Abstract

The scalar bidirectional reflectance distribution function (BRDF) due to a perfectly conducting surface with roughness and autocorrelation width comparable with the illumination wavelength is derived from coherence theory on the assumption of a random reflective phase screen and an expansion valid for large effective roughness. A general quadratic expansion of the two-dimensional isotropic surface autocorrelation function near the origin yields representative Cauchy and Gaussian BRDF solutions and an intermediate general solution as the sum of an incoherent component and a nonspecular coherent component proportional to an integral of the plasma dispersion function in the complex plane. Plots illustrate agreement of the derived general solution with original bistatic BRDF data due to a machined aluminum surface, and comparisons are drawn with previously published data in the examination of variations with incident angle, roughness, illumination wavelength, and autocorrelation coefficients in the bistatic and monostatic geometries. The general quadratic autocorrelation expansion provides a BRDF solution that smoothly interpolates between the well-known results of the linear and parabolic approximations.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. O. Steinvall, "Effects of target shape and reflection on laser radar cross sections," Appl. Opt. 39, 4381-4391 (2000).
    [Crossref]
  2. O. Steinvall and T. Carlsson, "Three-dimensional laser radar modelling," inLaser Radar Technology and Applications VI, G.W.Kamerman, ed., Proc. SPIE 4377, 23-24 (2001).
  3. A. I. Carswell, "Advances in laser ranging," in Laser Radar Technology for Remote Sensing, C.Werner, ed., Proc. SPIE 5240, 1-9 (2004).
  4. W. C. Snyder and Z. Wan, "BRDF models to predict spectral reflectance and emissivity in the thermal infrared," IEEE Trans. Geosci. Remote Sens. 36, 214-225 (1998).
    [Crossref]
  5. M. Kaasalainen, K. Muinonen, and T. Laakso, "Shapes and scattering properties of large irregular bodies from photometric data," Opt. Express 8, 296-301 (2001).
    [Crossref] [PubMed]
  6. S. Kaasalainen, J. Piironen, M. Kaasalainen, A. W. Harris, K. Muinonen, and A. Cellino, "Asteroid photometric and polarimetric phase curves: empirical interpretation," Icarus 161, 34-46 (2003).
    [Crossref]
  7. D. Fontani, F. Francini, G. Longobardi, and P. Sansoni, "Optical control of surface finish," Opt. Lasers Eng. 32, 459-472 (2000).
    [Crossref]
  8. P. A. Smith, D. A. van Veldhuizen, and K. S. Keppler, "Modeling and simulation tools for high-energy laser safety applications," in Enabling Technology for Simulation Science V, A.F.Sisti and D.A.Trevisani, eds., Proc. SPIE 4367, 478-485 (2001).
  9. S. H. C. P. McCall, "The importance of scatter in stray light analysis," Opt. Photonics News 12(11), 40-47 (2001).
    [Crossref]
  10. F. Drago and K. Myszkowski, "Validation proposal for global illumination and rendering techniques," Comput. Graphics 25, 511-518 (2001).
    [Crossref]
  11. M. S. Gilmore, R. Castaño, T. Mann, R. C. Anderson, E. D. Mjolsness, R. Manduchi, and R. S. Saunders, "Strategies for autonomous rovers at Mars," J. Geophys. Res. 105, 29223-29237 (2000).
    [Crossref]
  12. O. G. Cula and K. J. Dana, "3D texture recognition using bidirectional feature histograms," Int. J. Comput. Vis. 59, 33-60 (2004).
    [Crossref]
  13. T. Weyrich, H. Pfister, and M. Gross, "Rendering deformable surface reflectance fields," IEEE Trans. Vis. Comput. Graph. 11, 48-58 (2005).
    [Crossref] [PubMed]
  14. J. Dorsey and P. Hanrahan, "Digital materials and virtual weathering," Sci. Am. 282(2), 64-71 (2000).
    [Crossref] [PubMed]
  15. F. Bernardini, I. M. Martin, and H. Rushmeier, "High-quality texture reconstruction from multiple scans," IEEE Trans. Vis. Comput. Graph. 7, 318-332 (2001).
    [Crossref]
  16. C. Rocchini, P. Cignoni, C. Montani, and R. Scopigno, "Acquiring, stitching, and blending diffuse appearance attributes on 3D models," Visual Comput. 18, 186-204 (2002).
    [Crossref]
  17. J. Meseth, G. Müller, and R. Klein, "Reflectance field based real-time, high-quality rendering of bidirectional texture functions," Comput. Graphics 28, 105-112 (2004).
    [Crossref]
  18. A. Ishimaru and J. S. Chen, "Scattering from very rough metallic and dielectric surfaces: a theory based on the modified Kirchhoff approximation," Waves Random Media 1, 25-34 (1991).
    [Crossref]
  19. G. Macelloni, G. Nesti, P. Pampaloni, S. Sigismondi, D. Tarchi, and S. Lolli, "Experimental validation of surface scattering and emission models," IEEE Trans. Geosci. Remote Sens. 38, 459-469 (2000).
    [Crossref]
  20. O. P. Bruno, A. Sei, and M. Caponi, "High-order high-frequency solutions of rough surface scattering problems," Radio Sci. 37, 10.1029/2000 RS002551 (2002).
    [Crossref]
  21. M. Saillard and A. Sentenac, "Rigorous solutions for electromagnetic scattering from rough surfaces," Waves Random Media 11, R103-R137 (2001).
    [Crossref]
  22. T. M. Elfouhaily and C.-A. Guérin, "A critical survey of approximate scattering wave theories from random rough surfaces," Waves Random Media 14, R1-R40 (2004).
    [Crossref]
  23. S. O. Rice, "Reflection of electromagnetic waves from slightly rough surfaces," Commun. Pure Appl. Math. 4, 351-378 (1951).
    [Crossref]
  24. J. Stover, Optical Scattering: Measurement and Analysis (McGraw-Hill, 1990).
  25. J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger/IOP, 1991).
  26. G. S. Agarwal, "Scattering from rough surfaces," Opt. Commun. 14, 161-166 (1975).
    [Crossref]
  27. J. A. Sánchez-Gil and M. Nieto-Vesperinas, "Light scattering from random rough dielectric surfaces," J. Opt. Soc. Am. A 8, 1270-1286 (1991).
    [Crossref]
  28. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963).
  29. E. R. Méndez, E. E. Garcia-Guerrero, H. M. Escamilla, A. A. Maradudin, T. A. Leskova, and A. V. Shchegrov, "Photofabrication of random achromatic optical diffusers for uniform illumination," Appl. Opt. 40, 1098-1108 (2001).
    [Crossref]
  30. X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, "A comprehensive physical model for light reflection," Comput. Graph. 25, 175-186 (1991).
    [Crossref]
  31. S. Silver, "Scattering and diffraction," in Microwave Antenna Theory and Design, S.Silver, ed. (McGraw-Hill, 1949), Chap. 5.
  32. J. C. Leader, "Bidirectional scattering of electromagnetic waves from rough surfaces," J. Appl. Phys. 42, 4808-4816 (1971).
    [Crossref]
  33. J. C. Leader, "Analysis and prediction of laser scattering from rough-surface materials," J. Opt. Soc. Am. 69, 610-628 (1979).
    [Crossref]
  34. R. D. Kodis, "A note on the theory of scattering from an irregular surface," IEEE Trans. Antennas Propag. AP-14, 77-82 (1966).
    [Crossref]
  35. K. E. Torrance and E. M. Sparrow, "Theory for off-specular reflection from roughened surfaces," J. Opt. Soc. Am. 57, 1105-1114 (1967).
    [Crossref]
  36. D. E. Barrick, "Rough surface scattering based on the specular point theory," IEEE Trans. Antennas Propag. AP-16, 449-454 (1968).
    [Crossref]
  37. M. Ashikhmin, S. Premoze, and P. Shirley, "A microfacet-based BRDF generator," in Proceedings of ACM SIGGRAPH 2000 (www.siggraph.org), pp. 65-74.
  38. K. Ivanova, M. A. Michalev, and O. I. Yordanov, "Numerical study of scattering by rough surfaces with intermediate and large scale roughness," Radio Sci. 26, 505-510 (1991).
    [Crossref]
  39. F. Gori, "Matrix treatment for partially polarized, partially coherent beams," Opt. Lett. 23, 241-243 (1998).
    [Crossref]
  40. E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
    [Crossref]
  41. O. Korotkova, B. G. Hoover, V. L. Gamiz, and E. Wolf, "Coherence and polarization properties of far-fields generated by quasi-homogeneous electromagnetic sources," J. Opt. Soc. Am. A 22, 2547-2556 (2005).
    [Crossref]
  42. W. Gautschi, "Error function and Fresnel integrals," in Handbook of Mathematical Functions, 9th ed., M.Abramowitz and I.A.Stegun, eds. (Dover, 1972), Chap. 7.
  43. W. J. Thompson, "Numerous neat algorithms for the Voight profile function," Comput. Phys. 7, 627-631 (1993).
  44. R. J. Wells, "Rapid approximation to the Voight/Faddeeva function and its derivatives," J. Quant. Spectrosc. Radiat. Transf. 62, 29-48 (1999).
    [Crossref]
  45. D. L. Jordan, G. D. Lewis, and E. Jakeman, "Emission polarization of roughened glass and aluminum surfaces," Appl. Opt. 35, 3583-3590 (1996).
    [Crossref] [PubMed]
  46. A. K. Fung, "Theory of radar scatter from rough surfaces, bistatic and monostatic, with application to lunar radar return," J. Geophys. Res. 69, 1063-1073 (1964).
    [Crossref]
  47. P. Beckmann, "Scattering by composite rough surfaces," Proc. IEEE 53, 1012-1015 (1965).
    [Crossref]
  48. A. K. Fung and R. K. Moore, "Effects of structure size on moon and earth radar returns at various angles," J. Geophys. Res. 69, 1075-1081 (1964).
    [Crossref]
  49. W. K. Klemperer, "Angular scattering law for the moon at 6-meter wavelength," J. Geophys. Res. 70, 3798-3800 (1965).
    [Crossref]
  50. J. V. Evans, "Radar studies of planetary surfaces," Annu. Rev. Astron. Astrophys. 7, 201-248 (1969).
    [Crossref]
  51. D. E. Barrick, "Unacceptable height correlation coefficients and the quasi-specular component in rough surface scattering," Radio Sci. 5, 647-654 (1970).
    [Crossref]
  52. K. E. Warnick and D. V. Arnold, "Generalization of the geometrical-optics scattering limit for a rough conducting surface," J. Opt. Soc. Am. A 15, 2355-2361 (1998).
    [Crossref]
  53. J. W. Goodman, "Statistical properties of laser speckle patterns," in Laser Speckle and Related Phenomena, 2nd enlarged ed., J.C.Dainty, ed. (Springer-Verlag, 1984), Chap. 2.
  54. A review of the radiometric units and notation used in this paper is provided in E. L. Dereniak and D. G. Crowe, Optical Radiation Detectors (Wiley, 1984).
  55. A. Walther, "Radiometry and coherence," J. Opt. Soc. Am. 58, 1256-1259 (1968).
    [Crossref]
  56. E. Wolf and W. H. Carter, "Angular distribution of radiant intensity from sources of different degrees of spatial coherence," Opt. Commun. 13, 205-209 (1975).
    [Crossref]
  57. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), Chap. 5.7.
  58. J. C. Leader, "Similarities and distinctions between coherence theory relations and laser scattering phenomena," Opt. Eng. (Bellingham) 19, 593-601 (1980).
  59. D. D. Duncan, D. V. Hahn, and M. E. Thomas, "Physics-based polarimetric BRDF models," in Optical Diagnostic Methods for Inorganic Materials III, L.M.Hanssen, ed., Proc. SPIE 5192, 129-140 (2003).
  60. B. G. Hoover and V. L. Gamiz, "Diffractive bidirectional reflectance distributions of surfaces with large effective roughness in one dimension," in Laser Radar Techniques for Atmospheric Sensing, U.N.Singh, ed., Proc. SPIE 5575, 137-142 (2004).
  61. J. W. Goodman, Statistical Optics (Wiley, 1985), Chap. 5.
  62. x and Deltax represent two-dimensional vectors in the plane of the average surface, i.e., x=xx̂+yŷ.
  63. T. Karabacak, Y. Zhao, M. Stowe, B. Quayle, G.-C. Wang, and T.-M. Lu, "Large-angle in-plane light scattering from rough surfaces," Appl. Opt. 39, 4658-4668 (2000).
    [Crossref]
  64. See Ref. and references therein.
  65. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1961), Chap. 5.
  66. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), Chap. 9.
  67. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978), Chap. 9.
  68. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, corrected and enlarged ed. (Academic, 1980), 6.631(4).
  69. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, corrected and enlarged ed. (Academic, 1980), 6.623(2).
  70. B. D. Fried and S. D. Conte, The Plasma Dispersion Function: The Hilbert Transform of the Gaussian (Academic, 1961). Note that the function defined in this reference has its real and imaginary parts switched relative to the plasma dispersion function defined in Refs. .
  71. V. N. Faddeeva and N. M. Terent'ev, Tables of Values of the Function w(z)=e−z2(1+2i/pi∫0zet2dt) for Complex Argument, V.A.Fok ed., translated by D. G. Fry (Pergamon, 1961).
  72. V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa-Spagnolo, "Generalized Bessel-Gauss beams," J. Mod. Opt. 43, 1155-1166 (1996).
  73. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Chap. 2.
  74. W. H. Carter and E. Wolf, "Coherence properties of Lambertian and non-Lambertian sources," J. Opt. Soc. Am. 65, 1067-1071 (1975).
    [Crossref]
  75. H. P. Baltes, B. Steinle, and G. Antes, "Spectral coherence and the radiant intensity from statistically homogeneous and isotropic planar sources," Opt. Commun. 18, 242-246 (1976).
    [Crossref]
  76. The meanings of the terms "high frequency" and "low frequency" in coherence theory are unfortunately essentially opposite their meanings in scattering theory. Use of these terms in this context is therefore avoided.
  77. Derivatives of the delta function are considered in T. B. A. Senior and J. L. Volakis, Approximate Boundary Conditions in Electromagnetics (IEEE Press, 1995).
  78. E. W. Weisstein, "Erf," from MathWorld-A Wolfram Web Resource, http://mathworld.wolfram.com/Erf.html.
  79. K. A. O'Donnell and E. R. Méndez, "Experimental study of scattering from characterized random surfaces," J. Opt. Soc. Am. A 4, 1194-1205 (1987).
    [Crossref]
  80. J. Renau, P. K. Cheo, and H. G. Cooper, "Depolarization of linearly polarized EM waves backscattered from rough metals and inhomogeneous dielectrics," J. Opt. Soc. Am. 57, 459-466 (1967).
    [Crossref] [PubMed]
  81. P. K. Cheo and J. Renau, "Wavelength dependence of total and depolarized backscattered laser light from rough metallic surfaces," J. Opt. Soc. Am. 59, 821-826 (1969).
    [Crossref]
  82. E. Marx and T. V. Vorburger, "Direct and inverse problems for light scattered by rough surfaces," Appl. Opt. 29, 3613-3626 (1990).
    [Crossref] [PubMed]
  83. L. Brainard, W. Lynn, D. Ramer, and W. Shemano, "Optical measurements facility," Air Force Research Laboratory, Materials & Manufacturing Directorate, Rep. AFRL-ML-WP-TR-1999-4142 (1999).
  84. C. C. Sung and W. D. Eberhardt, "Explanation of the experimental results of light backscattered from a very rough surface," J. Opt. Soc. Am. 68, 323-328 (1978).
    [Crossref]
  85. A review of theory and measurements in the small-roughness regime is provided in J. M. Elson and J. M. Bennett, "Vector scattering theory," Opt. Eng. (Bellingham) 18, 116-124 (1979).
  86. A. J. Sant, J. C. Dainty, and M.-J. Kim, "Comparison of surface scattering between identical, randomly rough metal and dielectric diffusers," Opt. Lett. 14, 1183-1185 (1989).
    [Crossref] [PubMed]
  87. M.-J. Kim, J. C. Dainty, A. T. Friberg, and A. J. Sant, "Experimental study of enhanced backscattering from one- and two-dimensional random rough surfaces," J. Opt. Soc. Am. A 7, 569-577 (1990).
    [Crossref]
  88. N. C. Bruce and J. C. Dainty, "Multiple scattering from rough dielectric and metal surfaces using the Kirchhoff approximation," J. Mod. Opt. 38, 1471-1481 (1991).
    [Crossref]
  89. M. K. Shepard and B. A. Campbell, "Radar scattering from a self-affine fractal surface: near-nadir regime," Icarus 141, 156-171 (1999).
    [Crossref]
  90. W. C. Snyder, "Reciprocity of the bidirectional reflectance distribution function (BRDF) in measurements and models of structured surfaces," IEEE Trans. Geosci. Remote Sens. 36, 685-691 (1998).
    [Crossref]
  91. It is possible to dispense with the quasi-monochromatic conditions by using the cross-spectral density W(x1,x2,nu)=∫Gamma(x1,x2,tau)exp(2pijnutau)dtau rather than the coherence function, thus proceeding in the space-frequency rather than the space-time domain. Under the quasi-monochromatic conditions these two approaches are essentially equivalent.
  92. Some authors use the symbol µ to represent the normalized mutual intensity function. We reserve µ for the analogous correlation function in the space-frequency domain.

2005 (2)

2004 (3)

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

J. Meseth, G. Müller, and R. Klein, "Reflectance field based real-time, high-quality rendering of bidirectional texture functions," Comput. Graphics 28, 105-112 (2004).
[Crossref]

O. G. Cula and K. J. Dana, "3D texture recognition using bidirectional feature histograms," Int. J. Comput. Vis. 59, 33-60 (2004).
[Crossref]

2003 (2)

S. Kaasalainen, J. Piironen, M. Kaasalainen, A. W. Harris, K. Muinonen, and A. Cellino, "Asteroid photometric and polarimetric phase curves: empirical interpretation," Icarus 161, 34-46 (2003).
[Crossref]

E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
[Crossref]

2002 (2)

O. P. Bruno, A. Sei, and M. Caponi, "High-order high-frequency solutions of rough surface scattering problems," Radio Sci. 37, 10.1029/2000 RS002551 (2002).
[Crossref]

C. Rocchini, P. Cignoni, C. Montani, and R. Scopigno, "Acquiring, stitching, and blending diffuse appearance attributes on 3D models," Visual Comput. 18, 186-204 (2002).
[Crossref]

2001 (6)

F. Bernardini, I. M. Martin, and H. Rushmeier, "High-quality texture reconstruction from multiple scans," IEEE Trans. Vis. Comput. Graph. 7, 318-332 (2001).
[Crossref]

M. Saillard and A. Sentenac, "Rigorous solutions for electromagnetic scattering from rough surfaces," Waves Random Media 11, R103-R137 (2001).
[Crossref]

M. Kaasalainen, K. Muinonen, and T. Laakso, "Shapes and scattering properties of large irregular bodies from photometric data," Opt. Express 8, 296-301 (2001).
[Crossref] [PubMed]

S. H. C. P. McCall, "The importance of scatter in stray light analysis," Opt. Photonics News 12(11), 40-47 (2001).
[Crossref]

F. Drago and K. Myszkowski, "Validation proposal for global illumination and rendering techniques," Comput. Graphics 25, 511-518 (2001).
[Crossref]

E. R. Méndez, E. E. Garcia-Guerrero, H. M. Escamilla, A. A. Maradudin, T. A. Leskova, and A. V. Shchegrov, "Photofabrication of random achromatic optical diffusers for uniform illumination," Appl. Opt. 40, 1098-1108 (2001).
[Crossref]

2000 (6)

G. Macelloni, G. Nesti, P. Pampaloni, S. Sigismondi, D. Tarchi, and S. Lolli, "Experimental validation of surface scattering and emission models," IEEE Trans. Geosci. Remote Sens. 38, 459-469 (2000).
[Crossref]

M. S. Gilmore, R. Castaño, T. Mann, R. C. Anderson, E. D. Mjolsness, R. Manduchi, and R. S. Saunders, "Strategies for autonomous rovers at Mars," J. Geophys. Res. 105, 29223-29237 (2000).
[Crossref]

D. Fontani, F. Francini, G. Longobardi, and P. Sansoni, "Optical control of surface finish," Opt. Lasers Eng. 32, 459-472 (2000).
[Crossref]

O. Steinvall, "Effects of target shape and reflection on laser radar cross sections," Appl. Opt. 39, 4381-4391 (2000).
[Crossref]

J. Dorsey and P. Hanrahan, "Digital materials and virtual weathering," Sci. Am. 282(2), 64-71 (2000).
[Crossref] [PubMed]

T. Karabacak, Y. Zhao, M. Stowe, B. Quayle, G.-C. Wang, and T.-M. Lu, "Large-angle in-plane light scattering from rough surfaces," Appl. Opt. 39, 4658-4668 (2000).
[Crossref]

1999 (2)

R. J. Wells, "Rapid approximation to the Voight/Faddeeva function and its derivatives," J. Quant. Spectrosc. Radiat. Transf. 62, 29-48 (1999).
[Crossref]

M. K. Shepard and B. A. Campbell, "Radar scattering from a self-affine fractal surface: near-nadir regime," Icarus 141, 156-171 (1999).
[Crossref]

1998 (4)

W. C. Snyder, "Reciprocity of the bidirectional reflectance distribution function (BRDF) in measurements and models of structured surfaces," IEEE Trans. Geosci. Remote Sens. 36, 685-691 (1998).
[Crossref]

K. E. Warnick and D. V. Arnold, "Generalization of the geometrical-optics scattering limit for a rough conducting surface," J. Opt. Soc. Am. A 15, 2355-2361 (1998).
[Crossref]

W. C. Snyder and Z. Wan, "BRDF models to predict spectral reflectance and emissivity in the thermal infrared," IEEE Trans. Geosci. Remote Sens. 36, 214-225 (1998).
[Crossref]

F. Gori, "Matrix treatment for partially polarized, partially coherent beams," Opt. Lett. 23, 241-243 (1998).
[Crossref]

1996 (2)

D. L. Jordan, G. D. Lewis, and E. Jakeman, "Emission polarization of roughened glass and aluminum surfaces," Appl. Opt. 35, 3583-3590 (1996).
[Crossref] [PubMed]

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa-Spagnolo, "Generalized Bessel-Gauss beams," J. Mod. Opt. 43, 1155-1166 (1996).

1993 (1)

W. J. Thompson, "Numerous neat algorithms for the Voight profile function," Comput. Phys. 7, 627-631 (1993).

1991 (5)

J. A. Sánchez-Gil and M. Nieto-Vesperinas, "Light scattering from random rough dielectric surfaces," J. Opt. Soc. Am. A 8, 1270-1286 (1991).
[Crossref]

K. Ivanova, M. A. Michalev, and O. I. Yordanov, "Numerical study of scattering by rough surfaces with intermediate and large scale roughness," Radio Sci. 26, 505-510 (1991).
[Crossref]

X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, "A comprehensive physical model for light reflection," Comput. Graph. 25, 175-186 (1991).
[Crossref]

A. Ishimaru and J. S. Chen, "Scattering from very rough metallic and dielectric surfaces: a theory based on the modified Kirchhoff approximation," Waves Random Media 1, 25-34 (1991).
[Crossref]

N. C. Bruce and J. C. Dainty, "Multiple scattering from rough dielectric and metal surfaces using the Kirchhoff approximation," J. Mod. Opt. 38, 1471-1481 (1991).
[Crossref]

1990 (2)

1989 (1)

1987 (1)

1980 (1)

J. C. Leader, "Similarities and distinctions between coherence theory relations and laser scattering phenomena," Opt. Eng. (Bellingham) 19, 593-601 (1980).

1979 (2)

A review of theory and measurements in the small-roughness regime is provided in J. M. Elson and J. M. Bennett, "Vector scattering theory," Opt. Eng. (Bellingham) 18, 116-124 (1979).

J. C. Leader, "Analysis and prediction of laser scattering from rough-surface materials," J. Opt. Soc. Am. 69, 610-628 (1979).
[Crossref]

1978 (1)

1976 (1)

H. P. Baltes, B. Steinle, and G. Antes, "Spectral coherence and the radiant intensity from statistically homogeneous and isotropic planar sources," Opt. Commun. 18, 242-246 (1976).
[Crossref]

1975 (3)

W. H. Carter and E. Wolf, "Coherence properties of Lambertian and non-Lambertian sources," J. Opt. Soc. Am. 65, 1067-1071 (1975).
[Crossref]

E. Wolf and W. H. Carter, "Angular distribution of radiant intensity from sources of different degrees of spatial coherence," Opt. Commun. 13, 205-209 (1975).
[Crossref]

G. S. Agarwal, "Scattering from rough surfaces," Opt. Commun. 14, 161-166 (1975).
[Crossref]

1971 (1)

J. C. Leader, "Bidirectional scattering of electromagnetic waves from rough surfaces," J. Appl. Phys. 42, 4808-4816 (1971).
[Crossref]

1970 (1)

D. E. Barrick, "Unacceptable height correlation coefficients and the quasi-specular component in rough surface scattering," Radio Sci. 5, 647-654 (1970).
[Crossref]

1969 (2)

1968 (2)

A. Walther, "Radiometry and coherence," J. Opt. Soc. Am. 58, 1256-1259 (1968).
[Crossref]

D. E. Barrick, "Rough surface scattering based on the specular point theory," IEEE Trans. Antennas Propag. AP-16, 449-454 (1968).
[Crossref]

1967 (2)

1966 (1)

R. D. Kodis, "A note on the theory of scattering from an irregular surface," IEEE Trans. Antennas Propag. AP-14, 77-82 (1966).
[Crossref]

1965 (2)

W. K. Klemperer, "Angular scattering law for the moon at 6-meter wavelength," J. Geophys. Res. 70, 3798-3800 (1965).
[Crossref]

P. Beckmann, "Scattering by composite rough surfaces," Proc. IEEE 53, 1012-1015 (1965).
[Crossref]

1964 (2)

A. K. Fung and R. K. Moore, "Effects of structure size on moon and earth radar returns at various angles," J. Geophys. Res. 69, 1075-1081 (1964).
[Crossref]

A. K. Fung, "Theory of radar scatter from rough surfaces, bistatic and monostatic, with application to lunar radar return," J. Geophys. Res. 69, 1063-1073 (1964).
[Crossref]

1951 (1)

S. O. Rice, "Reflection of electromagnetic waves from slightly rough surfaces," Commun. Pure Appl. Math. 4, 351-378 (1951).
[Crossref]

Agarwal, G. S.

G. S. Agarwal, "Scattering from rough surfaces," Opt. Commun. 14, 161-166 (1975).
[Crossref]

Anderson, R. C.

M. S. Gilmore, R. Castaño, T. Mann, R. C. Anderson, E. D. Mjolsness, R. Manduchi, and R. S. Saunders, "Strategies for autonomous rovers at Mars," J. Geophys. Res. 105, 29223-29237 (2000).
[Crossref]

Antes, G.

H. P. Baltes, B. Steinle, and G. Antes, "Spectral coherence and the radiant intensity from statistically homogeneous and isotropic planar sources," Opt. Commun. 18, 242-246 (1976).
[Crossref]

Arnold, D. V.

Ashikhmin, M.

M. Ashikhmin, S. Premoze, and P. Shirley, "A microfacet-based BRDF generator," in Proceedings of ACM SIGGRAPH 2000 (www.siggraph.org), pp. 65-74.

Bagini, V.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa-Spagnolo, "Generalized Bessel-Gauss beams," J. Mod. Opt. 43, 1155-1166 (1996).

Baltes, H. P.

H. P. Baltes, B. Steinle, and G. Antes, "Spectral coherence and the radiant intensity from statistically homogeneous and isotropic planar sources," Opt. Commun. 18, 242-246 (1976).
[Crossref]

Barrick, D. E.

D. E. Barrick, "Unacceptable height correlation coefficients and the quasi-specular component in rough surface scattering," Radio Sci. 5, 647-654 (1970).
[Crossref]

D. E. Barrick, "Rough surface scattering based on the specular point theory," IEEE Trans. Antennas Propag. AP-16, 449-454 (1968).
[Crossref]

Beckmann, P.

P. Beckmann, "Scattering by composite rough surfaces," Proc. IEEE 53, 1012-1015 (1965).
[Crossref]

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

Bennett, J. M.

A review of theory and measurements in the small-roughness regime is provided in J. M. Elson and J. M. Bennett, "Vector scattering theory," Opt. Eng. (Bellingham) 18, 116-124 (1979).

Bernardini, F.

F. Bernardini, I. M. Martin, and H. Rushmeier, "High-quality texture reconstruction from multiple scans," IEEE Trans. Vis. Comput. Graph. 7, 318-332 (2001).
[Crossref]

Brainard, L.

L. Brainard, W. Lynn, D. Ramer, and W. Shemano, "Optical measurements facility," Air Force Research Laboratory, Materials & Manufacturing Directorate, Rep. AFRL-ML-WP-TR-1999-4142 (1999).

Bruce, N. C.

N. C. Bruce and J. C. Dainty, "Multiple scattering from rough dielectric and metal surfaces using the Kirchhoff approximation," J. Mod. Opt. 38, 1471-1481 (1991).
[Crossref]

Bruno, O. P.

O. P. Bruno, A. Sei, and M. Caponi, "High-order high-frequency solutions of rough surface scattering problems," Radio Sci. 37, 10.1029/2000 RS002551 (2002).
[Crossref]

Campbell, B. A.

M. K. Shepard and B. A. Campbell, "Radar scattering from a self-affine fractal surface: near-nadir regime," Icarus 141, 156-171 (1999).
[Crossref]

Caponi, M.

O. P. Bruno, A. Sei, and M. Caponi, "High-order high-frequency solutions of rough surface scattering problems," Radio Sci. 37, 10.1029/2000 RS002551 (2002).
[Crossref]

Carlsson, T.

O. Steinvall and T. Carlsson, "Three-dimensional laser radar modelling," inLaser Radar Technology and Applications VI, G.W.Kamerman, ed., Proc. SPIE 4377, 23-24 (2001).

Carswell, A. I.

A. I. Carswell, "Advances in laser ranging," in Laser Radar Technology for Remote Sensing, C.Werner, ed., Proc. SPIE 5240, 1-9 (2004).

Carter, W. H.

E. Wolf and W. H. Carter, "Angular distribution of radiant intensity from sources of different degrees of spatial coherence," Opt. Commun. 13, 205-209 (1975).
[Crossref]

W. H. Carter and E. Wolf, "Coherence properties of Lambertian and non-Lambertian sources," J. Opt. Soc. Am. 65, 1067-1071 (1975).
[Crossref]

Castaño, R.

M. S. Gilmore, R. Castaño, T. Mann, R. C. Anderson, E. D. Mjolsness, R. Manduchi, and R. S. Saunders, "Strategies for autonomous rovers at Mars," J. Geophys. Res. 105, 29223-29237 (2000).
[Crossref]

Cellino, A.

S. Kaasalainen, J. Piironen, M. Kaasalainen, A. W. Harris, K. Muinonen, and A. Cellino, "Asteroid photometric and polarimetric phase curves: empirical interpretation," Icarus 161, 34-46 (2003).
[Crossref]

Chen, J. S.

A. Ishimaru and J. S. Chen, "Scattering from very rough metallic and dielectric surfaces: a theory based on the modified Kirchhoff approximation," Waves Random Media 1, 25-34 (1991).
[Crossref]

Cheo, P. K.

Cignoni, P.

C. Rocchini, P. Cignoni, C. Montani, and R. Scopigno, "Acquiring, stitching, and blending diffuse appearance attributes on 3D models," Visual Comput. 18, 186-204 (2002).
[Crossref]

Conte, S. D.

B. D. Fried and S. D. Conte, The Plasma Dispersion Function: The Hilbert Transform of the Gaussian (Academic, 1961). Note that the function defined in this reference has its real and imaginary parts switched relative to the plasma dispersion function defined in Refs. .

Cooper, H. G.

Crowe, D. G.

A review of the radiometric units and notation used in this paper is provided in E. L. Dereniak and D. G. Crowe, Optical Radiation Detectors (Wiley, 1984).

Cula, O. G.

O. G. Cula and K. J. Dana, "3D texture recognition using bidirectional feature histograms," Int. J. Comput. Vis. 59, 33-60 (2004).
[Crossref]

Dainty, J. C.

Dana, K. J.

O. G. Cula and K. J. Dana, "3D texture recognition using bidirectional feature histograms," Int. J. Comput. Vis. 59, 33-60 (2004).
[Crossref]

Dereniak, E. L.

A review of the radiometric units and notation used in this paper is provided in E. L. Dereniak and D. G. Crowe, Optical Radiation Detectors (Wiley, 1984).

Dorsey, J.

J. Dorsey and P. Hanrahan, "Digital materials and virtual weathering," Sci. Am. 282(2), 64-71 (2000).
[Crossref] [PubMed]

Drago, F.

F. Drago and K. Myszkowski, "Validation proposal for global illumination and rendering techniques," Comput. Graphics 25, 511-518 (2001).
[Crossref]

Duncan, D. D.

D. D. Duncan, D. V. Hahn, and M. E. Thomas, "Physics-based polarimetric BRDF models," in Optical Diagnostic Methods for Inorganic Materials III, L.M.Hanssen, ed., Proc. SPIE 5192, 129-140 (2003).

Eberhardt, W. D.

Elfouhaily, T. M.

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

Elson, J. M.

A review of theory and measurements in the small-roughness regime is provided in J. M. Elson and J. M. Bennett, "Vector scattering theory," Opt. Eng. (Bellingham) 18, 116-124 (1979).

Escamilla, H. M.

Evans, J. V.

J. V. Evans, "Radar studies of planetary surfaces," Annu. Rev. Astron. Astrophys. 7, 201-248 (1969).
[Crossref]

Faddeeva, V. N.

V. N. Faddeeva and N. M. Terent'ev, Tables of Values of the Function w(z)=e−z2(1+2i/pi∫0zet2dt) for Complex Argument, V.A.Fok ed., translated by D. G. Fry (Pergamon, 1961).

Fontani, D.

D. Fontani, F. Francini, G. Longobardi, and P. Sansoni, "Optical control of surface finish," Opt. Lasers Eng. 32, 459-472 (2000).
[Crossref]

Francini, F.

D. Fontani, F. Francini, G. Longobardi, and P. Sansoni, "Optical control of surface finish," Opt. Lasers Eng. 32, 459-472 (2000).
[Crossref]

Frezza, F.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa-Spagnolo, "Generalized Bessel-Gauss beams," J. Mod. Opt. 43, 1155-1166 (1996).

Friberg, A. T.

Fried, B. D.

B. D. Fried and S. D. Conte, The Plasma Dispersion Function: The Hilbert Transform of the Gaussian (Academic, 1961). Note that the function defined in this reference has its real and imaginary parts switched relative to the plasma dispersion function defined in Refs. .

Fung, A. K.

A. K. Fung and R. K. Moore, "Effects of structure size on moon and earth radar returns at various angles," J. Geophys. Res. 69, 1075-1081 (1964).
[Crossref]

A. K. Fung, "Theory of radar scatter from rough surfaces, bistatic and monostatic, with application to lunar radar return," J. Geophys. Res. 69, 1063-1073 (1964).
[Crossref]

Gamiz, V. L.

O. Korotkova, B. G. Hoover, V. L. Gamiz, and E. Wolf, "Coherence and polarization properties of far-fields generated by quasi-homogeneous electromagnetic sources," J. Opt. Soc. Am. A 22, 2547-2556 (2005).
[Crossref]

B. G. Hoover and V. L. Gamiz, "Diffractive bidirectional reflectance distributions of surfaces with large effective roughness in one dimension," in Laser Radar Techniques for Atmospheric Sensing, U.N.Singh, ed., Proc. SPIE 5575, 137-142 (2004).

Garcia-Guerrero, E. E.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978), Chap. 9.

Gautschi, W.

W. Gautschi, "Error function and Fresnel integrals," in Handbook of Mathematical Functions, 9th ed., M.Abramowitz and I.A.Stegun, eds. (Dover, 1972), Chap. 7.

Gilmore, M. S.

M. S. Gilmore, R. Castaño, T. Mann, R. C. Anderson, E. D. Mjolsness, R. Manduchi, and R. S. Saunders, "Strategies for autonomous rovers at Mars," J. Geophys. Res. 105, 29223-29237 (2000).
[Crossref]

Goodman, J. W.

J. W. Goodman, "Statistical properties of laser speckle patterns," in Laser Speckle and Related Phenomena, 2nd enlarged ed., J.C.Dainty, ed. (Springer-Verlag, 1984), Chap. 2.

J. W. Goodman, Statistical Optics (Wiley, 1985), Chap. 5.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Chap. 2.

Gori, F.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, corrected and enlarged ed. (Academic, 1980), 6.631(4).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, corrected and enlarged ed. (Academic, 1980), 6.623(2).

Greenberg, D. P.

X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, "A comprehensive physical model for light reflection," Comput. Graph. 25, 175-186 (1991).
[Crossref]

Gross, M.

T. Weyrich, H. Pfister, and M. Gross, "Rendering deformable surface reflectance fields," IEEE Trans. Vis. Comput. Graph. 11, 48-58 (2005).
[Crossref] [PubMed]

Guérin, C.-A.

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

Hahn, D. V.

D. D. Duncan, D. V. Hahn, and M. E. Thomas, "Physics-based polarimetric BRDF models," in Optical Diagnostic Methods for Inorganic Materials III, L.M.Hanssen, ed., Proc. SPIE 5192, 129-140 (2003).

Hanrahan, P.

J. Dorsey and P. Hanrahan, "Digital materials and virtual weathering," Sci. Am. 282(2), 64-71 (2000).
[Crossref] [PubMed]

Harris, A. W.

S. Kaasalainen, J. Piironen, M. Kaasalainen, A. W. Harris, K. Muinonen, and A. Cellino, "Asteroid photometric and polarimetric phase curves: empirical interpretation," Icarus 161, 34-46 (2003).
[Crossref]

He, X. D.

X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, "A comprehensive physical model for light reflection," Comput. Graph. 25, 175-186 (1991).
[Crossref]

Hoover, B. G.

O. Korotkova, B. G. Hoover, V. L. Gamiz, and E. Wolf, "Coherence and polarization properties of far-fields generated by quasi-homogeneous electromagnetic sources," J. Opt. Soc. Am. A 22, 2547-2556 (2005).
[Crossref]

B. G. Hoover and V. L. Gamiz, "Diffractive bidirectional reflectance distributions of surfaces with large effective roughness in one dimension," in Laser Radar Techniques for Atmospheric Sensing, U.N.Singh, ed., Proc. SPIE 5575, 137-142 (2004).

Ishimaru, A.

A. Ishimaru and J. S. Chen, "Scattering from very rough metallic and dielectric surfaces: a theory based on the modified Kirchhoff approximation," Waves Random Media 1, 25-34 (1991).
[Crossref]

Ivanova, K.

K. Ivanova, M. A. Michalev, and O. I. Yordanov, "Numerical study of scattering by rough surfaces with intermediate and large scale roughness," Radio Sci. 26, 505-510 (1991).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), Chap. 9.

Jakeman, E.

Jordan, D. L.

Kaasalainen, M.

S. Kaasalainen, J. Piironen, M. Kaasalainen, A. W. Harris, K. Muinonen, and A. Cellino, "Asteroid photometric and polarimetric phase curves: empirical interpretation," Icarus 161, 34-46 (2003).
[Crossref]

M. Kaasalainen, K. Muinonen, and T. Laakso, "Shapes and scattering properties of large irregular bodies from photometric data," Opt. Express 8, 296-301 (2001).
[Crossref] [PubMed]

Kaasalainen, S.

S. Kaasalainen, J. Piironen, M. Kaasalainen, A. W. Harris, K. Muinonen, and A. Cellino, "Asteroid photometric and polarimetric phase curves: empirical interpretation," Icarus 161, 34-46 (2003).
[Crossref]

Karabacak, T.

Keppler, K. S.

P. A. Smith, D. A. van Veldhuizen, and K. S. Keppler, "Modeling and simulation tools for high-energy laser safety applications," in Enabling Technology for Simulation Science V, A.F.Sisti and D.A.Trevisani, eds., Proc. SPIE 4367, 478-485 (2001).

Kim, M.-J.

Klein, R.

J. Meseth, G. Müller, and R. Klein, "Reflectance field based real-time, high-quality rendering of bidirectional texture functions," Comput. Graphics 28, 105-112 (2004).
[Crossref]

Klemperer, W. K.

W. K. Klemperer, "Angular scattering law for the moon at 6-meter wavelength," J. Geophys. Res. 70, 3798-3800 (1965).
[Crossref]

Kodis, R. D.

R. D. Kodis, "A note on the theory of scattering from an irregular surface," IEEE Trans. Antennas Propag. AP-14, 77-82 (1966).
[Crossref]

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1961), Chap. 5.

Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1961), Chap. 5.

Korotkova, O.

Laakso, T.

Leader, J. C.

J. C. Leader, "Similarities and distinctions between coherence theory relations and laser scattering phenomena," Opt. Eng. (Bellingham) 19, 593-601 (1980).

J. C. Leader, "Analysis and prediction of laser scattering from rough-surface materials," J. Opt. Soc. Am. 69, 610-628 (1979).
[Crossref]

J. C. Leader, "Bidirectional scattering of electromagnetic waves from rough surfaces," J. Appl. Phys. 42, 4808-4816 (1971).
[Crossref]

Leskova, T. A.

Lewis, G. D.

Lolli, S.

G. Macelloni, G. Nesti, P. Pampaloni, S. Sigismondi, D. Tarchi, and S. Lolli, "Experimental validation of surface scattering and emission models," IEEE Trans. Geosci. Remote Sens. 38, 459-469 (2000).
[Crossref]

Longobardi, G.

D. Fontani, F. Francini, G. Longobardi, and P. Sansoni, "Optical control of surface finish," Opt. Lasers Eng. 32, 459-472 (2000).
[Crossref]

Lu, T.-M.

Lynn, W.

L. Brainard, W. Lynn, D. Ramer, and W. Shemano, "Optical measurements facility," Air Force Research Laboratory, Materials & Manufacturing Directorate, Rep. AFRL-ML-WP-TR-1999-4142 (1999).

Macelloni, G.

G. Macelloni, G. Nesti, P. Pampaloni, S. Sigismondi, D. Tarchi, and S. Lolli, "Experimental validation of surface scattering and emission models," IEEE Trans. Geosci. Remote Sens. 38, 459-469 (2000).
[Crossref]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), Chap. 5.7.

Manduchi, R.

M. S. Gilmore, R. Castaño, T. Mann, R. C. Anderson, E. D. Mjolsness, R. Manduchi, and R. S. Saunders, "Strategies for autonomous rovers at Mars," J. Geophys. Res. 105, 29223-29237 (2000).
[Crossref]

Mann, T.

M. S. Gilmore, R. Castaño, T. Mann, R. C. Anderson, E. D. Mjolsness, R. Manduchi, and R. S. Saunders, "Strategies for autonomous rovers at Mars," J. Geophys. Res. 105, 29223-29237 (2000).
[Crossref]

Maradudin, A. A.

Martin, I. M.

F. Bernardini, I. M. Martin, and H. Rushmeier, "High-quality texture reconstruction from multiple scans," IEEE Trans. Vis. Comput. Graph. 7, 318-332 (2001).
[Crossref]

Marx, E.

McCall, S. H. C. P.

S. H. C. P. McCall, "The importance of scatter in stray light analysis," Opt. Photonics News 12(11), 40-47 (2001).
[Crossref]

Méndez, E. R.

Meseth, J.

J. Meseth, G. Müller, and R. Klein, "Reflectance field based real-time, high-quality rendering of bidirectional texture functions," Comput. Graphics 28, 105-112 (2004).
[Crossref]

Michalev, M. A.

K. Ivanova, M. A. Michalev, and O. I. Yordanov, "Numerical study of scattering by rough surfaces with intermediate and large scale roughness," Radio Sci. 26, 505-510 (1991).
[Crossref]

Mjolsness, E. D.

M. S. Gilmore, R. Castaño, T. Mann, R. C. Anderson, E. D. Mjolsness, R. Manduchi, and R. S. Saunders, "Strategies for autonomous rovers at Mars," J. Geophys. Res. 105, 29223-29237 (2000).
[Crossref]

Montani, C.

C. Rocchini, P. Cignoni, C. Montani, and R. Scopigno, "Acquiring, stitching, and blending diffuse appearance attributes on 3D models," Visual Comput. 18, 186-204 (2002).
[Crossref]

Moore, R. K.

A. K. Fung and R. K. Moore, "Effects of structure size on moon and earth radar returns at various angles," J. Geophys. Res. 69, 1075-1081 (1964).
[Crossref]

Muinonen, K.

S. Kaasalainen, J. Piironen, M. Kaasalainen, A. W. Harris, K. Muinonen, and A. Cellino, "Asteroid photometric and polarimetric phase curves: empirical interpretation," Icarus 161, 34-46 (2003).
[Crossref]

M. Kaasalainen, K. Muinonen, and T. Laakso, "Shapes and scattering properties of large irregular bodies from photometric data," Opt. Express 8, 296-301 (2001).
[Crossref] [PubMed]

Müller, G.

J. Meseth, G. Müller, and R. Klein, "Reflectance field based real-time, high-quality rendering of bidirectional texture functions," Comput. Graphics 28, 105-112 (2004).
[Crossref]

Myszkowski, K.

F. Drago and K. Myszkowski, "Validation proposal for global illumination and rendering techniques," Comput. Graphics 25, 511-518 (2001).
[Crossref]

Nesti, G.

G. Macelloni, G. Nesti, P. Pampaloni, S. Sigismondi, D. Tarchi, and S. Lolli, "Experimental validation of surface scattering and emission models," IEEE Trans. Geosci. Remote Sens. 38, 459-469 (2000).
[Crossref]

Nieto-Vesperinas, M.

O'Donnell, K. A.

Ogilvy, J. A.

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger/IOP, 1991).

Pampaloni, P.

G. Macelloni, G. Nesti, P. Pampaloni, S. Sigismondi, D. Tarchi, and S. Lolli, "Experimental validation of surface scattering and emission models," IEEE Trans. Geosci. Remote Sens. 38, 459-469 (2000).
[Crossref]

Pfister, H.

T. Weyrich, H. Pfister, and M. Gross, "Rendering deformable surface reflectance fields," IEEE Trans. Vis. Comput. Graph. 11, 48-58 (2005).
[Crossref] [PubMed]

Piironen, J.

S. Kaasalainen, J. Piironen, M. Kaasalainen, A. W. Harris, K. Muinonen, and A. Cellino, "Asteroid photometric and polarimetric phase curves: empirical interpretation," Icarus 161, 34-46 (2003).
[Crossref]

Premoze, S.

M. Ashikhmin, S. Premoze, and P. Shirley, "A microfacet-based BRDF generator," in Proceedings of ACM SIGGRAPH 2000 (www.siggraph.org), pp. 65-74.

Quayle, B.

Ramer, D.

L. Brainard, W. Lynn, D. Ramer, and W. Shemano, "Optical measurements facility," Air Force Research Laboratory, Materials & Manufacturing Directorate, Rep. AFRL-ML-WP-TR-1999-4142 (1999).

Renau, J.

Rice, S. O.

S. O. Rice, "Reflection of electromagnetic waves from slightly rough surfaces," Commun. Pure Appl. Math. 4, 351-378 (1951).
[Crossref]

Rocchini, C.

C. Rocchini, P. Cignoni, C. Montani, and R. Scopigno, "Acquiring, stitching, and blending diffuse appearance attributes on 3D models," Visual Comput. 18, 186-204 (2002).
[Crossref]

Rushmeier, H.

F. Bernardini, I. M. Martin, and H. Rushmeier, "High-quality texture reconstruction from multiple scans," IEEE Trans. Vis. Comput. Graph. 7, 318-332 (2001).
[Crossref]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, corrected and enlarged ed. (Academic, 1980), 6.623(2).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, corrected and enlarged ed. (Academic, 1980), 6.631(4).

Saillard, M.

M. Saillard and A. Sentenac, "Rigorous solutions for electromagnetic scattering from rough surfaces," Waves Random Media 11, R103-R137 (2001).
[Crossref]

Sánchez-Gil, J. A.

Sansoni, P.

D. Fontani, F. Francini, G. Longobardi, and P. Sansoni, "Optical control of surface finish," Opt. Lasers Eng. 32, 459-472 (2000).
[Crossref]

Sant, A. J.

Santarsiero, M.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa-Spagnolo, "Generalized Bessel-Gauss beams," J. Mod. Opt. 43, 1155-1166 (1996).

Saunders, R. S.

M. S. Gilmore, R. Castaño, T. Mann, R. C. Anderson, E. D. Mjolsness, R. Manduchi, and R. S. Saunders, "Strategies for autonomous rovers at Mars," J. Geophys. Res. 105, 29223-29237 (2000).
[Crossref]

Schettini, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa-Spagnolo, "Generalized Bessel-Gauss beams," J. Mod. Opt. 43, 1155-1166 (1996).

Schirripa-Spagnolo, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa-Spagnolo, "Generalized Bessel-Gauss beams," J. Mod. Opt. 43, 1155-1166 (1996).

Scopigno, R.

C. Rocchini, P. Cignoni, C. Montani, and R. Scopigno, "Acquiring, stitching, and blending diffuse appearance attributes on 3D models," Visual Comput. 18, 186-204 (2002).
[Crossref]

Sei, A.

O. P. Bruno, A. Sei, and M. Caponi, "High-order high-frequency solutions of rough surface scattering problems," Radio Sci. 37, 10.1029/2000 RS002551 (2002).
[Crossref]

Senior, T. B. A.

Derivatives of the delta function are considered in T. B. A. Senior and J. L. Volakis, Approximate Boundary Conditions in Electromagnetics (IEEE Press, 1995).

Sentenac, A.

M. Saillard and A. Sentenac, "Rigorous solutions for electromagnetic scattering from rough surfaces," Waves Random Media 11, R103-R137 (2001).
[Crossref]

Shchegrov, A. V.

Shemano, W.

L. Brainard, W. Lynn, D. Ramer, and W. Shemano, "Optical measurements facility," Air Force Research Laboratory, Materials & Manufacturing Directorate, Rep. AFRL-ML-WP-TR-1999-4142 (1999).

Shepard, M. K.

M. K. Shepard and B. A. Campbell, "Radar scattering from a self-affine fractal surface: near-nadir regime," Icarus 141, 156-171 (1999).
[Crossref]

Shirley, P.

M. Ashikhmin, S. Premoze, and P. Shirley, "A microfacet-based BRDF generator," in Proceedings of ACM SIGGRAPH 2000 (www.siggraph.org), pp. 65-74.

Sigismondi, S.

G. Macelloni, G. Nesti, P. Pampaloni, S. Sigismondi, D. Tarchi, and S. Lolli, "Experimental validation of surface scattering and emission models," IEEE Trans. Geosci. Remote Sens. 38, 459-469 (2000).
[Crossref]

Sillion, F. X.

X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, "A comprehensive physical model for light reflection," Comput. Graph. 25, 175-186 (1991).
[Crossref]

Silver, S.

S. Silver, "Scattering and diffraction," in Microwave Antenna Theory and Design, S.Silver, ed. (McGraw-Hill, 1949), Chap. 5.

Smith, P. A.

P. A. Smith, D. A. van Veldhuizen, and K. S. Keppler, "Modeling and simulation tools for high-energy laser safety applications," in Enabling Technology for Simulation Science V, A.F.Sisti and D.A.Trevisani, eds., Proc. SPIE 4367, 478-485 (2001).

Snyder, W. C.

W. C. Snyder and Z. Wan, "BRDF models to predict spectral reflectance and emissivity in the thermal infrared," IEEE Trans. Geosci. Remote Sens. 36, 214-225 (1998).
[Crossref]

W. C. Snyder, "Reciprocity of the bidirectional reflectance distribution function (BRDF) in measurements and models of structured surfaces," IEEE Trans. Geosci. Remote Sens. 36, 685-691 (1998).
[Crossref]

Sparrow, E. M.

Spizzichino, A.

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

Steinle, B.

H. P. Baltes, B. Steinle, and G. Antes, "Spectral coherence and the radiant intensity from statistically homogeneous and isotropic planar sources," Opt. Commun. 18, 242-246 (1976).
[Crossref]

Steinvall, O.

O. Steinvall, "Effects of target shape and reflection on laser radar cross sections," Appl. Opt. 39, 4381-4391 (2000).
[Crossref]

O. Steinvall and T. Carlsson, "Three-dimensional laser radar modelling," inLaser Radar Technology and Applications VI, G.W.Kamerman, ed., Proc. SPIE 4377, 23-24 (2001).

Stover, J.

J. Stover, Optical Scattering: Measurement and Analysis (McGraw-Hill, 1990).

Stowe, M.

Sung, C. C.

Tarchi, D.

G. Macelloni, G. Nesti, P. Pampaloni, S. Sigismondi, D. Tarchi, and S. Lolli, "Experimental validation of surface scattering and emission models," IEEE Trans. Geosci. Remote Sens. 38, 459-469 (2000).
[Crossref]

Terent'ev, N. M.

V. N. Faddeeva and N. M. Terent'ev, Tables of Values of the Function w(z)=e−z2(1+2i/pi∫0zet2dt) for Complex Argument, V.A.Fok ed., translated by D. G. Fry (Pergamon, 1961).

Thomas, M. E.

D. D. Duncan, D. V. Hahn, and M. E. Thomas, "Physics-based polarimetric BRDF models," in Optical Diagnostic Methods for Inorganic Materials III, L.M.Hanssen, ed., Proc. SPIE 5192, 129-140 (2003).

Thompson, W. J.

W. J. Thompson, "Numerous neat algorithms for the Voight profile function," Comput. Phys. 7, 627-631 (1993).

Torrance, K. E.

X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, "A comprehensive physical model for light reflection," Comput. Graph. 25, 175-186 (1991).
[Crossref]

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

van Veldhuizen, D. A.

P. A. Smith, D. A. van Veldhuizen, and K. S. Keppler, "Modeling and simulation tools for high-energy laser safety applications," in Enabling Technology for Simulation Science V, A.F.Sisti and D.A.Trevisani, eds., Proc. SPIE 4367, 478-485 (2001).

Volakis, J. L.

Derivatives of the delta function are considered in T. B. A. Senior and J. L. Volakis, Approximate Boundary Conditions in Electromagnetics (IEEE Press, 1995).

Vorburger, T. V.

Walther, A.

Wan, Z.

W. C. Snyder and Z. Wan, "BRDF models to predict spectral reflectance and emissivity in the thermal infrared," IEEE Trans. Geosci. Remote Sens. 36, 214-225 (1998).
[Crossref]

Wang, G.-C.

Warnick, K. E.

Weisstein, E. W.

E. W. Weisstein, "Erf," from MathWorld-A Wolfram Web Resource, http://mathworld.wolfram.com/Erf.html.

Wells, R. J.

R. J. Wells, "Rapid approximation to the Voight/Faddeeva function and its derivatives," J. Quant. Spectrosc. Radiat. Transf. 62, 29-48 (1999).
[Crossref]

Weyrich, T.

T. Weyrich, H. Pfister, and M. Gross, "Rendering deformable surface reflectance fields," IEEE Trans. Vis. Comput. Graph. 11, 48-58 (2005).
[Crossref] [PubMed]

Wolf, E.

O. Korotkova, B. G. Hoover, V. L. Gamiz, and E. Wolf, "Coherence and polarization properties of far-fields generated by quasi-homogeneous electromagnetic sources," J. Opt. Soc. Am. A 22, 2547-2556 (2005).
[Crossref]

E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
[Crossref]

E. Wolf and W. H. Carter, "Angular distribution of radiant intensity from sources of different degrees of spatial coherence," Opt. Commun. 13, 205-209 (1975).
[Crossref]

W. H. Carter and E. Wolf, "Coherence properties of Lambertian and non-Lambertian sources," J. Opt. Soc. Am. 65, 1067-1071 (1975).
[Crossref]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), Chap. 5.7.

Yordanov, O. I.

K. Ivanova, M. A. Michalev, and O. I. Yordanov, "Numerical study of scattering by rough surfaces with intermediate and large scale roughness," Radio Sci. 26, 505-510 (1991).
[Crossref]

Zhao, Y.

Annu. Rev. Astron. Astrophys. (1)

J. V. Evans, "Radar studies of planetary surfaces," Annu. Rev. Astron. Astrophys. 7, 201-248 (1969).
[Crossref]

Appl. Opt. (5)

Commun. Pure Appl. Math. (1)

S. O. Rice, "Reflection of electromagnetic waves from slightly rough surfaces," Commun. Pure Appl. Math. 4, 351-378 (1951).
[Crossref]

Comput. Graph. (1)

X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, "A comprehensive physical model for light reflection," Comput. Graph. 25, 175-186 (1991).
[Crossref]

Comput. Graphics (2)

F. Drago and K. Myszkowski, "Validation proposal for global illumination and rendering techniques," Comput. Graphics 25, 511-518 (2001).
[Crossref]

J. Meseth, G. Müller, and R. Klein, "Reflectance field based real-time, high-quality rendering of bidirectional texture functions," Comput. Graphics 28, 105-112 (2004).
[Crossref]

Comput. Phys. (1)

W. J. Thompson, "Numerous neat algorithms for the Voight profile function," Comput. Phys. 7, 627-631 (1993).

Icarus (2)

S. Kaasalainen, J. Piironen, M. Kaasalainen, A. W. Harris, K. Muinonen, and A. Cellino, "Asteroid photometric and polarimetric phase curves: empirical interpretation," Icarus 161, 34-46 (2003).
[Crossref]

M. K. Shepard and B. A. Campbell, "Radar scattering from a self-affine fractal surface: near-nadir regime," Icarus 141, 156-171 (1999).
[Crossref]

IEEE Trans. Antennas Propag. (2)

R. D. Kodis, "A note on the theory of scattering from an irregular surface," IEEE Trans. Antennas Propag. AP-14, 77-82 (1966).
[Crossref]

D. E. Barrick, "Rough surface scattering based on the specular point theory," IEEE Trans. Antennas Propag. AP-16, 449-454 (1968).
[Crossref]

IEEE Trans. Geosci. Remote Sens. (3)

G. Macelloni, G. Nesti, P. Pampaloni, S. Sigismondi, D. Tarchi, and S. Lolli, "Experimental validation of surface scattering and emission models," IEEE Trans. Geosci. Remote Sens. 38, 459-469 (2000).
[Crossref]

W. C. Snyder, "Reciprocity of the bidirectional reflectance distribution function (BRDF) in measurements and models of structured surfaces," IEEE Trans. Geosci. Remote Sens. 36, 685-691 (1998).
[Crossref]

W. C. Snyder and Z. Wan, "BRDF models to predict spectral reflectance and emissivity in the thermal infrared," IEEE Trans. Geosci. Remote Sens. 36, 214-225 (1998).
[Crossref]

IEEE Trans. Vis. Comput. Graph. (2)

F. Bernardini, I. M. Martin, and H. Rushmeier, "High-quality texture reconstruction from multiple scans," IEEE Trans. Vis. Comput. Graph. 7, 318-332 (2001).
[Crossref]

T. Weyrich, H. Pfister, and M. Gross, "Rendering deformable surface reflectance fields," IEEE Trans. Vis. Comput. Graph. 11, 48-58 (2005).
[Crossref] [PubMed]

Int. J. Comput. Vis. (1)

O. G. Cula and K. J. Dana, "3D texture recognition using bidirectional feature histograms," Int. J. Comput. Vis. 59, 33-60 (2004).
[Crossref]

J. Appl. Phys. (1)

J. C. Leader, "Bidirectional scattering of electromagnetic waves from rough surfaces," J. Appl. Phys. 42, 4808-4816 (1971).
[Crossref]

J. Geophys. Res. (4)

A. K. Fung, "Theory of radar scatter from rough surfaces, bistatic and monostatic, with application to lunar radar return," J. Geophys. Res. 69, 1063-1073 (1964).
[Crossref]

M. S. Gilmore, R. Castaño, T. Mann, R. C. Anderson, E. D. Mjolsness, R. Manduchi, and R. S. Saunders, "Strategies for autonomous rovers at Mars," J. Geophys. Res. 105, 29223-29237 (2000).
[Crossref]

A. K. Fung and R. K. Moore, "Effects of structure size on moon and earth radar returns at various angles," J. Geophys. Res. 69, 1075-1081 (1964).
[Crossref]

W. K. Klemperer, "Angular scattering law for the moon at 6-meter wavelength," J. Geophys. Res. 70, 3798-3800 (1965).
[Crossref]

J. Mod. Opt. (2)

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa-Spagnolo, "Generalized Bessel-Gauss beams," J. Mod. Opt. 43, 1155-1166 (1996).

N. C. Bruce and J. C. Dainty, "Multiple scattering from rough dielectric and metal surfaces using the Kirchhoff approximation," J. Mod. Opt. 38, 1471-1481 (1991).
[Crossref]

J. Opt. Soc. Am. (7)

J. Opt. Soc. Am. A (5)

J. Quant. Spectrosc. Radiat. Transf. (1)

R. J. Wells, "Rapid approximation to the Voight/Faddeeva function and its derivatives," J. Quant. Spectrosc. Radiat. Transf. 62, 29-48 (1999).
[Crossref]

Opt. Commun. (3)

G. S. Agarwal, "Scattering from rough surfaces," Opt. Commun. 14, 161-166 (1975).
[Crossref]

H. P. Baltes, B. Steinle, and G. Antes, "Spectral coherence and the radiant intensity from statistically homogeneous and isotropic planar sources," Opt. Commun. 18, 242-246 (1976).
[Crossref]

E. Wolf and W. H. Carter, "Angular distribution of radiant intensity from sources of different degrees of spatial coherence," Opt. Commun. 13, 205-209 (1975).
[Crossref]

Opt. Eng. (Bellingham) (2)

J. C. Leader, "Similarities and distinctions between coherence theory relations and laser scattering phenomena," Opt. Eng. (Bellingham) 19, 593-601 (1980).

A review of theory and measurements in the small-roughness regime is provided in J. M. Elson and J. M. Bennett, "Vector scattering theory," Opt. Eng. (Bellingham) 18, 116-124 (1979).

Opt. Express (1)

Opt. Lasers Eng. (1)

D. Fontani, F. Francini, G. Longobardi, and P. Sansoni, "Optical control of surface finish," Opt. Lasers Eng. 32, 459-472 (2000).
[Crossref]

Opt. Lett. (2)

Opt. Photonics News (1)

S. H. C. P. McCall, "The importance of scatter in stray light analysis," Opt. Photonics News 12(11), 40-47 (2001).
[Crossref]

Phys. Lett. A (1)

E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
[Crossref]

Proc. IEEE (1)

P. Beckmann, "Scattering by composite rough surfaces," Proc. IEEE 53, 1012-1015 (1965).
[Crossref]

Radio Sci. (3)

K. Ivanova, M. A. Michalev, and O. I. Yordanov, "Numerical study of scattering by rough surfaces with intermediate and large scale roughness," Radio Sci. 26, 505-510 (1991).
[Crossref]

O. P. Bruno, A. Sei, and M. Caponi, "High-order high-frequency solutions of rough surface scattering problems," Radio Sci. 37, 10.1029/2000 RS002551 (2002).
[Crossref]

D. E. Barrick, "Unacceptable height correlation coefficients and the quasi-specular component in rough surface scattering," Radio Sci. 5, 647-654 (1970).
[Crossref]

Sci. Am. (1)

J. Dorsey and P. Hanrahan, "Digital materials and virtual weathering," Sci. Am. 282(2), 64-71 (2000).
[Crossref] [PubMed]

Visual Comput. (1)

C. Rocchini, P. Cignoni, C. Montani, and R. Scopigno, "Acquiring, stitching, and blending diffuse appearance attributes on 3D models," Visual Comput. 18, 186-204 (2002).
[Crossref]

Waves Random Media (3)

A. Ishimaru and J. S. Chen, "Scattering from very rough metallic and dielectric surfaces: a theory based on the modified Kirchhoff approximation," Waves Random Media 1, 25-34 (1991).
[Crossref]

M. Saillard and A. Sentenac, "Rigorous solutions for electromagnetic scattering from rough surfaces," Waves Random Media 11, R103-R137 (2001).
[Crossref]

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

Other (31)

P. A. Smith, D. A. van Veldhuizen, and K. S. Keppler, "Modeling and simulation tools for high-energy laser safety applications," in Enabling Technology for Simulation Science V, A.F.Sisti and D.A.Trevisani, eds., Proc. SPIE 4367, 478-485 (2001).

W. Gautschi, "Error function and Fresnel integrals," in Handbook of Mathematical Functions, 9th ed., M.Abramowitz and I.A.Stegun, eds. (Dover, 1972), Chap. 7.

M. Ashikhmin, S. Premoze, and P. Shirley, "A microfacet-based BRDF generator," in Proceedings of ACM SIGGRAPH 2000 (www.siggraph.org), pp. 65-74.

O. Steinvall and T. Carlsson, "Three-dimensional laser radar modelling," inLaser Radar Technology and Applications VI, G.W.Kamerman, ed., Proc. SPIE 4377, 23-24 (2001).

A. I. Carswell, "Advances in laser ranging," in Laser Radar Technology for Remote Sensing, C.Werner, ed., Proc. SPIE 5240, 1-9 (2004).

J. Stover, Optical Scattering: Measurement and Analysis (McGraw-Hill, 1990).

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger/IOP, 1991).

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

S. Silver, "Scattering and diffraction," in Microwave Antenna Theory and Design, S.Silver, ed. (McGraw-Hill, 1949), Chap. 5.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), Chap. 5.7.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Chap. 2.

D. D. Duncan, D. V. Hahn, and M. E. Thomas, "Physics-based polarimetric BRDF models," in Optical Diagnostic Methods for Inorganic Materials III, L.M.Hanssen, ed., Proc. SPIE 5192, 129-140 (2003).

B. G. Hoover and V. L. Gamiz, "Diffractive bidirectional reflectance distributions of surfaces with large effective roughness in one dimension," in Laser Radar Techniques for Atmospheric Sensing, U.N.Singh, ed., Proc. SPIE 5575, 137-142 (2004).

J. W. Goodman, Statistical Optics (Wiley, 1985), Chap. 5.

x and Deltax represent two-dimensional vectors in the plane of the average surface, i.e., x=xx̂+yŷ.

See Ref. and references therein.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1961), Chap. 5.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), Chap. 9.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978), Chap. 9.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, corrected and enlarged ed. (Academic, 1980), 6.631(4).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, corrected and enlarged ed. (Academic, 1980), 6.623(2).

B. D. Fried and S. D. Conte, The Plasma Dispersion Function: The Hilbert Transform of the Gaussian (Academic, 1961). Note that the function defined in this reference has its real and imaginary parts switched relative to the plasma dispersion function defined in Refs. .

V. N. Faddeeva and N. M. Terent'ev, Tables of Values of the Function w(z)=e−z2(1+2i/pi∫0zet2dt) for Complex Argument, V.A.Fok ed., translated by D. G. Fry (Pergamon, 1961).

J. W. Goodman, "Statistical properties of laser speckle patterns," in Laser Speckle and Related Phenomena, 2nd enlarged ed., J.C.Dainty, ed. (Springer-Verlag, 1984), Chap. 2.

A review of the radiometric units and notation used in this paper is provided in E. L. Dereniak and D. G. Crowe, Optical Radiation Detectors (Wiley, 1984).

The meanings of the terms "high frequency" and "low frequency" in coherence theory are unfortunately essentially opposite their meanings in scattering theory. Use of these terms in this context is therefore avoided.

Derivatives of the delta function are considered in T. B. A. Senior and J. L. Volakis, Approximate Boundary Conditions in Electromagnetics (IEEE Press, 1995).

E. W. Weisstein, "Erf," from MathWorld-A Wolfram Web Resource, http://mathworld.wolfram.com/Erf.html.

L. Brainard, W. Lynn, D. Ramer, and W. Shemano, "Optical measurements facility," Air Force Research Laboratory, Materials & Manufacturing Directorate, Rep. AFRL-ML-WP-TR-1999-4142 (1999).

It is possible to dispense with the quasi-monochromatic conditions by using the cross-spectral density W(x1,x2,nu)=∫Gamma(x1,x2,tau)exp(2pijnutau)dtau rather than the coherence function, thus proceeding in the space-frequency rather than the space-time domain. Under the quasi-monochromatic conditions these two approaches are essentially equivalent.

Some authors use the symbol µ to represent the normalized mutual intensity function. We reserve µ for the analogous correlation function in the space-frequency domain.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Geometry and notation relevant to derivation of the BRDF at point P from the coherence between the surface points x 1 and x 2 . Δ x = x 1 x 2 .

Fig. 2
Fig. 2

Illustration of the phase-screen approximation of the scattered field on the rough surface. The field is specified on the virtual surface S p s just above the actual surface, in contrast with the tangent-plane approximation in which the field is specified on the surface facets S t p , where the radius of curvature is sufficiently larger than the wavelength. The phase difference due to a lateral separation of surface points is k i x , while the phase difference due to the surface height h ( x ) is k h ( x ) ( 1 + cos θ i ) . The angle θ s is not relevant in the phase-screen approximation as it is in the tangent-plane approximation.

Fig. 3
Fig. 3

Contour plots of the integrand of the coherent component of the general BRDF solution, which is proportional to the imaginary part of the plasma dispersion function, for two surfaces, for scattering in the specular plane at two incident angles, as functions of the scattering angle θ s and the azimuthal correlation variable φ. The coherent component of the BRDF at θ s is proportional to the integral over the corresponding horizontal line. Fixed parameters for these plots are λ ¯ = 1 μ m , ρ 1 = 0.005 μ m 1 , ρ 2 = 0.005 μ m 2 , and σ = 2 [(a), (b)] or σ = 3 [(c), (d)]. White represents 0, and dark shades represent negative values in gray-scale coding.

Fig. 4
Fig. 4

Contour plots of the integrand of the coherent component of the general BRDF solution, which is proportional to the imaginary part of the plasma dispersion function, for the two surfaces specified in Fig. 3, for scattering normal to the specular plane, as functions of the scattering angle θ s and the azimuthal correlation variable φ. Fixed parameters for these plots are (a) λ ¯ = 1 μ m , ρ 1 = 0.005 μ m 1 , ρ 2 = 0.005 μ m 2 , and σ = 2 or (b) σ = 3 .

Fig. 5
Fig. 5

Variation with incident angle θ i of the general solution for the radiant intensity in the specular plane due to (a) a Gaussian-like ( ρ 1 = 0.0001 μ m 1 ) , (b) an intermediate ( ρ 1 = 0.005 μ m 1 ) , and (c) a Cauchy-like ( ρ 1 = 0.02 μ m 1 ) surface. The units on the vertical axes are normalized across the plots, and ρ 2 = 0.005 μ m 2 , σ h = 0.75 μ m , and λ ¯ = 1 μ m are fixed. The vertical dashed lines mark the specular direction for θ i = 30 ° .

Fig. 6
Fig. 6

Derived solution for the radiant intensity (solid curve) fit to data (squares) due to a milled, black-anodized aluminum surface with the illumination at θ i = 30 ° and the detection both in the s-polarization component. The dotted and dotted–dashed curves represent the derived radiant intensities due to Gaussian-like and Cauchy-like surfaces, respectively, with the same autocorrelation lower bound as that for the surface corresponding to the solid curve. Logarithms of the measured and derived radiant intensities are plotted following independent normalizations, and σ h = 0.35 μ m and λ ¯ = 1 μ m are fixed. The results imply that the surface autocorrelation function is non-Gaussian.

Fig. 7
Fig. 7

Derived monostatic radiant intensities due to three surfaces with the same autocorrelation lower bound and σ h = 7 μ m illuminated with λ ¯ = 0.6328 μ m . The logarithms of the derived radiant intensities are plotted following independent normalizations.

Fig. 8
Fig. 8

Variation with roughness σ h of the general solution for the radiant intensity in the specular plane ( θ i = 45 ° ) due to (a) a Gaussian-like ( ρ 1 = 0.0001 μ m 1 ) , (b) an intermediate ( ρ 1 = 0.005 μ m 1 ) , and (c) a Cauchy-like ( ρ 1 = 0.02 μ m 1 ) surface. The units on the vertical axes are normalized across the plots, and ρ 2 = 0.005 μ m 2 and λ ¯ = 1 μ m are fixed.

Fig. 9
Fig. 9

Full-hemisphere plots of the derived general solution for the radiant intensity due to two surfaces. The longitudinal lines correspond to scattered azimuth angle ϕ s , the latitudinal lines correspond to scattered elevation angle θ s , and the incident direction ( θ i = 30 ° ) is indicated by the vertical arrows. Fixed parameters are λ ¯ = 1 μ m , ρ 1 = 0.005 μ m 1 , ρ 2 = 0.005 μ m 2 , and σ h = 0.35 μ m (a) or σ h = 1 μ m (b). The logarithms of the radiant intensities are plotted over the range [ 3 , 0 ] following coupled normalization.

Fig. 10
Fig. 10

Derived monostatic radiant intensities due to a surface illuminated at three wavelengths. Fixed surface parameters are σ h = 7 μ m , ρ 1 = 0.00002 μ m 1 , and ρ 2 = 0.0006 μ m 2 . The logarithms of the derived radiant intensities are plotted following coupled normalizations.

Equations (36)

Equations on this page are rendered with MathJax. Learn more.

σ σ h λ ,
BRDF ( k s , k i ) = I ( k s , k i ) P i cos θ s ,
I ( k ¯ s ) = A M ¯ r cos 2 θ s λ ¯ 2 γ ( Δ x ) exp ( j k ¯ s Δ x ) d Δ x ,
u ( x ) = a ( x ) exp ( j k ¯ i x ) exp [ 2 π j λ ¯ ( 1 + cos θ i ) h ( x ) ] = a ( x ) exp ( j k ¯ i x ) exp [ j α ( x ) ] ,
σ α 2 = [ 2 π σ ( 1 + cos θ i ) ] 2 ,
R α ( Δ x ) = [ 2 π λ ¯ ( 1 + cos θ i ) ] 2 R h ( Δ x ) ,
γ ( Δ x ) u ( x ) u * ( x Δ x ) u 2 ( x ) u 2 ( x Δ x ) = exp ( j k ¯ i Δ x ) a 2 ( x ) a ( x ) a ( x Δ x ) exp { j [ α ( x ) α ( x Δ x ) ] } ,
γ ( Δ x ) = exp ( j k ¯ i Δ x ) exp { j [ α ( x ) α ( x Δ x ) ] } .
γ ( Δ x ) = exp ( j k ¯ i Δ x ) exp { σ α 2 [ 1 ρ h ( Δ x ) ] } ,
I ( k ¯ s , k ¯ i ) = A M ¯ r cos 2 θ s λ ¯ 2 exp { σ α 2 [ 1 ρ h ( Δ x ) ] } exp [ j ( k ¯ i k ¯ s ) Δ x ] d Δ x .
I ( k ¯ s , k ¯ i ) = A M ¯ r cos 2 θ s λ ¯ 2 0 0 2 π exp { σ α 2 [ 1 ρ h ( Δ x ) ] } exp [ j ( Δ k ¯ Δ x ) ] r d φ d r ,
Δ k ¯ Δ x = k ¯ r ( A cos φ + B sin φ ) ,
ρ h ( Δ x ) 1 + lim ε 0 [ [ Δ x ρ h ] ε Δ x + 1 2 [ ( Δ x ) ( Δ x ρ h ) ] ε Δ x ] .
ρ h ( Δ x ) 1 + r ( r ρ h ) + φ r ( φ ρ h ) + r 2 2 ( r r ρ h ) + φ 2 2 r 2 ( φ φ ρ h ) + φ ( r φ ρ h ) .
I ( k ¯ s , k ¯ i ) A M ¯ r cos 2 θ s λ ¯ 2 0 0 2 π exp [ j k ¯ r ( A cos φ + B sin φ ) ] d φ exp [ σ α 2 ( r ρ h ) r + σ α 2 2 ( r r ρ h ) r 2 ] r d r ,
0 2 π exp [ j k ¯ ξ r cos ( φ ψ ) ] d φ = 2 π J 0 ( k ¯ ξ r ) ,
ξ = sin 2 θ s + sin 2 θ i + 2 sin θ i sin θ s cos ϕ i cos ϕ s
I ( k ¯ s , k ¯ i ) 2 π A M ¯ r cos 2 θ s λ ¯ 2 exp ( β v 2 ) 0 r J 0 ( k ¯ ξ r ) exp [ β ( r + v ) 2 ] d r
v r ρ h r r ρ h ,
β σ α 2 r r ρ h 2 .
I ( k ¯ s , k ¯ i ) π A M ¯ r cos 2 θ s λ ¯ 2 β exp [ ( k ¯ ξ ) 2 4 β ] ,
I ( k ¯ s , k ¯ i ) 2 π A M ¯ r cos 2 θ s λ ¯ 2 σ α 2 ρ 1 [ σ α 4 ρ 1 2 + ( k ¯ ξ ) 2 ] 3 2 .
B { g ( r ) } = G ( R ) = 0 r J 0 ( R r ) g ( r ) d r .
B { exp [ β ( r + v ) 2 ] } = G ( R ) = 1 2 π 0 2 π 0 r exp ( j R r cos φ ) exp [ β ( r + v ) 2 ] d r d φ = 1 2 π 0 2 π G 1 ( R cos φ ) d φ ,
G 1 ( X ) = 1 β exp ( β v 2 Γ 2 ) [ j Γ t exp ( t 2 ) d t + β ( v + j X 2 β ) j Γ exp ( t 2 ) d t ] ,
Γ j v β X 2 β .
G 1 ( X ) = exp ( β v 2 ) 2 β [ 1 π β ( v + j X 2 β ) exp ( Γ 2 ) erfc ( j Γ ) ] = exp ( β v 2 ) 2 β [ 1 π β ( v + j X 2 β ) w ( Γ ) ] ,
G ( R ) = exp ( β v 2 ) 2 β [ 1 β π 0 π ( v + j k ¯ ξ cos φ 2 β ) w ( Γ ) d φ ] = exp ( β v 2 ) 2 β [ 1 + k ¯ ξ 2 π β 0 π L ( Γ ) ( cos φ ) d φ ] ,
I ( k ¯ s , k ¯ i ) π A M ¯ r cos 2 θ s λ ¯ 2 β [ 1 + k ¯ ξ 2 π β 0 π L ( Γ ) ( cos φ ) d φ ] .
u ( P ) = j λ S u exp ( j k r ) r ( n ̂ r ̂ ) d S ,
Γ ( x 1 , x 2 , τ ) Γ ( x 1 , x 2 , 0 ) exp ( 2 π j ν ¯ τ ) = J ( x 1 , x 2 ) exp ( 2 π j ν ¯ τ ) ,
J ( P 1 , P 2 ) u ( P 1 ) u * ( P 2 ) = 1 λ ¯ 2 S S u ( x 1 ) u * ( x 2 ) exp [ j k ¯ ( r 1 r 2 ) ] r 1 r 2 ( n ̂ 1 r ̂ 1 ) ( n ̂ 2 r ̂ 2 ) d x 1 d x 2 ,
I ( P ) J ( P , P ) cos 2 θ ( Z λ ¯ ) 2 S S J ( x 1 , x 2 ) exp [ j k ¯ s ̂ ( x 2 x 1 ) ] d x 1 d x 2 .
γ ( Δ x ) = J ( x 1 , x 2 ) I ( x 1 ) I ( x 2 ) ,
J ( x 1 , x 2 ) I ( x ¯ ) γ ( Δ x ) ,
I ( k ¯ s ) cos 2 θ λ ¯ 2 S I ( x ¯ ) d x ¯ S γ ( Δ x ) exp ( j k ¯ s Δ x ) d Δ x = A I ¯ r cos 2 θ λ ¯ 2 S γ ( Δ x ) exp ( j k ¯ s Δ x ) d Δ x ,

Metrics