.0014 & 0.0151 \\ - 0.0115 & 0.0229 & - 0.0159 & - 0.9857 \end{matrix}]$	$[\begin{matrix} 1.00 & - 0.20 & 0.00 & - 0.13 \\ - 0.20 & 0.35 & - 0.41 & 0.81 \\ 0.00 & 0.41 & - 0.75 & 0.50 \\ - 0.13 & 0.81 & - 0.50 & 0.03 \end{matrix}]$
Note	isotropic	anisotropic	anisotropic	anisotropic

	R change		ω change
	(S1, S4)	(S7, S10)	(S1, S7)	(S4, S10)
r1	85.6%	84.8%	99.5%	99.5%
r2	53.3%	48.4%	85.8%	84.3%
r3	8.88%	10.0%	36.6%	38.8%
r4	53.2%	49.1%	85.1%	83.8%

No.	P/A	ω	R	O1min^a	O2min	RMS1	RMS2	r (%)
S1	0/0	5:3	90	4.69	174.42	1.77	8.61	79.44
S3	39/141	5:3	90	3.47	55.04	0.58	2.82	79.43
S4	0/0	5:3	127	3.84	81.51	0.86	4.09	78.97
S6	39/141	5:3	127	2.62	26.01	0.27	1.35	80.00
S7	0/0	34:26	90	2.39	24.69	0.25	0.42	40.48
S9	178/171	34:26	90	2.44	26.61	0.27	0.39	30.77
S10	0/0	34:26	127	1.57	12.73	0.13	0.22	40.91
S12	178/171	34:26	127	1.64	13.59	0.14	0.19	26.32

Unevenly spaced continuous measurement approach for dual rotating–retarder Mueller matrix ellipsometry

Abstract

1. Introduction

2. Problem description and mathematical formulation

3. Optimization model

3.1. Optimization under perfect calibration

3.2. Optimization under no calibration

3.3. Multi-objective optimization

4. Optimization approach

5. Simulation and experiment

5.1. Test design

5.2. Simulation

5.3. Experimental results

6. Conclusion

Appendices

Appendix A: proof of Eq. (5)

Appendix B: proof of Eq. (12)

Appendix C

Appendix D

Funding

References

Cited By

Figures (7)

Tables (10)

Equations (28)

Optics Express

No.	S1	S2	S3	S4	S5	S6	S7	S8	S9	S10	S11	S12
P	P₁	P₂	P₃	P₁	P₂	P₃	P₁	P₂	P₃	P₁	P₂	P₃
A	A₁	A₂	A₃	A₁	A₂	A₃	A₁	A₂	A₃	A₁	A₂	A₃
ω	ω₁	ω₁	ω₁	ω₁	ω₁	ω₁	ω₂	ω₂	ω₂	ω₂	ω₂	ω₂
R	R₁	R₁	R₁	R₂	R₂	R₂	R₁	R₁	R₁	R₂	R₂	R₂

No.		1			2			3
Wavelength		402.6 nm			530.5nm			630.6nm
Retardance (deg)		72.36			97.30			97.30
SMM error		eM1	eM2	r (%)	eM1	eM2	r (%)	eM1	eM2	r (%)
Sample A	ω = 25:15	1.7718	1.9840	10.70	0.9867	2.1645	54.41	0.9538	2.2486	63.30
Sample A	ω = 34:26	0.5507	1.2237	55.00	0.4906	0.7245	52.28	0.7113	0.9602	25.92
Sample B	ω = 25:15	0.3499	2.3904	85.36	0.8877	5.1281	82.69	0.8204	4.4320	81.49
Sample B	ω = 34:26	0.3401	0.8610	60.50	0.1890	0.3487	45.80	0.2735	0.4395	37.77

No.	O1min^a	O2min	Optimal positions of Retarder 1 for model 1 ^b
1	4.69	174.42	13.07, 38.13, 65.37, 98.16, 124.91,142.55, 180.34, 208.70, 226.84, 245.26, 322.83, 355.51, 385.40, 411.48, 436.91, 450.00
2	4.69	174.51	0.00, 13.18, 38.74, 64.67, 94.58, 128.07, 205.29, 223.80, 242.66, 269.96, 307.10, 325.51, 352.55, 384.75, 412.02, 437.03
3	3.47	55.04	4.15, 30.72, 75.74, 116.34, 129.87, 156.15, 180.04, 196.31, 233.86, 245.58, 282.81, 328.91, 361.61, 401.53, 412.24, 449.99
4	3.84	81.51	12.85, 37.85, 64.73, 98.02, 126.36, 148.12, 185.47, 210.62, 227.26, 246.95, 325.10, 355.25, 384.61, 411.61, 437.18, 450.00
5	3.85	81.59	12.83, 38.59, 65.60, 97.99, 125.01, 185.89, 209.47, 228.15, 249.03, 304.40, 324.62, 355.26, 385.46, 412.41, 437.19, 450.00
6	2.62	26.01	2.69, 70.49, 87.80, 119.90, 148.90, 175.11, 196.66, 231.58, 245.68, 280.51, 307.37, 331.79, 364.48, 398.50, 413.20, 449.99
7	2.39	24.69	41.88, 105.09, 129.55, 196.27, 289.27, 329.42, 376.38, 417.19, 443.18, 511.71, 536.64, 572.73, 602.45, 624.83, 662.48, 711.42
8	2.46	26.43	60.63, 90.89, 118.13, 177.66, 198.30, 230.72, 272.84, 316.51, 366.56, 422.45, 504.26, 564.14, 609.73, 661.43, 701.55, 749.25
9	2.44	26.61	43.95, 108.67, 143.92, 174.29, 195.43, 265.16, 296.16, 377.65, 419.88, 444.08, 486.19, 513.34, 535.25, 575.77, 601.11, 693.43
10	1.57	12.73	44.19,104.75, 138.22, 196.60, 277.67, 302.82, 327.85, 373.08, 413.53, 438.80, 517.61, 571.65, 600.10, 622.15, 664.69, 712.33
11	1.61	13.25	107.65, 151.68, 201.88, 285.75, 309.44, 372.50, 399.18, 421.56, 446.20, 491.18, 528.83, 587.07, 617.28, 689.22, 716.70, 761.69
12	1.64	13.59	33.83, 61.00, 105.29, 146.90, 197.04, 247.33, 306.98, 379.00, 403.96, 442.07, 490.58, 530.45, 582.59, 616.29, 689.04, 726.60

No.	O3min ^a	O4min	Optimal positions of Retarder 1 for model 2 ^b
1	4.73	174.32	0.00, 13.40, 38.58, 65.07, 99.06, 124.18, 175.09, 194.62, 227.88, 253.42, 300.81, 326.87, 358.07, 384.83, 411.56, 436.48
2	4.72	174.24	0.00, 13.54, 38.86, 65.59, 94.10, 126.62, 199.49, 223.42, 260.31, 280.32, 307.06, 328.75, 355.54, 383.96, 411.47, 436.56
3	3.70	55.04	2.61, 31.82, 76.48, 108.53, 126.50, 150.88, 179.81, 201.47, 231.32, 247.65, 279.73, 331.45, 362.18, 397.24, 413.49, 450.00
4	4.31	81.57	0.00, 12.83, 38.08, 64.97, 101.97, 126.08, 170.76, 189.13, 225.15, 251.04, 296.09, 325.34, 358.10, 384.83, 412.60, 437.15
5	4.31	82.23	0.00, 12.79, 38.35, 66.94, 94.89, 124.06, 203.02, 226.97, 258.12, 281.53, 301.62, 325.95, 354.98, 384.10, 412.33, 437.15
6	3.31	26.11	2.29, 30.43, 75.29, 107.84, 130.56, 155.39, 184.25, 206.02, 232.37, 248.80, 280.61, 329.43, 359.52, 388.23, 412.77, 450.00
7	3.00	25.93	0.03, 28.36, 114.49, 163.95, 183.63, 220.91, 274.78, 308.05, 418.57, 498.33, 525.80, 549.35, 610.73, 657.91, 691.76, 765.00
8	2.99	25.33	11.10, 121.63, 152.74, 213.89, 280.25, 305.33, 401.19, 434.44, 525.03, 550.73, 593.88, 621.18, 647.99, 678.77, 707.14, 765.00
9	3.05	26.99	34.99, 93.34, 116.73, 142.33, 193.01, 227.62, 258.42, 282.05, 365.82, 390.81, 413.95, 505.70, 529.11, 610.10, 664.15, 694.92
10	2.70	13.19	32.70, 104.94, 150.72, 192.42, 247.25, 279.38, 304.93, 370.43, 408.07, 436.92, 489.94, 525.51, 576.00, 619.93, 682.91, 765.00
11	2.72	13.47	118.00, 167.22, 192.28, 228.75, 291.71, 316.82 368.52, 414.80, 448.33, 506.91, 533.33, 563.84, 606.09, 662.32, 705.33, 755.24
12	2.67	12.84	32.74, 105.04, 146.38, 189.25, 249.02, 276.64, 302.29, 373.02, 403.03, 439.01, 490.09, 526.17, 577.12, 615.87, 683.60, 764.99

No.	Sample 1			Sample 2			Sample 3			Sample 4
No.	RMS1	RMS2	r (%)	RMS1	RMS2	r (%)	RMS1	RMS2	r (%)	RMS1	RMS2	r (%)
S1	0.114	0.746	84.7	0.207	0.666	68.8	0.096	0.831	88.5	0.201	0.614	67.3
S3	0.054	0.275	80.5	0.041	0.288	85.6	0.056	0.269	79.1	0.048	0.226	78.7
S4	0.065	0.512	87.4	0.112	0.506	80.0	0.100	0.611	83.6	0.115	0.506	77.2
S6	0.033	0.166	80.1	0.029	0.180	84.1	0.039	0.181	78.5	0.032	0.136	76.3
S7	0.022	0.033	32.9	0.022	0.037	39.5	0.023	0.047	51.6	0.016	0.019	19.7
S9	0.022	0.031	30.4	0.026	0.031	17.6	0.025	0.040	37.7	0.018	0.019	8.3
S10	0.016	0.028	43.7	0.014	0.023	39.9	0.016	0.033	50.3	0.013	0.019	32.5
S12	0.015	0.025	37.4	0.013	0.017	25.9	0.015	0.027	43.5	0.013	0.016	14.2

No.	Sample 1			Sample 2			Sample 3			Sample 4
No.	RMS1	RMS2	r (%)	RMS1	RMS2	r (%)	RMS1	RMS2	r (%)	RMS1	RMS2	r (%)
S1	0.114	0.746	84.7	0.207	0.666	68.8	0.096	0.831	88.5	0.201	0.614	67.3
S3	0.054	0.275	80.5	0.041	0.288	85.6	0.056	0.269	79.1	0.048	0.226	78.7
S4	0.065	0.512	87.4	0.112	0.506	80.0	0.100	0.611	83.6	0.115	0.506	77.2
S6	0.033	0.166	80.1	0.029	0.180	84.1	0.039	0.181	78.5	0.032	0.136	76.3
S7	0.022	0.033	32.9	0.022	0.037	39.5	0.023	0.047	51.6	0.016	0.019	19.7
S9	0.022	0.031	30.4	0.026	0.031	17.6	0.025	0.040	37.7	0.018	0.019	8.3
S10	0.016	0.028	43.7	0.014	0.023	39.9	0.016	0.033	50.3	0.013	0.019	32.5
S12	0.015	0.025	37.4	0.013	0.017	25.9	0.015	0.027	43.5	0.013	0.016	14.2