Abstract

A popular class of BRDF models is the microfacet models, where geometric optics is assumed. In contrast, more complex physical optics models may more accurately predict the BRDF, but the calculation is more resource intensive. These seemingly disparate approaches are compared in detail for the rough and smooth surface approximations of the modified Beckmann-Kirchhoff BRDF model, assuming Gaussian surface statistics. An approximation relating standard Fresnel reflection with the semi-rough surface polarization term, Q, is presented for unpolarized light. For rough surfaces, the angular dependence of direction cosine space is shown to be identical to the angular dependence in the microfacet distribution function. For polished surfaces, the same comparison shows a breakdown in the microfacet models. Similarities and differences between microfacet BRDF models and the modified Beckmann-Kirchhoff model are identified. The rationale for the original Beckmann-Kirchhoff Fbk2 geometric term relative to both microfacet models and generalized Harvey-Shack model is presented. A modification to the geometric Fbk2 term in original Beckmann-Kirchhoff BRDF theory is proposed.

© 2015 Optical Society of America

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  5. D. R. Crow, C. F. Coker, D. L. Garbo, and E. M. Olson, “Closed-loop real-time infrared scene generator,” Proc. SPIE 3368, 342 (1998).
    [Crossref]
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    [Crossref]
  12. S. D. Butler, S. E. Nauyoks, and M. A. Marciniak, “Experimental analysis of bidirectional reflectance distribution function cross section conversion term in direction cosine space,” Opt. Lett. 40, 2445–2448 (2015).
    [Crossref] [PubMed]
  13. S. D. Butler, S. E. Nauyoks, and M. A. Marciniak, “Comparison of microfacet BRDF model elements to diffraction BRDF model elements,” Proc. SPIE 9472, 94720C (2015).
    [Crossref]
  14. J. E. Harvey and A. Krywonos, “Unified scatter model for rough surfaces at large incident and scatter angles,” Proc. SPIE 6672, 66720C (2007).
    [Crossref]
  15. A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles,” J. Opt. Sci. Am. A 28, 1121–1138 (2011).
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  26. S. Chakrabarti, A. A. Maradudin, and E. R. Mendez, “Reconstruction of the surface-height autocorrelation function of a randomly rough dielectric surface from incoherent light scattering,” Phys. Rev. A 88, 013812 (2013).
    [Crossref]
  27. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (MacMillan, 1963).
  28. J. E. Harvey, A. Krywonos, and C. L. Vernold, “Modified Beckmann-Kirchhoff scattering model for rough surfaces with large incident and scattering angles,” Opt. Eng. 46(7), 078002 (2007).
    [Crossref]
  29. A. Krywonos, “Predicting surface scatter using a linear systems formulation of non-paraxial scalar diffraction,” Ph.D. Dissertation, University of Central Florida (2006).
  30. J. C. Stover, Optical Scattering: Measurement and Analysis, 3rd ed. (SPIE, 2012).
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    [Crossref]
  32. S. Schröder, A. Duparré, L. Coriand, A. Tünnermann, D. H. Penalver, and J. E. Harvey, “Modeling of light scattering in different regimes of surface roughness,” Opt. Express 19, 9820–9835 (2011).
    [Crossref] [PubMed]
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2015 (2)

S. D. Butler, S. E. Nauyoks, and M. A. Marciniak, “Comparison of microfacet BRDF model elements to diffraction BRDF model elements,” Proc. SPIE 9472, 94720C (2015).
[Crossref]

S. D. Butler, S. E. Nauyoks, and M. A. Marciniak, “Experimental analysis of bidirectional reflectance distribution function cross section conversion term in direction cosine space,” Opt. Lett. 40, 2445–2448 (2015).
[Crossref] [PubMed]

2014 (2)

S. D. Butler and M. A. Marciniak, “Robust categorization of microfacet BRDF models to enable flexible application-specific BRDF adaptation,” Proc. SPIE 9205, 920506 (2014).
[Crossref]

E. Heitz, “Understanding the masking-shadowing function in microfacet-based BRDFs,” J. Comp. Graph. Tech. 3, 32–91 (2014).

2013 (1)

S. Chakrabarti, A. A. Maradudin, and E. R. Mendez, “Reconstruction of the surface-height autocorrelation function of a randomly rough dielectric surface from incoherent light scattering,” Phys. Rev. A 88, 013812 (2013).
[Crossref]

2011 (2)

S. Schröder, A. Duparré, L. Coriand, A. Tünnermann, D. H. Penalver, and J. E. Harvey, “Modeling of light scattering in different regimes of surface roughness,” Opt. Express 19, 9820–9835 (2011).
[Crossref] [PubMed]

A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles,” J. Opt. Sci. Am. A 28, 1121–1138 (2011).
[Crossref]

2009 (1)

2007 (3)

J. E. Harvey, A. Krywonos, and C. L. Vernold, “Modified Beckmann-Kirchhoff scattering model for rough surfaces with large incident and scattering angles,” Opt. Eng. 46(7), 078002 (2007).
[Crossref]

J. C. Stover and J. E. Harvey, “Unified scatter model for rough surfaces at large incident and scatter angles,” Proc. SPIE 6672, 66720B (2007).
[Crossref]

J. E. Harvey and A. Krywonos, “Unified scatter model for rough surfaces at large incident and scatter angles,” Proc. SPIE 6672, 66720C (2007).
[Crossref]

2006 (1)

A. Duer, “An improved normalization for the Ward reflectance model,” J. Graphics, GPU, and Game Tools 11, 51–59 (2006).
[Crossref]

2004 (1)

T. M. Elfouhaily and C.-A. Guerin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves in Random Media 14, R1–R40 (2004).
[Crossref]

2003 (1)

W. Matusik, H. Pfister, M. Brand, and L. McMillan, “A data-driven reflectance model,” ACM Trans. Graphics 22, 759–769 (2003).
[Crossref]

2002 (1)

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]

1998 (1)

D. R. Crow, C. F. Coker, D. L. Garbo, and E. M. Olson, “Closed-loop real-time infrared scene generator,” Proc. SPIE 3368, 342 (1998).
[Crossref]

1982 (1)

R. L. Cook and K. E. Torrance, “A reflectance model for computer graphics,” ACM Trans. Graphics 1, 7–24 (1982).
[Crossref]

1975 (1)

1967 (1)

Beard, J.

J. R. Maxwell, J. Beard, S. Weiner, D. Ladd, and S. Ladd, “Bidirectional reflectance model validation and utilization,” Environmental Research Institute of Michigan (ERIM) Technical Report AFAL-TR-73-303 (1973).

Beckmann, P.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (MacMillan, 1963).

Blinn, J. F.

J. F. Blinn, “Models of light reflection for computer synthesized pictures,” in Proceedings of the 4th Annual Conference on Computer Graphics and Interactive Techniques, (ACM, 1977), pp. 192–198.

Brand, M.

W. Matusik, H. Pfister, M. Brand, and L. McMillan, “A data-driven reflectance model,” ACM Trans. Graphics 22, 759–769 (2003).
[Crossref]

Butler, S. D.

S. D. Butler, S. E. Nauyoks, and M. A. Marciniak, “Experimental analysis of bidirectional reflectance distribution function cross section conversion term in direction cosine space,” Opt. Lett. 40, 2445–2448 (2015).
[Crossref] [PubMed]

S. D. Butler, S. E. Nauyoks, and M. A. Marciniak, “Comparison of microfacet BRDF model elements to diffraction BRDF model elements,” Proc. SPIE 9472, 94720C (2015).
[Crossref]

S. D. Butler and M. A. Marciniak, “Robust categorization of microfacet BRDF models to enable flexible application-specific BRDF adaptation,” Proc. SPIE 9205, 920506 (2014).
[Crossref]

Chakrabarti, S.

S. Chakrabarti, A. A. Maradudin, and E. R. Mendez, “Reconstruction of the surface-height autocorrelation function of a randomly rough dielectric surface from incoherent light scattering,” Phys. Rev. A 88, 013812 (2013).
[Crossref]

Choi, N.

A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles,” J. Opt. Sci. Am. A 28, 1121–1138 (2011).
[Crossref]

Coker, C. F.

D. R. Crow, C. F. Coker, D. L. Garbo, and E. M. Olson, “Closed-loop real-time infrared scene generator,” Proc. SPIE 3368, 342 (1998).
[Crossref]

Cook, R. L.

R. L. Cook and K. E. Torrance, “A reflectance model for computer graphics,” ACM Trans. Graphics 1, 7–24 (1982).
[Crossref]

Coriand, L.

Crow, D. R.

D. R. Crow, C. F. Coker, D. L. Garbo, and E. M. Olson, “Closed-loop real-time infrared scene generator,” Proc. SPIE 3368, 342 (1998).
[Crossref]

Dorsey, J.

J. Dorsey, H. Rushmeier, and F. Sillion, Digital Modeling of Material Appearance (Morgan Kaufmann, 2007).

Duer, A.

A. Duer, “An improved normalization for the Ward reflectance model,” J. Graphics, GPU, and Game Tools 11, 51–59 (2006).
[Crossref]

Duparré, A.

Durand, F.

A. Ngan, F. Durand, and W. Matusik, “Experimental analysis of BRDF models,” in Proceedings of the Euro-graphics Symposium on Rendering, (Eurographics Association) pp. 117–226 (2005).

Eismann, M. T.

M. T. Eismann, Hyperspectral Remote Sensing (SPIE, 2012).
[Crossref]

Elfouhaily, T. M.

T. M. Elfouhaily and C.-A. Guerin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves in Random Media 14, R1–R40 (2004).
[Crossref]

Garbo, D. L.

D. R. Crow, C. F. Coker, D. L. Garbo, and E. M. Olson, “Closed-loop real-time infrared scene generator,” Proc. SPIE 3368, 342 (1998).
[Crossref]

Germer, T. A.

R. G. Priest and T. A. Germer, “Polarimetric BRDF in the microfacet model: Theory and measurements,” inProceedings of 2000 Meeting of the MSS Specialty Sensors Group on Passive Sensors, (Naval Research Lab, 2000), pp. 169–182.

Ginsberg, I. W.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance,” National Bureau of Standards Monograph 160, Department of Commerce (1977).

Guerin, C.-A.

T. M. Elfouhaily and C.-A. Guerin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves in Random Media 14, R1–R40 (2004).
[Crossref]

Harvey, J. E.

A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles,” J. Opt. Sci. Am. A 28, 1121–1138 (2011).
[Crossref]

S. Schröder, A. Duparré, L. Coriand, A. Tünnermann, D. H. Penalver, and J. E. Harvey, “Modeling of light scattering in different regimes of surface roughness,” Opt. Express 19, 9820–9835 (2011).
[Crossref] [PubMed]

J. E. Harvey, A. Krywonos, and C. L. Vernold, “Modified Beckmann-Kirchhoff scattering model for rough surfaces with large incident and scattering angles,” Opt. Eng. 46(7), 078002 (2007).
[Crossref]

J. E. Harvey and A. Krywonos, “Unified scatter model for rough surfaces at large incident and scatter angles,” Proc. SPIE 6672, 66720C (2007).
[Crossref]

J. C. Stover and J. E. Harvey, “Unified scatter model for rough surfaces at large incident and scatter angles,” Proc. SPIE 6672, 66720B (2007).
[Crossref]

Havrilla, M. J.

Heitz, E.

E. Heitz, “Understanding the masking-shadowing function in microfacet-based BRDFs,” J. Comp. Graph. Tech. 3, 32–91 (2014).

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance,” National Bureau of Standards Monograph 160, Department of Commerce (1977).

Hyde, M. W.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley and Sons, Inc., 1999).

Krywonos, A.

A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles,” J. Opt. Sci. Am. A 28, 1121–1138 (2011).
[Crossref]

J. E. Harvey and A. Krywonos, “Unified scatter model for rough surfaces at large incident and scatter angles,” Proc. SPIE 6672, 66720C (2007).
[Crossref]

J. E. Harvey, A. Krywonos, and C. L. Vernold, “Modified Beckmann-Kirchhoff scattering model for rough surfaces with large incident and scattering angles,” Opt. Eng. 46(7), 078002 (2007).
[Crossref]

A. Krywonos, “Predicting surface scatter using a linear systems formulation of non-paraxial scalar diffraction,” Ph.D. Dissertation, University of Central Florida (2006).

Ladd, D.

J. R. Maxwell, J. Beard, S. Weiner, D. Ladd, and S. Ladd, “Bidirectional reflectance model validation and utilization,” Environmental Research Institute of Michigan (ERIM) Technical Report AFAL-TR-73-303 (1973).

Ladd, S.

J. R. Maxwell, J. Beard, S. Weiner, D. Ladd, and S. Ladd, “Bidirectional reflectance model validation and utilization,” Environmental Research Institute of Michigan (ERIM) Technical Report AFAL-TR-73-303 (1973).

Limperis, T.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance,” National Bureau of Standards Monograph 160, Department of Commerce (1977).

Maradudin, A. A.

S. Chakrabarti, A. A. Maradudin, and E. R. Mendez, “Reconstruction of the surface-height autocorrelation function of a randomly rough dielectric surface from incoherent light scattering,” Phys. Rev. A 88, 013812 (2013).
[Crossref]

Marciniak, M. A.

S. D. Butler, S. E. Nauyoks, and M. A. Marciniak, “Experimental analysis of bidirectional reflectance distribution function cross section conversion term in direction cosine space,” Opt. Lett. 40, 2445–2448 (2015).
[Crossref] [PubMed]

S. D. Butler, S. E. Nauyoks, and M. A. Marciniak, “Comparison of microfacet BRDF model elements to diffraction BRDF model elements,” Proc. SPIE 9472, 94720C (2015).
[Crossref]

S. D. Butler and M. A. Marciniak, “Robust categorization of microfacet BRDF models to enable flexible application-specific BRDF adaptation,” Proc. SPIE 9205, 920506 (2014).
[Crossref]

Matusik, W.

W. Matusik, H. Pfister, M. Brand, and L. McMillan, “A data-driven reflectance model,” ACM Trans. Graphics 22, 759–769 (2003).
[Crossref]

A. Ngan, F. Durand, and W. Matusik, “Experimental analysis of BRDF models,” in Proceedings of the Euro-graphics Symposium on Rendering, (Eurographics Association) pp. 117–226 (2005).

Maxwell, J. R.

J. R. Maxwell, J. Beard, S. Weiner, D. Ladd, and S. Ladd, “Bidirectional reflectance model validation and utilization,” Environmental Research Institute of Michigan (ERIM) Technical Report AFAL-TR-73-303 (1973).

McMillan, L.

W. Matusik, H. Pfister, M. Brand, and L. McMillan, “A data-driven reflectance model,” ACM Trans. Graphics 22, 759–769 (2003).
[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]

Mendez, E. R.

S. Chakrabarti, A. A. Maradudin, and E. R. Mendez, “Reconstruction of the surface-height autocorrelation function of a randomly rough dielectric surface from incoherent light scattering,” Phys. Rev. A 88, 013812 (2013).
[Crossref]

Nauyoks, S. E.

S. D. Butler, S. E. Nauyoks, and M. A. Marciniak, “Experimental analysis of bidirectional reflectance distribution function cross section conversion term in direction cosine space,” Opt. Lett. 40, 2445–2448 (2015).
[Crossref] [PubMed]

S. D. Butler, S. E. Nauyoks, and M. A. Marciniak, “Comparison of microfacet BRDF model elements to diffraction BRDF model elements,” Proc. SPIE 9472, 94720C (2015).
[Crossref]

Ngan, A.

A. Ngan, F. Durand, and W. Matusik, “Experimental analysis of BRDF models,” in Proceedings of the Euro-graphics Symposium on Rendering, (Eurographics Association) pp. 117–226 (2005).

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance,” National Bureau of Standards Monograph 160, Department of Commerce (1977).

Olson, E. M.

D. R. Crow, C. F. Coker, D. L. Garbo, and E. M. Olson, “Closed-loop real-time infrared scene generator,” Proc. SPIE 3368, 342 (1998).
[Crossref]

Penalver, D. H.

Pfister, H.

W. Matusik, H. Pfister, M. Brand, and L. McMillan, “A data-driven reflectance model,” ACM Trans. Graphics 22, 759–769 (2003).
[Crossref]

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]

R. G. Priest and T. A. Germer, “Polarimetric BRDF in the microfacet model: Theory and measurements,” inProceedings of 2000 Meeting of the MSS Specialty Sensors Group on Passive Sensors, (Naval Research Lab, 2000), pp. 169–182.

Reitz, K. P.

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance,” National Bureau of Standards Monograph 160, Department of Commerce (1977).

Rushmeier, H.

J. Dorsey, H. Rushmeier, and F. Sillion, Digital Modeling of Material Appearance (Morgan Kaufmann, 2007).

Rusinkiewicz, S.

S. Rusinkiewicz, “A new change of variables for efficient BRDF representation,” in Rendering Techniques ’98, (Springer Vienna, 1998), pp. 11–22.
[Crossref]

Schmidt, J. D.

Schott, J. R.

J. R. Schott, Fundamentals of Polarimetric Remote Sensing (SPIE, 2009).
[Crossref]

Schröder, S.

Sillion, F.

J. Dorsey, H. Rushmeier, and F. Sillion, Digital Modeling of Material Appearance (Morgan Kaufmann, 2007).

Sparrow, E. M.

Spizzichino, A.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (MacMillan, 1963).

Stover, J. C.

J. C. Stover and J. E. Harvey, “Unified scatter model for rough surfaces at large incident and scatter angles,” Proc. SPIE 6672, 66720B (2007).
[Crossref]

J. C. Stover, Optical Scattering: Measurement and Analysis, 3rd ed. (SPIE, 2012).

Torrance, K. E.

R. L. Cook and K. E. Torrance, “A reflectance model for computer graphics,” ACM Trans. Graphics 1, 7–24 (1982).
[Crossref]

K. E. Torrance and E. M. Sparrow, “Theory of off-specular reflection from roughened surfaces,” J. Opt. Soc. Am. 57, 1105–1114 (1967).
[Crossref]

Trowbridge, T. S.

Tünnermann, A.

Vernold, C. L.

J. E. Harvey, A. Krywonos, and C. L. Vernold, “Modified Beckmann-Kirchhoff scattering model for rough surfaces with large incident and scattering angles,” Opt. Eng. 46(7), 078002 (2007).
[Crossref]

Ward, G. J.

G. J. Ward, “Measuring and modeling anisotropic surfaces,” in Proceedings of SIGGRAPH Computer Graphics ’92, (ACM), 265–272. (1992).

Weiner, S.

J. R. Maxwell, J. Beard, S. Weiner, D. Ladd, and S. Ladd, “Bidirectional reflectance model validation and utilization,” Environmental Research Institute of Michigan (ERIM) Technical Report AFAL-TR-73-303 (1973).

ACM Trans. Graphics (2)

R. L. Cook and K. E. Torrance, “A reflectance model for computer graphics,” ACM Trans. Graphics 1, 7–24 (1982).
[Crossref]

W. Matusik, H. Pfister, M. Brand, and L. McMillan, “A data-driven reflectance model,” ACM Trans. Graphics 22, 759–769 (2003).
[Crossref]

J. Comp. Graph. Tech. (1)

E. Heitz, “Understanding the masking-shadowing function in microfacet-based BRDFs,” J. Comp. Graph. Tech. 3, 32–91 (2014).

J. Graphics, GPU, and Game Tools (1)

A. Duer, “An improved normalization for the Ward reflectance model,” J. Graphics, GPU, and Game Tools 11, 51–59 (2006).
[Crossref]

J. Opt. Sci. Am. A (1)

A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles,” J. Opt. Sci. Am. A 28, 1121–1138 (2011).
[Crossref]

J. Opt. Soc. Am. (2)

Opt. Eng. (2)

J. E. Harvey, A. Krywonos, and C. L. Vernold, “Modified Beckmann-Kirchhoff scattering model for rough surfaces with large incident and scattering angles,” Opt. Eng. 46(7), 078002 (2007).
[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]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. A (1)

S. Chakrabarti, A. A. Maradudin, and E. R. Mendez, “Reconstruction of the surface-height autocorrelation function of a randomly rough dielectric surface from incoherent light scattering,” Phys. Rev. A 88, 013812 (2013).
[Crossref]

Proc. SPIE (5)

J. C. Stover and J. E. Harvey, “Unified scatter model for rough surfaces at large incident and scatter angles,” Proc. SPIE 6672, 66720B (2007).
[Crossref]

D. R. Crow, C. F. Coker, D. L. Garbo, and E. M. Olson, “Closed-loop real-time infrared scene generator,” Proc. SPIE 3368, 342 (1998).
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S. D. Butler and M. A. Marciniak, “Robust categorization of microfacet BRDF models to enable flexible application-specific BRDF adaptation,” Proc. SPIE 9205, 920506 (2014).
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S. D. Butler, S. E. Nauyoks, and M. A. Marciniak, “Comparison of microfacet BRDF model elements to diffraction BRDF model elements,” Proc. SPIE 9472, 94720C (2015).
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J. E. Harvey and A. Krywonos, “Unified scatter model for rough surfaces at large incident and scatter angles,” Proc. SPIE 6672, 66720C (2007).
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Figures (2)

Fig. 1
Fig. 1 Relevant coordinate systems for the microfacet (geometric) BRDF model. (a) BRDF coordinate system orientation relative to the surface normal z ^. (b) Microfacet coordinates relative to the coordinates defined in (a). In all cases, ω ^ has a θ and ϕ component in spherical coordinates, and is normalized to a magnitude of 1 [11].

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Fig. 2
Fig. 2 Surface plot of relative difference Rd for five different indices of refraction: (a-d) ñ = 4+10i, (e-h) ñ = 1.5+i, (i-l) ñ = 1.4, (m-p) ñ = 1.7+5i, (q-t) ñ = 0.25+3i. For each index, four different incident angles are plotted: (a,e,i,m,q) θi = 15°; (b,f,j,n,r) θi = 30°; (c,g,k,o,s) θi = 45°; (d,h,l,p,t) θi = 60°. The plots are all symmetric about ϕs = 180°. The dotted black line at ϕs = 180° on the right side of each plot represents in-plane scatter. In each case, Rd = 0 when θi = θs and ϕs = 180°; Rd is generally small when θi and θs are small.

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Equations (31)

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f r = d L s ( ω ^ i , ω ^ s , λ ) d E i ( ω ^ i , λ ) ,
f μ ( ω ^ i , ω ^ s ) = ρ s F ( θ d ) D μ ( θ h ) G ( ω ^ i , ω ^ s ) σ ( θ i , θ s ) + ρ v V ( ω ^ i , ω ^ s ) + ρ d / π ,
σ ( θ i , θ s ) = 1 4 cos θ i cos θ s .
ω ^ h = ω ^ i + ω ^ s | | ω ^ i + ω ^ s | | , ω ^ d = ℝ y ( − θ h ) ℝ z ( − ϕ h ) ,
cos ( 2 θ d ) = cos θ i cos θ s + sin θ i sin θ s cos ϕ s ,
cos θ h = cos θ i + cos θ s 2 cos θ d .
D g ( θ h ) = 1 2 π σ g 2 cos 4 θ h exp [ − tan 2 θ h 2 σ g 2 ] ,
∫ 0 2 π ∫ 0 π / 2 D g ( θ h ) cos θ h sin θ h d θ h d ϕ h = 1 .
Δ α = α s − α i = λ ν y = sin θ s sin ( ϕ s − π ) Δ β = β s − β i = λ ν x = sin θ s cos ( ϕ s − π ) − sin θ i ,
η r 2 = ( Δ α ) 2 + ( Δ β ) 2 = sin 2 θ i + sin 2 θ s + 2 sin θ i sin θ s cos ϕ s = ( λ ν x y 2 π ) 2 ,
f a = π K l c 2 λ 2 exp ( − g ) ∑ m = 1 ∞ g m m ! m exp ( − ν x y 2 l c 2 4 m ) ,
g ( θ i , θ s ) = ( 2 π σ s λ ) 2 ( cos θ i + cos θ s ) 2 .
K ( β i ) = ∫ − ∞ ∞ ∫ − ∞ ∞ L s ( α s , β s − β i ) d β s d α s ∫ − 1 1 ∫ − 1 − α s 2 1 − α s 2 L s ( α s , β s − β i ) d β s d α s .
f p = 4 π 3 l c 2 σ s 2 Q ( cos θ i + cos θ s ) 2 λ 4 exp [ − ( π l c η r λ ) 2 ] ,
f v r = K Q l c 2 4 π σ s 2 ( cos θ i + cos θ s ) 2 exp [ − ( l c 2 σ s ) 2 ( η r cos θ i + cos θ s ) 2 ] ,
f μ f v r = ( ρ s 2 K ) ( 2 F G Q ) ( ( cos θ i + cos θ s ) 2 4 cos θ i cos θ s cos 4 θ h ) ( σ s 2 l c σ g ) 2 × exp [ − ( tan 2 θ h 2 σ g 2 − l c 2 4 σ s 2 η r 2 ( cos θ i + cos θ s ) 2 ) ] .
tan 2 θ h = 4 cos 2 θ d − ( cos θ i + cos θ s ) 2 ( cos θ i + cos θ s ) 2 .
4 cos 2 θ d − ( cos θ i + cos θ s ) 2 = 2 + 2 ( 2 cos 2 θ d − 1 ) − ( cos θ i + cos θ s ) 2 = 2 + 2 cos ( 2 θ d ) − ( cos θ i + cos θ s ) 2 = sin 2 θ i + cos 2 θ i + sin 2 θ s + cos 2 θ s + 2 cos ( 2 θ d ) − ( cos θ i + cos θ s ) 2 = sin 2 θ i + sin 2 θ s + 2 sin θ i sin θ s cos ϕ s = η r 2 ,
tan 2 θ h = ( η r cos θ i + cos θ s ) 2 .
σ g , v r = σ s 2 l c .
f μ f v r = ( ρ s 2 K ) ( 2 F G Q ) ( ( cos θ i + cos θ s ) 2 4 cos θ i cos θ s cos 4 θ h ) .
S = 4 cos θ i cos θ s cos 4 θ h ( cos θ i + cos θ s ) 2 ≈ 2 F Q ,
R d = | 2 F / Q − S | 2 F / Q .
f μ f p = ( ρ s λ 2 8 π 2 σ s 2 ) ( 2 F G Q ) ( 1 4 cos θ i cos θ s cos 4 θ h ( cos θ i + cos θ s ) 2 ) × ( λ 2 2 π 2 σ g 2 l c 2 ) exp [ − ( tan 2 θ h 2 σ g 2 − π 2 l c 2 η r 2 λ 2 ) ] .
σ g , p = λ π l c 2 ,
F b k 2 = ( 1 + ( cos θ i cos θ s + sin θ i sin θ s cos ϕ s ) cos θ i ( cos θ i + cos θ s ) ) 2 .
F b k 2 = cos 2 θ d cos 2 θ i cos 2 θ h .
1 S = cos 2 θ d cos θ i cos θ s cos 2 θ h ≈ Q 2 F .
f v r = K F G 4 cos θ i cos θ s ( l c 2 2 π σ s 2 cos 4 θ h ) exp [ − ( l c 2 σ s ) 2 tan 2 θ h ] ,
f p = ( 2 π ) 3 F G ( cos θ i + cos θ s ) 4 4 cos θ i cos θ s cos 4 θ h ( σ s 2 l c 2 λ 4 ) × exp [ − ( π l c λ ) 2 tan 2 θ h ( cos θ i + cos θ s ) 2 ] .
f M B K = π F G ( cos θ i + cos θ s ) 2 K l c 2 2 λ 2 cos θ i cos θ s cos 4 θ h exp ( − g ) × ∑ m = 1 ∞ g m m ! m exp ( − l c 2 4 m λ 2 tan 2 θ h ( cos θ i + cos θ s ) 2 ) ,

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