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Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editor: Gregory W. Faris
  • Vol. 4, Iss. 2 — Feb. 10, 2009
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On the statistics of the entropy-depolarization relation in random light scattering

H. Di Lorenzo Pires and C. H. Monken  »View Author Affiliations


Optics Express, Vol. 16, Issue 25, pp. 21059-21068 (2008)
http://dx.doi.org/10.1364/OE.16.021059


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Abstract

We analyze an apparent disagreement between simulational and experimental results in a recent work of Puentes et al. [Opt. Lett., 30(23):3216, 2005] on the universality in depolarized light scattering. We show that the distribution of experimental points in the allowed region of the index of depolarization versus entropy diagram is ultimately determined by the statistics on the Mueller matrices, rather than on the eigenvalues of an associated Hermitian matrix. We propose a reasonable criterion that distinguishes the class of physically admissible from the physically realizable scattering media. This strategy yields further insight into the depolarization properties of media.

© 2008 Optical Society of America

1. Introduction

In the process of light scattering, the state evolution in the polarization subspace is described by a non-unitary transformation, which, for linear media, can be represented by a 4×4 matrix. Although we may not be able to describe the scattering phenomenon microscopically, the transformation matrix can be readily obtained from a tomographic procedure. When the polarization state of the field is characterized by means of the Stokes parameters, the transformation matrix is known as Mueller matrix. From the properties of the Mueller matrix one can draw insightful information about the underlying system.

The exploitation of polarimetric properties of light has a wide range of applications in a variety of fields, such as photonics technology, astrophysics, biological optics [1

1. D. P. Cubián, L. A. José, R. Diego, and Rentmeesters, “Characterization of depolarizing optical media by means of the entropy factor: application to biological tissues”, Appl. Opt. 44, 358–365 (2005). [CrossRef]

], and quantum communication [2

2. A. Aiello, G. Puentes, and J. P. Woerdman, “Linear optics and quantum maps”, Phys. Rev. A 76, 032323 (2007). [CrossRef]

], where scattering is linked to decoherence and loss of entanglement.

Recently, Aiello and Woerdman [6

6. A. Aiello and J. P. Woerdman, “Physical bounds to the entropy-depolarization relation in random light scattering”, Phys. Rev. Lett. 94, 090406 (2005). [CrossRef] [PubMed]

] showed that all possible scattering media satisfy some universal constraints, in the sense that all physically acceptable Mueller matrices are bounded to a relatively small region in the plane D M×E M. Each Mueller matrix M has an associated Hermitian matrix H, whose eigenvalues are sufficient to calculate D M and E M. Aiello and Woerdman showed that a uniform distribution of eigenvalues leads to a uniform distribution of points in the physically accessible domain of the D M×E M plane. In order to test the predicted universal behavior in depolarized light scattering, Puentes et al. performed a series of experiments, where a broad class of scattering media were characterized [5

5. G. Puentes, D. Voigt, A. Aiello, and J. P. Woerdman, “Experimental observation of universality in depolarized light scattering”, Opt. Lett. 30, 3216–3218 (2005). [CrossRef] [PubMed]

]. By gathering a relatively large ensemble of Mueller matrices, they were able to cover a significant part of the theoretically allowed domain, but with a rather different occupation density than expected from a uniform distribution of eigenvalues. An intriguing feature, though, was the lack of experimental data inside a specific subregion of the physically accessible domain.

In this paper we present an alternative approach to characterize the universality in depolarized light scattering. By focusing on the statistical properties of the Mueller matrices rather than on the eigenvalues of the associated matrices H, we show that the experimental distribution can indeed be recovered. We introduce a simple physical criterion to justify why the empty subregion of the D M×E M plane, though allowed, is almost statistically forbidden for some classes of scattering media. Furthermore, with the help of the Lu-Chipman decomposition of Mueller matrices [7

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

], we attempt to attribute a more operational physical meaning to the few sub-domains defined in [6

6. A. Aiello and J. P. Woerdman, “Physical bounds to the entropy-depolarization relation in random light scattering”, Phys. Rev. Lett. 94, 090406 (2005). [CrossRef] [PubMed]

].

The paper is organized as follows: In Sec. 2 we review the theoretical formulation of the problem, introduce the physically accessible region of the D M×E M plane and compare with previously obtained experimental results. In Sec. 3 we analyze the depolarization properties of a class of isotropic media and show how to recover the experimental distribution of points. In Sec. 4 we extend the analysis for more general media, via the Lu-Chipman decomposition. We conclude with Sec. 5

2. Entropy-depolarization relation in random light scattering

To any given Mueller matrixM a nonnegative Hermitian matrix H is associated according to

H=14i,j=03Mij(σiσj*),
(1)

where M ij are the elements of the Mueller matrix, σk (k=0…3) are the standard Pauli matrices and the superscript * denotes complex conjugation. H is normalized so that Tr H=1. The nonnegativeness of H assures that its eigenvalues satisfy 0 ≤λ i≤1, (i=0, …,3). The depolarization strength of a medium can be measured, for instance, by the index of depolarization (D M) and the entropy (E M), which are more easily written as a function of the eigenvalues of H:

EM=i=03λilog4(λi),
(2)
DM=[(4i=04λi21)3]12.
(3)

The physically accessible region in the D M×E M diagram can be obtained by randomly choosing the eigenvalues λ i constrained to Σ3 i=0 λ i=1 from a uniform distribution, i.e., from points distributed on the surface of a 4-dimensional hyper-sphere. By following this simulation strategy we can recover the results predicted in [6

6. A. Aiello and J. P. Woerdman, “Physical bounds to the entropy-depolarization relation in random light scattering”, Phys. Rev. Lett. 94, 090406 (2005). [CrossRef] [PubMed]

] for the physical region in the D M×E M plane. The result is shown in Fig. 1(a). The curves connecting the cusp points are plotted according to the equations also presented in [6

6. A. Aiello and J. P. Woerdman, “Physical bounds to the entropy-depolarization relation in random light scattering”, Phys. Rev. Lett. 94, 090406 (2005). [CrossRef] [PubMed]

]. Notice how the simulated points uniformly fill the allowed bounded region in the plane.

The predicted behavior of the quantities D M and E M was experimentally verified by Puentes et al. [5

5. G. Puentes, D. Voigt, A. Aiello, and J. P. Woerdman, “Experimental observation of universality in depolarized light scattering”, Opt. Lett. 30, 3216–3218 (2005). [CrossRef] [PubMed]

], where they conclude that their experimental data was able to cover the allowed domain almost completely, except for the region below the inner curve connecting points A and C.

Fig. 1. (a) Numerical simulation of the physically accessible region in the plane D M×E M. The points were obtained from an ensemble of uniformly distributed eigenvalues λ. The analytical expressions for the boundaries curves can be be found in [6]. (b) Experimental results obtained by G. Puentes et al. in Opt. Lett. 30, 3216 (2005) for the depolarization properties of a broad class of optically scattering media.

The experimental results obtained by Puentes et al. [5

5. G. Puentes, D. Voigt, A. Aiello, and J. P. Woerdman, “Experimental observation of universality in depolarized light scattering”, Opt. Lett. 30, 3216–3218 (2005). [CrossRef] [PubMed]

], reproduced in Fig. 1(b), seem to indicate that even for a wide choice of random depolarizing media, some regions in the D M×E M diagram, though allowed, are empty. While their results confirm the predicted “depolarization universality in light scattering”, they also raise some questions about the nature of the “forbidden” region below curve AC.

We will argue that the uniform filling is, in fact, not to be expected. By following a simulation criterion grounded on operational assumptions rather then on eigenvalues distributions, we will show that the unoccupied region below curve AC is, in some cases, almost statistically forbidden.

3. Depolarization properties of an ensemble of randomly oriented particles

The problem of simulating Mueller matrices is considerably more involved than selecting a list of acceptable eigenvalues, which is basically a set of four positive numbers whose sum is normalized to one. The Mueller matrix has sixteen free parameters that are restricted by nontrivial constraints. The relationship between elements of the Mueller matrix have been extensively studied and different, but equivalent, sets of conditions have been obtained [10

10. E. S. Fry and G. W. Kattawar, “Relationships between elements of the Stokes matrix”, Appl. Opt. 20, 2811–2814 (1981). [CrossRef] [PubMed]

, 11

11. C. Brosseau, C. R. Givens, and A. B. Kotinski, “Generalized trace condition on the Mueller-Jones polarization matrix”, J. Opt. Soc. Am. A 10, 2248–2251 (1993). [CrossRef]

, 12

12. J. W. Hovenier, H. C. van de Hulst, and C. V. M. van der Mee, “Conditions for the elements of the scattering matrix”, Astron. Astrophys. 157, 301–310 (1986).

]. In addition, since every sort of depolarizing and non-depolarizing media have their particular Mueller description, it is not necessarily meaningful (nor practicable) to proceed with a fully general analysis. We will consider, instead, a class of scattering media defined by its microscopic properties and then derive the entropy E M and the index of depolarization D M for this class of media.

Our study will be based on an ensemble of randomly oriented microscopic particles, which are not too small in comparison with the wavelength of the incident field. We will assume that the particle concentration is not large so we can neglect dense media effects. Within these considerations, the particles can have any size, shape, orientation and distribution; but we will assume further that the medium is isotropic, an isotropy that may be only in the statistical sense. This class of scattering media have been studied extensively and several applications have been discussed [13

13. F. Perrin, “Polarization of light scattered by opalescent media”, J. Chem. Phys. 10, 415–427 (1942). [CrossRef]

, 14

14. R. S. Krishnan, “The reciprocity theorem in colloid optics and its generalization”, Proc. Indian Acad. Sci. fA,21–35 (1938).

, 15

15. J. B. A. Card and A. R. Jones, “An investigation of the potential of polarized light scattering for the characterization of irregular particles”, J. Phys. D: Appl. Phys. 32, 2467–2474 (1999). [CrossRef]

, 16

16. A. A. Kokhanovsky and A. R. Jones, “The cross-polarization of light by large non-spherical particles”, J. Phys. D: Appl. Phys. 35, 1903–1906 (2002). [CrossRef]

, 17

17. A. Shi and W. M. McClain, “Closed-form Mueller scattering matrix for a random ensemble of long, thin cylinders”, J. Chem. Phys. 98, 1695–1711 (1993). [CrossRef]

, 18

18. A. Macke, “Scattering of light by polyhedral ice crystals”, Appl. Opt. 32, 2780–2788 (1993). [CrossRef] [PubMed]

].

Fig. 2. Scattering geometry. A monochromatic light beam propagating along the z axis is scattered by a medium located in its path. The observed scattering direction and the z axis define the scattering plane. The scattered “horizontal” and “vertical” polarization directions H and V are defined as the directions perpendicular and parallel to the scattering plane, respectively, both perpendicular to the scattering direction.

S0=M00S0+M01S1+M02S2+M03S3,
S1=M10S0+M11S1+M12S2+M13S3,
S2=M20S0+M21S1+M22S2+M23S3,
S3=M30S0+M31S1+M32S2+M33S3,
(4)

For a symmetrical medium the scattering plane is also a symmetry plane, i.e., if we replace the input beam by the symmetrically polarized beam (S 0,S 1,-S 2,-S 3), the new scattered beam will be symmetrically polarized with respect to the first scattered beam and it will be described by (S0,S1,-S2,-S3). This requires that eight of the Mueller coefficients are identically zero, M 20=M 21=M 30=M 31=M 02=M 03=M 12=M 13=0. Two additional constraints can be obtained by considering the reciprocity theorem [13

13. F. Perrin, “Polarization of light scattered by opalescent media”, J. Chem. Phys. 10, 415–427 (1942). [CrossRef]

, 14

14. R. S. Krishnan, “The reciprocity theorem in colloid optics and its generalization”, Proc. Indian Acad. Sci. fA,21–35 (1938).

], namely, M 01=M 10 and M 23=-M 32. The Mueller matrix necessary to describe an isotropic media composed by a low-density collection of microscopic non-spherical particles thus becomes

M=[M00M0100M01M110000M22M2300M23M33].
(5)

For the special case of spherical particles we also have M 22=M 33 [13

13. F. Perrin, “Polarization of light scattered by opalescent media”, J. Chem. Phys. 10, 415–427 (1942). [CrossRef]

, 19

19. M. Hofer and O. Glatter, “Mueller matrix calculations for randomly oriented rotationally symmetric objects with low contrast”, Appl. Opt. 28, 2389–2400 (1989). [CrossRef] [PubMed]

]. For the general case of non-spherical particles, one usually finds that these two coefficients have at least the same sign [4

4. F. Le Roy-Brehonnet and B. Le Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties”, Prog. Quantum Electron. 21, 109–151 (1997). [CrossRef]

]. In the simulations to be described below, these two parameters will be varied independently, but we will require their signs to be the same. The form of the Mueller matrix with M 22=M 33 is also adequate to describe depolarization by scattering from random rough surfaces [20

20. E. R. Méndez, A. G. Navarrette, and R. E. Luna, “Statistics of the polarization properties of one-dimensional randomly rough surfaces”, J. Opt. Soc. Am. A 12, 2507–2516 (1995). [CrossRef]

].

We define now our simulation strategy. First we normalize the Mueller matrix with respect to the element M 00 and define m ij=M ij/M 00, so that |m i j|≤1. To give an operational meaning to the strategy, we resort to one of the oldest depolarization criteria: the cross-polarization ratio. In a series of papers published in the 1930’s [14

14. R. S. Krishnan, “The reciprocity theorem in colloid optics and its generalization”, Proc. Indian Acad. Sci. fA,21–35 (1938).

, 21

21. R. S. Krishnan, “Optical evidence for molecular clustering in fluids”, Proc. Indian Acad. Sci. fA, 211–216 (1934).

, 22

22. R. S. Krishnan, “Reciprocity theorem in colloid optics”, Proc. Indian Acad. Sci. fA, 782–789 (1935).

] R. Krishnan employed the two easily measurable quantities

ρh=VhHhandρv=HvVv
(6)

to characterize the depolarizing power, where H h and V h are the intensities of the horizontal and vertical scattered field when the input field is horizontally polarized. The factors H v and V v are similarly defined. These ratios are known as the cross-polarization ratios, which can be rewritten in terms of the normalized Mueller components as

ρh=1m111+2m01+m11andρv=1m1112m01+m11.
(7)

It is natural to expect that both ratios should be limited by ρ h≤1 and ρ v≤1, whereas a higher value for these parameters would mean that the medium not only depolarizes but tends to swap the initial H and V intensities. A value greater than unity for these parameters is termed “anomalous depolarization” [21

21. R. S. Krishnan, “Optical evidence for molecular clustering in fluids”, Proc. Indian Acad. Sci. fA, 211–216 (1934).

]. As a matter of fact, many experimental and theoretical results up to date show that for disordered media the cross-polarization ratios are never simultaneously greater than unity [14

14. R. S. Krishnan, “The reciprocity theorem in colloid optics and its generalization”, Proc. Indian Acad. Sci. fA,21–35 (1938).

, 15

15. J. B. A. Card and A. R. Jones, “An investigation of the potential of polarized light scattering for the characterization of irregular particles”, J. Phys. D: Appl. Phys. 32, 2467–2474 (1999). [CrossRef]

, 16

16. A. A. Kokhanovsky and A. R. Jones, “The cross-polarization of light by large non-spherical particles”, J. Phys. D: Appl. Phys. 35, 1903–1906 (2002). [CrossRef]

, 17

17. A. Shi and W. M. McClain, “Closed-form Mueller scattering matrix for a random ensemble of long, thin cylinders”, J. Chem. Phys. 98, 1695–1711 (1993). [CrossRef]

].

We proceed our simulation by randomly choosing the free parameters m 01, m 11, m 22, m 23 and m 33 in the range -1≤m ij≤1. Naturally, non-physical matrices will be part of that ensemble, as the nontrivial constraints that delimit the physical Mueller matrices are not taken into consideration. We will post-select our results with the following purposes:

1. To assure that the Mueller matrix is physically admissible. This can be done by checking whether the associated Hermitian matrix H is nonnegative [9

9. D. G. M. Anderson and R. Barakat, “Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix”, J. Opt. Soc. Am. A 11, 2305–2319 (1994). [CrossRef]

].

2. To further select the physically realizable matrices by means of the cross-polarization ratio criterion, i.e., to assure that both ρ h≤1 and ρ v≤1.

For each Mueller matrix of the post-selected ensemble we obtain the associated Hermitian H matrix according to Eq. (1). After computing its eigenvalues, the entropy of the medium E M and the index of depolarization D M can be calculated using Eq. (2) and Eq. (3) respectively. The thus obtained pairs of points are then plotted in the D M×E M diagram and shown in Fig. 3(a). One can immediately notice that this strategy yields a distribution of points more consistent with the experimental results of Puentes et al. In particular, the region below the curve AC (previously defined) is scarcely filled. This region, though allowed, is not physically likely to be obtained. This “incomplete” filling is a consequence of the additional physical constraints that we have imposed. In order to check this statement we repeat the same procedure, but post-selecting only media with “anomalous depolarization”, i.e., the set of media for which both cross-polarization ratios are greater than one (ρ h>1 and ρ v>1). The result is shown in Fig. 3(b). It is remarkable to notice how this approach provides a set of points that are upper bounded by a curve that was independently obtained in terms of the nature of the eigenvalues. According to these observations, we may provide a physical interpretation to the AC curve: it distinguishes the region of normal depolarization from the “anomalous depolarization” (according to cross-polarization ratio criterion). It should be stressed however, that this criterion does not define an exact partition of the D M×E M diagram, as can be verified by the presence of a few points below the AC curve in Fig. 3(a).

Fig. 3. Depolarization properties of an ensemble of randomly oriented non-spherical microscopic particles. D M is the index of depolarization of the medium and E M the entropy of the medium. After obtaining a set of physically acceptable Mueller matrices we post select the results according to (a) physically realizable matrices as specified by the cross-polarization ratio criterion; both ρ h≤1 and ρ v≤1. (b) Matrices for media with “anomalous depolarization”, that is, ρ h>1 and ρ v>1.

4. Depolarization properties via polar decomposition of Mueller matrices

We will now make use of the polar decomposition of Mueller matrices, originally proposed by Lu and Chipman [7

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

]. Any (normalized) Mueller matrix can be written in the generic form

M=(1dTpm),
(8)

where d and p are three-component vectors known as diattenuation and polarizance, and m is a 3×3 submatrix. It is possible to decompose it into a sequence of three factors, M=MΔMRMD, each one representing an elementary operation on the Stokes vector. In this factorization we can split the contributions of the diattenuation (MD) and retardance (MR) from the depolarization (MΔ), with

MR=(10T0mR),MD=(1dTdmd)andMΔ=(10TpmΔ).
(9)

The matrix MR represents a purely birefringent element. MD, which is completely determined by the diattenuation vector d, represents the action of a dichroic element. The modulus of d defines the amount of diattenuation, and its direction defines the polarization direction (in the Poincaré sphere) for maximum transmission. The elements of the submatrix md are defined as μij=didj+1d2(δijdidj), where d i(i=1…3) are the components of d. According to Eq. (8), MD is completely defined by the first line of the normalized Mueller matrix. MΔ represents a purely depolarizing element, with mΔ being a symmetric submatrix. A detailed derivation of these formulas can be found in [7

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

]. Since MR represents a birefringent element, the submatrix mR simply implements a rotation on the Poincaré sphere. In addition, the symmetric submatrix mΔ can be diagonalized by retarder Mueller matrices. These properties allow us to write

M=MRMΔdiagMRTMRMD,
=MRMΔdiagMRMD.
(10)

M R=M′T RMR also represents a rotation and Mdiag Δ is a depolarizing Mueller matrix whose submatrix mΔ=diag{a,b,c} is diagonal.

Although in a different form, we still have the most general Mueller matrix, describing non-depolarizing and depolarizing media of any kind. We will consider only the cases in which the singular values of mΔ are all equal (a=b=c). The factor MΔ will therefore represent an isotropic depolarizing media. With certain freedom, we define “anomalous depolarization” in this context those transformations for which the singular value a is negative. This is intuitively reasonable, as the effect of the purely depolarizing Mueller matrix MΔ=diag{1,-|a|,-|a|,-|a|}, with |a|<1, is to reduce the norm of the Stokes vector (that is, to depolarize) but also to invert the Stokes vector through the origin, for every possible input state. This is not likely to happen in a random medium. If a, b and c are not equal, the Stokes vector undergoes additional operations besides shrinking and inversion. Depending on the values of a, b and c, “anomalous depolarization ” may or may not occur. The restriction a=b=c was made in order to focus the present analysis on the role of “anomalous depolarization”.

We need to establish now a simulation criterion. A physically reasonable criterion is to consider the vectors d and p, the three-dimensional rotation matrix m R and the singular value a the essential attributes concerning the depolarization properties of a Mueller matrix. The ensemble of Mueller matrices is constructed by randomly choosing these parameters from an uniform distribution. Naturally, weighted distributions could be employed if we had a specific class of scattering systems in mind. Since this is not the case, we proceed with a very regular distribution: (i) we select the vectors d and p from a three-dimensional unit ball (sphere of unitary radius and its interior), (ii) the singular value a is obtained from the interval [0,1

1. D. P. Cubián, L. A. José, R. Diego, and Rentmeesters, “Characterization of depolarizing optical media by means of the entropy factor: application to biological tissues”, Appl. Opt. 44, 358–365 (2005). [CrossRef]

], so as to rule out “anomalous depolarization” and (iii) the submatrix m R is chosen from an ensemble of uniformly distributed three-dimensional real special orthogonal matrices (rotation matrices) [8

8. M. Pozniak, K. Zyczkowski, and M. Kus, “Composed ensembles of random unitary matrices”, J. Phys. A 31, 1059–1071 (1998). [CrossRef]

]. Once we have this ensemble we can use again Eq. (1), (2) and (3) to compute the index of depolarization and the entropy of the medium. We postselect the results by checking whether the associated Hermitian matrix H is nonnegative [9

9. D. G. M. Anderson and R. Barakat, “Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix”, J. Opt. Soc. Am. A 11, 2305–2319 (1994). [CrossRef]

].

Fig. 4. Numerical simulations of the depolarization properties of random Mueller matrices obtained according to the polar decomposition criterion (see text). (a) Singular values of the submatrix mΔ are all equal, a=b=c, and nonnegative. We impose further restrictions: (b) modulus of the polarizance vector |p|≤0.1, (c) diattenuation vector|d|≤0.1 and (d) diattenuation vector |d|≥0.9.

This approach, despite not having a direct connection with a specific class of scattering media, is less restrictive than the one discussed in Sec. 3. The physical interpretation of the domain below curve AC as a region of “anomalous depolarization” remains valid in a more general context and explains the incomplete covering of the D M×E M domain obtained by Puentes et al.

5. Conclusion

In this work we analyzed the problem of universality in depolarized light scattering in the entropy versus index of depolarization representation. In particular, we extended the theoretical analysis of Aiello and Woerdman [2

2. A. Aiello, G. Puentes, and J. P. Woerdman, “Linear optics and quantum maps”, Phys. Rev. A 76, 032323 (2007). [CrossRef]

] in order to better understand the experimental results of Puentes et al. [5

5. G. Puentes, D. Voigt, A. Aiello, and J. P. Woerdman, “Experimental observation of universality in depolarized light scattering”, Opt. Lett. 30, 3216–3218 (2005). [CrossRef] [PubMed]

]. We showed how a physically grounded criterion, based on the Mueller matrix representation, leads to a statistical distribution of points in the D M×E M diagram more consistent with the experimental observations. We explain the origin of of the “inaccessible” region in the depolarization-entropy plane as a region of “anomalous depolarization”. For an ensemble of randomly oriented microscopic particles this concept can be well understood in terms of one the earliest depolarization measures: the cross-polarization ratio. Following, we extended the concept of “anomalous depolarization” to more general media through the polar decomposition of Mueller matrices. The results are self-consistent. Finally, we provided some physical meaning to the sub-domains in the D M×E M diagram, as they were originally defined only mathematically.

In this work we did not intend to give a full description of the statistical properties of random Mueller matrices, but to highlight the role of the simulation criteria on the distribution of some depolarization measures. Our analysis could be extended for different media with different depolarization properties by following basically the same strategy.

Acknowledgments

This work has been supported by the Stichting voor Fundamenteel Onderzoek der Materie and the Brazilian agencies CNPq and CAPES. We thank M. P. van Exter for stimulating discussions.

References and links

1.

D. P. Cubián, L. A. José, R. Diego, and Rentmeesters, “Characterization of depolarizing optical media by means of the entropy factor: application to biological tissues”, Appl. Opt. 44, 358–365 (2005). [CrossRef]

2.

A. Aiello, G. Puentes, and J. P. Woerdman, “Linear optics and quantum maps”, Phys. Rev. A 76, 032323 (2007). [CrossRef]

3.

J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system”, J. Mod. Opt. 33, 185–189 (1986).

4.

F. Le Roy-Brehonnet and B. Le Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties”, Prog. Quantum Electron. 21, 109–151 (1997). [CrossRef]

5.

G. Puentes, D. Voigt, A. Aiello, and J. P. Woerdman, “Experimental observation of universality in depolarized light scattering”, Opt. Lett. 30, 3216–3218 (2005). [CrossRef] [PubMed]

6.

A. Aiello and J. P. Woerdman, “Physical bounds to the entropy-depolarization relation in random light scattering”, Phys. Rev. Lett. 94, 090406 (2005). [CrossRef] [PubMed]

7.

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

8.

M. Pozniak, K. Zyczkowski, and M. Kus, “Composed ensembles of random unitary matrices”, J. Phys. A 31, 1059–1071 (1998). [CrossRef]

9.

D. G. M. Anderson and R. Barakat, “Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix”, J. Opt. Soc. Am. A 11, 2305–2319 (1994). [CrossRef]

10.

E. S. Fry and G. W. Kattawar, “Relationships between elements of the Stokes matrix”, Appl. Opt. 20, 2811–2814 (1981). [CrossRef] [PubMed]

11.

C. Brosseau, C. R. Givens, and A. B. Kotinski, “Generalized trace condition on the Mueller-Jones polarization matrix”, J. Opt. Soc. Am. A 10, 2248–2251 (1993). [CrossRef]

12.

J. W. Hovenier, H. C. van de Hulst, and C. V. M. van der Mee, “Conditions for the elements of the scattering matrix”, Astron. Astrophys. 157, 301–310 (1986).

13.

F. Perrin, “Polarization of light scattered by opalescent media”, J. Chem. Phys. 10, 415–427 (1942). [CrossRef]

14.

R. S. Krishnan, “The reciprocity theorem in colloid optics and its generalization”, Proc. Indian Acad. Sci. fA,21–35 (1938).

15.

J. B. A. Card and A. R. Jones, “An investigation of the potential of polarized light scattering for the characterization of irregular particles”, J. Phys. D: Appl. Phys. 32, 2467–2474 (1999). [CrossRef]

16.

A. A. Kokhanovsky and A. R. Jones, “The cross-polarization of light by large non-spherical particles”, J. Phys. D: Appl. Phys. 35, 1903–1906 (2002). [CrossRef]

17.

A. Shi and W. M. McClain, “Closed-form Mueller scattering matrix for a random ensemble of long, thin cylinders”, J. Chem. Phys. 98, 1695–1711 (1993). [CrossRef]

18.

A. Macke, “Scattering of light by polyhedral ice crystals”, Appl. Opt. 32, 2780–2788 (1993). [CrossRef] [PubMed]

19.

M. Hofer and O. Glatter, “Mueller matrix calculations for randomly oriented rotationally symmetric objects with low contrast”, Appl. Opt. 28, 2389–2400 (1989). [CrossRef] [PubMed]

20.

E. R. Méndez, A. G. Navarrette, and R. E. Luna, “Statistics of the polarization properties of one-dimensional randomly rough surfaces”, J. Opt. Soc. Am. A 12, 2507–2516 (1995). [CrossRef]

21.

R. S. Krishnan, “Optical evidence for molecular clustering in fluids”, Proc. Indian Acad. Sci. fA, 211–216 (1934).

22.

R. S. Krishnan, “Reciprocity theorem in colloid optics”, Proc. Indian Acad. Sci. fA, 782–789 (1935).

OCIS Codes
(260.5430) Physical optics : Polarization
(290.5820) Scattering : Scattering measurements
(290.5855) Scattering : Scattering, polarization

ToC Category:
Scattering

History
Original Manuscript: October 22, 2008
Revised Manuscript: November 21, 2008
Manuscript Accepted: November 29, 2008
Published: December 4, 2008

Virtual Issues
Vol. 4, Iss. 2 Virtual Journal for Biomedical Optics

Citation
H. Di Lorenzo Pires and C. H. Monken, "On the statistics of the entropy-depolarization relation in random light scattering," Opt. Express 16, 21059-21068 (2008)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-16-25-21059


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References

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  12. J. W. Hovenier, H. C. van de Hulst and C. V. M. van der Mee, "Conditions for the elements of the scattering matrix," Astron. Astrophys. 157, 301-310 (1986).
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  14. R. S. Krishnan, "The reciprocity theorem in colloid optics and its generalization," Proc. Indian Acad. Sci. fA, 21-35 (1938).
  15. J. B. A. Card and A. R. Jones, "An investigation of the potential of polarized light scattering for the characterization of irregular particles," J. Phys. D: Appl. Phys. 32, 2467-2474 (1999). [CrossRef]
  16. A. A. Kokhanovsky and A. R. Jones, "The cross-polarization of light by large non-spherical particles," J. Phys. D: Appl. Phys. 35, 1903-1906 (2002). [CrossRef]
  17. A. Shi and W. M. McClain, "Closed-form Mueller scattering matrix for a random ensemble of long, thin cylinders," J. Chem. Phys. 98, 1695-1711 (1993). [CrossRef]
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  20. E. R. Méndez, A. G. Navarrette and R. E. Luna, "Statistics of the polarization properties of one-dimensional randomly rough surfaces," J. Opt. Soc. Am. A 12, 2507-2516 (1995). [CrossRef]
  21. R. S. Krishnan, "Optical evidence for molecular clustering in fluids," Proc. Indian Acad. Sci. fA, 211-216 (1934).
  22. R. S. Krishnan, "Reciprocity theorem in colloid optics," Proc. Indian Acad. Sci. fA, 782-789 (1935).

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