## Degree of polarization surfaces and maps for analysis of depolarization

Optics Express, Vol. 12, Issue 20, pp. 4941-4958 (2004)

http://dx.doi.org/10.1364/OPEX.12.004941

Acrobat PDF (1938 KB)

### Abstract

The concept of degree of polarization surfaces is introduced as an aid to classifying the depolarization properties of Mueller matrices. Degree of polarization surfaces provide a visualization of the dependence of depolarization on incident polarization state. The surfaces result from a non-uniform contraction of the Poincaré sphere corresponding to the depolarization properties encoded in a Mueller matrix. For a given Mueller matrix, the degree of polarization surface is defined by moving each point on the unit Poincaré sphere radially inward until its distance from the origin equals the output state degree of polarization for the corresponding input state. Of the sixteen elements in a Mueller matrix, twelve contribute to the shape of the degree of polarization surface, yielding a complex family of surfaces. The surface shapes associated with the numerator and denominator of the degree of polarization function are analyzed separately. Protrusion of the numerator surface through the denominator surface at any point indicates non-physical Mueller matrices. Degree of polarization maps are plots of the degree of polarization on flat projections of the sphere. These maps reveal depolarization patterns in a manner well suited for quantifying the degree of polarization variations, making degree of polarization surfaces and maps valuable tools for categorizing and classifying the depolarization properties of Mueller matrices.

© 2004 Optical Society of America

## 1. Inroduction

1. R.C. Jones, “A new calculus for the treatment of optical systems: I. Description and discussion of the calculus,” J. Opt. Soc. Am. **31**, 488–493 (1941). [CrossRef]

4. K.D. Abhyankar and A.L. Fymat, “Relations between the elements of the phase matrix for scattering,” J. Math Phys. **10**, 1935–1938 (1969). [CrossRef]

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

8. J.L. Pezzaniti, S.C. McClain, R.A. Chipman, and S.Y. Lu, “Depolarization in liquid crystal TV’s,” Opt. Lett. **18**, 2071–2073 (1993). [CrossRef] [PubMed]

9. S.-M.F. Nee and T.-W. Nee, “Principal Mueller matrix of reflection and scattering for a one-dimensional rough surface,” Opt. Eng. **41**, 994–1001 (2002). [CrossRef]

21. J.J. Gil and E. Bernabeau, “Depolarization and polarization indices of an optical system,” Opt. Acta **33**, 185–189 (1986). [CrossRef]

*) of incident states. Here,*

**DoP***surfaces and maps are introduced to quantify and visualize the complete depolarizing properties of a Mueller matrix for all incident states.*

**DoP***is a measure of the randomness of polarization in a light beam, a property characterized by how much of this beam may be blocked by a polarizer. Mathematically, on the Poincaré sphere, the*

**DoP***represents the distance of a normalized Stokes vector’s last three components from the origin. The surface of the unit Poincaré sphere has*

**DoP***=1 and represents all fully polarized states [23*

**DoP**23. D. Goldstein, *Polarized Light, Second Ed., Revised and Expanded*, (Marcel Dekker, New York,2003). [CrossRef]

*≤1. The*

**DoP***surface plots these exiting*

**DoP***values along the same radial vectors from the origin that define the corresponding input states. A similar plot was discussed in a different context by Williams [10*

**DoP**10. M.W. Williams, “Depolarization and cross polarization in ellipsometry of rough surfaces.” Appl. Opt. **25**, 3616–3621 (1986). [CrossRef] [PubMed]

*map is a two-dimensional contour plot of the*

**DoP***surface versus two coordinates defining input polarization states (i.e. polarization ellipse orientation and ellipticity).*

**DoP***of polarized states offers significant advantages over single-valued methods.*

**DoP***surfaces and maps provide an insightful visualization of how a Mueller matrix depolarizes all incident polarized states. The information content of*

**DoP***surfaces and maps is greater than other depolarization metrics. For example, maxima and minima in the output*

**DoP***are readily observed with*

**DoP***surfaces, and the values of these extremes easily quantified with*

**DoP***maps.*

**DoP***surfaces and maps are thorough in describing the behaviors of depolarizing Mueller matrices and are examined in detail here.*

**DoP**## 2. The degree of polarization

*) characterizes the randomness of a polarization state. The*

**DoP***of Stokes vector*

**DoP****S**=(S

_{0},S

_{1},S

_{2},S

_{3}) is

*=0, the light is unpolarized and all ideal polarizers block half the beam. When*

**DoP***=1, the beam is completely polarized and some ideal polarizer, either linear, circular, or elliptical, will completely block the beam. Thus, (1+*

**DoP***)/2 is the fraction of a beam that can be blocked by an ideal polarizer.*

**DoP***= 1 are fully polarized states which lie on the surface of the unit Poincaré sphere. The Stokes vectors on the surface of the Poincaré sphere can be parameterized as*

**DoP***φ*is the latitude on the Poincaré sphere. For example, (

*θ*,

*φ*)=(0°, 0°) is horizontal linearly polarized light, (45°, 0°) is 45° polarized light, and (

*θ*, 90°) represents right circularly polarized light for all

*θ*. The Degree of Circular Polarization (

*) of a Stokes vector is defined as*

**DoCP***= S*

**DoCP**_{3}/S

_{0}= Sin(

*φ*). A circular polarizer will block (1+|

*|)/2 of a beam; the polarizer’s helicity is right or left depending on whether the state is in the upper or lower hemisphere of the Poincaré sphere, respectively.*

**DoCP**## 3. Degree of polarization surfaces and maps

*surface for a Mueller matrix,*

**DoP****M**, is formed by moving normalized Stokes vectors,

**S**, on the surface of the Poincaré sphere radially inward to a distance

*(*

**DoP****S**′=

**M**-

**S**) from the origin, plotted for all incident

**S**on the surface of the Poincaré sphere, given in Eq. (2). The

*surface results from the product of a scalar, the*

**DoP***, and a vector, (S*

**DoP**_{1}, S

_{2}, S

_{3}), formed from the last three elements of the normalized Stokes vector,

^{1/2}=1.

*map for a Mueller matrix is a contour plot of the*

**DoP***of exiting light as a function of the incident polarized state and represents a “flattened”*

**DoP***surface. In this paper, the*

**DoP***map is plotted with axes*

**DoP***(polarization ellipse major axis orientation) and DoCP, but there is some flexibility in the choice of parameterization of the polarized Stokes vectors. In general the*

**θ***map provides easier visualization of maxima, minima, saddles, and other features of the depolarization variation than the*

**DoP***surface.*

**DoP***surface and its corresponding*

**DoP***map for an example Mueller matrix with depolarization,*

**DoP***surface is the output*

**DoP***for the corresponding incident state on the Poincaré sphere. Remember that the output polarization state is in general different from the input polarization state, and this output state information is not contained in the*

**DoP***Surface. Where the surface is pinched toward the origin, those incident Stokes states are more depolarized by the Mueller matrix. In this example, (160°, 0.1) is depolarized the most and (175°, -0.7) is depolarized the least.*

**DoP***surface is analyzed here by decomposing the surface into terms relating to specific polarization properties of the Mueller matrix. The output Stokes vectors for*

**DoP****M**as a function of the input Stokes vector,

**S**, are

## 4. The denominator of the *DoP* surface

*S*

_{0}′, for input

**S**. Note that

*S*

_{0}′ depends only on the first row of

**M**. The m

_{00}element is the intensity throughput of the Mueller matrix for unpolarized light and describes losses associated with absorption, reflection or scattering. The elements m

_{01}, m

_{02}, and m

_{03}characterize the diattenuation, the tendency of

**M**to act as a partial polarizer [23

23. D. Goldstein, *Polarized Light, Second Ed., Revised and Expanded*, (Marcel Dekker, New York,2003). [CrossRef]

*ξ*varies. LimaÇons belong to a more general family of curves, the botanic curves, with polar form

**M**

_{0}constructed from the sum of horizontal linear polarizer and 45° linear polarizer Mueller matrices,

*surface for*

**DoP****M**

_{0}is

_{01}, m

_{02}, and m

_{03}define three degrees of freedom, two of which determine the diattenuation axis orientation and latitude as

*denominator surface, as well as the Poincaré sphere points yielding these transmittance extema, T*

**DoP**_{max}=((2θ)

_{axis}, ϕ

_{axis}) and T

_{min}= (π+(2θ)

_{axis}, -ϕ

_{axis}), respectively. The third degree of freedom defined by Mueller elements m

_{01}, m

_{02}, and m

_{03}describes the diattenuation (also “diattenuation magnitude”),

*d*, as

23. D. Goldstein, *Polarized Light, Second Ed., Revised and Expanded*, (Marcel Dekker, New York,2003). [CrossRef]

*d*=1), the dimple in the DoP denominator surface becomes a cusp with its point at the origin.

## 5. The degree of polarization numerator

*numerator surface, described in Eq. (6), is more complex than the denominator surface because it depends on S*

**DoP**_{1}′,S

_{2}′, and S

_{3}′ and includes a square root of the sum of three terms. The

*numerator depends on the twelve elements in the bottom three rows of the Mueller matrix, which encode the polarizance, retardance, and depolarization properties of the matrix [7, 23*

**DoP***Polarized Light, Second Ed., Revised and Expanded*, (Marcel Dekker, New York,2003). [CrossRef]

**M**is the polarizance vector,

**P**, the output Stokes vector when unpolarized light is incident,

**M**,

**Q**.

**Q**can be decomposed into the product of a unitary matrix and a Hermitian matrix in two ways [26, 27

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

**Q**matrices for retarders are unitary.

**H**or

**H**’, of submatrix

**Q**(see Eq. (18)) is diagonalizable into the form of a similarity transformation [25]

**D**

_{1}shrinks the Poincaré sphere uniformly into a smaller sphere. The corresponding Mueller matrix,

**UD**, given by

**UD**depolarizes all incident states equally.

**D**

_{2}and

**D**

_{3}indicate nonuniform contractions along orthogonal axes associated with nonuniform depolarization. Figures 4–6 show the depolarization-contracted surfaces and cross-sections associated with

**D**

_{1},

**D**

_{2}, and

**D**

_{3}, respectively. The cross sections all belong to the family of botanic curves of order c=2 (see Eq. (10)), having two petals, where the circle in Fig. 4 is the special case of

*ξ*=0.

**U**

_{1}nor

**U**

_{2}affects the

*numerator shape, and only*

**DoP****U**

_{2}affects the orientation. The contracting effect of

**D**

_{1}(along axes first rotated by

**U**

_{2}) is shown in Fig. 7. The Poincaré sphere is contracted along three orthogonal axes to radii of 0.7477, 0.5769 and 0.3981. The first contraction occurs along an axis specified by the top row of

**U**

_{2}, and so forth. The resulting surface is not always convex.

**Q**

_{0}of matrix

**M**

_{0}in Eq. (11)).

## 6. Effect of polarizance on **DoP** numerator

**DoP**

*numerator surface depends on the sum of*

**DoP****Q**and the polarizance as shown in Eqs. (6) and (15). This combination drags and distorts the

**Q**-surface in the direction of the polarizance vector (Eq. (16)), moving the origin a distance equal to the magnitude of the polarizance vector,

**P**. Before the effect of polarizance, the

*numerator surface coincides with the*

**DoP****Q**-surface, and the square root in the

*numerator represents the magnitudes of the radial vectors to this surface. Because the summation of*

**DoP****P**with

**Q**occurs inside the

*numerator square root, the induced effect on the*

**DoP***numerator surface is not merely a pure translation, but rather a translation and distortion. This transformation is animated for*

**DoP****M**

_{0}(Eq. (11)) in Fig. 8, and for

**M**

_{1}(Eq. (4)) in Fig. 9, where the final shapes in Figs. 8 and 9 are the

*numerator surfaces for*

**DoP****M**

_{0}and

**M**

_{1}, respectively.

*numerator surface is visualized in three steps, a rotation by*

**DoP****U**

_{2}, the subsequent contraction of the Poincaré sphere in three orthogonal Stokes dimensions by

**D**, and the shift and distortion (associated with the DoP numerator square root) in the direction of the three-component polarizance vector.

## 7. *DoP* surface and *DoP* map

*surface is the quotient of the*

**DoP***numerator and*

**DoP***denominator surfaces. For example, Fig. 1(a) is the*

**DoP***surface for*

**DoP****M**

_{1}(Eq. (4)), and Fig. 10 is the

*surface for*

**DoP****M**

_{0}(Eq. (11)).

*maps and surfaces may exhibit one or two maxima and one or two minima. These maxima or minima may be degenerate for entire circles of incident states around the Poincaré sphere, as in the minimum of Fig. 11 for*

**DoP****M**

_{0}. Two local maxima also appear in the

*map for*

**DoP****M**

_{0}.

**M**

_{1}has two minima and two maxima. After an extensive search, cases where the

*maps contain three or more maxima or minima have not been found.*

**DoP**## 8. Physical realizability of a Mueller matrix

*for any incident state is less than zero or greater than one, that matrix is not a physically realizable Mueller matrix. Many relationships among Mueller matrix elements to ensure physical realizability have been published [28–30]. Such relationships can be understood geometrically in terms of the numerator and denominator surfaces. The numerator surface must not protrude from the denominator surface at any point or the*

**DoP***at that point is greater than one. The two surfaces may be tangent and the*

**DoP***is one at points of tangency. Similarly, the origin of the Poincaré sphere must lie within or on both surfaces, or the*

**DoP***will be negative. When the numerator surface passes through the origin, the corresponding state is completely depolarized.*

**DoP***numerator and denominator surfaces together for a given Mueller matrix aids the visualization of these geometrical relationships and is useful for establishing physical realizability. For example, the numerator and denominator surfaces for*

**DoP****M**

_{1}(Eq. (4)) are plotted in Fig. 12. The denominator surface surrounds the numerator surface, so

**M**

_{1}is physical.

**M**

_{0}parameterized by coefficient

*,*

**i***increases the numerator surface uniformly grows starting as a single point at the origin. Figure 13 animates the growth of the numerator surface within the denominator surface for*

**i***ranging from 0.0 to 0.7. When*

**i***= 0.5 the numerator surface is tangent to the denominator surface at the two maxima of Fig. 11. When*

**i***exceeds 0.5, the numerator surface protrudes from the denominator surface and the Mueller matrices are non-physical.*

**i**## 9. A family of *DoP* maps

*surfaces and maps. Observing the metamorphosis of*

**DoP***surfaces and maps clarifies the depolarizing effects of the matrix family. Consider the family of Mueller matrices*

**DoP****LP**(α), with its polarization axis at an angle α to horizontal. Figure 14 shows the

*surfaces for each member of this family plotted within the Poincaré sphere, and Fig. 15 the corresponding*

**DoP***maps. Figure 16 animates the evolution of*

**DoP***surfaces.*

**DoP***= 1 which occur for Stokes vectors*

**DoP***orthogonal*to the two polarizer axes. An incident state orthogonal to one of the polarizers is completely blocked by that polarizer and light only transmits through the other polarizer, thus emerging with

*= 1. States aligned with either polarizer also partially transmit through the second polarizer (unless the two polarizers are orthogonal) and the contribution of these two beams reduces the*

**DoP***. Since one of the polarizers forming every member of Eq. (25) is a horizontal linear polarizer, every*

**DoP***surface and map in this family shows a maximum*

**DoP***= 1 for vertically polarized light. The other*

**DoP***maximum rotates around the Poincaré sphere as the second polarizer rotates. Half way between these maxima is a valley of increasing width and depth (see Fig. 14). When the second polarizer axis is at 90° to the first, the output*

**DoP***maxima occur for input vertically and horizontally polarized light and the valley reaches a*

**DoP***= 0. The*

**DoP***surface in this case separates into two tangent spheres. The*

**DoP***= 0 output states occur for input states on the great circle of the Poincaré sphere through 45°, right circular, 135°, and left circular states.*

**DoP**## 10. Conclusion

*maps and surfaces represent the variation of depolarization of a Mueller matrix for all fully polarized incident states. For non-depolarizing Mueller matrices, the*

**DoP***surface is the unit sphere and the*

**DoP***map is unity everywhere. For depolarizing Mueller matrices the*

**DoP***surface contracts toward the origin by an amount equal to the depolarization for that incident state. Thus, the maps and surfaces indicate the variation of depolarization with incident state.*

**DoP***numerator surface function has nine degrees of freedom relating to depolarization, retardance, and polarizance. The unitary matrix*

**DoP****U**

_{1}in the singular value decomposition of Mueller submatrix

**Q**is a rotation of the output polarization state due to retardance which does not change the

*surface. Three depolarization degrees of freedom in matrix*

**DoP****D**contract the Poincaré sphere along three axes rotated from the Stokes basis states by the retardance effects of unitary matrix

**U**

_{2}. The polarizance vector acts via an additive term that drags and distorts the surface in the direction of the polarizance vector.

*surfaces may also be plotted in terms of the output Stokes states rather than input Stokes states (making visible the rotation effects of*

**DoP****U**

_{1}) but this representation is not explored here.

*numerator surface must lie entirely within the denominator surface and must incorporate the Poincaré sphere origin (either on or within the surface). Physically realizable*

**DoP***maps have only been observed with one or two maxima and one or two minima. It is postulated that the*

**DoP***maps cannot have more than two maxima or minima, but such a conjecture is not proven here.*

**DoP***maps yield a detailed picture of depolarization and are of useful for understanding and classifying depolarizing Mueller matrices.*

**DoP**## Acknowledgments

## References and Links

1. | R.C. Jones, “A new calculus for the treatment of optical systems: I. Description and discussion of the calculus,” J. Opt. Soc. Am. |

2. | R.C. Jones, “New calculus for the treatment of optical systems: VIII. Electromagnetic theory,” J. Opt. Soc. Am. |

3. | M. Izdebski, W. Kucharczyk, and R.E. Raab, “Application of the Jones calculus for a modulated double-refracted light beam propagating in a homogeneous and nondepolarizing electro-optic uniaxial crystal,” J. Opt. Soc. Am. A |

4. | K.D. Abhyankar and A.L. Fymat, “Relations between the elements of the phase matrix for scattering,” J. Math Phys. |

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

6. | W. Swindell, ed., |

7. | R.A. Chipman, “Polarimetry,” in |

8. | J.L. Pezzaniti, S.C. McClain, R.A. Chipman, and S.Y. Lu, “Depolarization in liquid crystal TV’s,” Opt. Lett. |

9. | S.-M.F. Nee and T.-W. Nee, “Principal Mueller matrix of reflection and scattering for a one-dimensional rough surface,” Opt. Eng. |

10. | M.W. Williams, “Depolarization and cross polarization in ellipsometry of rough surfaces.” Appl. Opt. |

11. | B. Laude-Boulesteix, A. DeMartino, B. Drévilon, and L. Schwartz, “Mueller polarimetric imaging system with liquid crystals,” Appl. Opt. |

12. | N. Ghosh, H.S. Patel, and P.K. Gupta, “Depolarization of light in tissue phantoms—effect of a distribution in the size of scatterers,” Opt. Express |

13. | K. Sassen, “Cirrus cloud iridescence: a rare case study,” Appl. Opt. |

14. | G.E. Jellison, C.O. Griffiths, D.E. Holcomb, and C.M. Rouleau, “Transmission two-modulator generalized ellipsometry measurements,” Appl. Opt. |

15. | W.-N. Chen, C.-W. Chiang, and J.-B. Nee, “Lidar ratio and depolarization ratio for cirrus clouds,” Appl. Opt. |

16. | V.A. Ruiz-Cortés and J.C. Dainty, “Experimental light-scattering measurements from large-scale composite randomly rough surfaces,” J. Opt. Soc. Am. A |

17. | I. Shoji, Y. Sato, S. Kurimura, V. Lupei, T. Taira, A. Ikesue, and K. Yoshida, “Thermal-birefringence-induced depolarization in NdYAG ceramics,” Opt. Lett. |

18. | S.-M. F. Nee, “Depolarization and principal Mueller matrix measured by null ellipsometry,” Appl. Opt. |

19. | M. Moscoso, J.B. Keller, and G. Papanicolaou, “Depolarization and blurring of optical images by biological tissue,” J. Opt. Soc. Am. A |

20. | S.-M.F. Nee, “Depolarization and retardation of a birefringent slab,” J. Opt. Soc. Am. A |

21. | J.J. Gil and E. Bernabeau, “Depolarization and polarization indices of an optical system,” Opt. Acta |

22. | B. DeBoo, “Investigation of polarization scatter properties using active imaging polarimetry,” PhD dissertation, Optical Sciences Center, University of Arizona (2004). |

23. | D. Goldstein, |

24. | R.C. Yates, |

25. | G.B. Arfken and H.J. Weber, |

26. | S.-Y. Lu, “An interpretation of polarization matrices,” PhD dissertation, Dept. of Physics, University of Alabama at Huntsville (1995). |

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

28. | C. Brosseau, |

29. | C.R. Givens and A.B. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt. |

30. | J. Cariou, B. Le Jeune, J. Lotrian, and Y. Guern, “Polarization effects of seawater and underwater targets,” Appl. Opt. |

**OCIS Codes**

(120.5410) Instrumentation, measurement, and metrology : Polarimetry

(230.5440) Optical devices : Polarization-selective devices

(260.5430) Physical optics : Polarization

**ToC Category:**

Research Papers

**History**

Original Manuscript: August 20, 2004

Revised Manuscript: September 24, 2004

Published: October 4, 2004

**Citation**

Brian DeBoo, J. Sasian, and R. Chipman, "Degree of polarization surfaces and maps for analysis of depolarization," Opt. Express **12**, 4941-4958 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-20-4941

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### References

- R.C. Jones, �??A new calculus for the treatment of optical systems: I. Description and discussion of the calculus,�?? J. Opt. Soc. Am. 31, 488-493 (1941). [CrossRef]
- R.C. Jones, �??New calculus for the treatment of optical systems: VIII. Electromagnetic theory,�?? J. Opt. Soc. Am. 46, 126-131 (1956). [CrossRef]
- M. Izdebski, W. Kucharczyk, and R.E. Raab, �??Application of the Jones calculus for a modulated doublerefracted light beam propagating in a homogeneous and nondepolarizing electro-optic uniaxial crystal,�?? J. Opt. Soc. Am. A 21, 132-139 (2004). [CrossRef]
- K.D. Abhyankar and A.L. Fymat, �??Relations between the elements of the phase matrix for scattering,�?? J. Math Phys. 10, 1935-1938 (1969). [CrossRef]
- E.S. Fry and G.W. Kattawar, �??Relationships between elements of the Stokes matrix,�?? Appl. Opt. 20, 2811-2814 (1981). [CrossRef] [PubMed]
- W. Swindell, ed., Benchmark Papers in Optics: Polarized Light, (Hutchinson and Ross, Stroudsburg, 1975).
- R.A. Chipman, �??Polarimetry,�?? in Handbook of Optics, Vol. II, (McGraw Hill, New York, 1995).
- J.L. Pezzaniti, S.C. McClain, R.A. Chipman, and S.Y. Lu, �??Depolarization in liquid crystal TV�??s,�?? Opt. Lett. 18, 2071-2073 (1993). [CrossRef] [PubMed]
- S.-M.F. Nee and T.-W. Nee, �??Principal Mueller matrix of reflection and scattering for a one-dimensional rough surface,�?? Opt. Eng. 41, 994-1001 (2002). [CrossRef]
- M.W. Williams, �??Depolarization and cross polarization in ellipsometry of rough surfaces.�?? Appl. Opt. 25, 3616-3621 (1986). [CrossRef] [PubMed]
- B. Laude-Boulesteix, A. DeMartino, B. Drévilon, and L. Schwartz, �??Mueller polarimetric imaging system with liquid crystals,�?? Appl. Opt. 43, 2824-2832 (2004). [CrossRef] [PubMed]
- N. Ghosh, H.S. Patel, and P.K. Gupta, �??Depolarization of light in tissue phantoms�??effect of a distribution in the size of scatterers,�?? Opt. Express 11, 2198-2205 (2003). [CrossRef] [PubMed]
- K. Sassen, �??Cirrus cloud iridescence: a rare case study,�?? Appl. Opt. 42, 486-491 (2003). [CrossRef] [PubMed]
- G.E. Jellison, C.O. Griffiths, D.E. Holcomb, and C.M. Rouleau, �??Transmission two-modulator generalized ellipsometry measurements,�?? Appl. Opt. 41, 6555-6566 (2002). [CrossRef] [PubMed]
- W.-N. Chen, C.-W. Chiang, J.-B. Nee, �??Lidar ratio and depolarization ratio for cirrus clouds,�?? Appl. Opt. 41, 6470-6476 (2002). [CrossRef] [PubMed]
- V.A. Ruiz-Cortés, and J.C. Dainty, �??Experimental light-scattering measurements from large-scale composite randomly rough surfaces,�?? J. Opt. Soc. Am. A 19, 2043-2052 (2002). [CrossRef]
- I. Shoji, Y. Sato, S. Kurimura, V. Lupei, T. Taira, A. Ikesue, and K. Yoshida, �??Thermal-birefringenceinduced depolarization in NdYAG ceramics,�?? Opt. Lett. 27, 234-236 (2002). [CrossRef]
- S.-M. F. Nee, �??Depolarization and principal Mueller matrix measured by null ellipsometry,�?? Appl. Opt. 40, 4933-4939 (2001). [CrossRef]
- M. Moscoso, J.B. Keller, and G. Papanicolaou, �??Depolarization and blurring of optical images by biological tissue,�?? J. Opt. Soc. Am. A 18, 948-960 (2001). [CrossRef]
- S.-M.F. Nee, �??Depolarization and retardation of a birefringent slab,�?? J. Opt. Soc. Am. A 17, 2067-2073 (2000). [CrossRef]
- J.J. Gil and E. Bernabeau, �??Depolarization and polarization indices of an optical system,�?? Opt. Acta 33, 185-189 (1986). [CrossRef]
- B. DeBoo, �??Investigation of polarization scatter properties using active imaging polarimetry,�?? PhD dissertation, Optical Sciences Center, University of Arizona (2004).
- D. Goldstein, Polarized Light, Second Ed., Revised and Expanded, (Marcel Dekker, New York, 2003). [CrossRef]
- R.C. Yates, A Handbook on Curves and Their Properties, (J. W. Edwards, Ann Arbor, 1952).
- G.B. Arfken, and H.J. Weber, Mathematical Methods for Physicists, (Academic Press, San Diego, 2001).
- S.-Y. Lu, �??An interpretation of polarization matrices,�?? PhD dissertation, Dept. of Physics, University of Alabama at Huntsville (1995).
- S.-Y. Lu and R.A. Chipman, �??Interpretation of Mueller matrices based on polar decomposition,�?? J. Opt. Soc. Am. A 13, 1106-1113 (1996). [CrossRef]
- C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach, (John Wiley and Sons, Inc., New York, 1998).
- C.R. Givens and A.B. Kostinski, �??A simple necessary and sufficient condition on physically realizable Mueller matrices,�?? J. Mod. Opt. 41, 471-481 (1994).
- J. Cariou, B. Le Jeune, J. Lotrian, and Y. Guern, �??Polarization effects of seawater and underwater targets,�?? Appl. Opt. 29, 1689- (1990) [CrossRef] [PubMed]

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### Supplementary Material

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