## On the statistics of the entropy-depolarization relation in random light scattering

Optics Express, Vol. 16, Issue 25, pp. 21059-21068 (2008)

http://dx.doi.org/10.1364/OE.16.021059

Acrobat PDF (1699 KB)

### 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

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]

*predominant*depolarizing behavior of the sample media over the set of all possible input states. Among them, we focus on two popular metrics, namely, the index of depolarization of the medium (

*D*

*) [3] and the entropy of the medium (*

_{M}*E*

*) [4*

_{M}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]

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]

*D*

*×*

_{M}*E*

*diagram was introduced by Roy-Brehonnet and Le Jeune [4*

_{M}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]

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]

*D*

*×*

_{M}*E*

*. Each Mueller matrix M has an associated Hermitian matrix H, whose eigenvalues are sufficient to calculate*

_{M}*D*

*and*

_{M}*E*

*. Aiello and Woerdman showed that a uniform distribution of eigenvalues leads to a uniform distribution of points in the physically accessible domain of the*

_{M}*D*

*×*

_{M}*E*

*plane. In order to test the predicted universal behavior in depolarized light scattering, Puentes*

_{M}*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]

*D*

*×*

_{M}*E*

*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*

_{M}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]

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]

*D*

*×*

_{M}*E*

*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*

_{M}## 2. Entropy-depolarization relation in random light scattering

*S*

_{0},

*S*

_{1},

*S*

_{2},

*S*

_{3}), where

*S*

_{0}=

*I*

_{0}is the intensity of the field,

*S*

_{1}=

*I*

*-*

_{H}*I*

*is the difference of intensities in the horizontal and vertical polarization components,*

_{V}*S*

_{2}=

*I*

_{+45}-

*I*

_{-45}, is the difference of intensities in the +45°and -45°polarization components, and

*S*

_{3}=

*I*

*-*

_{L}*I*

*has an analogous definition for left and right circular polarization. Any possible transformation on this state can be completely described by a 4×4 Mueller matrix, which links input states to output states via the relation*

_{R}*S*

^{′}=MS.

*M*

*are the elements of the Mueller matrix, σ*

_{ij}*(*

_{k}*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*

*≤1, (*

_{i}*i*=0, …,3). The depolarization strength of a medium can be measured, for instance, by the index of depolarization (

*D*

*) and the entropy (*

_{M}*E*

*), which are more easily written as a function of the eigenvalues of H:*

_{M}*D*

*×*

_{M}*E*

*diagram can be obtained by randomly choosing the eigenvalues*

_{M}*λ*

*constrained to Σ*

_{i}^{3}

_{i=0}

*λ*

*=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*

_{i}**94**, 090406 (2005). [CrossRef] [PubMed]

*D*

*×*

_{M}*E*

*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*

_{M}**94**, 090406 (2005). [CrossRef] [PubMed]

*D*

*and*

_{M}*E*

*was experimentally verified by Puentes*

_{M}*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]

*et al.*[5

**30**, 3216–3218 (2005). [CrossRef] [PubMed]

*D*

*×*

_{M}*E*

*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.*

_{M}## 3. Depolarization properties of an ensemble of randomly oriented particles

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]

*E*

*and the index of depolarization*

_{M}*D*

*for this class of media.*

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

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]

*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 (

*S*′

_{0},

*S*′

_{1},-

*S*′

_{2},-

*S*′

_{3}). 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]

*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*

_{22}=

*M*

_{33}[13

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

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]

**21**, 109–151 (1997). [CrossRef]

*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]

*M*

_{00}and define

*m*

*=*

_{ij}*M*

*/*

_{ij}*M*

_{00}, so that |

*m*

*|≤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, 21, 22] R. Krishnan employed the two easily measurable quantities*

_{i j}*H*

*and*

_{h}*V*

*are the intensities of the horizontal and vertical scattered field when the input field is horizontally polarized. The factors*

_{h}*H*

*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*

_{v}*ρ*

*≤1 and*

_{h}*ρ*

*≤1, whereas a higher value for these parameters would mean that the medium not only depolarizes but tends to swap the initial*

_{v}*H*and

*V*intensities. A value greater than unity for these parameters is termed “anomalous depolarization” [21]. 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, 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. 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]

*m*

_{01},

*m*

_{11},

*m*

_{22},

*m*

_{23}and

*m*

_{33}in the range -1≤

*m*

*≤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:*

_{ij}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]

*ρ*

*≤1 and*

_{h}*ρ*

*≤1.*

_{v}*E*

*and the index of depolarization*

_{M}*D*

*can be calculated using Eq. (2) and Eq. (3) respectively. The thus obtained pairs of points are then plotted in the*

_{M}*D*

*×*

_{M}*E*

*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*

_{M}*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*

*>1 and*

_{h}*ρ*

*>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*

_{v}*D*

*×*

_{M}*E*

*diagram, as can be verified by the presence of a few points below the AC curve in Fig. 3(a).*

_{M}*et al.*in terms of physical parameters of the Lu-Chipman decomposition of Mueller matrices. We also provide a criterion for “anomalous depolarization” in terms of those parameters.

## 4. Depolarization properties via polar decomposition of Mueller matrices

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]

**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=M

_{Δ}M

_{R}M

*, each one representing an elementary operation on the Stokes vector. In this factorization we can split the contributions of the diattenuation (M*

_{D}_{D}) and retardance (M

*) from the depolarization (M*

_{R}_{Δ}), with

*represents a purely birefringent element. M*

_{R}*, which is completely determined by the diattenuation vector*

_{D}**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 m

*are defined as*

_{d}*d*

*(*

_{i}*i*=1…3) are the components of

**d**. According to Eq. (8), M

*is completely defined by the first line of the normalized Mueller matrix. M*

_{D}_{Δ}represents a purely depolarizing element, with m

_{Δ}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]

*represents a birefringent element, the submatrix m*

_{R}*simply implements a rotation on the Poincaré sphere. In addition, the symmetric submatrix m*

_{R}_{Δ}can be diagonalized by retarder Mueller matrices. These properties allow us to write

^{″}

*=M*

_{R}^{′T}

_{R}M

*also represents a rotation and M*

_{R}

^{diag}_{Δ}is a depolarizing Mueller matrix whose submatrix m

_{Δ}=diag{

*a*,

*b*,

*c*} is diagonal.

_{Δ}are all equal (

*a*=

*b*=

*c*). The factor M

_{Δ}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 M

_{Δ}=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”.

**d**and

**p**, the three-dimensional rotation matrix m

^{′}

*and the singular value*

_{R}*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]

^{′}

*is chosen from an ensemble of uniformly distributed three-dimensional real special orthogonal matrices (rotation matrices) [8*

_{R}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]

**p**|≤0.1), and figures 4(c) and 4(d) elements where the losses due to dichroism are very small (|

**d**|≤0.1) and very large (|

**d**|≥0.9) respectively. Notice how the restrictions imposed on the Mueller matrices affect the density of points through the theoretically allowed region. For example, large dichroism leads to states with a greater degree of polarization, concentrated on the right-lower sub-domain of the plot. The most interesting conclusion, however, is that the region below the AC curve is almost empty, which is again the same pattern experimentally observed. If we relax the restriction on parameter

*a*, allowing it to assume negative values, the region below curve

*AC*will be filled.

*D*

*×*

_{M}*E*

*domain obtained by Puentes*

_{M}*et al.*

## 5. Conclusion

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

*et al.*[5

**30**, 3216–3218 (2005). [CrossRef] [PubMed]

*D*

*×*

_{M}*E*

*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*

_{M}*D*

*×*

_{M}*E*

*diagram, as they were originally defined only mathematically.*

_{M}## Acknowledgments

## 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. |

2. | A. Aiello, G. Puentes, and J. P. Woerdman, “Linear optics and quantum maps”, Phys. Rev. A |

3. | J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system”, J. Mod. Opt. |

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. |

5. | G. Puentes, D. Voigt, A. Aiello, and J. P. Woerdman, “Experimental observation of universality in depolarized light scattering”, Opt. Lett. |

6. | A. Aiello and J. P. Woerdman, “Physical bounds to the entropy-depolarization relation in random light scattering”, Phys. Rev. Lett. |

7. | S. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition”, J. Opt. Soc. Am. A |

8. | M. Pozniak, K. Zyczkowski, and M. Kus, “Composed ensembles of random unitary matrices”, J. Phys. A |

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 |

10. | E. S. Fry and G. W. Kattawar, “Relationships between elements of the Stokes matrix”, Appl. Opt. |

11. | C. Brosseau, C. R. Givens, and A. B. Kotinski, “Generalized trace condition on the Mueller-Jones polarization matrix”, J. Opt. Soc. Am. A |

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. |

13. | F. Perrin, “Polarization of light scattered by opalescent media”, J. Chem. Phys. |

14. | R. S. Krishnan, “The reciprocity theorem in colloid optics and its generalization”, Proc. Indian Acad. Sci. |

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. |

16. | A. A. Kokhanovsky and A. R. Jones, “The cross-polarization of light by large non-spherical particles”, J. Phys. D: Appl. Phys. |

17. | A. Shi and W. M. McClain, “Closed-form Mueller scattering matrix for a random ensemble of long, thin cylinders”, J. Chem. Phys. |

18. | A. Macke, “Scattering of light by polyhedral ice crystals”, Appl. Opt. |

19. | M. Hofer and O. Glatter, “Mueller matrix calculations for randomly oriented rotationally symmetric objects with low contrast”, Appl. Opt. |

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 |

21. | R. S. Krishnan, “Optical evidence for molecular clustering in fluids”, Proc. Indian Acad. Sci. |

22. | R. S. Krishnan, “Reciprocity theorem in colloid optics”, Proc. Indian Acad. Sci. |

**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

Sort: Year | Journal | Reset

### References

- D. P. Cubián, J. L. A. Diego and R . Rentmeesters, "Characterization of depolarizing optical media by means of the entropy factor: application to biological tissues," Appl. Opt. 44, 358-365 (2005). [CrossRef]
- A. Aiello, G. Puentes and J. P. Woerdman, "Linear optics and quantum maps," Phys. Rev. A 76, 032323 (2007). [CrossRef]
- J. J. Gil and E. Bernabeu, "Depolarization and polarization indices of an optical system," J. Mod. Opt. 33, 185-189 (1986).
- 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]
- 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]
- 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]
- S. Lu and R. A. Chipman, "Interpretation of Mueller matrices based on polar decomposition," J. Opt. Soc. Am. A 13, 1106-1113 (1996). [CrossRef]
- M. Pozniak, K. Zyczkowski and M. Kus, "Composed ensembles of random unitary matrices," J. Phys. A 31, 1059-1071 (1998). [CrossRef]
- 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]
- E. S. Fry and G. W. Kattawar, "Relationships between elements of the Stokes matrix," Appl. Opt. 20, 2811-2814 (1981). [CrossRef] [PubMed]
- 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]
- 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).
- F. Perrin, "Polarization of light scattered by opalescent media," J. Chem. Phys. 10, 415-427 (1942). [CrossRef]
- R. S. Krishnan, "The reciprocity theorem in colloid optics and its generalization," Proc. Indian Acad. Sci. fA, 21-35 (1938).
- 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]
- 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]
- 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]
- A. Macke, "Scattering of light by polyhedral ice crystals," Appl. Opt. 32, 2780- 2788 (1993). [CrossRef] [PubMed]
- 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]
- 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]
- R. S. Krishnan, "Optical evidence for molecular clustering in fluids," Proc. Indian Acad. Sci. fA, 211-216 (1934).
- R. S. Krishnan, "Reciprocity theorem in colloid optics," Proc. Indian Acad. Sci. fA, 782-789 (1935).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.