Abstract

A polarimetric bidirectional reflectance distribution function (pBRDF), based on geometrical optics, is presented. The pBRDF incorporates a visibility (shadowing/masking) function and a Lambertian (diffuse) component which distinguishes it from other geometrical optics pBRDFs in literature. It is shown that these additions keep the pBRDF bounded (and thus a more realistic physical model) as the angle of incidence or observation approaches grazing and better able to model the behavior of light scattered from rough, reflective surfaces. In this paper, the theoretical development of the pBRDF is shown and discussed. Simulation results of a rough, perfect reflecting surface obtained using an exact, electromagnetic solution and experimental Mueller matrix results of two, rough metallic samples are presented to validate the pBRDF.

© 2009 Optical Society of America

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    [CrossRef]
  2. C. Eckart, "The scattering of sound from the sea surface," J. Acoust. Soc. Am. 25, 566-570 (1953).
    [CrossRef]
  3. E. Y. Harper and F. M. Labianca, "Scattering of sound from a point source by a rough surface progressing over an isovelocity ocean," J. Acoust. Soc. Am. 58(2), 349-364 (1975).
    [CrossRef]
  4. K. Krishen, "Correlation of radar backscattering cross sections with ocean wave height and wind velocity," J. Geophys. Res. 76, 6528-6539 (1971).
    [CrossRef]
  5. B. W. Hapke, "A theoretical photometric function for the lunar surface," J. Geophys. Res. 68(15), 4571-4586 (1963).
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    [CrossRef]
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  11. R. L. Cook and K. E. Torrance, "A reflectance model for computer graphics," in SIGGRAPH 1981 Proceedings, vol. 15, pp. 307-316, Special Interest Group on Graphics and Interactive Techniques (Computer Graphics, 1981).
  12. X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, "A comprehensive physical model for light reflection," in SIGGRAPH 1991 Proceedings, vol. 25, pp. 175-186, Special Interest Group on Graphics and Interactive Techniques (Computer Graphics, 1991).
  13. M. A. Greiner, B. D. Duncan, and M. P. Dierking, "Bidirectional scattering distribution functions of maple and cottonwood leaves," Appl. Opt. 46(25), 6485-6494 (2007).
    [CrossRef]
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    [CrossRef]
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  16. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, Inc., Norwood, MA, 1963).
  17. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, New York, NY, 1997).
  18. C.-H. An and K. J. Zeringue, "Polarization scattering from rough surfaces based on the vector Kirchoff diffraction model," in Proc. SPIE, vol. 5158, pp. 205-216 (The International Society for Optical Engineering (SPIE), 2003).
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    [CrossRef]
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  23. R. G. Priest and T. A. Germer, "Polarimetric BRDF in the microfacet model: theory and measurements," in Proceedings of the 2000 Meeting of the Military Sensing Symposia Specialty Group on Passive Sensors, pp. 169-181 (Infrared Information Analysis Center, 2000).
  24. R. G. Priest and S. R. Meier, "Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces," Opt. Eng. 41(5), 988-993 (2002).
    [CrossRef]
  25. D. Wellems, S. Ortega, D. Bowers, J. Boger, and M. Fetrow, "Long wave infrared polarimetric model: theory, measurements and parameters," J. Opt. A: Pure Appl. Opt. 8(10), 914-925 (2006).
    [CrossRef]
  26. D. Wellems, M. Serna, S. H. Sposato, M. P. Fetrow, K. P. Bishop, S. A. Arko, and T. R. Caudill, "Spectral polarimetric BRDF model and comparison to measurements from isotropic roughened glass," in Workshop on Multi/Hyperspectral Sensors, Measurements, Modeling and Simulation (U.S. Army Aviation and Missile Command, Huntsville, AL, 2000).
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    [CrossRef]
  29. R. Anderson, "Matrix description of radiometric quantities," Appl. Opt. 30(7), 858-867 (1991).
    [CrossRef]
  30. D. S. Flynn and C. Alexander, "Polarized surface scattering expressed in terms of a bidirectional reflectance distribution function," Opt. Eng. 34(6), 1646-1650 (1995).
    [CrossRef]
  31. F. E. Nicodemus, "Radiance," Am. J. Phys. 31, 368-377 (1963).
    [CrossRef]
  32. F. E. Nicodemus, "Directional reflectance and emissivity of an opaque surface," Appl. Opt. 4(7), 368-377 (1965).
  33. J. R. Schott, Fundamentals of Polarimetric Remote Sensing (SPIE Press, Bellingham, WA, 2009).
    [CrossRef]
  34. J. R. Shell, "Polarimetric Remote Sensing in the Visible to Near Infrared," Ph.D. dissertation, Chester F. Carslon Center for Imaging Science, Rochester Institute of Technology, Rochester, NY (2005).
  35. Y. Sun, "Statistical ray method for deriving reflection models of rough surfaces," J. Opt. Soc. Am. A 24(3), 724-744 (2007).
    [CrossRef]
  36. W. S. Bickel and W. M. Bailey, "Stokes vectors, Mueller matrices, and polarized scattered light," Am. J. Phys. 53(5), 468-478 (1985).
    [CrossRef]
  37. M. G. Gartley, S. D. Brown, and J. R. Schott, "Micro-scale surface and contaminate modeling for polarimetric signature prediction," in Proc. SPIE, vol. 6972 (The International Society for Optical Engineering (SPIE), 2008).
    [CrossRef]
  38. J. R. Maxwell, J. Beard, S. Weiner, D. Ladd, and S. Ladd, "Bidirectional Reflectance Model Validation and Utilization," Tech. Rep. AFAL-TR-73-303, Air Force Avionics Laboratory, Wright-Patterson Air Force Base, OH (1973).
  39. M. G. Gartley, "Polarimetric Modeling of Remotely Sensed Scenes in the Thermal Infrared," Ph.D. dissertation, Chester F. Carslon Center for Imaging Science, Rochester Institute of Technology, Rochester, NY (2007).
  40. A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics (IEEE Press, New York, NY, 1998).
  41. R. M. Axline and A. K. Fung, "Numerical computation of scattering from a perfectly conducting random surface," IEEE Trans. Antennas Propag. AP-26(3), 482-488 (1978).
    [CrossRef]
  42. E. I. Thorsos, "The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum," J. Acoust. Soc. Am. 83(1), 78-92 (1988).
    [CrossRef]
  43. A. K. Fung and M. F. Chen, "Numerical simulation of scattering from simple and composite random surfaces," J. Opt. Soc. Am. A 2(12), 2274-2284 (1985).
    [CrossRef]
  44. M. F. Chen and S. Y. Bai, "Computer simulation of wave scattering from a dielectric random surface in two dimensions—cylindrical case," J. Electromagn. Waves Appl. 4(10), 963-982 (1990).
    [CrossRef]
  45. E. Compain, S. Poirier, and B. Drevillon, "General and self-consistent method for the calibration of polarization modulators, polarimeters, and Mueller-matrix ellipsometers," Appl. Opt. 38(16), 3490-3502 (1999).
    [CrossRef]
  46. LabSphere, Inc., "A guide to reflectance coatings and materials," http://www.labsphere.com/tecdocs.aspx.
  47. Luxpop, Inc.http://www.luxpop.com/.

2008

2007

2006

D. Wellems, S. Ortega, D. Bowers, J. Boger, and M. Fetrow, "Long wave infrared polarimetric model: theory, measurements and parameters," J. Opt. A: Pure Appl. Opt. 8(10), 914-925 (2006).
[CrossRef]

2002

R. G. Priest and S. R. Meier, "Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces," Opt. Eng. 41(5), 988-993 (2002).
[CrossRef]

1999

1996

1995

D. S. Flynn and C. Alexander, "Polarized surface scattering expressed in terms of a bidirectional reflectance distribution function," Opt. Eng. 34(6), 1646-1650 (1995).
[CrossRef]

1991

1990

M. F. Chen and S. Y. Bai, "Computer simulation of wave scattering from a dielectric random surface in two dimensions—cylindrical case," J. Electromagn. Waves Appl. 4(10), 963-982 (1990).
[CrossRef]

1988

E. I. Thorsos, "The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum," J. Acoust. Soc. Am. 83(1), 78-92 (1988).
[CrossRef]

1985

A. K. Fung and M. F. Chen, "Numerical simulation of scattering from simple and composite random surfaces," J. Opt. Soc. Am. A 2(12), 2274-2284 (1985).
[CrossRef]

W. S. Bickel and W. M. Bailey, "Stokes vectors, Mueller matrices, and polarized scattered light," Am. J. Phys. 53(5), 468-478 (1985).
[CrossRef]

1984

1978

R. M. Axline and A. K. Fung, "Numerical computation of scattering from a perfectly conducting random surface," IEEE Trans. Antennas Propag. AP-26(3), 482-488 (1978).
[CrossRef]

1975

E. Y. Harper and F. M. Labianca, "Scattering of sound from a point source by a rough surface progressing over an isovelocity ocean," J. Acoust. Soc. Am. 58(2), 349-364 (1975).
[CrossRef]

1971

K. Krishen, "Correlation of radar backscattering cross sections with ocean wave height and wind velocity," J. Geophys. Res. 76, 6528-6539 (1971).
[CrossRef]

D. E. Barrick, "Theory of HF and VHF propagation across the rough sea—parts I and II," Radio Sci. 6, 517-533 (1971).
[CrossRef]

1967

1965

F. E. Nicodemus, "Directional reflectance and emissivity of an opaque surface," Appl. Opt. 4(7), 368-377 (1965).

1963

F. E. Nicodemus, "Radiance," Am. J. Phys. 31, 368-377 (1963).
[CrossRef]

B. W. Hapke, "A theoretical photometric function for the lunar surface," J. Geophys. Res. 68(15), 4571-4586 (1963).

1953

C. Eckart, "The scattering of sound from the sea surface," J. Acoust. Soc. Am. 25, 566-570 (1953).
[CrossRef]

Alexander, C.

D. S. Flynn and C. Alexander, "Polarized surface scattering expressed in terms of a bidirectional reflectance distribution function," Opt. Eng. 34(6), 1646-1650 (1995).
[CrossRef]

Anderson, R.

Axline, R. M.

R. M. Axline and A. K. Fung, "Numerical computation of scattering from a perfectly conducting random surface," IEEE Trans. Antennas Propag. AP-26(3), 482-488 (1978).
[CrossRef]

Bai, S. Y.

M. F. Chen and S. Y. Bai, "Computer simulation of wave scattering from a dielectric random surface in two dimensions—cylindrical case," J. Electromagn. Waves Appl. 4(10), 963-982 (1990).
[CrossRef]

Bailey, W. M.

W. S. Bickel and W. M. Bailey, "Stokes vectors, Mueller matrices, and polarized scattered light," Am. J. Phys. 53(5), 468-478 (1985).
[CrossRef]

Barrick, D. E.

D. E. Barrick, "Theory of HF and VHF propagation across the rough sea—parts I and II," Radio Sci. 6, 517-533 (1971).
[CrossRef]

Barta, A.

Bassukas, I.

Bernáth, B.

Bickel, W. S.

W. S. Bickel and W. M. Bailey, "Stokes vectors, Mueller matrices, and polarized scattered light," Am. J. Phys. 53(5), 468-478 (1985).
[CrossRef]

Boger, J.

D. Wellems, S. Ortega, D. Bowers, J. Boger, and M. Fetrow, "Long wave infrared polarimetric model: theory, measurements and parameters," J. Opt. A: Pure Appl. Opt. 8(10), 914-925 (2006).
[CrossRef]

Bowers, D.

D. Wellems, S. Ortega, D. Bowers, J. Boger, and M. Fetrow, "Long wave infrared polarimetric model: theory, measurements and parameters," J. Opt. A: Pure Appl. Opt. 8(10), 914-925 (2006).
[CrossRef]

Chen, M. F.

M. F. Chen and S. Y. Bai, "Computer simulation of wave scattering from a dielectric random surface in two dimensions—cylindrical case," J. Electromagn. Waves Appl. 4(10), 963-982 (1990).
[CrossRef]

A. K. Fung and M. F. Chen, "Numerical simulation of scattering from simple and composite random surfaces," J. Opt. Soc. Am. A 2(12), 2274-2284 (1985).
[CrossRef]

Compain, E.

Dierking, M. P.

Dimou, A.

Drevillon, B.

Duncan, B. D.

Eckart, C.

C. Eckart, "The scattering of sound from the sea surface," J. Acoust. Soc. Am. 25, 566-570 (1953).
[CrossRef]

Ellis, K. K.

Fetrow, M.

D. Wellems, S. Ortega, D. Bowers, J. Boger, and M. Fetrow, "Long wave infrared polarimetric model: theory, measurements and parameters," J. Opt. A: Pure Appl. Opt. 8(10), 914-925 (2006).
[CrossRef]

Flynn, D. S.

D. S. Flynn and C. Alexander, "Polarized surface scattering expressed in terms of a bidirectional reflectance distribution function," Opt. Eng. 34(6), 1646-1650 (1995).
[CrossRef]

Fung, A. K.

A. K. Fung and M. F. Chen, "Numerical simulation of scattering from simple and composite random surfaces," J. Opt. Soc. Am. A 2(12), 2274-2284 (1985).
[CrossRef]

R. M. Axline and A. K. Fung, "Numerical computation of scattering from a perfectly conducting random surface," IEEE Trans. Antennas Propag. AP-26(3), 482-488 (1978).
[CrossRef]

Greiner, M. A.

Hapke, B. W.

B. W. Hapke, "A theoretical photometric function for the lunar surface," J. Geophys. Res. 68(15), 4571-4586 (1963).

Harper, E. Y.

E. Y. Harper and F. M. Labianca, "Scattering of sound from a point source by a rough surface progressing over an isovelocity ocean," J. Acoust. Soc. Am. 58(2), 349-364 (1975).
[CrossRef]

Hegedüs, R.

Horváth, G.

Kimes, D. S.

Krishen, K.

K. Krishen, "Correlation of radar backscattering cross sections with ocean wave height and wind velocity," J. Geophys. Res. 76, 6528-6539 (1971).
[CrossRef]

Labianca, F. M.

E. Y. Harper and F. M. Labianca, "Scattering of sound from a point source by a rough surface progressing over an isovelocity ocean," J. Acoust. Soc. Am. 58(2), 349-364 (1975).
[CrossRef]

Meier, S. R.

R. G. Priest and S. R. Meier, "Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces," Opt. Eng. 41(5), 988-993 (2002).
[CrossRef]

Meyer-Rochow, V. B.

Nicodemus, F. E.

F. E. Nicodemus, "Directional reflectance and emissivity of an opaque surface," Appl. Opt. 4(7), 368-377 (1965).

F. E. Nicodemus, "Radiance," Am. J. Phys. 31, 368-377 (1963).
[CrossRef]

Ortega, S.

D. Wellems, S. Ortega, D. Bowers, J. Boger, and M. Fetrow, "Long wave infrared polarimetric model: theory, measurements and parameters," J. Opt. A: Pure Appl. Opt. 8(10), 914-925 (2006).
[CrossRef]

Poirier, S.

Priest, R. G.

R. G. Priest and S. R. Meier, "Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces," Opt. Eng. 41(5), 988-993 (2002).
[CrossRef]

Sparrow, E. M.

Sun, Y.

Thorsos, E. I.

E. I. Thorsos, "The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum," J. Acoust. Soc. Am. 83(1), 78-92 (1988).
[CrossRef]

Torrance, K. E.

Wellems, D.

D. Wellems, S. Ortega, D. Bowers, J. Boger, and M. Fetrow, "Long wave infrared polarimetric model: theory, measurements and parameters," J. Opt. A: Pure Appl. Opt. 8(10), 914-925 (2006).
[CrossRef]

Xia, J.

Yao, G.

Zonios, G.

Am. J. Phys.

F. E. Nicodemus, "Radiance," Am. J. Phys. 31, 368-377 (1963).
[CrossRef]

W. S. Bickel and W. M. Bailey, "Stokes vectors, Mueller matrices, and polarized scattered light," Am. J. Phys. 53(5), 468-478 (1985).
[CrossRef]

Appl. Opt.

IEEE Trans. Antennas Propag.

R. M. Axline and A. K. Fung, "Numerical computation of scattering from a perfectly conducting random surface," IEEE Trans. Antennas Propag. AP-26(3), 482-488 (1978).
[CrossRef]

J. Acoust. Soc. Am.

E. I. Thorsos, "The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum," J. Acoust. Soc. Am. 83(1), 78-92 (1988).
[CrossRef]

C. Eckart, "The scattering of sound from the sea surface," J. Acoust. Soc. Am. 25, 566-570 (1953).
[CrossRef]

E. Y. Harper and F. M. Labianca, "Scattering of sound from a point source by a rough surface progressing over an isovelocity ocean," J. Acoust. Soc. Am. 58(2), 349-364 (1975).
[CrossRef]

J. Electromagn. Waves Appl.

M. F. Chen and S. Y. Bai, "Computer simulation of wave scattering from a dielectric random surface in two dimensions—cylindrical case," J. Electromagn. Waves Appl. 4(10), 963-982 (1990).
[CrossRef]

J. Geophys. Res.

K. Krishen, "Correlation of radar backscattering cross sections with ocean wave height and wind velocity," J. Geophys. Res. 76, 6528-6539 (1971).
[CrossRef]

B. W. Hapke, "A theoretical photometric function for the lunar surface," J. Geophys. Res. 68(15), 4571-4586 (1963).

J. Opt. A: Pure Appl. Opt.

D. Wellems, S. Ortega, D. Bowers, J. Boger, and M. Fetrow, "Long wave infrared polarimetric model: theory, measurements and parameters," J. Opt. A: Pure Appl. Opt. 8(10), 914-925 (2006).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

D. S. Flynn and C. Alexander, "Polarized surface scattering expressed in terms of a bidirectional reflectance distribution function," Opt. Eng. 34(6), 1646-1650 (1995).
[CrossRef]

R. G. Priest and S. R. Meier, "Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces," Opt. Eng. 41(5), 988-993 (2002).
[CrossRef]

Radio Sci.

D. E. Barrick, "Theory of HF and VHF propagation across the rough sea—parts I and II," Radio Sci. 6, 517-533 (1971).
[CrossRef]

Other

D. Wellems, M. Serna, S. H. Sposato, M. P. Fetrow, K. P. Bishop, S. A. Arko, and T. R. Caudill, "Spectral polarimetric BRDF model and comparison to measurements from isotropic roughened glass," in Workshop on Multi/Hyperspectral Sensors, Measurements, Modeling and Simulation (U.S. Army Aviation and Missile Command, Huntsville, AL, 2000).

H. G. Tompkins and E. A. Irene, Handbook of Ellipsometry (William Andrew, Inc., Norwich, NY, 2005).
[CrossRef]

J. R. Schott, Fundamentals of Polarimetric Remote Sensing (SPIE Press, Bellingham, WA, 2009).
[CrossRef]

J. R. Shell, "Polarimetric Remote Sensing in the Visible to Near Infrared," Ph.D. dissertation, Chester F. Carslon Center for Imaging Science, Rochester Institute of Technology, Rochester, NY (2005).

M. G. Gartley, S. D. Brown, and J. R. Schott, "Micro-scale surface and contaminate modeling for polarimetric signature prediction," in Proc. SPIE, vol. 6972 (The International Society for Optical Engineering (SPIE), 2008).
[CrossRef]

J. R. Maxwell, J. Beard, S. Weiner, D. Ladd, and S. Ladd, "Bidirectional Reflectance Model Validation and Utilization," Tech. Rep. AFAL-TR-73-303, Air Force Avionics Laboratory, Wright-Patterson Air Force Base, OH (1973).

M. G. Gartley, "Polarimetric Modeling of Remotely Sensed Scenes in the Thermal Infrared," Ph.D. dissertation, Chester F. Carslon Center for Imaging Science, Rochester Institute of Technology, Rochester, NY (2007).

A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics (IEEE Press, New York, NY, 1998).

J. F. Blinn, "Models of light reflection for computer synthesized pictures," in SIGGRAPH 1977 Proceedings, vol. 11, pp. 192-198, Special Interest Group on Graphics and Interactive Techniques (Computer Graphics, 1977).

R. L. Cook and K. E. Torrance, "A reflectance model for computer graphics," in SIGGRAPH 1981 Proceedings, vol. 15, pp. 307-316, Special Interest Group on Graphics and Interactive Techniques (Computer Graphics, 1981).

X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, "A comprehensive physical model for light reflection," in SIGGRAPH 1991 Proceedings, vol. 25, pp. 175-186, Special Interest Group on Graphics and Interactive Techniques (Computer Graphics, 1991).

B. P. Sandford and D. C. Robertson, "Infrared reflectance properties of aircraft paints," in Proceedings of IRIS Targets, Backgrounds and Discrimination (1985).

M. P. Fetrow, D. Wellems, S. H. Sposato, K. P. Bishop, T. R. Caudill, M. L. Davis, and E. R. Simrell, "Results of a new polarization simulation," in Proc. SPIE, vol. 4481, pp. 149-162 (The International Society for Optical Engineering (SPIE), 2002).

R. G. Priest and T. A. Germer, "Polarimetric BRDF in the microfacet model: theory and measurements," in Proceedings of the 2000 Meeting of the Military Sensing Symposia Specialty Group on Passive Sensors, pp. 169-181 (Infrared Information Analysis Center, 2000).

P. Y. Ufimtsev, Fundamentals of the Physical Theory of Diffraction (John Wiley & Sons, Inc., Hoboken, NJ, 2007).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, New York, NY, 1999).

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, Inc., Norwood, MA, 1963).

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, New York, NY, 1997).

C.-H. An and K. J. Zeringue, "Polarization scattering from rough surfaces based on the vector Kirchoff diffraction model," in Proc. SPIE, vol. 5158, pp. 205-216 (The International Society for Optical Engineering (SPIE), 2003).

D. A. McNamara, C. W. I. Pistorius, and J. A. G. Malherbe, Introduction to the Uniform Geometrical Theory of Diffraction (Artech House, Inc., Norwood, MA, 1990).

LabSphere, Inc., "A guide to reflectance coatings and materials," http://www.labsphere.com/tecdocs.aspx.

Luxpop, Inc.http://www.luxpop.com/.

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Figures (9)

Fig. 1.
Fig. 1.

Macroscopic surface scattering geometry. Light subtending solid angle dωi is incident from the (θi ,ϕi ) direction on a small area dA of a much larger rough surface with complex index of refraction η=n-jκ. Light is scattered and observed within solid angle dωr in the (θr ,ϕr ) direction.

Fig. 2.
Fig. 2.

Scattering geometry of a single microfacet. The angle α is the polar angle from the mean surface normal to the microfacet normal n. The angle β is the incident angle onto and reflected angle from a microfacet as measured from the microfacet normal. The angle γi is the angle between the macroscopic plane of incidence and the scattering plane of the microfacet (depicted in the figure as the plane containing the vectors n and t). Likewise, the angle γr is the angle between the macroscopic plane of reflection and the scattering plane of the microfacet.

Fig. 3.
Fig. 3.

Scattering geometry of a v-shaped groove. The top subfigure depicts shadowing while the bottom subfigure depicts masking. Shadowing occurs when the angle of incidence approaches grazing. Similarly, masking occurs when the angle of observation nears grazing.

Fig. 4.
Fig. 4.

Comparisons of the F00 elements of the Priest and Germer pBRDF [23, 24] and the pBRDF in Eq. (13) for θi =45°, 60°, 75°, and 85° with 21/2σ h /=0.3. The pBRDFs are evaluated in the specular plane (ϕ=π) and using a perfect reflecting surface.

Fig. 5.
Fig. 5.

Scattering geometry of the MoM solutions. The surface is a 15,000λ long, random (surface height is Gaussian distributed) PEC surface. The surface is assumed to be invariant in the z direction.

Fig. 6.
Fig. 6.

Comparisons of the reflectance distributions predicted by MoM solutions of a 15,000λ long, random (surface height is Gaussian distributed) PEC surface with those of the pBRDF in Eq. (17) for θi =10°, 30°, 45°, 60°, and 75° and 21/2σ h /=0.3. Note that the reflectance distributions in the figure are normalized with respect to their values at the specular angles (θi =θr ). Observation for both the MoM and the pBRDF predictions is in the specular plane (ϕ=π).

Fig. 7.
Fig. 7.

Photograph of the Mueller matrix ellipsometer used in this experiment. The ellipsometer is located at the Air Force Research Laboratory, Wright-Patterson Air Force Base, Ohio.

Fig. 8.
Fig. 8.

Mueller matrix measurement results for LabSphere Infragold [46] compared to predictions made using the pBRDF. The measurement results are plotted as symbols; the pBRDF predictions are plotted as solid lines. Note that the measurements are made in the specular plane (ϕ=π). The complex index of refraction used for gold is η=0.285-j7.3523 [47] and 21/2σ h /=0.44. The plotted values for the measured Mueller matrix elements of LabSphere Infragold are the means of 256 irradiance measurements. The bars on the figure represent ±1σ, i.e., one standard deviation of those 256 measurements.

Fig. 9.
Fig. 9.

Mueller matrix measurement results for flame sprayed aluminum (FSA) compared to predictions made using the pBRDF. The measurement results are plotted as symbols; the pBRDF predictions are plotted as solid lines. Note that the measurements are made in the specular plane (ϕ=π). The complex index of refraction used for aluminum is η=1.226-j10.413 [47] and 21/2σ h /=0.43. The plotted values for the measured Mueller matrix elements of FSA are the means of 256 irradiance measurements. The bars on the figure represent ±1σ, i.e., one standard deviation of those 256 measurements.

Equations (34)

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f ( θ i , θ r , ϕ ) = d L r ( θ r , ϕ ) d E i ( θ i ) = d L r ( θ r , ϕ ) L i ( θ i ) cos θ i d ω i
F ( θ i , θ r , ϕ ) = d L r ( θ r , ϕ ) L i ( θ i ) cos θ i d ω i .
L r ( θ r , ϕ ) = L r sin gle ( θ r , ϕ ) + L r multiple ( θ r , ϕ )
F = F sin gle + F multiple .
F s = F sin gle , F d = F multiple
F = F s + F d .
F s ( θ i , θ r , ϕ ; σ h , ; η ) = P ( α ; σ h , ) M ( β ; η ) G ( θ i , θ r , ϕ ) 4 cos θ i cos θ r cos α
cos α = ( cos θ i + cos θ r ) ( 2 cos β )
cos 2 β = cos θ i cos θ r + sin θ i sin θ r cos ϕ .
P ( α ; σ h , ) = 2 exp ( 2 tan 2 α 4 σ h 2 ) 4 π σ h 2 cos 3 α .
[ E r s E r p ] = [ cos γ r sin γ r sin γ r cos γ r ] [ r s 0 0 r p ] [ cos γ i sin γ i sin γ i cos γ i ] [ E i s E i p ]
[ E r s E r p ] = [ T ss T ps T sp T pp ] [ E i s E i p ]
cos γ i = ( cos α cos θ i cos β ) ( sin θ i sin β )
cos γ r = ( cos α cos θ r cos β ) ( sin θ r sin β ) .
M = 1 2 [ M 00 M 01 0 M 01 M 00 0 0 0 M 22 j M 23 0 0 j M 23 M 22 ]
M 00 = T ss 2 + T sp 2 + T ps 2 + T pp 2
M 01 = T ss 2 + T sp 2 T ps 2 T pp 2
M 22 = T ss T pp * + T ss * T pp + T ps T sp * + T ps * T sp
M 23 = T ps T sp * T ps * T sp T ss T pp * + T ss * T pp
G ( θ i , θ r , ϕ ) = min ( 1 ; 2 cos α cos θ r cos β ; 2 cos α cos θ i cos β ) .
F jk s ( θ i , θ r , ϕ ; σ h , ; η ) = 2 exp ( 2 tan 2 α 4 σ h 2 ) 16 π σ h 2 cos θ i cos θ r cos 4 α G ( θ i , θ r , ϕ ) M jk ( β ; η ) .
ρ DHR ( θ i ; σ h , ) = 0 2 π 0 π 2 F 00 ( θ i , θ r , ϕ ; σ h , ; η ) cos θ r sin θ r d θ r d ϕ .
1 = 0 2 π 0 π 2 F 00 s , PEC cos θ r sin θ r d θ r d ϕ + 0 2 π 0 π 2 F 00 d , PEC cos θ r sin θ r d θ r d ϕ .
F 00 d , PEC ( θ i ; σ h , ) = 1 π ( 1 0 2 π 0 π 2 F 00 s , PEC cos θ r sin θ r d θ r d ϕ ) .
F 00 d , PEC ( θ i ; σ h , ) = 1 π [ 1 ρ DHR s , PEC ( θ i ; σ h , ) ]
F 00 ( θ i , θ r , ϕ ; σ h , ; η ) = F 00 s ( θ i , θ r , ϕ ; σ h , ; η ) + 1 π [ 1 ρ DHR s , PEC ( θ i ; σ h , ) ] M 00 ( β ; η ) .
F jk ( θ i , θ r , ϕ ; σ h , ; η ) = F jk s ( θ i , θ r , ϕ ; σ h , ; η ) j , k 0
π Z 0 2 λ c J z ( ρ ) H 0 ( 2 ) ( 2 π λ ρ ρ ) d C ' = exp [ j 2 π λ ( k i · ρ ) ] ρ C
J z ( ρ ) = Σ n = 1 N α n p n ( ρ ) .
[ α 11 α 12 α 1 N α 21 α 22 α 2 N α N 1 α N 2 α NN ] [ α 1 α 2 α N ] = [ E i , 1 z E i , 2 z E i , N N ] .
E r z ( x , y ) = π Z 0 2 λ Σ n = 1 N α n C n H 0 ( 2 ) ( 2 π λ ρ ρ ) d C n lim ρ E r z ( ρ , θ r ) = Z 0 2 ρλ exp [ j ( 2 π λ ρ π 4 ) ] Σ n = 1 N α n C n exp [ j 2 π λ ( x sin θ r + y cos θ r ) ] d C n
W m ( x ) = exp [ ( x x m w ) 2 ]
σ ( θ r ) = 1 w π / 2 ( 1 M lim ρ 2 π ρ Σ m = 1 M E r , m z 2 )
M = A 1 SW 1

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