Experimental validation of the symmetric decomposition of Mueller matrices

Clément Fallet; Angelo Pierangelo; Razvigor Ossikovski; Antonello De Martino

doi:10.1364/OE.18.000831

1. Introduction

With the continuous development of experimental Mueller polarimetry, the physical interpretation of measured Mueller matrices is also of increasing importance. This is by no means a trivial task, as diattenuation, polarizance, mean transmittance or reflectance (directly from m₁₁) and the depolarization index [1] are the only properties which can be directly and unambiguously read from the matrix elements themselves. The other properties can be retrieved by treatments which decompose the input matrices into sums [2] or products of “simple” polarimetric elements. One widely used such treatment is the polar decomposition proposed by Lu and Chipman [3], which allows to recast any Mueller matrix M into the product

M = M (Δ) \cdot M (R) \cdot M (D)

where D is a general diattenuator, R a generic retarder and Δ a depolarizer. As discussed in [4], the other two decompositions with the depolarizer set after the diattenuator are readily deduced from Eq. (1) and form a first family of “forward decompositions”. The “reverse decomposition”, introduced by Ossikovski et al. [5], completes the family with the three decompositions with the depolarizer set before the diattenuator.

The Mueller matrix M(D) of a generic diattenuator D is of the form:

M (D) = [\begin{matrix} 1 & {\vec{D}}^{T} \\ \vec{D} & m_{D} \end{matrix}]

where the diattenuation vector

\vec{D}

and the submatrix

m_{D}

have the form defined in [3].

The retarder matrix has the form

M (R) = [\begin{matrix} 1 & {\vec{0}}^{T} \\ \vec{0} & m_{R} \end{matrix}]

where

m_{R}

is a three dimensional rotation matrix. The scalar retardance R is defined as the phase difference (or phase shift) between the two eigenmodes resulting from transmission or reflection:

R = \cos^{- 1} [\frac{1}{2} Tr M (R) - 1]

In the following we consider only linear retarders, so the coordinates of the fast axis on the Poincaré sphere are

[\begin{matrix} \cos 2 φ & \sin 2 φ & 0 \end{matrix}]

whereφ is the orientation of the fast axis of the retarder.

The Mueller matrix of a pure depolarizer expressed in its eigenvector basis is given below. This element has zero diattenuation and retardance.

M (Δ) = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & a & 0 & 0 \\ 0 & 0 & b & 0 \\ 0 & 0 & 0 & c \end{matrix}] \begin{matrix}  \end{matrix} w i t h \begin{matrix}  \end{matrix} | a |, | b |, | c | \leq 1

However, in the most general case, the Lu-Chipman decomposition involves a depolarizer with nonzero polarizance, while for the reverse decomposition this element may have nonzero diattenuation, which is somewhat at odds with respect to “real’ depolarizers.

The symmetric decomposition [6] of any input matrix M is expressed by

M = M (D_{2}) \cdot M (R_{2}) \cdot M (Δ) \cdot M {(R_{1})}^{T} \cdot M (D_{1})

where

M (D_{i})

and

M (R_{i})

are the matrices of diattenuators and retarders, while the depolarizer

M (Δ)

is actually diagonal as in Eq. (5). One should notice and be aware that Eq. (6) only holds for Stockes-diagonalizable matrices [7] whereas this symmetric decomposition theory still needs to be extended for Stokes-non-diagonalizable matrices [8] as stated in [6] and will be developed in another article yet to be published.

This is a special case of the normal form of Mueller matrices described previously in refs [7,9]

M = M_{2} \cdot M (Δ) \cdot M_{1}

where

M_{1}

and

M_{1}

are two non-depolarizing (ie. which do not reduce the degree of polarization of any totally polarized Stokes vector) Mueller matrices bracketing the diagonal depolarizer matrix

M (Δ)

.

The central position of the depolarizer in the symmetric decomposition can be very useful when trying to localize the various effects of the sample without any a priori modeling, as most depolarizers occurring in practice are diagonal indeed.

The purpose of this paper is to propose an experimental investigation of the accuracy of the symmetric decomposition on real measured Mueller matrices. The experimental setup, the samples and the measurement procedures are briefly described in section 2. Section 3 is devoted to the results of the symmetric decomposition of the measured polarimetric images, with particular attention to “degenerate” depolarizers with two equal eigenvalues. Section 4 concludes the paper.

2. Experimental

2.1 The polarimetric set-up

Our Mueller imaging polarimeter, outlined in Fig. 1 , is an upgraded version of the instrument described in [10]. The illumination part of the set-up comprises a halogen lamp (Olympus CLH-SC 150W), a fiber bundle with its output at the focus of an aspherical condenser L₁ (Newport KPA046, f ₁ = 37 mm) followed by an achromatic lens L₂ (Edmund Optics NT-32-886, f ₂ = 150mm) whose focal point F coincides with the condenser's one. This combination is both telecentric in the object and image space which allows us to neglect the angle-dependency of the following liquid crystals: each point is illuminated with the same angular aperture defined by the iris at F. This system provides a homogeneous illumination over the field of view up to 2cm diameter. A second achromatic lens (Edmund Optics NT45-353, f = 100mm) images the source on the sample.

Fig. 1 Scheme of the Mueller polarimeter.

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The Polarization State Generator (PSG) was developed according to the optimized design described in [11], with a linear polarizer (Melles Griot, 03 FPG 007) followed by two nematic liquid crystals (NLC) variable retarders (Meadowlark LVR 300). Four polarization states are sequentially generated by switching the liquid crystals between two suitably chosen retardations.

The Polarization State Analyzer (PSA), which analyzes the light emerging from the sample, is composed of the same elements as those of the PSG, but assembled in the reverse order. As a result, the PSA is also optimized with the same parameters as the PSG.

The sample was imaged on a fast CCD camera (Dalsa CAD1, 256x256 pixels, 12 bits) by means of a 12.5-75mm zoom with an additional 350mm achromatic close-up lens. The working wavelength on the camera was set to 650nm by means of an interference filter (20 nm spectral bandwidth). The dark current on the camera was measured and subtracted from the signal for each pixel. In order to increase robustness and repeatability of the measurement, four acquisitions were averaged for each polarization state. The calibration was carried out frequently by using a procedure quite similar to that described in ref [12]. With this calibration, the Mueller matrix of a mirror is the unit matrix, as for propagation in vacuum, and not a diagonal one with values (1,1,-1,-1) which is the most widely used convention.

2.2 The samples

The samples imaged in this study were combinations of different elements assembled in various fashions. We used two different diattenuators (that we will call D1 and D2), two different retarders (R1 and R2) and three different depolarizers (Δ_s, Δ_p. and Δ_t). Considering all the possible configurations of these 7 elements, we could make a huge set of measurements. However, as shown in the following section, only a few of such measurements are sufficient for the validation of the decomposition.

Diattenuators. Like Lu-Chipman or reverse algorithms, the symmetric decomposition requires partial polarizers, with scalar diattenuations D <1, to be applicable [6], thus excluding usual polarizers. The partial polarizers used in this study are schematized in Fig. 2.

Fig. 2 Diattenuators used in transmission. Left: 1 mm thick glass plate coated with a thin film of amorphous silicon (D₁). Right: pair of uncoated thick glass plates (D₂).

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The diattenuator D₁ consisted of a tilted 1 mm thick glass plate coated with a thin layer of amorphous Si on one side to increase the plate diattenuation with respect to uncoated glass. This diattenuation obviously increased with the tilt angle, which was limited to about 45° to maintain full coverage of the field of view. As a result, the scalar diattenuation D ₁ was limited to 0,15 with, however, the possibility to “spread” this value among the first two components of the diattenuation vector by rotating the tilted plate around the beam direction.

The second diattenuator, D₂, consisted of two uncoated glass plates tilted at about 45°. The scalar diattenuation value D₂ was about 0,20, slightly larger than the previous one. However, as this more bulky device could not be easily rotated about the light propagation direction, the high transmission polarization state was always in the incidence plane.

Retarders. Both retarders were commercially available zero order mica retardation plates (CVI MWPS-532-20-4) whose orientation could easily be varied with standard mounts.
Depolarizers. We used two types of depolarizers:
- • The Δs depolarizer was an aqueous suspension of latex microspheres observed in backscattering. In such systems, depolarization occurs due to incoherent superposition of many scattering paths (with both spatial and temporal integration, due to the Brownian and convective motions of the microspheres). As a consequence of axial symmetry, under uniform illumination the Mueller matrix of such a sample is of the form [10,13]
  $M_{Δ s} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & a & 0 & 0 \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & b \end{matrix}]$
- • The Δp depolarizer was an optically biaxial polytethylene terephtalate (PET) sheet operated in transmission. When inserted in the beam path close to normal incidence, such a sample features essentially no diattenuation, and wavelength and incidence dependent retardation. As a result, to obtain the matrix M _Δp the elementary Mueller matrix M ^δ of a pure retarder with retardation δ (expressed in its polarization axes)
  $M^{δ} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos δ & \sin δ \\ 0 & 0 & - \sin δ & \cos δ \end{matrix}]$

must be integrated over an interval Δδ centered on δ to account for the spectral bandwidth of the filter, and to a lesser extent, to the beam angular aperture:

M_{Δ p} = \frac{1}{Δ δ} \int_{δ - Δ δ / 2}^{δ + Δ δ / 2} M^{δ} d δ = [\begin{matrix} ¨ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & a \end{matrix}] \cdot M^{δ}

with

a = sinc (\frac{Δ δ}{2})

the central retardation δ could be modified by tilting the sheet. Thus, to get a “pure” depolarizer, the PET sheet had to be tilted so that δ vanished, an adjustment which turned out to be quite critical in practice.

Now it is clear that in spite of their practical relevance, the depolarizers Δ_s and Δ_p are “degenerate” ones, as they feature two identical eigenvalues. A “nondegenenerate” depolarizer Δ_t, with three different eigenvalues, could be obtained by piling up two degenerate Δ_s and Δ_p depolarizers

2.3 Method

Our polarimeter gives sixteen 256x256 pixels images, each image being a Mueller coefficient. Those images were then averaged on a selected area where they were very homogeneous to eventually retrieve a single 4x4 matrix. The same window was kept throughout the study.

The various samples described above were measured as shown in Fig. 3 , either in combinations, or “one by one”, by removing all the elements but the one to be characterized and the bottom metallic plate, whose Mueller matrix is essentially identity.

Fig. 3 Relative positions of the different available components. D1,D2: diattenuators described in Fig. 2 . R1, R2: mica retarders. Δp: plastic sheet depolarizer. Δs: suspension of latex microspheres (second depolarizer).

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The experimental matrices of the combinations were normalized by their m ₁₁ element and then decomposed following the symmetric decomposition algorithm described in [6], with an optional filtering procedure to make sure all decomposed matrices are physically realizable. Actually, for nondepolarizing samples even small experimental errors or noise may render the measured matrices unphysical (overpolarizing), making the decomposition algorithm inapplicable. The filtering procedure then provides the small corrections needed to remove this difficulty without significantly distorting the final results.

To avoid excessively lengthy presentations of complete Mueller matrices, in the following we will present the essential results in a condensed fashion, i.e. the diattenuation vectors $\vec{D}$ for diattenuators, the scalar retardances R and the azimuths ϕ for retarders, and the depolarization vectors $\vec{Δ}$ for the depolarizers (the $\vec{Δ}$ components are the last three diagonal elements of $M (Δ)$ ).

It is then natural to evaluate the algorithm by comparing the parameters obtained from direct measurements of single components with those provided by the decomposition of the combined measurement. The simplest estimators are thus the root mean square differences for all one-dimensional quantities. For diattenuators, the relevant estimator $δ ({\vec{D}}_{i})$ ) is defined as:

δ {\vec{D}}_{i} = \frac{1}{3} \sqrt{\sum_{k = 1}^{3} {D_{i, k}^{m} - D_{i, k}^{d e c}}^{2}}

where the superscripts m and dec respectively identify the parameters coming form direct measurements of individual components and from the decomposition. An analogous formula is used the error estimators

δ ({\vec{Δ}}_{i})

of the depolarizers. In contrast, for the two one-dimensional parameters R and φ defining the retarders, the error estimators

δ (R)

and

δ (φ)

are simply the absolute values of the differences between measured and retrieved parameter values.

3. Results

3.1 Characterization of the individual components

As mentioned above, the Mueller matrices of the single polarization components have been separately measured and make up the reference measurements.

As typical examples, we first show the experimental matrices of the degenerate depolarizers.

Table 1. Experimental matrices M^m(Δ_i) of the degenerate depolarizers measured alone.

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M ^m(Δ_s) doesn’t exhibit any significant diattenuation, and its retardance is less than of 1°. For M ^m(Δ_p) the diattenuation and polarizance are slightly larger, up to 0.025, and its retardance is close to 3° (as explained above, this residual retardance critically depends on the tilt of the plastic sheet, and is difficult to adjust below this typical value of 3°). These components can thus be considered as pure depolarizers. M ^m(Δ_p) features two negative diagonal elements. This is not of a problem for the following, as the determinant remains positive [14]. However one should be very careful when decomposing such matrices because of the sign inversion it requires for the study of the retarders.

For the nondegenerate depolarizer Δ_t, we measured:

M_{}^{m} (Δ_{t}) = [\begin{matrix} 1 & 0.010 & 0.013 & 0.0057 \\ 0.009 & 0.579 & 0.017 & 0.023 \\ 0.018 & 0.012 & - 0.472 & 0.004 \\ - 0.001 & 0.002 & - 0.000 & - 0.379 \end{matrix}]

This matrix is not the product of the other two. In fact, to obtain adequate values of the diagonal elements and avoid too a strong depolarization, for the synthesis of the nondegenerate depolarizer Δ_t the concentration of latex microspheres was decreased with respect to that used for Δ_s alone. Again, for Δ_t the off-diagonal elements are close to 0. The symmetric decomposition of this matrix provides 0.5° residual retardance and diattenuation below 0.03. Thus we will consider this element too as a pure nondegenerate depolarizer.

The measured characteristics of all individual components are summarized in Table 2 .

Table 2. Characteristics of individual polarization elements from direct measurements.

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3.2 Validation of the algorithm with a nondegenerate depolarizer

The matrix $M_{0}^{m}$ measured with all elements inserted in the optical path is given below:

M_{0}^{m} = [\begin{matrix} 1 & 0.193 & - 0.02 & 0.021 \\ 0.023 & - 0.312 & - 0.023 & 0.026 \\ 0.018 & 0.008 & - 0.410 & 0.183 \\ 0.042 & 0.122 & 0.204 & 0.422 \end{matrix}]

The rms distances between the results of the symmetric decomposition of

M_{0}^{m}

and the values listed in Table 2 are given in Table 3 . All these results are compatible with an experimental rms error of the order of 0.02 in the individual elements of the raw measured matrices.

Table 3. Rms distances between the directly measured parameters of the individual components listed in Table 2 and the results of the symmetric decomposition of $M_{0}^{m}$ .

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Comparison with other decompositions

To assess the improvement brought by the symmetric decomposition with respect to the previously known algorithms, we also computed the Lu-Chipman and reverse decompositions of $M_{0}^{m}$ . The LC decomposition is expected to provide reasonable approximations of D₁ and possibly R₁ elements whereas D₂ and R₂ should appear in the reverse decomposition. However, we point out that the depolarizers retrieved by these decompositions are not “pure”: the Lu-Chipman depolarizer has nonzero polarizance and the reverse one has nonzero diattenuation.

The same error estimators as above are used for these decompositions. The diagonal coefficients of the depolarizers found by the two decompositions being very close, only the depolarizer given by the Lu-Chipman decomposition is shown.

Table 4. Error estimators for the polar (LC) and reverse (Rev) decompositions with the nondegenerate depolarizer

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In this case the diattenuators are pretty well recovered; while gross errors are observed for both the retarders and the depolarizer. The good evaluation of the diattenuators is certainly due to the good “decoupling” of diattenuation and polarizance due to the significant depolarization introduced by the compound nondegenerate depolarizer: due to this decoupling the diattenuation of the whole system is then essentially that of D₁, which is retrieved by the polar decomposition, while the overall polarizance is very close to the diattenuation of D₂, which is obtained via the reverse algorithm. Of course, with weaker depolarizations this “decoupling” would not be as effective, and the diattenuators would be less accurately retrieved by these two procedures.

3.2 Evaluation of the algorithm with degenerate depolarizers

Degenerate depolarizers, with two equal eigenvalues, may seem very special cases from a mathematical point of view, and were not explicitly considered in the general theory presented in ref [6]. However, in practice such depolarizers naturally occur in many instances like in the two examples Δ_s and Δ_p presented above. Nondegenerate depolarizers are actually more difficult to experimentally realize than degenerate ones. Unfortunately, with degenerate depolarizers the symmetric decomposition algorithm cannot unambiguously retrieve the retarders.

As explained in detail in Ref. [6], the first step of the symmetric decomposition determines the input and output diattenuators D₁ and D₂ as the unique eigenvectors of M^TGMG and MGM^TG matrices (where M is the input matrix and G = diag(1,-1,-1,-1) representing physically realizable Stokes vectors.. This step is independent of the nature of the depolarizer and is thus quite robust.

Next, a matrix M’ with zero diattenuation and zero polarizance is readily obtained as

M' = {M (D_{2})}^{- 1} \cdot M \cdot {M (D_{1})}^{- 1}

The last step merely consists of decomposing M’ by the well known Singular Value Decomposition (SVD):

M' = U \cdot Σ \cdot V^{†}

where the unitary matrices U and V are nothing else but the retrieved M ^dec(R₂) and M ^dec(R₁) retarders and the diagonal matrix Σ is that of the depolarizer. If all the eigenvalues of Σ are different, U and V are uniquely determined. But this is no longer true if Σ is degenerate. For example, if Σ is of the form given by Eq. (8), corresponding to then any retarder matrix of the form

R_{α} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \cos α & \sin α & 0 \\ 0 & - \sin α & \cos α & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

commutes with D, implying that

M' = U \cdot Σ \cdot V^{†} = U \cdot R_{α} \cdot Σ \cdot R_{- α} \cdot V^{†} = (U \cdot R_{α}) \cdot Σ \cdot {(V \cdot R_{α})}^{†}

and U and V can be replaced by U R _α and V R _α without any change in the final result. Of course the same holds for degenerate depolarizers of the type defined by Eq. (10), with rotation matrices of the form defined by Eq. (9).

It is important to realize that this is not a shortcoming of the proposed algorithm, but an intrinsic ambiguity of the decomposition when the depolarizer is degenerate. Only the product $U \cdot V^{†}$ is then unambiguously defined. As a result, the decomposition recovers its uniqueness only with additional assumptions about the retarders. We tested the algorithm with Δ_p and Δ_s degenerate depolarizers and only one retarder in the optical path, placed either before or after the depolarizer. We thus measured and decomposed the four $M_{i}^{m}$ matrices corresponding to the sequences of elements listed in Table 5 . The experimental $M {}_{i}^{m}$ are listed in Table 6 .

Table 5. Experimental configurations with a single retarder and a degenerate depolarizer.

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Table 6. Experimental matrices measured in the configurations listed in Table 5.

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The error estimators listed in Table 7 show that all elements are accurately recovered with degenerate depolarizers provided only one retarder is present and its position, before or after the depolarizer, is known to be properly taken into account.

Table 7. Error estimators for symmetric decomposition of the data listed in Table 6.

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Comparison with the other decompositions

The polar and reverse decompositions were also applied to the matrices $M_{i}^{m}$ . The retarders found in both decompositions are very close so only the one given by the Lu-Chipman decomposition is studied. The error estimators listed below indicate again that the Lu-Chipman and reverse decompositions provide results with reasonable accuracy if the depolarizing power of Δ is sufficiently high, which is the case for Δ_s (matrix $M_{4}^{m}$ ) but less so for Δ_p (matrix $M_{1}^{m}$ ).

Table 8. Error estimators for the Lu-Chipman and reverse decompositions of the data listed in Table 6.

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4. Conclusions

The product decomposition of depolarizing Mueller matrices named ‘symmetric’ has been experimentally validated on various arrangements of pure polarization components. The possibility to fully recover the five factors of the decomposition in the general (non-degenerate) case has been exposed; the decomposition errors are then close to experimental errors. We have then exposed two special cases where the depolarizer has one multiple singular value (degenerate case): we have demonstrated that only the product of the two retarders could be recovered. In both cases, we also have shown that the symmetric decomposition has smaller errors than Lu-Chipman or reverse decompositions when decomposing combinations of polarization elements where the depolarizer is set in the middle. We believe that this decomposition could be very valuable because the depolarizer is given in its diagonal form; especially in the biomedical field where light undergoes surface effects (diattenuation) and volume effects (depolarization): the symmetric decomposition should be particularly well adapted to the interpretation of polarimetric images of isotropic tissues where diattenuation and retardation are likely to occur at the interfaces, while scattering in the bulk is expected to provide diagonal depolarization.

Acknowledgements

Financial support from European Union and French Agence Nationale pour la Recherche via NANOCHARM and RNTS-POLARIMETRIE contracts is gratefully acknowledged.

References and links

1. J. J. Gil and E. Bernabeu, “Depolarization and polarization indexes of an optical-system,” Opt. Acta (Lond.) 33, 185–189 (1986).

2. S. R. Cloude, “Physical Realisability of Matrix Operators in Polarimetry,” Proc. SPIE 1166, 177–185 (1989).

3. S.-Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13(5), 1106–1113 (1996). [CrossRef]

4. J. Morio and F. Goudail, “Influence of the order of diattenuator, retarder, and polarizer in polar decomposition of Mueller matrices,” Opt. Lett. 29(19), 2234–2236 (2004). [CrossRef] [PubMed]

5. R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. 32(6), 689–691 (2007). [CrossRef] [PubMed]

6. R. Ossikovski, “Analysis of depolarizing Mueller matrices through a symmetric decomposition,” J. Opt. Soc. Am. A 26(5), 1109–1118 (2009). [CrossRef]

7. R. Sridhar and R. Simon, “Normal form for Mueller matrices in polarization optics,” J. Mod. Opt. 41(10), 1903–1915 (1994). [CrossRef]

8. A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics I. Identifying a Mueller matrix from its N matrix,” J. Mod. Opt. 45, 955–987 (1998).

9. Z. Xing, “On the deterministic and non-deterministic Mueller matrices,” J. Mod. Opt. 39(3), 461–484 (1992). [CrossRef]

10. M. Anastasiadou, S. Ben Hatit, R. Ossikovski, S. Guyot, and A. De Martino, “Experimental validation of the reverse polar decomposition of depolarizing Mueller matrices,” J. Europ. Opt. Soc. Rap. Public 2, 07018 (2007). [CrossRef]

11. B. Laude-Boulesteix, A. De Martino, B. Drevillon, and L. Schwartz, “Mueller polarimetric imaging system with liquid crystals,” Appl. Opt. 43, 2224–2832 (2004). [CrossRef]

12. E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and mueller-matrix ellipsometers,” Appl. Opt. 38(16), 3490–3502 (1999). [CrossRef]

13. C. Brosseau, Polarized Light: A Statistical Optics Approach, (Wiley, 1998), Chap 4.1 and 4.4.

14. R. Ossikovski, M. Anastasiadou, and A. De Martino, “Product decompositions of depolarizing Mueller matrices with negative determinants,” Opt. Commun. 281(9), 2406–2410 (2008). [CrossRef]

$M_{1}^{m} = [\begin{matrix} 1 & 0.022 & 0.134 & 0.226 \\ 0.223 & - 0.261 & 0.025 & 0.910 \\ - 0.089 & - 0.058 & - 0.744 & - 0.010 \\ 0.060 & 0.664 & - 0.050 & 0.215 \end{matrix}]$	$M_{2}^{m} = [\begin{matrix} 1 & 0.203 & 0.065 & 0.168 \\ 0.254 & 0.501 & - 0.252 & 0.620 \\ - 0.042 & 0.272 & - 0.579 & - 0.394 \\ 0.098 & 0.701 & 0.362 & - 0.192 \end{matrix}]$
$M_{3}^{m} = [\begin{matrix} 1 & 0.073 & 0.104 & 0.105 \\ 0.220 & - 0.078 & 0.004 & 0.420 \\ 0.038 & 0.010 & 0.409 & 0.011 \\ - 0.026 & - 0.263 & 0.008 & - 0.071 \end{matrix}]$	$M_{4}^{m} = [\begin{matrix} 1 & 0.136 & 0.140 & - 0.058 \\ 0.258 & 0.193 & 0.166 & - 0.271 \\ 0.044 & 0.162 & 0.351 & 0.155 \\ 0.013 & 0.354 & - 0.196 & 0.047 \end{matrix}]$

	$δ ({\vec{D}}_{1})$	$δ ({\vec{D}}_{2})$	$δ (R_{1})$	$δ (R_{2})$	$δ (\vec{Δ})$
$M_{1}^{m}$	0.020	0.020	3,66°	0.34°	0.016
$M_{2}^{m}$	0.023	0.024	1.63 °	3.31 °	0.019
$M_{3}^{m}$	0.003	0.005	0.26°	1.26°	0.011
$M_{4}^{m}$	0.002	0.005	7.5 °	1.25 °	0.005

	$δ ({\vec{D}}_{1} / LC)$	$δ ({\vec{D}}_{2} / Rev)$	$δ (R_{1} / LC)$	$δ (R_{2} / Rev)$	$δ (\vec{Δ} /LC)$
$M_{1}^{m}$	0.080	0.030	3.16°	0.098°	0.02
$M_{4}^{m}$	0.035	0.020	0.0012°	1.19°	0.005

$M_{1}^{m} = [\begin{matrix} 1 & 0.022 & 0.134 & 0.226 \\ 0.223 & - 0.261 & 0.025 & 0.910 \\ - 0.089 & - 0.058 & - 0.744 & - 0.010 \\ 0.060 & 0.664 & - 0.050 & 0.215 \end{matrix}]$	$M_{2}^{m} = [\begin{matrix} 1 & 0.203 & 0.065 & 0.168 \\ 0.254 & 0.501 & - 0.252 & 0.620 \\ - 0.042 & 0.272 & - 0.579 & - 0.394 \\ 0.098 & 0.701 & 0.362 & - 0.192 \end{matrix}]$
$M_{3}^{m} = [\begin{matrix} 1 & 0.073 & 0.104 & 0.105 \\ 0.220 & - 0.078 & 0.004 & 0.420 \\ 0.038 & 0.010 & 0.409 & 0.011 \\ - 0.026 & - 0.263 & 0.008 & - 0.071 \end{matrix}]$	$M_{4}^{m} = [\begin{matrix} 1 & 0.136 & 0.140 & - 0.058 \\ 0.258 & 0.193 & 0.166 & - 0.271 \\ 0.044 & 0.162 & 0.351 & 0.155 \\ 0.013 & 0.354 & - 0.196 & 0.047 \end{matrix}]$

	$δ ({\vec{D}}_{1})$	$δ ({\vec{D}}_{2})$	$δ (R_{1})$	$δ (R_{2})$	$δ (\vec{Δ})$
$M_{1}^{m}$	0.020	0.020	3,66°	0.34°	0.016
$M_{2}^{m}$	0.023	0.024	1.63 °	3.31 °	0.019
$M_{3}^{m}$	0.003	0.005	0.26°	1.26°	0.011
$M_{4}^{m}$	0.002	0.005	7.5 °	1.25 °	0.005

Experimental validation of the symmetric decomposition of Mueller matrices

Abstract

1. Introduction

2. Experimental

2.1 The polarimetric set-up

2.2 The samples

2.3 Method

3. Results

3.1 Characterization of the individual components

3.2 Validation of the algorithm with a nondegenerate depolarizer

Comparison with other decompositions

3.2 Evaluation of the algorithm with degenerate depolarizers

Comparison with the other decompositions

4. Conclusions

Acknowledgements

References and links

Cited By

Figures (3)

Tables (8)

Equations (17)

Optics Express

D₁	${\vec{D}}_{1}^{m}^{T}$ = [0.10 0.11 0.002]
D₂	${\vec{D}}_{2}^{m}^{T}$ = [0.22 −0.02 0.002]
R₁	$R_{1}^{m} = 104 ° φ_{1}^{m} = 46 °$
R₂	$R_{2}^{m} = 72 ° φ_{2}^{m} = 32 °$
Δ_s	${\vec{Δ}}_{s}^{m}^{T}$ = [0.44 0.44 0.30]
Δ_p	${\vec{Δ}}_{p}^{m}^{T}$ = [0.98 −0.80 −0.77]
Δ_t	${\vec{Δ}}_{t}^{m}^{T}$ = [0.58 −0.47 −0.38]