1. Introduction

Polarimetric measurements may in some practical situations lead to depolarizing Mueller matrices. In a previous work we studied the depolarization appearing in the experimental Mueller matrix of a sample consisting of a diffraction grating, where measurements were done by positioning the beam on the boundary between the grating and the silicon substrate [1]. In practice, depolarization can appear in measurements of samples with lateral dimensions of tens to hundreds of micrometers [2, 3] when is difficult to position the beam so that it does not illuminate area around sample. Nevertheless, even if the measured Mueller matrix has a certain degree of depolarization, it contains enough information to allow the recovery of one of the Mueller matrices corresponding to the homogeneous region contributing to the total optical response of the sample.

The possibility of the retrieval of a non-depolarizing component from a depolarizing Mueller matrix has been pointed out theoretically in the framework of the theory of Mueller matrix algebra introduced by Gil and Correas [4, 5]. Subsequent application of this approach allows subtracting theoretically up to three known non-depolarizing components from measured matrix and getting the last unknown component. Limits are only given by the internal structure of individual Mueller matrices and the presence of noise or systematic errors in experimental data. Our method is based on the same theoretical background as the decomposition of an arbitrary Mueller matrix into the linear combination of four non-depolarizing matrices published by Cloude [6], though there is infinite number of such matrix foursomes without direct connection to the physical matrices.

The purpose of this paper is to show a practical method to retrieve a non-depolarizing component from an experimental depolarizing Mueller matrix provided that all the other non-depolarizing components are known. In addition we present in this paper the application of the method to two examples representative of common practical situations that experimentalists encounter in their work. Practical situation suitable for the application of our method happens when the beam spot is larger then the sample and simultaneous illumination of the sample and its surroundings is inevitable. The method allows the separation of the pure sample component from the measured Mueller matrix, which is a mixture of the optical responses of the sample and its surroundings, provided that the Mueller matrix of the surroundings is known, e.g. from independent measurement outside the sample area. Section 2 of this article summarizes briefly the theoretical background and describes our method of subtraction. In next section the experimental conditions (sample and optical instrumentation) used to measure the Mueller matrix are described. In Section 4 we present the two examples of subtraction. The first example shows the simplest subtraction procedure consisting of the retrieval a non-depolarizing matrix from a depolarizing matrix containing two components assuming that the second component is known. The second example shows the implementation of a generalized version of the method to obtain one non-depolarizing Mueller matrix from a depolarizing matrix constituted of three non-depolarizing components. In this case two other components are considered to be known.

2. Theory

Polarized light can be described by four components Stokes vectors [7] in the form:

where 〈x〉 represents time averaging over the measurement time and E_x and E_y denote the orthogonal field components perpendicular to the direction of propagation. The polarimetric response of any sample can be fully described by 4×4 Mueller matrix formalism [7] describing the relation between incident and emerging Stokes vectors:

In practice, polarimeters usually measure normalized Mueller matrices, where all the elements are divided by the element M ₁₁. The necessary and sufficient condition for a physically realizable Mueller matrix to represent a non-depolarizing optical system, introduced by Gil and Bernabeu [8], can be written as

From the arguments given in [8] it follows that a Mueller matrix M resulting from the linear combination of two (or more) non-depolarizing linearly independent Mueller matrices is depolarizing. The analysis of depolarization of such matrices is usually simpler when they are rewritten in terms of coherency matrices [6, 9]. Any Mueller matrix M can be directly related to the corresponding coherency matrix H using relation [5]:

Henceforth we denote the linear transformation of a Mueller matrix M to the corresponding coherency matrix H by the linear operator 𝓗 such that H=𝓗(M). The rank of the coherency matrix H, i.e. the number of non-zero eigenvalues, determines the number of non-depolarizing components contained in measured Mueller matrix M. Moreover, for the physically realizable matrices all eigenvalues have to be non-negative since H is hermitian positive semi-definite matrix. We have developed our method for the retrieval of non-depolarizing components from measured matrices in the framework of coherency matrices. In the following, we will show how the method works applying it to the simplest case, i.e. the retrieval of a component from a depolarizing Mueller matrix produced by the sum of two non-depolarizing components provided that one of the components is known. After that we will show how to generalize the method to treat more complex cases.

A normalized depolarizing Mueller matrix M′ made of the sum of two normalized non-depolarizing matrix components M ⁽¹⁾ and M ⁽²⁾ can be written in form

where the coefficient p denotes the relative proportion of the two non-depolarizing matrices. In this conditions the coherency matrix H′=𝓗(M′) associated to M′ has two non-zero eigenvalues (rank(H′)=2) if the matrices M ⁽¹⁾ and M ⁽²⁾ are linearly independent. Our method is based on the assumption that one of the non-depolarizing matrices of Eq. (5), M ⁽¹⁾ for instance, is known. Under this condition it is possible to find unique real number α such that [5]

The value of the parameter α can be obtained either using an analytical approach with the expression

based on the condition in Eq. (3), or using a numerical approach. The numerical approach is based on a least square minimization procedure that searches the value of parameter α satisfying expression in Eq. (6) by minimizing the values of three eigenvalues associated to 𝓗 (M′-αM ⁽¹⁾) which are supposed to be zero. Once the value of the parameter α is known, the normalized Mueller matrix M ⁽²⁾ can be written in terms of M′ and M ⁽¹⁾ as

As a result, the method allows the extraction of the unknown non-depolarizing component M ⁽²⁾ from the original depolarizing matrix M′ by subtracting the right proportion of the known component M ⁽¹⁾.

This method can be further generalized to subtract a non-depolarizing component from depolarizing Mueller matrices composed of up to four non-depolarizing components. The procedure is very similar to the simpler case. The depolarizing matrix constituted by the contribution of four non-depolarizing matrices can be written as

We are interested in the retrieval of one non-depolarizing component, which is supposed to be unknown, from depolarizing matrix M′ under the condition that the rest of non-depolarizing matrices are known. To illustrate the procedure we will take the fourth component M ⁽⁴⁾ in Eq. (9) to be unknown. The first step is to find the value of parameter α, which reduces the rank of the matrix 𝓗(M′-αM ⁽¹⁾) by one. Once the parameter α is known, the same algorithm is repeated to find the value of parameter β that reduces rank of the matrix 𝓗(M′-αM ⁽¹⁾-βM ⁽²⁾). Last step is to obtain the value of parameter γ by further decreasing rank of the matrix 𝓗(M′-αM ⁽¹⁾-βM ⁽²⁾-γM ⁽³⁾). The resulting matrix M′-αM ⁽¹⁾-βM ⁽²⁾-γM ⁽³⁾ is after normalization equal to the unknown matrix M ⁽⁴⁾. The procedure for the special case of the rank three depolarizing matrix is similar since we can put r=0 and reduce one step.

Necessary and sufficient condition to apply this generalized method is that the rank of the matrix M′ is equal to the number of non-depolarizing components. In the simplest case of two components this condition is equal with linear independence of the matrices. On the other hand, the linear independence is not sufficient condition to obtain rank three or rank four matrices. In following we will illustrate necessary condition for matrix of rank higher than two, which is that at most two of the non-depolarizing components can be block diagonal, the rest of the components has to have non-zero block off-diagonal elements. This condition is of relevance for practical purposes because block-diagonal matrices of isotropic samples are frequently encountered in experimental applications.

Let us consider n>2 different non-depolarizing normalized block-diagonal Mueller matrices, which can be written in the form

where ψ and Δ are the ellipsometric angles [7]. Linear combination of n matrices from Eq. (10) gives Mueller matrix M′ with the following relations between its non-zero elements:

The transformation of the Mueller matrix M′ into the related coherency matrix and use of the relations in Eq. (11) lead to the following result:

It is immediately obvious that matrix 𝓗(M′) is of the rank two at most, because only two rows in matrix have non-zero elements. This shows that any linear combination of two or more Mueller matrices of the form in Eq. (10) leads to a coherency matrix of rank two at most, therefore only two non-depolarizing components can be in principle distinguished.

3. Experimental configuration

Figure 1(a) shows a picture of the sample that we used to measure depolarizing Mueller matrices. The sample is a silicon wafer coated with a photoresist whose surface has been structured by etching a given pattern by photolithography. Each pattern contains an ensemble of 49 square boxes. Each box is either uniformly etched to the silicon substrate or it is an one-dimensional photoresist diffraction grating with a well defined pitch. We have already used this sample in some of our previous studies, therefore more detailed information concerning the etching procedure as well as the structure and the optical properties of some of the gratings can be found elsewhere [1]. Figure 1(b) shows a portion of the surface of the wafer with few boxes of the size of 3×3 mm. The shape of the probe beam used in our experiment was elliptic because of oblique incidence (see details below). The major and minor diameters of the ellipse have sizes of approximatively 3 mm and 1.5 mm, which means that it fitted perfectly inside the rotated boxes. The yellow ellipses in Fig. 1(b) represent schematically the size and the shape of the beam spot. When the spot size covered only a single box we obtained a non-depolarizing Mueller matrix, however if the spot was placed in between two or more boxes we measured different optical responses at the same time and as a consequence the resulting Mueller matrix was depolarizing.

In order to illustrate our method we have carried out two experiments. For the first one we placed the spot between the substrate and a grating of 600 nm pitch that we will call “grating 1”. The ellipse in Fig. 1(b) labeled EC1 schematically shows the position of the beam. The matrix obtained in this way is depolarizing and has two components, one from the substrate and one from grating 1. In addition to this measurement, we have also measured the single substrate and the single grating 1 by placing the entire beam spot inside the corresponding boxes of the substrate and the grating 1, respectively. For the second experiment we placed the beam spot between the grating 1 and a second grating, with a pitch of 500 nm, that we will further call “grating 2”. The ellipse in Fig. 1(b) labeled EC2 schematically shows the position of the beam to measure the depolarizing matrix. It is worth to note that between the gratings 1 and 2 there is a uniformly etched space of the substrate. Therefore, the depolarizing matrix obtained in this way has three contribution: grating 1, grating 2 and the silicon substrate. Apart from this measurement we have additionally measured the single Mueller matrix of the grating 2 by positioning the entire beam spot inside the corresponding box.

 

Fig. 1. (a) Left image: photography of the patterned wafer. (b) Right image: detailed image of the grating 1 and grating 2 boxes used for the experiments. The yellow ellipses represent the positions of the beam spot during measurements of the depolarizing Mueller matrices in our experiments labeled as EC1 and EC2, respectively.

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Polarimetric measurements were performed with a commercial spectroscopic Mueller matrix polarimeter (MM16 from HORIBA Jobin-Yvon), operating in the visible range (450–850 nm with a spectral step of 1.5 nm). More detailed description of this polarimeter including its operation and calibration method can be found in [10]. All measurements were done in reflection with an angle of incidence of 60°. We chose this angle to have a projected spot size on the sample surface small enough to perfectly fit inside a single box. Gratings 1 and 2 were measured in a conical configurations, where the lines of the grating are neither parallel nor perpendicular to the plane of incidence and corresponding Mueller matrices are not block-diagonal. The angle between the grating lines and the plane of incidence was for both gratings equal to 45 degrees. This configuration maximized the values of the off-diagonal elements of the matrices and the differences between the optical responses of the gratings.

4. Results and discussions

In this section we present the results of the two experiments that we conducted to illustrate the application of our method of retrieval of non-depolarizing matrix component.

4.1. Retrieval of one of two non-depolarizing components from a depolarizing Mueller matrix

The goal of this experiment was to extract one non-depolarizing component from a depolarizing Mueller matrix obtained by placing the beam spot between the grating 1 and the silicon substrate. This matrix, here called M′, is shown in Fig. 2 together with the non-depolarizing matrix of the substrate (M ⁽¹⁾).

 

Fig. 2. Spectral dependence of the measured normalized depolarizing Mueller matrix M′ (solid red line) and the normalized non-depolarizing Mueller matrix of the substrate M ⁽¹⁾ (dashed blue line). Different boxes arranged in 4×4 array correspond to each from sixteen elements of the normalized Mueller matrices.

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According to our method we have decided to extract the matrix corresponding to the grating 1 assuming that the matrix of the substrate is known. We have thus calculated the coherency matrices corresponding to M′ and M ⁽¹⁾. Figure 3 shows the spectral dependence of the eigenvalues of the coherency matrix corresponding to M′. This matrix has two non-zero eigenvalues, which proves that the rank of this matrix is equal to two as expected. It is worth to note that there is a narrow spectral region around 525 nm in Fig. 3 where the values of one of the non-null eigenvalues become very small. This is not without some consequences in terms of the accuracy of the subtraction.

The next step was to retrieve the value of the parameter α allowing the extraction of the unknown matrix (expressions (6)–(8)). We applied both, the analytical and the numerical approaches to evaluate the value of the parameter α and plotted comparison between values in Fig. 4. The spectroscopic values of the parameter α almost coincide for both approaches for wavelengths between 600 and 750 nm. Conversely, for the wavelengths below 600 nm and over 750 nm the oscillations are observed in the values of α obtained from the analytical approach, while a relatively smooth behavior holds for values from the numerical approach. An explanation for the oscillatory behavior of α lies in the higher sensitivity of the analytical approach to the experimental errors which is enhanced by the small values of the second non-zero eigenvalue shown in Fig. 3. Mathematically, very small value of one of the non-zero eigenvalues makes the numerator and the denominator in Eq. (7) tend to zero bringing instability into the evaluation of α explaining the oscillations. The oscillations observed for wavelengths over 750 nm are due to the higher experimental errors present in near-infrared part of measured spectrum shown in Fig. 2. In contrast, the values of α evaluated with the numerical approach show more stable spectroscopic behavior, which means that this approach is more robust than the analytic approach.

 

Fig. 3. Spectral dependence of the eigenvalues of the coherency matrix 𝓗(M′).

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Fig. 4. Numerically obtained (solid red line) and algebraically calculated (dashed blue line) values of the parameter α depending on the wavelength of an incident light.

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Beyond the fact that one of the approaches appears to be more robust than the other, both of them have limits to evaluate the value of α when one of the non-zero eigenvalues is very close to zero. In our case this situation arises because the optical behavior of M′ is very similar to the one of the non-depolarizing matrices leading to an ill-conditioned problem to find value of α. In the limiting case when M′ became non-depolarizing, the assumptions of Correas et al. [5] leading to an unique value of α, which are the basis of our subtraction method, would no longer be valid. In other words, the value of the parameter α would be indefinite. In condition of ill-conditioned problem, the presence of the experimental errors in the matrices M′ and M ⁽¹⁾ makes the determination of α inaccurate. In our case we had to apply a Savitzky-Golay filter [11] to the measured matrices to decrease influence of the noise in the experimental data and to make the evaluation of α for all the wavelengths below 600 nm possible. We can conclude that use of the analytical formula in Eq. (7) should be done with caution as it may provide less accurate results than numerical approach in the presence of experimental errors. For this reason we prefer to use the numerical approach to perform matrix subtraction.

The spectroscopic values of α obtained analytically and numerically were used to subtract the Mueller matrix of the grating 1. The resulting matrices are plotted in Fig. 5 and compared with the experimental matrix of the grating 1 measured alone. The overall correspondence between measured and numerically reconstructed data is very good apart from some slight differences around wavelength of 700 nm in the element M ₄₄. Using analytically calculated coefficient α leads to the significant differences between the measured and reconstructed data in spectral regions, where the coefficient itself oscillates (see Fig. 4). For a more detailed inspection of the quality of the result we present in Fig. 6 the difference between the reconstructed and directly measured Mueller matrices. Relatively high differences for the longer wavelengths (close to 850 nm) are caused by a depolarization in the measured matrices coming from the finite spectral resolution of the monochromator of the polarimeter. This effect averages Mueller matrices over the wavelengths included in narrow region of few nanometers corresponding to the spectral resolution of the monochromator. The average has a washing-out effect on sharp spectral features and introduces additional depolarization into the measured Mueller matrices as we discussed in a previous article [1]. Due to the grating’s Rayleigh anomaly at wavelength around 850 nm and discussed peak washing-out effect, certain depolarization is observed in this spectral region. Because we did not consider this additional source of depolarization, the subtraction method based on Eq. (6) is no longer exact when depolarization caused by finite resolution is noticeable and its application leads to lack of accuracy in the subtracted matrix. The rest of the spectrum in Fig. 6 shows differences smaller than 0.03. Figures 5 and 6 confirm that decomposition of the depolarizing Mueller matrices is possible even when noise is present in the measured matrices.

 

Fig. 5. Spectral values of the analytically (dashed red line) and numerically (dash-dotted blue line) retrieved Mueller matrices compared with the directly measured matrix corresponding to grating 1 (solid black line).

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4.2. Retrieval of the third non-depolarizing component from a depolarizing Mueller matrix

The goal of this experiment was to retrieve one non-depolarizing component from a depolarizing matrix consisting of the combination of three different components. To measure corresponding matrix we positioned the beam between the grating 1 and grating 2 as it is demonstrated in Fig. 1 by ellipse labeled EC2. To implement our method we suppose that the non-depolarizing matrices of the substrate and grating 1 are known. Moreover, we have measured the gratings in conical diffraction in order to ensure that their matrices are not block diagonal, which is one of the necessary conditions for application of the generalized subtraction method. Figure 7 shows the spectroscopic Mueller matrices of the substrate alone, grating 1 and grating 2 alone as well as the depolarizing matrix corresponding to the “mixture” of the substrate, grating 1 and grating 2.

 

Fig. 6. Difference between the numerically retrieved and the experimental non-depolarizing Mueller matrices corresponding to grating 1 (both are plotted in Fig. 5).

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Fig. 7. Measured normalized Mueller matrices of substrate, grating 1, grating 2, and their “mixture”. Sixteen boxes correspond to different elements of Mueller matrices.

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As in the previous experiment, the experimental depolarizing Mueller matrix was converted into the corresponding coherency matrix and we calculated the eigenvalues represented in Fig. 8. There are three non-zero eigenvalues confirming that the rank of the coherency matrix associated to the depolarizing matrix equals three as expected. A very important issue concerns the contrast between the eigenvalues that is the ratio between the smallest non-zero and zero eigenvalues. First and second eigenvalues are relatively high, with spectral values around 0.8 and 0.2 respectively, and therefore they are well defined. However, the values of the third eigenvalue are considerably smaller (less than 0.05) and in a small spectral region around the wavelength of 600 nm they can be considered as zero.

 

Fig. 8. Eigenvalues of the coherency matrix obtained from measured Mueller matrix on the boundary between the three sample regions. Spectral values of all four eigenvalues are plotted in the left plot with a detailed view of the three smallest eigenvalues on the right.

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We applied the generalized method detailed in Section 2 to the case of two known and one unknown non-depolarizing components mixed together. The values of the parameters α and β plotted in Fig. 9 were determined numerically for each wavelength. The parameters α and β are related to the relative contribution of the substrate and the grating 1, respectively. The spectral dependence of the parameters is relatively smooth except for a narrow band between 550 and 650 nm.

 

Fig. 9. Spectral dependence of the resulting values of the parameters α (solid red line) and β (dashed blue line).

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Once the values of α and β were known, we obtained the unknown normalized non-depolarizing matrix corresponding to grating 2 as presented before in Section 2. The accuracy of the subtraction procedure is illustrated by comparison of the retrieved Mueller matrix with the directly measured Mueller matrix in Fig. 10. Both matrices are almost identical for whole spectral range except for the mentioned narrow band between 550 and 650 nm. As previously pointed out, the origin of those differences, i.e. the lack of accuracy of the method in this spectral region, is due to the small difference existing between the smallest non-zero and the zero eigenvalue of the matrix 𝓗(M′) (see Fig. 8). The third eigenvalue is very small in the critical spectral region due to the weak anisotropic optical response of the structures resulting in small values of block off-diagonal elements of the grating Mueller matrices.

The anisotropic behavior of the optical response is revealed by the element M ₂₂ in Fig. 7. The value of M ₂₂ equals one for a sample with zero block off-diagonal elements in corresponding Mueller matrix. Element M ₂₂ in Fig. 7 shows value very close to one for all Mueller matrices entering the subtraction procedure (substrate, grating 1 and grating 2) in the region between 550 and 650 nm. This situation is very similar to the pathological case of three matrices without non-zero block off-diagonal elements, discussed in Section 2, that invalidates the generalized subtraction procedure. In such case, it is not possible to distinguish accurately the three independent non-depolarizing components. As a conclusion we can say that the generalized subtraction procedure can be implemented satisfactorily except for the case where all matrices are almost block-diagonal. For the pathological case, however, the closer the matrices are to block-diagonal, the more inaccurate the decomposition becomes.

 

Fig. 10. Spectral dependence of the retrieved Mueller matrix (solid red line) compared with the directly measured matrix corresponding to grating 2 (dashed blue line).

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5. Conclusions

This work demonstrates the feasibility of the retrieval of an unknown non-depolarizing component of a depolarizing Mueller matrix, under the condition that all other non-depolarizing components are known. The robustness and the weaknesses of the method are illustrated in two examples concerning depolarizing Mueller matrices containing two and three components, respectively. In the case of a depolarizing matrix constituted of two components, the subtraction method can be successfully applied, provided that the two components are different enough to be considered as linearly independent in numerical terms. We tested two different approaches to implement our method: the analytical and the numerical one. Despite of the fact that both approaches should be equivalent when working with ideal matrices, the numerical approach ap-peared to provide more accurate results when it was applied to experimental Mueller matrices affected by random noise and small systematic errors. It was also shown that the subtraction procedure can be carried out successfully with depolarizing matrices having three components if at least one of the non-depolarizing components is not block-diagonal.