Abstract

The polarization properties of light scattered or diffusely reflected from seven different man-made samples are studied. For each diffusely reflecting sample an in-plane Mueller matrix bidirectional reflectance distribution function is measured at a fixed bistatic angle using a Mueller matrix imaging polarimeter. The measured profile of depolarization index with changing scattering geometry for most samples is well approximated by an inverted Gaussian function. Depolarization is minimum for specular reflection and increases asymptotically in a Gaussian fashion as the angles of incidence and scatter increase. Parameters of the Gaussian profiles fitted to the depolarization data are used to compare samples. The dependence of depolarization on the incident polarization state is compared for each Stokes basis vector: horizontal, vertical, 45°, 135°, and right- and left-circular polarized light. Linear states exhibit similar depolarization profiles that typically differ in value by less than 0.06 (where 1.0 indicates complete depolarization). Circular polarization states are depolarized more than linear states for all samples tested, with the output degree of polarization reduced from that of linear states by as much as 0.15. The depolarization difference between linear and circular states varies significantly between samples.

© 2005 Optical Society of America

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References

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2005

2004

2003

2002

M. H. Smith, “Optimization of a dual-rotating-retarder Mueller matrix imaging polarimeter,” Appl. Opt. 41, 2488–2493 (2002).
[CrossRef] [PubMed]

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

1998

1996

1995

1992

1991

1986

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

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

1985

J. J. Gil, E. Bernabeu, “A depolarization criterion in Mueller matrices,” Opt. Acta 32, 259–261 (1985).
[CrossRef]

Beckman, P.

P. Beckman, A. Spizzichino, Scattering of Electromagnetic Radiation from Rough Surfaces (Franklin, 1963).

Bernabeu, E.

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

J. J. Gil, E. Bernabeu, “A depolarization criterion in Mueller matrices,” Opt. Acta 32, 259–261 (1985).
[CrossRef]

Bickel, W. S.

Bjork, D. R.

F. Schiff, J. C. Stover, D. R. Bjork, B. D. Swimley, “Mueller matrix measurements of scattered light,” in Polarization Analysis and Measurement, R. Chipman, D. Goldstein, eds., Proc. SPIE1746, 311–318 (1992).

Chipman, R. A.

R. A. Chipman, “Depolarization index and the average degree of polarization,” Appl. Opt. 44, 2490–2495 (2005).
[CrossRef] [PubMed]

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

J. L. Pezzaniti, R. A. Chipman, “Mueller matrix scatter polarimetry of a diamond-turned mirror,” Opt. Eng. 34, 1593–1598 (1995).
[CrossRef]

E. A. Sornsin, R. A. Chipman, “Polarization BRDF of satellite materials,” presented at the Workshop on Infrared and Millimeter Wave Polarimetry, Redstone Arsenal, Alabama, 5–7 December 1995.

R. A. Chipman, “Polarimetry,” in Handbook of Optics (McGraw-Hill, 1995), Vol. 2.

Dayton, D. C.

B. G. Hoover, D. C. Dayton, J. E. Havey, “Active detection of off-diagonal Mueller elements of rough targets,” in Polarization Science and Remote Sensing, J. A. Shaw, J. S. Tyo, eds., Proc. SPIE5158, 226–238 (2003).
[CrossRef]

Germer, T. A.

T. A. Germer, E. Marx, “Ray model of light scattering by flake pigments or rough surfaces with smooth transparent coatings,” Appl. Opt. 43, 1266–1274 (2004).
[CrossRef] [PubMed]

T. A. Germer, M. E. Nadal, “Modeling the appearance of special effect pigment coatings,” in Surface Scattering and Diffraction for Advanced Metrology, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE4447, 77–86 (2001).
[CrossRef]

Gil, J. J.

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

J. J. Gil, E. Bernabeu, “A depolarization criterion in Mueller matrices,” Opt. Acta 32, 259–261 (1985).
[CrossRef]

Havey, J. E.

B. G. Hoover, D. C. Dayton, J. E. Havey, “Active detection of off-diagonal Mueller elements of rough targets,” in Polarization Science and Remote Sensing, J. A. Shaw, J. S. Tyo, eds., Proc. SPIE5158, 226–238 (2003).
[CrossRef]

Hoover, B. G.

B. G. Hoover, D. C. Dayton, J. E. Havey, “Active detection of off-diagonal Mueller elements of rough targets,” in Polarization Science and Remote Sensing, J. A. Shaw, J. S. Tyo, eds., Proc. SPIE5158, 226–238 (2003).
[CrossRef]

Hovenier, J. W.

Hsu, J.-Y.

Jakeman, E.

Jordan, D. L.

Jun, K. H.

Knotts, M. E.

Kwak, J. H.

Lewis, G. D.

Lim, K. S.

Lu, S.-Y.

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

S.-Y. Lu, “An interpretation of polarization matrices,” Ph.D. dissertation (Department of Physics, University of Alabama at Huntsville, 1995).

Luna, R. E.

Marx, E.

Mendez, E. R.

Michael, T. R.

Mishchenko, M. I.

Nadal, M. E.

T. A. Germer, M. E. Nadal, “Modeling the appearance of special effect pigment coatings,” in Surface Scattering and Diffraction for Advanced Metrology, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE4447, 77–86 (2001).
[CrossRef]

Navarrete, A. G.

Nee, S.-M. F.

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

S.-M. F. Nee, T.-W. Nee, “Polarization of scattering by rough surfaces,” in Scattering and Surface Roughness II, Z. H. Gu, A. A. Maradudin, eds., Proc. SPIE3426, 169–180 (1998).
[CrossRef]

Nee, T.-W.

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

S.-M. F. Nee, T.-W. Nee, “Polarization of scattering by rough surfaces,” in Scattering and Surface Roughness II, Z. H. Gu, A. A. Maradudin, eds., Proc. SPIE3426, 169–180 (1998).
[CrossRef]

O’Donnell, K. A.

Pezzaniti, J. L.

J. L. Pezzaniti, R. A. Chipman, “Mueller matrix scatter polarimetry of a diamond-turned mirror,” Opt. Eng. 34, 1593–1598 (1995).
[CrossRef]

Schiff, F.

F. Schiff, J. C. Stover, D. R. Bjork, B. D. Swimley, “Mueller matrix measurements of scattered light,” in Polarization Analysis and Measurement, R. Chipman, D. Goldstein, eds., Proc. SPIE1746, 311–318 (1992).

Smith, M. H.

Sornsin, E. A.

E. A. Sornsin, R. A. Chipman, “Polarization BRDF of satellite materials,” presented at the Workshop on Infrared and Millimeter Wave Polarimetry, Redstone Arsenal, Alabama, 5–7 December 1995.

Spizzichino, A.

P. Beckman, A. Spizzichino, Scattering of Electromagnetic Radiation from Rough Surfaces (Franklin, 1963).

Stover, J. C.

J. C. Stover, Optical Scatter Measurement and Analysis, 2nd ed. (SPIE Press, 1995).
[CrossRef]

F. Schiff, J. C. Stover, D. R. Bjork, B. D. Swimley, “Mueller matrix measurements of scattered light,” in Polarization Analysis and Measurement, R. Chipman, D. Goldstein, eds., Proc. SPIE1746, 311–318 (1992).

Swimley, B. D.

F. Schiff, J. C. Stover, D. R. Bjork, B. D. Swimley, “Mueller matrix measurements of scattered light,” in Polarization Analysis and Measurement, R. Chipman, D. Goldstein, eds., Proc. SPIE1746, 311–318 (1992).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

Videen, G.

Williams, M. W.

Wolfe, W.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Acta

J. J. Gil, E. Bernabeu, “A depolarization criterion in Mueller matrices,” Opt. Acta 32, 259–261 (1985).
[CrossRef]

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

Opt. Eng.

J. L. Pezzaniti, R. A. Chipman, “Mueller matrix scatter polarimetry of a diamond-turned mirror,” Opt. Eng. 34, 1593–1598 (1995).
[CrossRef]

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

Opt. Lett.

Other

F. Schiff, J. C. Stover, D. R. Bjork, B. D. Swimley, “Mueller matrix measurements of scattered light,” in Polarization Analysis and Measurement, R. Chipman, D. Goldstein, eds., Proc. SPIE1746, 311–318 (1992).

S.-M. F. Nee, T.-W. Nee, “Polarization of scattering by rough surfaces,” in Scattering and Surface Roughness II, Z. H. Gu, A. A. Maradudin, eds., Proc. SPIE3426, 169–180 (1998).
[CrossRef]

B. G. Hoover, D. C. Dayton, J. E. Havey, “Active detection of off-diagonal Mueller elements of rough targets,” in Polarization Science and Remote Sensing, J. A. Shaw, J. S. Tyo, eds., Proc. SPIE5158, 226–238 (2003).
[CrossRef]

T. A. Germer, M. E. Nadal, “Modeling the appearance of special effect pigment coatings,” in Surface Scattering and Diffraction for Advanced Metrology, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE4447, 77–86 (2001).
[CrossRef]

E. A. Sornsin, R. A. Chipman, “Polarization BRDF of satellite materials,” presented at the Workshop on Infrared and Millimeter Wave Polarimetry, Redstone Arsenal, Alabama, 5–7 December 1995.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

P. Beckman, A. Spizzichino, Scattering of Electromagnetic Radiation from Rough Surfaces (Franklin, 1963).

J. C. Stover, Optical Scatter Measurement and Analysis, 2nd ed. (SPIE Press, 1995).
[CrossRef]

R. A. Chipman, “Polarimetry,” in Handbook of Optics (McGraw-Hill, 1995), Vol. 2.

S.-Y. Lu, “An interpretation of polarization matrices,” Ph.D. dissertation (Department of Physics, University of Alabama at Huntsville, 1995).

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Figures (14)

Fig. 1
Fig. 1

MMIP used for in-plane MmBRDF measurements, with bistatic angle β = 14° between the polarization generator arm and the analyzer arm. The sample rotation angle θ is measured with respect to the specular reflection condition.

Fig. 2
Fig. 2

Normalized in-plane MmBRDF plot for a green painted metal sample. Each of the 4 × 4 arrays of graphs contains a different Mueller matrix element for the light-scattering process as the sample rotates from −70° to 70° with respect to the specular condition (0°).

Fig. 3
Fig. 3

Normalized in-plane MmBRDF plot for gold-coated diffuser as a function of sample rotation angle θ.

Fig. 4
Fig. 4

Normalized in-plane MmBRDF plot for screen mesh fabric as a function of sample rotation angle θ.

Fig. 5
Fig. 5

Normalized in-plane MmBRDF plot for a glass diffuser as a function of sample rotation angle θ. The spike at θ = 0 in the m11 element indicates the increase in specular reflection relative to the diffusely scattering background.

Fig. 6
Fig. 6

Normalized in-plane MmBRDF plot for canvas material as a function of sample rotation angle θ.

Fig. 7
Fig. 7

Normalized in-plane MmBRDF plot for concrete as a function of sample rotation angle θ.

Fig. 8
Fig. 8

Normalized in-plane MmBRDF plot for green nylon (plastic) material as a function of sample rotation angle θ.

Fig. 9
Fig. 9

Depolarization index versus sample rotation angle θ for five man-made scattering samples tends to follow an inverted Gaussian functional form. Labeled plot styles represent (a) painted metal, (b) canvas, (c) concrete, (d) glass diffuser, and (e) gold-coated diffuser. The depolarization index indicates the DOP of the exiting light averaged over all possible polarized incident states. A value of zero indicates no depolarization, whereas a value of one indicates that the scattered light is completely unpolarized for polarized illumination.

Fig. 10
Fig. 10

Average depolarization index versus sample rotation angle for the glass diffuser, with inverted Gaussian fit.

Fig. 11
Fig. 11

Depolarization profiles as a function of sample rotation angle for the nylon plastic and screen mesh samples.

Fig. 12
Fig. 12

DOP in response to the six Stokes basis states for six samples. Circularly polarized light is consistently depolarized more than linearly polarized light. Solid curves, H, horizontal; dotted curves with triangles, V, vertical; dotted–dashed curves, 45°; dashed curves with circles, 135°; dotted–dotted–dashed curves, R, right-circular; solid curves with squares, L, left-circular.

Fig. 13
Fig. 13

Mueller matrix image of scattered light from a glass diffuser illuminated with a collimated beam. Normalized Mueller matrix images range from −1 to 1. Here the positive Mueller matrix values are displayed in the left panel and the negative values are displayed in the right panel with black indicating zero in both image arrays since gray-scale printing does not lend itself well to displaying the positive and negative values in a single image.

Fig. 14
Fig. 14

Coordinate system for scattered light measurement. θi is the angle of incidence of a beam of solid angle Ωi on a surface element with surface normal z. θs is the angle from the sample surface normal at which a solid angle Ωs of diffusely reflected light is collected. The planes of incidence and scatter are rotated from the xz plane of reference by angles ϕi and ϕs, respectively.

Tables (2)

Tables Icon

Table 1 Man-Made Samples Characterized by Scatter Polarimetry with a MMIP

Tables Icon

Table 2 Shape and Quality Parameters of Gaussian Profiles Fitted to the Average Depolarization Data

Equations (14)

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Dep ( M ) = 1 - ( i , j m i j 2 ) - m 11 2 3 m 11 ,
f ( x ) = K - A exp ( - θ 2 / σ 2 ) ,
[ S 0 S 1 S 2 S 3 ] = [ m 11 m 12 m 13 m 14 m 21 m 22 m 23 m 24 m 31 m 32 m 33 m 34 m 41 m 42 m 43 m 44 ] [ S 0 S 1 S 2 S 3 ] .
M ˜ = [ m 11 m 12 / m 11 m 13 / m 11 m 14 / m 11 m 21 / m 11 m 22 / m 11 m 23 / m 11 m 24 / m 11 m 31 / m 11 m 32 / m 11 m 33 / m 11 m 34 / m 11 m 41 / m 11 m 42 / m 11 m 43 / m 11 m 44 / m 11 ] .
M R ( θ ) = R ( θ ) · M · R ( θ ) = [ 1 0 0 0 0 cos ( θ ) - sin ( θ ) 0 0 sin ( θ ) cos ( θ ) 0 0 0 0 1 ] [ m 11 m 12 m 13 m 14 m 21 m 22 m 23 m 24 m 31 m 32 m 33 m 34 m 41 m 42 m 43 m 44 ] × [ 1 0 0 0 0 cos ( θ ) - sin ( θ ) 0 0 sin ( θ ) cos ( θ ) 0 0 0 0 1 ]
M T ( θ ) = R ( - θ ) · M · R ( θ )
M refl = [ 1 0 0 0 0 1 0 0 0 0 - 1 0 0 0 0 - 1 ] .
M = M depol M ret M diat .
DOP ( S ) = S 1 2 + S 2 2 + S 3 2 S 0 .
ID = [ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] , UD = [ 1 0 0 0 0 a 0 0 0 0 - a 0 0 0 0 - a ] , DNUD = [ 1 0 0 0 0 a 0 0 0 0 - b 0 0 0 0 - c ] .
NUD = [ 1 0 0 0 m 21 m 22 0 0 m 31 0 m 33 0 m 42 0 0 m 44 ] ,
Dep ( M ) = 1 - ( i , j m i j 2 ) - m 11 2 3 m 11 .
BRDR ( φ i , θ i , φ s , θ s ) = d L s ( φ s , θ s ) d E i ( φ i , θ i ) .
MmBRDF ( φ i , θ i , φ s , θ s ) = [ m 11 ( φ i , θ i , φ s , θ s ) m 14 ( φ i , θ i , φ s , θ s ) m 41 ( φ i , θ i , φ s , θ s ) m 44 ( φ i , θ i , φ s , θ s ) ] .

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