## Retrieval of a non-depolarizing component of experimentally determined depolarizing Mueller matrices

Optics Express, Vol. 17, Issue 15, pp. 12794-12806 (2009)

http://dx.doi.org/10.1364/OE.17.012794

Acrobat PDF (517 KB)

### Abstract

The measurement of the Mueller matrix when the probing beam is placed on the boundary between two (or more) regions of the sample with different optical properties may lead to a depolarization in the Mueller matrix. The depolarization is due to the incoherent superposition of the optical responses of different sample regions in the probe beam. Despite of the depolarization, the measured Mueller matrix has information enough to subtract a Mueller matrix corresponding to one of the regions of sample provided that this subtracted matrix is non-depolarizing. For clarity, we will call these non-depolarizing Mueller matrices of one individual region of the sample simply as the non-depolarizing components. In the framework of the theory of Mueller matrix algebra, we have implemented a procedure allowing the retrieval of a non-depolarizing component from a depolarizing Mueller matrix constituted by the sum of several non-depolarizing components. In order to apply the procedure, the Mueller matrices of the rest of the non-depolarizing components have to be known. Here we present a numerical and algebraic approaches to implement the subtraction method. To illustrate our method as well as the performance of the two approaches, we present two practical examples. In both cases we have measured depolarizing Mueller matrices by positioning an illumination beam on the boundary between two and three different regions of a sample, respectively. The goal was to retrieve the non-depolarizing Mueller matrix of one of those regions from the measured depolarizing Mueller matrix. In order to evaluate the performance of the method we compared the subtracted matrix with the Mueller matrix of the selected region measured separately.

© 2009 Optical Society of America

## 1. Introduction

1. M. Foldyna, A. De Martino, R. Ossikovski, E. Garcia-Caurel, and C. Licitra, “Characterization of grating structures by Mueller polarimetry in presence of strong depolarization due to finite spot size and spectral resolution,” Opt. Commun. **282**, 735–741 (2009).
[CrossRef]

2. G. Subramania, Y.-J. Lee, I. Brener, T. Luk, and P. Clem, “Nano-lithographically fabricated titanium dioxide based visible frequency three dimensional gap photonic crystal,” Opt. Express **15(20)**, 13049–13057 (2007).
[CrossRef]

3. M. Roussey, M.-P. Bernal, N. Courjal, and F. I. Baida, “Experimental and theoretical characterization of a lithium niobate photonic crystal,” Appl. Phys. Lett. **87**, 241101 (2005).
[CrossRef]

4. J. J. Gil, “Polarimetric characterization of light and media Physical quantities involved in polarimetric phenomena,” Eur. Phys. J. Appl. Phys. **40**, 1–47 (2007).
[CrossRef]

## 2. Theory

*x*〉 represents time averaging over the measurement time and

*E*and

_{x}*E*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:

_{y}*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

*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.

*M*′ made of the sum of two normalized non-depolarizing matrix components

*M*

^{(1)}and

*M*

^{(2)}can be written in form

*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]

*α*can be obtained either using an analytical approach with the expression

*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

*M*

^{(2)}from the original depolarizing matrix

*M*′ by subtracting the right proportion of the known component

*M*

^{(1)}.

*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.

*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.

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

*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:

*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

1. M. Foldyna, A. De Martino, R. Ossikovski, E. Garcia-Caurel, and C. Licitra, “Characterization of grating structures by Mueller polarimetry in presence of strong depolarization due to finite spot size and spectral resolution,” Opt. Commun. **282**, 735–741 (2009).
[CrossRef]

## 4. Results and discussions

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

*M*′, is shown in Fig. 2 together with the non-depolarizing matrix of the substrate (

*M*

^{(1)}).

*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.

*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.

*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

11. A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. **36**, 1627–1639 (1964).
[CrossRef]

*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.

*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

1. M. Foldyna, A. De Martino, R. Ossikovski, E. Garcia-Caurel, and C. Licitra, “Characterization of grating structures by Mueller polarimetry in presence of strong depolarization due to finite spot size and spectral resolution,” Opt. Commun. **282**, 735–741 (2009).
[CrossRef]

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

*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.

*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.

## 5. Conclusions

## Acknowledgments

## References and links

1. | M. Foldyna, A. De Martino, R. Ossikovski, E. Garcia-Caurel, and C. Licitra, “Characterization of grating structures by Mueller polarimetry in presence of strong depolarization due to finite spot size and spectral resolution,” Opt. Commun. |

2. | G. Subramania, Y.-J. Lee, I. Brener, T. Luk, and P. Clem, “Nano-lithographically fabricated titanium dioxide based visible frequency three dimensional gap photonic crystal,” Opt. Express |

3. | M. Roussey, M.-P. Bernal, N. Courjal, and F. I. Baida, “Experimental and theoretical characterization of a lithium niobate photonic crystal,” Appl. Phys. Lett. |

4. | J. J. Gil, “Polarimetric characterization of light and media Physical quantities involved in polarimetric phenomena,” Eur. Phys. J. Appl. Phys. |

5. | J. M. Correas, P. Melero, and J. J. Gil, “Decomposition of Mueller matrices in pure optical media,” Mon. Sem. Mat. García de Galdeano |

6. | S. R. Cloude, “Group theory and polarisation algebra,” Optik (Stuttgart) |

7. | R. M. A. Azzam and N. M. Bashara, |

8. | J. J. Gil and E. Bernabeu, “A depolarization criterion in Mueller matrices,” J. Mod. Opt. |

9. | S. R. Cloude, “Conditions for the physical realisability of matrix operators in polarimetry,” vol. 1166 of Proc. of SPIE, pp. 177–185 (1989). |

10. | E. Garcia-Caurel, A. De Martino, and B. Drévillon, “Spectroscopic Mueller polarimeter based on liquid crystal devices ,” Thin Solid Films |

11. | A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. |

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology

(120.2130) Instrumentation, measurement, and metrology : Ellipsometry and polarimetry

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: April 20, 2009

Revised Manuscript: May 27, 2009

Manuscript Accepted: May 29, 2009

Published: July 13, 2009

**Citation**

M. Foldyna, E. Garcia-Caurel, R. Ossikovski, A. De Martino, and J. J. Gil, "Retrieval of a non-depolarizing
component of experimentally
determined depolarizing Mueller
matrices," Opt. Express **17**, 12794-12806 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-12794

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### References

- M. Foldyna, A. De Martino, R. Ossikovski, E. Garcia-Caurel, and C. Licitra, "Characterization of grating structures by Mueller polarimetry in presence of strong depolarization due to finite spot size and spectral resolution," Opt. Commun. 282, 735-741 (2009). [CrossRef]
- G. Subramania, Y.-J. Lee, I. Brener, T. Luk, and P. Clem, "Nano-lithographically fabricated titanium dioxide based visible frequency three dimensional gap photonic crystal," Opt. Express 15(20), 13049-13057 (2007). [CrossRef]
- M. Roussey, M.-P. Bernal, N. Courjal, and F. I. Baida, "Experimental and theoretical characterization of a lithium niobate photonic crystal," Appl. Phys. Lett. 87, 241101 (2005). [CrossRef]
- J. J. Gil, "Polarimetric characterization of light and media Physical quantities involved in polarimetric phenomena," Eur. Phys. J. Appl. Phys. 40, 1-47 (2007). [CrossRef]
- J. M. Correas, P. Melero, and J. J. Gil, "Decomposition of Mueller matrices in pure optical media," Mon. Sem. Mat. García de Galdeano 27, 233-240 (2003).
- S. R. Cloude, "Group theory and polarisation algebra," Optik (Stuttgart) 75, 26-36 (1986).
- R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, 2nd ed. (North-Holland, Amsterdam, 1987).
- J. J. Gil and E. Bernabeu, "A depolarization criterion in Mueller matrices," J. Mod. Opt. 32, 259-261 (1985).
- S. R. Cloude, "Conditions for the physical realisability of matrix operators in polarimetry," vol. 1166 of Proc. of SPIE, pp. 177-185 (1989).
- E. Garcia-Caurel, A. De Martino, and B. Drévillon, "Spectroscopic Mueller polarimeter based on liquid crystal devices," Thin Solid Films 455-456, 120-123 (2004).
- A. Savitzky and M. J. E. Golay, "Smoothing and differentiation of data by simplified least squares procedures," Anal. Chem. 36, 1627-1639 (1964). [CrossRef]

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