Abstract

Scattering effects from microtopographic surface roughness are merely nonparaxial diffraction phenomena resulting from random phase variations in the reflected or transmitted wavefront. Rayleigh–Rice, Beckmann–Kirchhoff. or Harvey–Shack surface scatter theories are commonly used to predict surface scatter effects. Smooth-surface and/or paraxial approximations have severely limited the range of applicability of each of the above theoretical treatments. A recent linear systems formulation of nonparaxial scalar diffraction theory applied to surface scatter phenomena resulted first in an empirically modified Beckmann–Kirchhoff surface scatter model, then a generalized Harvey–Shack theory that produces accurate results for rougher surfaces than the Rayleigh–Rice theory and for larger incident and scattered angles than the classical Beckmann–Kirchhoff and the original Harvey–Shack theories. These new developments simplify the analysis and understanding of nonintuitive scattering behavior from rough surfaces illuminated at arbitrary incident angles.

© 2011 Optical Society of America

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  1. S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
    [CrossRef]
  2. E. L. Church, H. A. Jenkinson, and J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).
  3. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963).
  4. J. E. Harvey, “Light-scattering characteristics of optical surfaces,” Ph.D. dissertation (University of Arizona, 1976).
  5. J. E. Harvey, “Surface scatter phenomena: a linear, shift-invariant process,” Proc. SPIE 1165, 87–99 (1989).
  6. C. L. Vernold and J. E. Harvey, “A modified Beckmann-Kirchhoff scattering theory,” Proc. SPIE 3426, 51–56 (1998).
    [CrossRef]
  7. J. E. Harvey, A. Krywonos, and C. L. Vernold, “Modified Beckmann-Kirchhoff scattering model for rough surfaces with large incident and scattered angles,” Opt. Eng. 46, 078002 (2007). One of the ten most frequently downloaded SPIE papers and articles (from SPIE Digital Library) in August 2007.
    [CrossRef]
  8. H. Ragheb and E. Hancock, “The modified Beckmann-Kirchhoff scattering theory for rough surface analysis,” Pattern Recog. 40, 2004–2020 (2007).
    [CrossRef]
  9. Y. Sun, “Statistical ray method for deriving reflection models of rough surfaces,” J. Opt. Soc. Am. A 24, 724–744 (2007).
    [CrossRef]
  10. H. Ragheb and E. R. Hancock, “Testing new variants of the Beckmann-Kirchhoff model against radiance data,” Comput. Vis. Image Understanding 102, 145–168 (2006).
    [CrossRef]
  11. A. Robles-Kelly and E. R. Hancock, “Estimating the surface radiance function from single images,” Graph. Models 67, 518–548 (2005).
    [CrossRef]
  12. H. Ragheb and E. R. Hancock, “Surface radiance correction for shape from shading,” Pattern Recog. 38, 1574–1595 (2005).
    [CrossRef]
  13. H. Ragheb and E. R. Hancock, “Adding subsurface attenuation to the Beckmann-Kirchhoff theory,” in Pattern Recognition and Image Analysis, Part 2, Vol.  3523 of Lecture Notes in Computer Science (Springer-Verlag, 2005), pp. 247–254.
    [CrossRef]
  14. A. Robles-Kelly and E. R. Hancock, “Radiance function estimation for object classification,” in Progress in Pattern Recognition, Image Analysis and Applications, Vol.  3287 of Lecture Notes in Computer Science (Springer-Verlag, 2004), pp. 67–75.
    [CrossRef]
  15. P. Hermansson, G. Forssell, and J. Fagerstrom, “A review of models for scattering from rough Surfaces,” Scientific Report FOI-R-0988-SE (Swedish Defense Research Agency, Nov. 2003).
  16. H. Ragheb and E. R. Hancock, “Rough surface correction and re-illumination using the modified Beckmann model,” in Computer Analysis of Images and Patterns: 10th International Conference, Proceedings, Vol.  2756 of Lecture Notes in Computer Science (Springer-Verlag, 2003), pp. 98–106.
  17. H. Ragheb and E. R. Hancock, “Rough surface estimation using the Kirchhoff model,” in Image Analysis, Proceedings, Vol.  2749 of Lecture Notes in Computer Science (Springer-Verlag, 2003), pp. 477–484.
  18. T. M. Elfouhaily and C. A. Guerin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14, R1–R40 (2004).
    [CrossRef]
  19. J. E. Harvey, C. L. Vernold, A. Krywonos, and P. L. Thompson, “Diffracted radiance: a fundamental quantity in a non-paraxial scalar diffraction theory,” Appl. Opt. 38, 6469–6481 (1999).
    [CrossRef]
  20. J. E. Harvey, C. L. Vernold, A. Krywonos, and P. L. Thompson, “Diffracted radiance: a fundamental quantity in a nonparaxial scalar diffraction theory: errata,” Appl. Opt. 39, 6374–6375(2000).
    [CrossRef]
  21. J. E. Harvey, A. Krywonos, and Dijana Bogunovic, “Non-paraxial scalar treatment of sinusoidal phase gratings,” J. Opt. Soc. Am. A 23, 858–865 (2006).
    [CrossRef]
  22. R. J. Noll, “Effect of mid and high spatial frequencies on optical performance,” Opt. Eng. 18, 137–142 (1979).
  23. J. E. Harvey, “Bridging the gap between “figure” and “finish”,” presented at the Optical Society of America Optical Fabrication and Testing Meeting, Boston, Massachusetts, May 3, 1996.
  24. E. L. Church, “Fractal surface finish,” Appl. Opt. 27, 1518–1526(1988).
    [CrossRef] [PubMed]
  25. J. C. Stover, Optical Scattering, Measurement and Analysis, 2nd ed. (SPIE, 1995).
    [CrossRef]
  26. J. E. Harvey, N. Choi, A. Krywonos, S. Schroder, and D. H. Penalver, “Scattering from moderately rough interfaces between two arbitrary media,” Proc. SPIE 7794, 77940V (2010).
    [CrossRef]
  27. E. L. Church and P. Z. Takacs, “Instrumental effects in surface finish measurements,” Proc. SPIE 1009, 46–55 (1988).
  28. E. L. Church and P. Z. Takacs, “Effects of the optical transfer function in surface profile measurements,” Proc. SPIE 1164, 46–59 (1989).
  29. A. Duparré, J. Ferre-Borrull, S. Gliech, G. Notni, J. Steinert, and J. M. Bennett, “Surface characterization techniques for determining the root-mean-square roughness and power spectral densities of optical components,” Appl. Opt. 41, 154–171(2002).
    [CrossRef] [PubMed]
  30. F. E. Nicodemus, “Reflectance nomenclature and directional reflectance and emissivity,” Appl. Opt. 9, 1474–1475 (1970).
    [CrossRef] [PubMed]
  31. J. E. Harvey, “Scattering effects in x-ray imaging systems,” Proc. SPIE 2515, 246–272 (1995).
    [CrossRef]
  32. A. Krywonos, “Predicting surface scatter using a linear systems formulation of nonparaxial scalar diffraction,” Ph.D. dissertation (University of Central Florida, 2006).
  33. J. A. Ratcliff, “Some aspects of diffraction theory and their application to the ionosphere,” in Reports of Progress in Physics, A.C.Strickland, ed. (The Physical Society, 1956), Vol.  XIX.
  34. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  35. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978).
  36. J. E. Harvey and C. L. Vernold, “Description of diffraction grating behavior in direction cosine space,” Appl. Opt. 37, 8158–8160 (1998).
    [CrossRef]
  37. J. E. Harvey and A. Krywonos, “Radiance: the natural quantity for describing diffraction and propagation,” Proc. SPIE 6285, 628503 (2006).
    [CrossRef]
  38. Users Manual for APART/PADE, Version 8.6 (Breault Research Organization, 4601 East First Street, Tucson, Ariz., 1987), p. 5-2.
  39. ASAP Reference Manual (Breault Research Organization, 4601 East First Street, Tucson, Ariz., 1990), pp. 3–43.
  40. TracePro User’s Manual, Release 3.0 (Lambda Research Corporation, 80 Taylor Street, Littleton, Mass. (1998), p. 7.12.
  41. ZEMAX User’s Guide, August 2007 (ZEMAX Development Corp., 3001 112th Avenue NE, Suite 202, Bellevue, Wash., 2007), p. 391.
  42. FRED User’s Manual, Version 9.110 (Photon Engineering, 440 S. Williams Blvd. #106, Tucson, Ariz., 2010).
  43. http://www.opticsinfobase.org/submit/ocis/.
  44. J. E. Harvey, E. C. Moran, and W. P. Zmek, “Transfer function characterization of grazing incidence optical systems,” Appl. Opt. 27, 1527–1533 (1988).
    [CrossRef] [PubMed]
  45. P. Glenn, P. Reid, A. Slomba, and L. P. Van Speybroeck, “Performance prediction of AXAF technology mirror assembly using measured mirror surface errors,” Appl. Opt. 27, 1539–1543(1988).
    [CrossRef] [PubMed]
  46. J. E. Harvey and P. L. Thompson, “Generalized Wolter Type I design for the solar x-ray imager (SXI),” Proc. SPIE 3766, 173–183 (1999).
    [CrossRef]
  47. J. E. Harvey, A. Krywonos, G. Peterson, and M. Bruner, “Image Degradation due to scattering effects in two-mirror telescopes,” Opt. Eng. 49, 063202 (2010).
    [CrossRef]
  48. H. Davies, “The reflection of electromagnetic waves from a rough surface,” Proc. IEE IV 101, 209–214 (1954).
    [CrossRef]
  49. H. E. Bennett and J. O. Porteus, “Relation between surface roughness and specular reflectance at normal incidence,” J. Opt. Soc. Am. 51, 123–129 (1961).
    [CrossRef]
  50. J. M. Bennett and L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, 1989).
  51. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics(Wiley, 1991).
    [CrossRef]
  52. S. Fay, S. Dubail, U. Kroll, J. Meier, Y. Ziegler, and A. Shah, “Light trapping enhancement for thin-film silicon solar cells by roughness improvement of the ZnC front TCO,” in Proceedings of the 16th European Photovoltaic Solar Energy Conference & Exhibition (WIP Wirtschaft und Infrastruktur Planungs-KG, 2000), pp. 361–364.
  53. D. Domine, F. J. Haug, C. Battaglia, and C. Ballif, “Modeling of light scattering from micro- and nanotextured surfaces,” J. Appl. Phys. 107, 044504 (2010).
    [CrossRef]
  54. S. Schröder, A. Duparré, K. Füchsel, N. Kaiser, A. Tünnermann, and J. E. Harvey, “Scattering of roughened TCO films—modeling and measurement,” in OSA Topical Meeting on Optical Interference Coatings, OSA Technical Digest (CD) (2010).
  55. M. Guzar-Sicairos and J. C. Gutierrez-Vega, “Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields,” J. Opt. Soc. Am. A 21, 53–58(2004).
    [CrossRef]
  56. K. A. O’Donnell and E. R. Mendez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
    [CrossRef]
  57. E. L. Church and P. Z. Takacs, “The optimal estimation of finish parameters,” Proc SPIE 1530, 71–86 (1991).
    [CrossRef]
  58. J. M. Elson, J. M. Bennett, and J. C. Stover, “Wavelength and Angular Dependence of light scattering from beryllium: comparison of theory and experiment,” Appl. Opt. 32, 3362–3376(1993).
    [CrossRef] [PubMed]
  59. M. G. Dittman, “K-correlation power spectral density and surface scatter model,” Proc. SPIE 6291, 62910P (2006).
    [CrossRef]
  60. A. J. S. Hamilton, “Uncorrelated modes of nonlinear power spectrum,” Mon. Not. R. Astron. Soc. 312, 257–284 (2000).
    [CrossRef]
  61. J. E. Harvey, N. Choi, A. Krywonos, and J. Marcen, “Calculating BRDFs from surface PSDs for moderately rough optical surfaces,” Proc. SPIE 7426, 742601 (2009).
    [CrossRef]
  62. J. C. Stover and J. E. Harvey, “Limitations of Rayleigh-Rice perturbation theory for describing surface scatter,” Proc. SPIE 6672, 66720B (2007).
    [CrossRef]
  63. J. E. Harvey, A. Krywonos, and J. C. Stover “Unified scatter model for rough surfaces at large incident and scatter angles,” Proc. SPIE 6672, 66720C (2007).
    [CrossRef]
  64. R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983).
  65. J. J. Murrey, F. E. Nicodemus, and I. Wunderman, “Proposed supplement to the SI nomenclature for radiometry and photometry,” Appl. Opt. 10, 1465–1468 (1971).
    [CrossRef]
  66. The NIST Reference on Constants, Units, and Uncertainty, http://physics.nist.gov/cuu/Units/units.html.
  67. S. Schröder, S. Gliech, and A. Duparre, “Measurement system to determine the total and angle-resolved light scattering of optical components in the deep-ultraviolet and vacuum-ultraviolet spectral regions,” Appl. Opt. 44, 6093–6107 (2005).
    [CrossRef] [PubMed]
  68. E. L. Church, P. Z. Takacs, and T. A. Leonard, “The prediction of BRDFs from surface profile measurements,” Proc. SPIE 1165, 136–150 (1989).
  69. S. Schröder, “Light scattering of optical components at 193 nm and 13.5 nm,” Ph.D. dissertation (Friedrich-Schiller-Universitat, Jena, Germany, 2008).

2010

J. E. Harvey, N. Choi, A. Krywonos, S. Schroder, and D. H. Penalver, “Scattering from moderately rough interfaces between two arbitrary media,” Proc. SPIE 7794, 77940V (2010).
[CrossRef]

J. E. Harvey, A. Krywonos, G. Peterson, and M. Bruner, “Image Degradation due to scattering effects in two-mirror telescopes,” Opt. Eng. 49, 063202 (2010).
[CrossRef]

D. Domine, F. J. Haug, C. Battaglia, and C. Ballif, “Modeling of light scattering from micro- and nanotextured surfaces,” J. Appl. Phys. 107, 044504 (2010).
[CrossRef]

2009

J. E. Harvey, N. Choi, A. Krywonos, and J. Marcen, “Calculating BRDFs from surface PSDs for moderately rough optical surfaces,” Proc. SPIE 7426, 742601 (2009).
[CrossRef]

2007

J. C. Stover and J. E. Harvey, “Limitations of Rayleigh-Rice perturbation theory for describing surface scatter,” Proc. SPIE 6672, 66720B (2007).
[CrossRef]

J. E. Harvey, A. Krywonos, and J. C. Stover “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 scattered angles,” Opt. Eng. 46, 078002 (2007). One of the ten most frequently downloaded SPIE papers and articles (from SPIE Digital Library) in August 2007.
[CrossRef]

H. Ragheb and E. Hancock, “The modified Beckmann-Kirchhoff scattering theory for rough surface analysis,” Pattern Recog. 40, 2004–2020 (2007).
[CrossRef]

Y. Sun, “Statistical ray method for deriving reflection models of rough surfaces,” J. Opt. Soc. Am. A 24, 724–744 (2007).
[CrossRef]

2006

J. E. Harvey, A. Krywonos, and Dijana Bogunovic, “Non-paraxial scalar treatment of sinusoidal phase gratings,” J. Opt. Soc. Am. A 23, 858–865 (2006).
[CrossRef]

H. Ragheb and E. R. Hancock, “Testing new variants of the Beckmann-Kirchhoff model against radiance data,” Comput. Vis. Image Understanding 102, 145–168 (2006).
[CrossRef]

M. G. Dittman, “K-correlation power spectral density and surface scatter model,” Proc. SPIE 6291, 62910P (2006).
[CrossRef]

J. E. Harvey and A. Krywonos, “Radiance: the natural quantity for describing diffraction and propagation,” Proc. SPIE 6285, 628503 (2006).
[CrossRef]

2005

A. Robles-Kelly and E. R. Hancock, “Estimating the surface radiance function from single images,” Graph. Models 67, 518–548 (2005).
[CrossRef]

H. Ragheb and E. R. Hancock, “Surface radiance correction for shape from shading,” Pattern Recog. 38, 1574–1595 (2005).
[CrossRef]

S. Schröder, S. Gliech, and A. Duparre, “Measurement system to determine the total and angle-resolved light scattering of optical components in the deep-ultraviolet and vacuum-ultraviolet spectral regions,” Appl. Opt. 44, 6093–6107 (2005).
[CrossRef] [PubMed]

2004

M. Guzar-Sicairos and J. C. Gutierrez-Vega, “Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields,” J. Opt. Soc. Am. A 21, 53–58(2004).
[CrossRef]

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

2002

2000

1999

1998

C. L. Vernold and J. E. Harvey, “A modified Beckmann-Kirchhoff scattering theory,” Proc. SPIE 3426, 51–56 (1998).
[CrossRef]

J. E. Harvey and C. L. Vernold, “Description of diffraction grating behavior in direction cosine space,” Appl. Opt. 37, 8158–8160 (1998).
[CrossRef]

1995

J. E. Harvey, “Scattering effects in x-ray imaging systems,” Proc. SPIE 2515, 246–272 (1995).
[CrossRef]

1993

1991

E. L. Church and P. Z. Takacs, “The optimal estimation of finish parameters,” Proc SPIE 1530, 71–86 (1991).
[CrossRef]

1989

E. L. Church, P. Z. Takacs, and T. A. Leonard, “The prediction of BRDFs from surface profile measurements,” Proc. SPIE 1165, 136–150 (1989).

E. L. Church and P. Z. Takacs, “Effects of the optical transfer function in surface profile measurements,” Proc. SPIE 1164, 46–59 (1989).

J. E. Harvey, “Surface scatter phenomena: a linear, shift-invariant process,” Proc. SPIE 1165, 87–99 (1989).

1988

1987

1979

R. J. Noll, “Effect of mid and high spatial frequencies on optical performance,” Opt. Eng. 18, 137–142 (1979).

E. L. Church, H. A. Jenkinson, and J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).

1971

1970

1961

1954

H. Davies, “The reflection of electromagnetic waves from a rough surface,” Proc. IEE IV 101, 209–214 (1954).
[CrossRef]

1951

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
[CrossRef]

Ballif, C.

D. Domine, F. J. Haug, C. Battaglia, and C. Ballif, “Modeling of light scattering from micro- and nanotextured surfaces,” J. Appl. Phys. 107, 044504 (2010).
[CrossRef]

Battaglia, C.

D. Domine, F. J. Haug, C. Battaglia, and C. Ballif, “Modeling of light scattering from micro- and nanotextured surfaces,” J. Appl. Phys. 107, 044504 (2010).
[CrossRef]

Beckmann, P.

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

Bennett, H. E.

Bennett, J. M.

Bogunovic, Dijana

Boyd, R. W.

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983).

Bruner, M.

J. E. Harvey, A. Krywonos, G. Peterson, and M. Bruner, “Image Degradation due to scattering effects in two-mirror telescopes,” Opt. Eng. 49, 063202 (2010).
[CrossRef]

Choi, N.

J. E. Harvey, N. Choi, A. Krywonos, S. Schroder, and D. H. Penalver, “Scattering from moderately rough interfaces between two arbitrary media,” Proc. SPIE 7794, 77940V (2010).
[CrossRef]

J. E. Harvey, N. Choi, A. Krywonos, and J. Marcen, “Calculating BRDFs from surface PSDs for moderately rough optical surfaces,” Proc. SPIE 7426, 742601 (2009).
[CrossRef]

Church, E. L.

E. L. Church and P. Z. Takacs, “The optimal estimation of finish parameters,” Proc SPIE 1530, 71–86 (1991).
[CrossRef]

E. L. Church, P. Z. Takacs, and T. A. Leonard, “The prediction of BRDFs from surface profile measurements,” Proc. SPIE 1165, 136–150 (1989).

E. L. Church and P. Z. Takacs, “Effects of the optical transfer function in surface profile measurements,” Proc. SPIE 1164, 46–59 (1989).

E. L. Church and P. Z. Takacs, “Instrumental effects in surface finish measurements,” Proc. SPIE 1009, 46–55 (1988).

E. L. Church, “Fractal surface finish,” Appl. Opt. 27, 1518–1526(1988).
[CrossRef] [PubMed]

E. L. Church, H. A. Jenkinson, and J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).

Davies, H.

H. Davies, “The reflection of electromagnetic waves from a rough surface,” Proc. IEE IV 101, 209–214 (1954).
[CrossRef]

Dittman, M. G.

M. G. Dittman, “K-correlation power spectral density and surface scatter model,” Proc. SPIE 6291, 62910P (2006).
[CrossRef]

Domine, D.

D. Domine, F. J. Haug, C. Battaglia, and C. Ballif, “Modeling of light scattering from micro- and nanotextured surfaces,” J. Appl. Phys. 107, 044504 (2010).
[CrossRef]

Dubail, S.

S. Fay, S. Dubail, U. Kroll, J. Meier, Y. Ziegler, and A. Shah, “Light trapping enhancement for thin-film silicon solar cells by roughness improvement of the ZnC front TCO,” in Proceedings of the 16th European Photovoltaic Solar Energy Conference & Exhibition (WIP Wirtschaft und Infrastruktur Planungs-KG, 2000), pp. 361–364.

Duparre, A.

Duparré, A.

A. Duparré, J. Ferre-Borrull, S. Gliech, G. Notni, J. Steinert, and J. M. Bennett, “Surface characterization techniques for determining the root-mean-square roughness and power spectral densities of optical components,” Appl. Opt. 41, 154–171(2002).
[CrossRef] [PubMed]

S. Schröder, A. Duparré, K. Füchsel, N. Kaiser, A. Tünnermann, and J. E. Harvey, “Scattering of roughened TCO films—modeling and measurement,” in OSA Topical Meeting on Optical Interference Coatings, OSA Technical Digest (CD) (2010).

Elfouhaily, T. M.

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

Elson, J. M.

Fagerstrom, J.

P. Hermansson, G. Forssell, and J. Fagerstrom, “A review of models for scattering from rough Surfaces,” Scientific Report FOI-R-0988-SE (Swedish Defense Research Agency, Nov. 2003).

Fay, S.

S. Fay, S. Dubail, U. Kroll, J. Meier, Y. Ziegler, and A. Shah, “Light trapping enhancement for thin-film silicon solar cells by roughness improvement of the ZnC front TCO,” in Proceedings of the 16th European Photovoltaic Solar Energy Conference & Exhibition (WIP Wirtschaft und Infrastruktur Planungs-KG, 2000), pp. 361–364.

Ferre-Borrull, J.

Forssell, G.

P. Hermansson, G. Forssell, and J. Fagerstrom, “A review of models for scattering from rough Surfaces,” Scientific Report FOI-R-0988-SE (Swedish Defense Research Agency, Nov. 2003).

Füchsel, K.

S. Schröder, A. Duparré, K. Füchsel, N. Kaiser, A. Tünnermann, and J. E. Harvey, “Scattering of roughened TCO films—modeling and measurement,” in OSA Topical Meeting on Optical Interference Coatings, OSA Technical Digest (CD) (2010).

Gaskill, J. D.

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

Glenn, P.

Gliech, S.

Goodman, J. W.

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

Guerin, C. A.

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

Gutierrez-Vega, J. C.

Guzar-Sicairos, M.

Hamilton, A. J. S.

A. J. S. Hamilton, “Uncorrelated modes of nonlinear power spectrum,” Mon. Not. R. Astron. Soc. 312, 257–284 (2000).
[CrossRef]

Hancock, E.

H. Ragheb and E. Hancock, “The modified Beckmann-Kirchhoff scattering theory for rough surface analysis,” Pattern Recog. 40, 2004–2020 (2007).
[CrossRef]

Hancock, E. R.

H. Ragheb and E. R. Hancock, “Testing new variants of the Beckmann-Kirchhoff model against radiance data,” Comput. Vis. Image Understanding 102, 145–168 (2006).
[CrossRef]

A. Robles-Kelly and E. R. Hancock, “Estimating the surface radiance function from single images,” Graph. Models 67, 518–548 (2005).
[CrossRef]

H. Ragheb and E. R. Hancock, “Surface radiance correction for shape from shading,” Pattern Recog. 38, 1574–1595 (2005).
[CrossRef]

H. Ragheb and E. R. Hancock, “Rough surface correction and re-illumination using the modified Beckmann model,” in Computer Analysis of Images and Patterns: 10th International Conference, Proceedings, Vol.  2756 of Lecture Notes in Computer Science (Springer-Verlag, 2003), pp. 98–106.

H. Ragheb and E. R. Hancock, “Rough surface estimation using the Kirchhoff model,” in Image Analysis, Proceedings, Vol.  2749 of Lecture Notes in Computer Science (Springer-Verlag, 2003), pp. 477–484.

H. Ragheb and E. R. Hancock, “Adding subsurface attenuation to the Beckmann-Kirchhoff theory,” in Pattern Recognition and Image Analysis, Part 2, Vol.  3523 of Lecture Notes in Computer Science (Springer-Verlag, 2005), pp. 247–254.
[CrossRef]

A. Robles-Kelly and E. R. Hancock, “Radiance function estimation for object classification,” in Progress in Pattern Recognition, Image Analysis and Applications, Vol.  3287 of Lecture Notes in Computer Science (Springer-Verlag, 2004), pp. 67–75.
[CrossRef]

Harvey, J. E.

J. E. Harvey, A. Krywonos, G. Peterson, and M. Bruner, “Image Degradation due to scattering effects in two-mirror telescopes,” Opt. Eng. 49, 063202 (2010).
[CrossRef]

J. E. Harvey, N. Choi, A. Krywonos, S. Schroder, and D. H. Penalver, “Scattering from moderately rough interfaces between two arbitrary media,” Proc. SPIE 7794, 77940V (2010).
[CrossRef]

J. E. Harvey, N. Choi, A. Krywonos, and J. Marcen, “Calculating BRDFs from surface PSDs for moderately rough optical surfaces,” Proc. SPIE 7426, 742601 (2009).
[CrossRef]

J. E. Harvey, A. Krywonos, and J. C. Stover “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, “Limitations of Rayleigh-Rice perturbation theory for describing surface scatter,” Proc. SPIE 6672, 66720B (2007).
[CrossRef]

J. E. Harvey, A. Krywonos, and C. L. Vernold, “Modified Beckmann-Kirchhoff scattering model for rough surfaces with large incident and scattered angles,” Opt. Eng. 46, 078002 (2007). One of the ten most frequently downloaded SPIE papers and articles (from SPIE Digital Library) in August 2007.
[CrossRef]

J. E. Harvey and A. Krywonos, “Radiance: the natural quantity for describing diffraction and propagation,” Proc. SPIE 6285, 628503 (2006).
[CrossRef]

J. E. Harvey, A. Krywonos, and Dijana Bogunovic, “Non-paraxial scalar treatment of sinusoidal phase gratings,” J. Opt. Soc. Am. A 23, 858–865 (2006).
[CrossRef]

J. E. Harvey, C. L. Vernold, A. Krywonos, and P. L. Thompson, “Diffracted radiance: a fundamental quantity in a nonparaxial scalar diffraction theory: errata,” Appl. Opt. 39, 6374–6375(2000).
[CrossRef]

J. E. Harvey, C. L. Vernold, A. Krywonos, and P. L. Thompson, “Diffracted radiance: a fundamental quantity in a non-paraxial scalar diffraction theory,” Appl. Opt. 38, 6469–6481 (1999).
[CrossRef]

J. E. Harvey and P. L. Thompson, “Generalized Wolter Type I design for the solar x-ray imager (SXI),” Proc. SPIE 3766, 173–183 (1999).
[CrossRef]

C. L. Vernold and J. E. Harvey, “A modified Beckmann-Kirchhoff scattering theory,” Proc. SPIE 3426, 51–56 (1998).
[CrossRef]

J. E. Harvey and C. L. Vernold, “Description of diffraction grating behavior in direction cosine space,” Appl. Opt. 37, 8158–8160 (1998).
[CrossRef]

J. E. Harvey, “Scattering effects in x-ray imaging systems,” Proc. SPIE 2515, 246–272 (1995).
[CrossRef]

J. E. Harvey, “Surface scatter phenomena: a linear, shift-invariant process,” Proc. SPIE 1165, 87–99 (1989).

J. E. Harvey, E. C. Moran, and W. P. Zmek, “Transfer function characterization of grazing incidence optical systems,” Appl. Opt. 27, 1527–1533 (1988).
[CrossRef] [PubMed]

S. Schröder, A. Duparré, K. Füchsel, N. Kaiser, A. Tünnermann, and J. E. Harvey, “Scattering of roughened TCO films—modeling and measurement,” in OSA Topical Meeting on Optical Interference Coatings, OSA Technical Digest (CD) (2010).

J. E. Harvey, “Light-scattering characteristics of optical surfaces,” Ph.D. dissertation (University of Arizona, 1976).

J. E. Harvey, “Bridging the gap between “figure” and “finish”,” presented at the Optical Society of America Optical Fabrication and Testing Meeting, Boston, Massachusetts, May 3, 1996.

Haug, F. J.

D. Domine, F. J. Haug, C. Battaglia, and C. Ballif, “Modeling of light scattering from micro- and nanotextured surfaces,” J. Appl. Phys. 107, 044504 (2010).
[CrossRef]

Hermansson, P.

P. Hermansson, G. Forssell, and J. Fagerstrom, “A review of models for scattering from rough Surfaces,” Scientific Report FOI-R-0988-SE (Swedish Defense Research Agency, Nov. 2003).

Jenkinson, H. A.

E. L. Church, H. A. Jenkinson, and J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).

Kaiser, N.

S. Schröder, A. Duparré, K. Füchsel, N. Kaiser, A. Tünnermann, and J. E. Harvey, “Scattering of roughened TCO films—modeling and measurement,” in OSA Topical Meeting on Optical Interference Coatings, OSA Technical Digest (CD) (2010).

Kroll, U.

S. Fay, S. Dubail, U. Kroll, J. Meier, Y. Ziegler, and A. Shah, “Light trapping enhancement for thin-film silicon solar cells by roughness improvement of the ZnC front TCO,” in Proceedings of the 16th European Photovoltaic Solar Energy Conference & Exhibition (WIP Wirtschaft und Infrastruktur Planungs-KG, 2000), pp. 361–364.

Krywonos, A.

J. E. Harvey, A. Krywonos, G. Peterson, and M. Bruner, “Image Degradation due to scattering effects in two-mirror telescopes,” Opt. Eng. 49, 063202 (2010).
[CrossRef]

J. E. Harvey, N. Choi, A. Krywonos, S. Schroder, and D. H. Penalver, “Scattering from moderately rough interfaces between two arbitrary media,” Proc. SPIE 7794, 77940V (2010).
[CrossRef]

J. E. Harvey, N. Choi, A. Krywonos, and J. Marcen, “Calculating BRDFs from surface PSDs for moderately rough optical surfaces,” Proc. SPIE 7426, 742601 (2009).
[CrossRef]

J. E. Harvey, A. Krywonos, and J. C. Stover “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 scattered angles,” Opt. Eng. 46, 078002 (2007). One of the ten most frequently downloaded SPIE papers and articles (from SPIE Digital Library) in August 2007.
[CrossRef]

J. E. Harvey and A. Krywonos, “Radiance: the natural quantity for describing diffraction and propagation,” Proc. SPIE 6285, 628503 (2006).
[CrossRef]

J. E. Harvey, A. Krywonos, and Dijana Bogunovic, “Non-paraxial scalar treatment of sinusoidal phase gratings,” J. Opt. Soc. Am. A 23, 858–865 (2006).
[CrossRef]

J. E. Harvey, C. L. Vernold, A. Krywonos, and P. L. Thompson, “Diffracted radiance: a fundamental quantity in a nonparaxial scalar diffraction theory: errata,” Appl. Opt. 39, 6374–6375(2000).
[CrossRef]

J. E. Harvey, C. L. Vernold, A. Krywonos, and P. L. Thompson, “Diffracted radiance: a fundamental quantity in a non-paraxial scalar diffraction theory,” Appl. Opt. 38, 6469–6481 (1999).
[CrossRef]

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

Leonard, T. A.

E. L. Church, P. Z. Takacs, and T. A. Leonard, “The prediction of BRDFs from surface profile measurements,” Proc. SPIE 1165, 136–150 (1989).

Marcen, J.

J. E. Harvey, N. Choi, A. Krywonos, and J. Marcen, “Calculating BRDFs from surface PSDs for moderately rough optical surfaces,” Proc. SPIE 7426, 742601 (2009).
[CrossRef]

Mattsson, L.

J. M. Bennett and L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, 1989).

Meier, J.

S. Fay, S. Dubail, U. Kroll, J. Meier, Y. Ziegler, and A. Shah, “Light trapping enhancement for thin-film silicon solar cells by roughness improvement of the ZnC front TCO,” in Proceedings of the 16th European Photovoltaic Solar Energy Conference & Exhibition (WIP Wirtschaft und Infrastruktur Planungs-KG, 2000), pp. 361–364.

Mendez, E. R.

Moran, E. C.

Murrey, J. J.

Nicodemus, F. E.

Noll, R. J.

R. J. Noll, “Effect of mid and high spatial frequencies on optical performance,” Opt. Eng. 18, 137–142 (1979).

Notni, G.

O’Donnell, K. A.

Penalver, D. H.

J. E. Harvey, N. Choi, A. Krywonos, S. Schroder, and D. H. Penalver, “Scattering from moderately rough interfaces between two arbitrary media,” Proc. SPIE 7794, 77940V (2010).
[CrossRef]

Peterson, G.

J. E. Harvey, A. Krywonos, G. Peterson, and M. Bruner, “Image Degradation due to scattering effects in two-mirror telescopes,” Opt. Eng. 49, 063202 (2010).
[CrossRef]

Porteus, J. O.

Ragheb, H.

H. Ragheb and E. Hancock, “The modified Beckmann-Kirchhoff scattering theory for rough surface analysis,” Pattern Recog. 40, 2004–2020 (2007).
[CrossRef]

H. Ragheb and E. R. Hancock, “Testing new variants of the Beckmann-Kirchhoff model against radiance data,” Comput. Vis. Image Understanding 102, 145–168 (2006).
[CrossRef]

H. Ragheb and E. R. Hancock, “Surface radiance correction for shape from shading,” Pattern Recog. 38, 1574–1595 (2005).
[CrossRef]

H. Ragheb and E. R. Hancock, “Rough surface correction and re-illumination using the modified Beckmann model,” in Computer Analysis of Images and Patterns: 10th International Conference, Proceedings, Vol.  2756 of Lecture Notes in Computer Science (Springer-Verlag, 2003), pp. 98–106.

H. Ragheb and E. R. Hancock, “Rough surface estimation using the Kirchhoff model,” in Image Analysis, Proceedings, Vol.  2749 of Lecture Notes in Computer Science (Springer-Verlag, 2003), pp. 477–484.

H. Ragheb and E. R. Hancock, “Adding subsurface attenuation to the Beckmann-Kirchhoff theory,” in Pattern Recognition and Image Analysis, Part 2, Vol.  3523 of Lecture Notes in Computer Science (Springer-Verlag, 2005), pp. 247–254.
[CrossRef]

Ratcliff, J. A.

J. A. Ratcliff, “Some aspects of diffraction theory and their application to the ionosphere,” in Reports of Progress in Physics, A.C.Strickland, ed. (The Physical Society, 1956), Vol.  XIX.

Reid, P.

Rice, S. O.

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
[CrossRef]

Robles-Kelly, A.

A. Robles-Kelly and E. R. Hancock, “Estimating the surface radiance function from single images,” Graph. Models 67, 518–548 (2005).
[CrossRef]

A. Robles-Kelly and E. R. Hancock, “Radiance function estimation for object classification,” in Progress in Pattern Recognition, Image Analysis and Applications, Vol.  3287 of Lecture Notes in Computer Science (Springer-Verlag, 2004), pp. 67–75.
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics(Wiley, 1991).
[CrossRef]

Schroder, S.

J. E. Harvey, N. Choi, A. Krywonos, S. Schroder, and D. H. Penalver, “Scattering from moderately rough interfaces between two arbitrary media,” Proc. SPIE 7794, 77940V (2010).
[CrossRef]

Schröder, S.

S. Schröder, S. Gliech, and A. Duparre, “Measurement system to determine the total and angle-resolved light scattering of optical components in the deep-ultraviolet and vacuum-ultraviolet spectral regions,” Appl. Opt. 44, 6093–6107 (2005).
[CrossRef] [PubMed]

S. Schröder, “Light scattering of optical components at 193 nm and 13.5 nm,” Ph.D. dissertation (Friedrich-Schiller-Universitat, Jena, Germany, 2008).

S. Schröder, A. Duparré, K. Füchsel, N. Kaiser, A. Tünnermann, and J. E. Harvey, “Scattering of roughened TCO films—modeling and measurement,” in OSA Topical Meeting on Optical Interference Coatings, OSA Technical Digest (CD) (2010).

Shah, A.

S. Fay, S. Dubail, U. Kroll, J. Meier, Y. Ziegler, and A. Shah, “Light trapping enhancement for thin-film silicon solar cells by roughness improvement of the ZnC front TCO,” in Proceedings of the 16th European Photovoltaic Solar Energy Conference & Exhibition (WIP Wirtschaft und Infrastruktur Planungs-KG, 2000), pp. 361–364.

Slomba, A.

Spizzichino, A.

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

Steinert, J.

Stover, J. C.

J. E. Harvey, A. Krywonos, and J. C. Stover “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, “Limitations of Rayleigh-Rice perturbation theory for describing surface scatter,” Proc. SPIE 6672, 66720B (2007).
[CrossRef]

J. M. Elson, J. M. Bennett, and J. C. Stover, “Wavelength and Angular Dependence of light scattering from beryllium: comparison of theory and experiment,” Appl. Opt. 32, 3362–3376(1993).
[CrossRef] [PubMed]

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

Sun, Y.

Takacs, P. Z.

E. L. Church and P. Z. Takacs, “The optimal estimation of finish parameters,” Proc SPIE 1530, 71–86 (1991).
[CrossRef]

E. L. Church, P. Z. Takacs, and T. A. Leonard, “The prediction of BRDFs from surface profile measurements,” Proc. SPIE 1165, 136–150 (1989).

E. L. Church and P. Z. Takacs, “Effects of the optical transfer function in surface profile measurements,” Proc. SPIE 1164, 46–59 (1989).

E. L. Church and P. Z. Takacs, “Instrumental effects in surface finish measurements,” Proc. SPIE 1009, 46–55 (1988).

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics(Wiley, 1991).
[CrossRef]

Thompson, P. L.

Tünnermann, A.

S. Schröder, A. Duparré, K. Füchsel, N. Kaiser, A. Tünnermann, and J. E. Harvey, “Scattering of roughened TCO films—modeling and measurement,” in OSA Topical Meeting on Optical Interference Coatings, OSA Technical Digest (CD) (2010).

Van Speybroeck, L. P.

Vernold, C. L.

J. E. Harvey, A. Krywonos, and C. L. Vernold, “Modified Beckmann-Kirchhoff scattering model for rough surfaces with large incident and scattered angles,” Opt. Eng. 46, 078002 (2007). One of the ten most frequently downloaded SPIE papers and articles (from SPIE Digital Library) in August 2007.
[CrossRef]

J. E. Harvey, C. L. Vernold, A. Krywonos, and P. L. Thompson, “Diffracted radiance: a fundamental quantity in a nonparaxial scalar diffraction theory: errata,” Appl. Opt. 39, 6374–6375(2000).
[CrossRef]

J. E. Harvey, C. L. Vernold, A. Krywonos, and P. L. Thompson, “Diffracted radiance: a fundamental quantity in a non-paraxial scalar diffraction theory,” Appl. Opt. 38, 6469–6481 (1999).
[CrossRef]

J. E. Harvey and C. L. Vernold, “Description of diffraction grating behavior in direction cosine space,” Appl. Opt. 37, 8158–8160 (1998).
[CrossRef]

C. L. Vernold and J. E. Harvey, “A modified Beckmann-Kirchhoff scattering theory,” Proc. SPIE 3426, 51–56 (1998).
[CrossRef]

Wunderman, I.

Zavada, J. M.

E. L. Church, H. A. Jenkinson, and J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).

Ziegler, Y.

S. Fay, S. Dubail, U. Kroll, J. Meier, Y. Ziegler, and A. Shah, “Light trapping enhancement for thin-film silicon solar cells by roughness improvement of the ZnC front TCO,” in Proceedings of the 16th European Photovoltaic Solar Energy Conference & Exhibition (WIP Wirtschaft und Infrastruktur Planungs-KG, 2000), pp. 361–364.

Zmek, W. P.

Appl. Opt.

E. L. Church, “Fractal surface finish,” Appl. Opt. 27, 1518–1526(1988).
[CrossRef] [PubMed]

J. E. Harvey, E. C. Moran, and W. P. Zmek, “Transfer function characterization of grazing incidence optical systems,” Appl. Opt. 27, 1527–1533 (1988).
[CrossRef] [PubMed]

P. Glenn, P. Reid, A. Slomba, and L. P. Van Speybroeck, “Performance prediction of AXAF technology mirror assembly using measured mirror surface errors,” Appl. Opt. 27, 1539–1543(1988).
[CrossRef] [PubMed]

J. M. Elson, J. M. Bennett, and J. C. Stover, “Wavelength and Angular Dependence of light scattering from beryllium: comparison of theory and experiment,” Appl. Opt. 32, 3362–3376(1993).
[CrossRef] [PubMed]

J. E. Harvey, C. L. Vernold, A. Krywonos, and P. L. Thompson, “Diffracted radiance: a fundamental quantity in a non-paraxial scalar diffraction theory,” Appl. Opt. 38, 6469–6481 (1999).
[CrossRef]

J. E. Harvey, C. L. Vernold, A. Krywonos, and P. L. Thompson, “Diffracted radiance: a fundamental quantity in a nonparaxial scalar diffraction theory: errata,” Appl. Opt. 39, 6374–6375(2000).
[CrossRef]

A. Duparré, J. Ferre-Borrull, S. Gliech, G. Notni, J. Steinert, and J. M. Bennett, “Surface characterization techniques for determining the root-mean-square roughness and power spectral densities of optical components,” Appl. Opt. 41, 154–171(2002).
[CrossRef] [PubMed]

S. Schröder, S. Gliech, and A. Duparre, “Measurement system to determine the total and angle-resolved light scattering of optical components in the deep-ultraviolet and vacuum-ultraviolet spectral regions,” Appl. Opt. 44, 6093–6107 (2005).
[CrossRef] [PubMed]

J. J. Murrey, F. E. Nicodemus, and I. Wunderman, “Proposed supplement to the SI nomenclature for radiometry and photometry,” Appl. Opt. 10, 1465–1468 (1971).
[CrossRef]

F. E. Nicodemus, “Reflectance nomenclature and directional reflectance and emissivity,” Appl. Opt. 9, 1474–1475 (1970).
[CrossRef] [PubMed]

J. E. Harvey and C. L. Vernold, “Description of diffraction grating behavior in direction cosine space,” Appl. Opt. 37, 8158–8160 (1998).
[CrossRef]

Commun. Pure Appl. Math.

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
[CrossRef]

Comput. Vis. Image Understanding

H. Ragheb and E. R. Hancock, “Testing new variants of the Beckmann-Kirchhoff model against radiance data,” Comput. Vis. Image Understanding 102, 145–168 (2006).
[CrossRef]

Graph. Models

A. Robles-Kelly and E. R. Hancock, “Estimating the surface radiance function from single images,” Graph. Models 67, 518–548 (2005).
[CrossRef]

J. Appl. Phys.

D. Domine, F. J. Haug, C. Battaglia, and C. Ballif, “Modeling of light scattering from micro- and nanotextured surfaces,” J. Appl. Phys. 107, 044504 (2010).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Mon. Not. R. Astron. Soc.

A. J. S. Hamilton, “Uncorrelated modes of nonlinear power spectrum,” Mon. Not. R. Astron. Soc. 312, 257–284 (2000).
[CrossRef]

Opt. Eng.

R. J. Noll, “Effect of mid and high spatial frequencies on optical performance,” Opt. Eng. 18, 137–142 (1979).

E. L. Church, H. A. Jenkinson, and J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).

J. E. Harvey, A. Krywonos, and C. L. Vernold, “Modified Beckmann-Kirchhoff scattering model for rough surfaces with large incident and scattered angles,” Opt. Eng. 46, 078002 (2007). One of the ten most frequently downloaded SPIE papers and articles (from SPIE Digital Library) in August 2007.
[CrossRef]

J. E. Harvey, A. Krywonos, G. Peterson, and M. Bruner, “Image Degradation due to scattering effects in two-mirror telescopes,” Opt. Eng. 49, 063202 (2010).
[CrossRef]

Pattern Recog.

H. Ragheb and E. Hancock, “The modified Beckmann-Kirchhoff scattering theory for rough surface analysis,” Pattern Recog. 40, 2004–2020 (2007).
[CrossRef]

H. Ragheb and E. R. Hancock, “Surface radiance correction for shape from shading,” Pattern Recog. 38, 1574–1595 (2005).
[CrossRef]

Proc SPIE

E. L. Church and P. Z. Takacs, “The optimal estimation of finish parameters,” Proc SPIE 1530, 71–86 (1991).
[CrossRef]

Proc. IEE IV

H. Davies, “The reflection of electromagnetic waves from a rough surface,” Proc. IEE IV 101, 209–214 (1954).
[CrossRef]

Proc. SPIE

J. E. Harvey and A. Krywonos, “Radiance: the natural quantity for describing diffraction and propagation,” Proc. SPIE 6285, 628503 (2006).
[CrossRef]

J. E. Harvey, N. Choi, A. Krywonos, S. Schroder, and D. H. Penalver, “Scattering from moderately rough interfaces between two arbitrary media,” Proc. SPIE 7794, 77940V (2010).
[CrossRef]

E. L. Church and P. Z. Takacs, “Instrumental effects in surface finish measurements,” Proc. SPIE 1009, 46–55 (1988).

E. L. Church and P. Z. Takacs, “Effects of the optical transfer function in surface profile measurements,” Proc. SPIE 1164, 46–59 (1989).

J. E. Harvey, “Scattering effects in x-ray imaging systems,” Proc. SPIE 2515, 246–272 (1995).
[CrossRef]

J. E. Harvey, “Surface scatter phenomena: a linear, shift-invariant process,” Proc. SPIE 1165, 87–99 (1989).

C. L. Vernold and J. E. Harvey, “A modified Beckmann-Kirchhoff scattering theory,” Proc. SPIE 3426, 51–56 (1998).
[CrossRef]

M. G. Dittman, “K-correlation power spectral density and surface scatter model,” Proc. SPIE 6291, 62910P (2006).
[CrossRef]

J. E. Harvey, N. Choi, A. Krywonos, and J. Marcen, “Calculating BRDFs from surface PSDs for moderately rough optical surfaces,” Proc. SPIE 7426, 742601 (2009).
[CrossRef]

J. C. Stover and J. E. Harvey, “Limitations of Rayleigh-Rice perturbation theory for describing surface scatter,” Proc. SPIE 6672, 66720B (2007).
[CrossRef]

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

E. L. Church, P. Z. Takacs, and T. A. Leonard, “The prediction of BRDFs from surface profile measurements,” Proc. SPIE 1165, 136–150 (1989).

J. E. Harvey and P. L. Thompson, “Generalized Wolter Type I design for the solar x-ray imager (SXI),” Proc. SPIE 3766, 173–183 (1999).
[CrossRef]

Waves Random Media

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

Other

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

S. Schröder, A. Duparré, K. Füchsel, N. Kaiser, A. Tünnermann, and J. E. Harvey, “Scattering of roughened TCO films—modeling and measurement,” in OSA Topical Meeting on Optical Interference Coatings, OSA Technical Digest (CD) (2010).

J. E. Harvey, “Bridging the gap between “figure” and “finish”,” presented at the Optical Society of America Optical Fabrication and Testing Meeting, Boston, Massachusetts, May 3, 1996.

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983).

The NIST Reference on Constants, Units, and Uncertainty, http://physics.nist.gov/cuu/Units/units.html.

S. Schröder, “Light scattering of optical components at 193 nm and 13.5 nm,” Ph.D. dissertation (Friedrich-Schiller-Universitat, Jena, Germany, 2008).

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

J. E. Harvey, “Light-scattering characteristics of optical surfaces,” Ph.D. dissertation (University of Arizona, 1976).

H. Ragheb and E. R. Hancock, “Adding subsurface attenuation to the Beckmann-Kirchhoff theory,” in Pattern Recognition and Image Analysis, Part 2, Vol.  3523 of Lecture Notes in Computer Science (Springer-Verlag, 2005), pp. 247–254.
[CrossRef]

A. Robles-Kelly and E. R. Hancock, “Radiance function estimation for object classification,” in Progress in Pattern Recognition, Image Analysis and Applications, Vol.  3287 of Lecture Notes in Computer Science (Springer-Verlag, 2004), pp. 67–75.
[CrossRef]

P. Hermansson, G. Forssell, and J. Fagerstrom, “A review of models for scattering from rough Surfaces,” Scientific Report FOI-R-0988-SE (Swedish Defense Research Agency, Nov. 2003).

H. Ragheb and E. R. Hancock, “Rough surface correction and re-illumination using the modified Beckmann model,” in Computer Analysis of Images and Patterns: 10th International Conference, Proceedings, Vol.  2756 of Lecture Notes in Computer Science (Springer-Verlag, 2003), pp. 98–106.

H. Ragheb and E. R. Hancock, “Rough surface estimation using the Kirchhoff model,” in Image Analysis, Proceedings, Vol.  2749 of Lecture Notes in Computer Science (Springer-Verlag, 2003), pp. 477–484.

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

J. A. Ratcliff, “Some aspects of diffraction theory and their application to the ionosphere,” in Reports of Progress in Physics, A.C.Strickland, ed. (The Physical Society, 1956), Vol.  XIX.

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

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

Users Manual for APART/PADE, Version 8.6 (Breault Research Organization, 4601 East First Street, Tucson, Ariz., 1987), p. 5-2.

ASAP Reference Manual (Breault Research Organization, 4601 East First Street, Tucson, Ariz., 1990), pp. 3–43.

TracePro User’s Manual, Release 3.0 (Lambda Research Corporation, 80 Taylor Street, Littleton, Mass. (1998), p. 7.12.

ZEMAX User’s Guide, August 2007 (ZEMAX Development Corp., 3001 112th Avenue NE, Suite 202, Bellevue, Wash., 2007), p. 391.

FRED User’s Manual, Version 9.110 (Photon Engineering, 440 S. Williams Blvd. #106, Tucson, Ariz., 2010).

http://www.opticsinfobase.org/submit/ocis/.

J. M. Bennett and L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, 1989).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics(Wiley, 1991).
[CrossRef]

S. Fay, S. Dubail, U. Kroll, J. Meier, Y. Ziegler, and A. Shah, “Light trapping enhancement for thin-film silicon solar cells by roughness improvement of the ZnC front TCO,” in Proceedings of the 16th European Photovoltaic Solar Energy Conference & Exhibition (WIP Wirtschaft und Infrastruktur Planungs-KG, 2000), pp. 361–364.

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Figures (21)

Fig. 1
Fig. 1

Schematic diagram of a surface profile and its relevant statistical parameters.

Fig. 2
Fig. 2

Different spatial-frequency regimes and their resulting effects upon image quality.

Fig. 3
Fig. 3

(a) Illustration of the two-dimensional boundary of the appropriate band-limited portion of the surface PSD for an arbitrary incident angle, θ i . (b) Illustration of the relevant portion of the surface PSD, whose integral yields the square of the relevant rms surface roughness.

Fig. 4
Fig. 4

Illustration of the OPD for a specularly reflected ray.

Fig. 5
Fig. 5

Illustration of both forward (transmitted) and backward (reflected) scattering from a moderately rough interface between two media with arbitrary refractive indices.

Fig. 6
Fig. 6

Comparison of scattered intensity predictions from the OHS, MHS, and GHS theories for different incident angles. Experimental data is also displayed for incident angles of 20 ° and 70 ° . The difference between the MHS and GHS theories is modest but significant, and the experimental data provides excellent agreement with the GHS predictions.

Fig. 7
Fig. 7

GHS predictions compared to MHS and classical Beckmann–Kirchhoff predictions. Excellent agreement is indicated between the GHS theory and the O’Donnell–Mendez experimental data for this rough surface with a large incident angle.

Fig. 8
Fig. 8

Composite surface PSD function determined from four different metrology instruments. An ABC, or K-correlation, function has been fit to the experimental data to characterize the surface over the entire range of relevant spatial frequencies.

Fig. 9
Fig. 9

These BRDF profiles were numerically calculated from the real metrology data (surface PSD) illustrated in Fig. 8. The GHS surface scatter theory was used for this moder ately rough surface. The FFTLog algorithm was implemented in the calculations.

Fig. 10
Fig. 10

The BRDF profiles illustrated in Fig. 9 are plotted here on a log–log scale as a function of β = sin ( θ ) . They now exhibit the shape of the surface PSD as expected from the Rayleigh–Rice theory for the long wavelengths.

Fig. 11
Fig. 11

Comparison of the BRDF profile predicted by the GHS and Rayleigh–Rice surface scatter theories shows that, for λ = 93.9 A , the Rayleigh–Rice theory results in a peak BRDF value 122% higher than the GHS theory (lower left inset). Errors for other wavelengths are tabulated. Upper right inset shows distinctly finite value of the GHS curve at a 90 ° scattering angle.

Fig. 12
Fig. 12

The GHS surface scatter theory and its smooth-surface approximation are numerically validated by the well-known Rayleigh–Rice surface scatter theory for smooth surfaces and σ rel σ s .

Fig. 13
Fig. 13

Even for smooth surfaces, the well-known Rayleigh–Rice theory increasingly disagrees with the GHS theory as the ACV width decreases and σ rel becomes smaller than σ s . The GHS Smooth theory continues to agree with the GHS theory as long as the surface is smooth.

Fig. 14
Fig. 14

For moderately rough surfaces where the ACV width is large enough such that σ rel σ s , the Rayleigh–Rice theory and the GHS Smooth theory agree but are both inaccurate due to their explicit smooth-surface approximations. The GHS theory is presumed to be accurate.

Fig. 15
Fig. 15

For moderately rough surfaces where σ rel < σ s , the Rayleigh–Rice and the GHS Smooth theory both produce predictions for the scattered intensity that are too high, with the worst offender depending upon the specific value of the renormalization constant and the ratio σ rel / σ s .

Fig. 16
Fig. 16

Comparison of BRDF predictions from the Rayleigh–Rice theory and the smooth-surface approximation to the GHS surface scatter theories for smooth surfaces with (a) a Gaussian PSD and (b) a K-correlation (inverse power law) PSD. Note the difference in the two predictions for θ s 90   degrees .

Fig. 17
Fig. 17

Meticulously measured, high-angular-resolution, relative intensity measurements, and the corresponding BRDF profile, extending to a scatter angle of 89.6 ° .

Fig. 18
Fig. 18

Illustration of the very prominent “hook” in the surface PSD predicted by the Rayleigh–Rice theory, and the virtual absence of any such hook predicted by the smooth-surface approximation to the GHS surface scatter theory.

Fig. 19
Fig. 19

BRDF scans were taken from the back side of a clean silicon wafer for incident angles of 5 ° , 45 ° . 70 ° , and 80 ° . A shorter wavelength of 488 nm was used at θ i = 5 ° to increase the apparent “roughness” as expressed by Eq. (58).

Fig. 20
Fig. 20

Surface PSD calculated from Eq. (57) illustrated along with a fitting function to be used as input for the GHS surface scatter theory to calculate BRDFs at different wavelengths and incident angles.

Fig. 21
Fig. 21

BRDFs predicted with both the GHS and the Rayleigh–Rice surface scatter theories are compared to experimental measurements for three cases that do not satisfy the smooth-surface criterion.

Equations (58)

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f o = sin θ o λ , θ o = θ i .
σ rel ( λ , θ i ) = 1 / λ + f o 1 / λ + f o 1 / λ 2 ( f x f o ) 2 1 / λ 2 ( f x f o ) 2 PSD ( f x , f y ) d f x d f y .
σ rel ( λ ) = 2 π f = 0 1 / λ PSD ( f ) f d f .
H s ( x ^ , y ^ ) = exp { ( 4 π σ ^ s ) 2 [ 1 C s ( x ^ , y ^ ) / σ s 2 ] } ,
H ( x ^ , y ^ ) = A + B G ( x ^ , y ^ ) ,
A = exp [ ( 4 π σ ^ s ) 2 ] ,
B = 1 exp [ ( 4 π σ ^ s ) 2 ] ,
G ( x ^ , y ^ ) = exp [ ( 4 π ) 2 C s ( x ^ , y ^ ) ] 1 exp ( 4 π σ ^ s ) 2 1 .
ASF ( α , β ) = F { H ( x ^ , y ^ ) } = A δ ( α , β ) + S ( α , β ) ,
S ( α , β ) = B F { G ( x ^ , y ^ ) } .
BRDF = f ( θ i , ϕ i , θ s , ϕ s ) = d L ( θ i , ϕ i , θ s , ϕ s ) d E ( θ i , ϕ i ) .
BRDF ( θ i , ϕ i , θ s , ϕ s ) = θ i , ϕ i R ASF ( α , β ) .
BRDF ( θ i , ϕ i , θ s , ϕ s ) | θ i , ϕ i = R ASF ( α , β ) .
α = sin θ cos ϕ , β = sin θ sin ϕ , γ = cos θ .
OPD = ( γ i + γ o ) h ( x ^ , y ^ ) = 2 γ i h ( x ^ , y ^ ) ,
ϕ ( x ^ , y ^ ) = ( 2 π / λ ) OPD = 4 π γ i h ^ ( x ^ , y ^ ) .
H s ( x ^ , y ^ ; γ i ) = exp { ( 4 π γ i σ ^ ref ) 2 [ 1 C s ( x ^ , y ^ ) / σ s 2 ] } .
H s ( x ^ , y ^ ; γ i ) = A ( γ i ) + B ( γ i ) G ( x ^ , y ^ ; γ i ) ,
A ( γ i ) = exp [ ( 4 π γ i σ ^ rel ) 2 ] , B ( γ i ) = 1 exp [ ( 4 π γ i σ ^ rel ) 2 ] ,
G ( x ^ , y ^ ; γ i ) = exp [ ( 4 π γ i ) 2 C s ( x ^ , y ^ ) ] 1 exp [ ( 4 π γ i σ ^ rel ) 2 ] 1 .
ASF ( α , β β o ; γ i ) = F { H s ( x ^ , y ^ ; γ i ) exp ( i 2 π y ^ β o ) } = A ( γ i ) δ ( α , β β o ) + S ( α , β β o ; γ i ) ,
S ( α , β β o ; γ i ) = B ( γ i ) F { G ( x ^ , y ^ ; γ i ) exp ( i 2 π y ^ β o ) } , β o = β i .
K ( γ i ) = B ( γ i ) ( α = 1 1 β = 1 α 2 1 α 2 S ( α , β β o ; γ i ) d α d β ) 1 ,
I ( α , β β o ; γ i ) = R P i ASF ( α , β β o ; γ i ) cos θ s .
OPD ( x ^ , y ^ ) = ( n 1 cos θ i n 2 cos θ s ) h ( x ^ , y ^ ) ,
ϕ ( x ^ , y ^ ; γ i , γ s ) = ( 2 π / λ ) OPD = 2 π ( n 1 cos θ i n 2 cos θ s ) h ^ ( x ^ , y ^ ) .
H s ( x ^ , y ^ ; γ i , γ s ) = exp { [ 2 π σ ^ rel ( n 1 γ i n 2 γ s ) ] 2 [ 1 C s ( x ^ , y ^ ) / σ s 2 ] } ,
γ i = cos θ i , γ s = cos θ s .
H s ( x ^ , y ^ ; γ i , γ s ) = exp { [ 2 π σ ^ rel ( γ i + γ s ) ] 2 [ 1 C s ( x ^ , y ^ ) / σ s 2 ] } .
ASF ( α s , β s ; γ i , γ s ) = F { H s ( x ^ , y ^ ; γ i , γ s ) exp ( i 2 π β o y ^ ) } | α = α s , β = β s .
γ s = 1 α s 2 β s 2 .
H s ( x ^ , y ^ ; γ i , γ s ) = A ( γ i , γ s ) + B ( γ i , γ s ) G ( x ^ , y ^ ; γ i , γ s ) ,
A ( γ i , γ s ) = exp { [ 2 π ( γ i + γ s ) σ ^ rel ] 2 } ,
B ( γ i , γ s ) = 1 exp { [ 2 π ( γ i + γ s ) σ ^ rel ] 2 } ,
G ( x ^ , y ^ ; γ i , γ s ) = exp { [ 2 π ( γ i + γ s ) ] 2 σ rel 2 σ s 2 C s ( x ^ , y ^ ) } 1 exp [ 2 π ( γ i + γ s ) ] 2 σ ^ rel 2 1 .
ASF ( α s , β s ; γ i , γ s ) = [ A ( γ i , γ s ) δ ( α , β β o ) + S ( α , β ; γ i , γ s ) ] | α = α s , β = β s ,
S ( α , β ; γ i , γ s ) = B ( γ i , γ s ) F { G ( x ^ , y ^ ; γ i , γ s ) exp ( i 2 π β o y ^ ) } .
S j k ( α , β ; γ i ) = K ( γ i ) j k S ( α j , β k ; γ i , γ j k ) ,
S ( α j , β k ; γ i , γ j k ) = F { G ( x ^ , y ^ ; γ i , γ j k ) exp ( i 2 π β o y ^ ) } | α s = α j , β s = β k ,
γ j k = 1 α j 2 β k 2 ,
C s ( x ^ , y ^ ) = σ s 2 exp [ ( r 2 / c 2 ) ] ,
PSD ( f x ) 1 D = A [ 1 + ( B f x ) 2 ] C / 2 .
PSD ( f ) 2 D = K A B [ 1 + ( B f ) 2 ] ( C + 1 ) / 2 , K = 1 2 π Γ ( ( C + 1 ) / 2 ) Γ ( C / 2 ) .
ACV ( r ) = ( 2 π ) 1 / 2 A B 2 C / 2 Γ ( C / 2 ) ( 2 π r B ) ( C 1 ) / 2 K ( C 1 ) / 2 ( 2 π r B ) .
f x = sin θ s cos ϕ s sin θ i λ , f y = sin θ s sin ϕ s λ .
σ total 2 = 2 π K A B ( C 1 ) B 2 , for     C > 1.0.
BRDF ( θ s ) = 16 π 2 λ 4 cos θ i cos θ s Q PSD ( f ) , f = sin θ s / λ .
TIS R R = ( 4 π cos θ i σ s / λ ) 2 .
A ( γ i , γ s ) 1 [ 2 π σ ^ rel ( γ i + γ s ) ] 2 ,
B ( γ i , γ s ) [ 2 π σ ^ rel ( γ i + γ s ) ] 2 ,
G ( x ^ , y ^ ) C s ( x ^ , y ^ ) / σ s 2 .
BRDF = 4 π 2 λ 4 K ( cos θ i + cos θ s ) 2 Q σ rel 2 σ s 2 PSD ( f x , f y ) .
Irradiance = E = d P d A ( radiant power / area ) , Radiant Intensity = I = d P d ω ( radiant power / steradian ) , Radiance = L = d 2 P d ω d A cos θ ( radiant power / steradian projected area ) .
ARS ( θ s , ϕ s ) d ω s = 1 P i ( d P d ω ) s d ω s = 16 π 2 λ 4 cos θ i cos 2 θ s Q PSD ( f x , f y ) d ω s ,
ARS ( θ s , ϕ s ) d ω s = 1 P s ( d P d ω ) s d ω s = 4 π 2 λ 4 K ( cos θ i + cos θ s ) 2 cos θ s Q σ rel 2 σ s 2 PSD ( f x , f y ) d ω s ,
PSD ( f x , f y ) = λ 4 4 π 2 σ s 2 σ rel 2 BRDF K ( cos θ i + cos θ s ) 2 Q ,
PSD ( f x , f y ) = λ 4 16 π 2 BRDF cos θ i cos θ s Q .
4 π σ rel cos θ i / λ 1.

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