Abstract

In April of 1972, Professor Roland Shack presented a series of four colloquium talks at the Optical Sciences Center at the University of Arizona in which he reformulated scalar diffraction theory in terms of the direction cosines of the propagation vectors of the angular spectrum of plane waves described by the Fourier integral transform of the diffracting aperture. The fourth lecture, entitled Radiometry and Lambert’s Law, described diffuse reflectance and surface scatter phenomena as merely a diffraction phenomenon caused by random phase variations in the system pupil function. In 1974, he elegantly condensed these four lectures into a single colloquium talk entitled A Global View of Diffraction. This paper is intended to provide a compilation showing the further development of that work over the last 46 years.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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  1. R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, 1965).
  2. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  3. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978).
  4. S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
    [Crossref]
  5. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963).
  6. R. V. Shack, Colloquium talk at the Optical Sciences Center at the University of Arizona in 1974.
  7. J. E. Harvey, “Light-scattering characteristics of optical surfaces,” Ph.D. dissertation (University of Arizona, 1976).
  8. J. E. Harvey and R. V. Shack, “Aberrations of diffracted wave fields,” Appl. Opt. 17, 3003–3009 (1978).
    [Crossref]
  9. J. E. Harvey, “A Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
    [Crossref]
  10. 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.
  11. J. E. Harvey and C. L. Vernold, “Description of diffraction grating behavior in direction cosine space,” Appl. Opt. 37, 8158–8160 (1998).
    [Crossref]
  12. J. E. Harvey, “Surface scatter phenomena: a linear, shift-invariant process,” Proc. SPIE 1165, 87–99 (1989).
    [Crossref]
  13. F. E. Nicodemus, “Reflectance nomenclature and directional reflectance and emissivity,” Appl. Opt. 9, 1474–1475 (1970).
    [Crossref]
  14. R. Hufnagel, “Random wavefront effects,” Photo. Sci. Eng. 9, 244–247 (1965).
  15. P. J. Chandley and W. T. Welford, “A re-formulation of some results of P. Beckmann for scattering from rough surfaces,” Opt. Quantum Electron. 7, 393–397 (1975).
    [Crossref]
  16. 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]
  17. 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]
  18. 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]
  19. A. Krywonos, “Predicting surface scatter using a linear systems formulation of non–paraxial scalar diffraction,” Ph.D. dissertation (University of Central Florida, 2006).
  20. J. E. Harvey, S. Schroder, N. Choi, and A. Duparre, “Total integrated scatter from surfaces with arbitrary roughness, correlation widths, and incident angles,” Opt. Eng. 51, 013402 (2012).
    [Crossref]
  21. 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]
  22. J. E. Harvey, Understanding Surface Scatter Phenomena (SPIE, 2019).
  23. J. J. Muray, F. E. Nicodemus, and I. Wunderman, “Proposed supplement to the SI nomenclature for radiometry and photometry,” Appl. Opt. 10, 1465–1468 (1971).
    [Crossref]
  24. J. M. Palmer, “Getting intense about intensity,” Opt. Photon. News 30, 4 (1995).
  25. E. L. Dereniak and G. D. Boreman, Infrared Detectors and Systems (Wiley, 1996).
  26. R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983).
  27. J. E. Harvey, C. L. Vernold, A. Krywonos, and P. L. Thompson, “Diffracted radiance: a fundamental quantity in non-paraxial scalar diffraction theory,” Appl. Opt. 38, 6469–6481 (1999).
    [Crossref]
  28. J. E. Harvey, C. L. Vernold, A. Krywonos, and P. L. Thompson, “Diffracted radiance: a fundamental quantity in non-paraxial scalar diffraction theory: eratta,” Appl. Opt. 39, 6374–6375 (2000).
    [Crossref]
  29. J. E. Harvey, A. Krywonos, and D. Bogunovic, “Non-paraxial scalar treatment of sinusoidal phase gratings,” J. Opt. Soc. Am. A 23, 858–865 (2006).
    [Crossref]
  30. R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, 1980), p. 98.
  31. E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997).
  32. J. E. Harvey, A. Krywonos, and D. Bogunovic, “A tolerance on defocus precisely locates the far field (exactly where is that far field anyway?),” Appl. Opt. 41, 2586–2588 (2002).
    [Crossref]
  33. V. N. Mahajan, Optical Imaging and Aberrations: Part I (SPIE, 1998).
  34. H. H. Hopkins, Wave Theory of Aberrations (Clarendon, 1950).
  35. J. E. Harvey, A. Krywonos, and C. L. Vernold, “A modified Beckmann-Kirchhoff surface scatter model for rough surfaces with large incident and scattering angles,” Opt. Eng. 46, 078002 (2007).
    [Crossref]
  36. 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]
  37. A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of surface scatter theory for rough surfaces with arbitrary incident and scattering angles,” J. Opt. Soc. Am. A 28, 1121–1138 (2011).
    [Crossref]
  38. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
  39. 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 Proc. 16th EU-PVSEC, Glasgow, Scotland (2000), pp. 361–364.
  40. 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]
  41. S. Schröder, A. Duparré, K. Füchsel, N. Kaiser, A. Tünnermann, and J. E. Harvey, “Scattering of roughened TCO films—modeling and measurement,” in Presented at OSA Topical Meeting on Optical Interference Coatings, Arizona, USA, June7–9, 2010.
  42. 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]
  43. J. E. Harvey, “Parametric analysis of the effect of scattered light upon the modulation transfer function,” Opt. Eng. 52, 073110 (2013).
    [Crossref]
  44. N. Choi and J. E. Harvey, “Numerical validation of the Generalized Harvey-Shack surface scatter theory,” Opt. Eng. 52, 115103 (2013).
    [Crossref]
  45. N. Choi and J. E. Harvey, “Image degradation due to surface scatter in the presence of aberrations,” Appl. Opt. 51, 535–546 (2012).
    [Crossref]
  46. J. E. Harvey, “Integrating optical fabrication and metrology into the optical design process,” Appl. Opt. 54, 2224–2233 (2015).
    [Crossref]

2015 (1)

2013 (2)

J. E. Harvey, “Parametric analysis of the effect of scattered light upon the modulation transfer function,” Opt. Eng. 52, 073110 (2013).
[Crossref]

N. Choi and J. E. Harvey, “Numerical validation of the Generalized Harvey-Shack surface scatter theory,” Opt. Eng. 52, 115103 (2013).
[Crossref]

2012 (2)

N. Choi and J. E. Harvey, “Image degradation due to surface scatter in the presence of aberrations,” Appl. Opt. 51, 535–546 (2012).
[Crossref]

J. E. Harvey, S. Schroder, N. Choi, and A. Duparre, “Total integrated scatter from surfaces with arbitrary roughness, correlation widths, and incident angles,” Opt. Eng. 51, 013402 (2012).
[Crossref]

2011 (1)

2010 (2)

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. 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]

2007 (1)

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

2006 (1)

2004 (1)

2002 (1)

2000 (1)

1999 (2)

J. E. Harvey, C. L. Vernold, A. Krywonos, and P. L. Thompson, “Diffracted radiance: a fundamental quantity in 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]

1998 (1)

1995 (1)

J. M. Palmer, “Getting intense about intensity,” Opt. Photon. News 30, 4 (1995).

1989 (1)

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

1988 (2)

1987 (1)

1979 (1)

J. E. Harvey, “A Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[Crossref]

1978 (1)

1975 (1)

P. J. Chandley and W. T. Welford, “A re-formulation of some results of P. Beckmann for scattering from rough surfaces,” Opt. Quantum Electron. 7, 393–397 (1975).
[Crossref]

1971 (1)

1970 (1)

1965 (1)

R. Hufnagel, “Random wavefront effects,” Photo. Sci. Eng. 9, 244–247 (1965).

1951 (1)

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).

Bogunovic, D.

Boreman, G. D.

E. L. Dereniak and G. D. Boreman, Infrared Detectors and Systems (Wiley, 1996).

Boyd, R. W.

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

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, 1965).

Chandley, P. J.

P. J. Chandley and W. T. Welford, “A re-formulation of some results of P. Beckmann for scattering from rough surfaces,” Opt. Quantum Electron. 7, 393–397 (1975).
[Crossref]

Choi, N.

N. Choi and J. E. Harvey, “Numerical validation of the Generalized Harvey-Shack surface scatter theory,” Opt. Eng. 52, 115103 (2013).
[Crossref]

J. E. Harvey, S. Schroder, N. Choi, and A. Duparre, “Total integrated scatter from surfaces with arbitrary roughness, correlation widths, and incident angles,” Opt. Eng. 51, 013402 (2012).
[Crossref]

N. Choi and J. E. Harvey, “Image degradation due to surface scatter in the presence of aberrations,” Appl. Opt. 51, 535–546 (2012).
[Crossref]

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

Dereniak, E. L.

E. L. Dereniak and G. D. Boreman, Infrared Detectors and Systems (Wiley, 1996).

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 Proc. 16th EU-PVSEC, Glasgow, Scotland (2000), pp. 361–364.

Duparre, A.

J. E. Harvey, S. Schroder, N. Choi, and A. Duparre, “Total integrated scatter from surfaces with arbitrary roughness, correlation widths, and incident angles,” Opt. Eng. 51, 013402 (2012).
[Crossref]

Duparré, A.

S. Schröder, A. Duparré, K. Füchsel, N. Kaiser, A. Tünnermann, and J. E. Harvey, “Scattering of roughened TCO films—modeling and measurement,” in Presented at OSA Topical Meeting on Optical Interference Coatings, Arizona, USA, June7–9, 2010.

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 Proc. 16th EU-PVSEC, Glasgow, Scotland (2000), pp. 361–364.

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 Presented at OSA Topical Meeting on Optical Interference Coatings, Arizona, USA, June7–9, 2010.

Gaskill, J. D.

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

Glenn, P.

Goodman, J. W.

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

Gutierrez-Vega, J. C.

Guzar-Sicairos, M.

Harvey, J. E.

J. E. Harvey, “Integrating optical fabrication and metrology into the optical design process,” Appl. Opt. 54, 2224–2233 (2015).
[Crossref]

J. E. Harvey, “Parametric analysis of the effect of scattered light upon the modulation transfer function,” Opt. Eng. 52, 073110 (2013).
[Crossref]

N. Choi and J. E. Harvey, “Numerical validation of the Generalized Harvey-Shack surface scatter theory,” Opt. Eng. 52, 115103 (2013).
[Crossref]

J. E. Harvey, S. Schroder, N. Choi, and A. Duparre, “Total integrated scatter from surfaces with arbitrary roughness, correlation widths, and incident angles,” Opt. Eng. 51, 013402 (2012).
[Crossref]

N. Choi and J. E. Harvey, “Image degradation due to surface scatter in the presence of aberrations,” Appl. Opt. 51, 535–546 (2012).
[Crossref]

A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of surface scatter theory for rough surfaces with arbitrary incident and scattering angles,” J. Opt. Soc. Am. A 28, 1121–1138 (2011).
[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, A. Krywonos, and C. L. Vernold, “A modified Beckmann-Kirchhoff surface scatter model for rough surfaces with large incident and scattering angles,” Opt. Eng. 46, 078002 (2007).
[Crossref]

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

J. E. Harvey, A. Krywonos, and D. Bogunovic, “A tolerance on defocus precisely locates the far field (exactly where is that far field anyway?),” Appl. Opt. 41, 2586–2588 (2002).
[Crossref]

J. E. Harvey, C. L. Vernold, A. Krywonos, and P. L. Thompson, “Diffracted radiance: a fundamental quantity in non-paraxial scalar diffraction theory: eratta,” 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 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]

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, “Surface scatter phenomena: a linear, shift-invariant process,” Proc. SPIE 1165, 87–99 (1989).
[Crossref]

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]

J. E. Harvey, “A Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[Crossref]

J. E. Harvey and R. V. Shack, “Aberrations of diffracted wave fields,” Appl. Opt. 17, 3003–3009 (1978).
[Crossref]

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

J. E. Harvey, Understanding Surface Scatter Phenomena (SPIE, 2019).

S. Schröder, A. Duparré, K. Füchsel, N. Kaiser, A. Tünnermann, and J. E. Harvey, “Scattering of roughened TCO films—modeling and measurement,” in Presented at OSA Topical Meeting on Optical Interference Coatings, Arizona, USA, June7–9, 2010.

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]

Hopkins, H. H.

H. H. Hopkins, Wave Theory of Aberrations (Clarendon, 1950).

Hufnagel, R.

R. Hufnagel, “Random wavefront effects,” Photo. Sci. Eng. 9, 244–247 (1965).

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 Presented at OSA Topical Meeting on Optical Interference Coatings, Arizona, USA, June7–9, 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 Proc. 16th EU-PVSEC, Glasgow, Scotland (2000), pp. 361–364.

Krywonos, A.

A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of surface scatter theory for rough surfaces with arbitrary incident and scattering angles,” J. Opt. Soc. Am. A 28, 1121–1138 (2011).
[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, A. Krywonos, and C. L. Vernold, “A modified Beckmann-Kirchhoff surface scatter model for rough surfaces with large incident and scattering angles,” Opt. Eng. 46, 078002 (2007).
[Crossref]

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

J. E. Harvey, A. Krywonos, and D. Bogunovic, “A tolerance on defocus precisely locates the far field (exactly where is that far field anyway?),” Appl. Opt. 41, 2586–2588 (2002).
[Crossref]

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

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

Loewen, E. G.

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997).

Mahajan, V. N.

V. N. Mahajan, Optical Imaging and Aberrations: Part I (SPIE, 1998).

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 Proc. 16th EU-PVSEC, Glasgow, Scotland (2000), pp. 361–364.

Mendez, E. R.

Moran, E. C.

Muray, J. J.

Nicodemus, F. E.

O’Donnell, K. A.

Palmer, J. M.

J. M. Palmer, “Getting intense about intensity,” Opt. Photon. News 30, 4 (1995).

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]

Petit, R.

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, 1980), p. 98.

Popov, E.

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997).

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]

Saleh, B. E. A.

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

Schroder, S.

J. E. Harvey, S. Schroder, N. Choi, and A. Duparre, “Total integrated scatter from surfaces with arbitrary roughness, correlation widths, and incident angles,” Opt. Eng. 51, 013402 (2012).
[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]

Schröder, S.

S. Schröder, A. Duparré, K. Füchsel, N. Kaiser, A. Tünnermann, and J. E. Harvey, “Scattering of roughened TCO films—modeling and measurement,” in Presented at OSA Topical Meeting on Optical Interference Coatings, Arizona, USA, June7–9, 2010.

Shack, R. V.

J. E. Harvey and R. V. Shack, “Aberrations of diffracted wave fields,” Appl. Opt. 17, 3003–3009 (1978).
[Crossref]

R. V. Shack, Colloquium talk at the Optical Sciences Center at the University of Arizona in 1974.

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 Proc. 16th EU-PVSEC, Glasgow, Scotland (2000), pp. 361–364.

Slomba, A.

Spizzichino, A.

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

Teich, M. C.

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

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 Presented at OSA Topical Meeting on Optical Interference Coatings, Arizona, USA, June7–9, 2010.

Van Speybroeck, L. P.

Vernold, C. L.

Welford, W. T.

P. J. Chandley and W. T. Welford, “A re-formulation of some results of P. Beckmann for scattering from rough surfaces,” Opt. Quantum Electron. 7, 393–397 (1975).
[Crossref]

Wunderman, I.

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 Proc. 16th EU-PVSEC, Glasgow, Scotland (2000), pp. 361–364.

Zmek, W. P.

Am. J. Phys. (1)

J. E. Harvey, “A Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[Crossref]

Appl. Opt. (11)

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

J. E. Harvey and R. V. Shack, “Aberrations of diffracted wave fields,” Appl. Opt. 17, 3003–3009 (1978).
[Crossref]

J. E. Harvey, C. L. Vernold, A. Krywonos, and P. L. Thompson, “Diffracted radiance: a fundamental quantity in 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]

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Figures (26)

Fig. 1.
Fig. 1. Illustration of the physical situation for the case of a spherical wavefront incident upon a diffraction grating, and converging to a hemispherical observation space (adapted from Ref. [7]).
Fig. 2.
Fig. 2. Aberrations inherent to the diffraction process are illustrated here as magnified diffraction orders from a 10-line per millimeter Ronchi Ruling placed in a converging cone of light ($r$ is the radius of the observation hemisphere, $d$ is the aperture diameter in which grating is placed, and $\beta$ is the sin of diffracted angle) (adapted from Ref. [7]).
Fig. 3.
Fig. 3. Geometrical configuration of Planes ${P_o}$ and $P$ (adapted from Ref. [7]).
Fig. 4.
Fig. 4. Unit circle in direction cosine space. The planewave components inside the circle will propagate, and the planewave components outside the circle do not propagate and constitute the evanescent wave (adapted from Ref. [7]).
Fig. 5.
Fig. 5. (a) Geometric relationship between the normally incident beam, diffracting aperture, and observation hemisphere. (b) Geometric configuration when the incident beam strikes the diffracting aperture at an oblique angle (adapted from Ref. [7]).
Fig. 6.
Fig. 6. (a) Illustration of the position of the diffracted orders in real space and in direction cosine space, showing that the diffracted orders strike the observation hemisphere in a latitude slice, not a great circle (adapted from Ref. [7]). (b) Relative position of the diffracted orders and the incident beam (for a reflection grating) in direction cosine space. Diffracted orders outside the unit circle are evanescent (adapted from Ref. [11]).
Fig. 7.
Fig. 7. Direction cosine diagrams for four different orientations of a diffraction grating with period $d = {3}\lambda$ illuminated with an obliquely incident beam ($\alpha_i = - 0.3$, $\beta_i = - 0.4$) (adapted from Ref. [11]).
Fig. 8.
Fig. 8. Schematic representation of reflectance from a rough surface (adapted from Ref. [7]).
Fig. 9.
Fig. 9. (a) Experimentally measured scattered intensity plotted versus scattered angle. (b) Scattered radiance plotted versus $\beta - {\beta _o}$ in direction cosine space (adapted from Ref. [7]).
Fig. 10.
Fig. 10. Illustration of the geometrical configuration for scatter measurements (adapted from Ref. [7]).
Fig. 11.
Fig. 11. Geometrical configuration of the two principal planes in which the scattered light distribution was sampled (adapted from Ref. [7]).
Fig. 12.
Fig. 12. (a) Illustration of shift-invariance, relative to incident angle, when displayed in direction cosine space. (b) illustrates that the forward, backward, left, and right surface scatter profiles (for a fixed 45° incident angle) are also superposed (adapted from Ref. [7]).
Fig. 13.
Fig. 13. Illustration of the OPD for a specularly reflected ray (adapted from Ref. [37]).
Fig. 14.
Fig. 14. Comparison of scattered intensity predictions from the OHS and the MHS surface scatter theories for different incident angles. Experimental data is also displayed for $\theta_i = {70}^\circ$ (this figure previously published as Fig. 3–23 in Ref. [22]).
Fig. 15.
Fig. 15. Geometrical configuration used to demonstrate the fundamental theorem of radiometry from which the quantity radiance is obtained.
Fig. 16.
Fig. 16. Illustration of a moderately broad Gaussian diffracted radiance distribution. (a) Normal incidence, (b) 64° incident angle (adapted from Ref. [27]).
Fig. 17.
Fig. 17. Diffraction grating efficiency of the first order of a perfectly conducting sinusoidal phase grating in the Littrow condition (for two different values of $h/d$). Courtesy of OSA, first appeared in Ref. [29].
Fig. 18.
Fig. 18. Illustration of the axial regions of validity for the Rayleigh–Sommerfeld, Fresnel and Fraunhofer diffraction integrals. Courtesy of OSA; this figure previously published as Fig. 1 in Ref. [32].
Fig. 19.
Fig. 19. Non-intuitive surface scatter effects. Solid lines represent classical Beckmann–Kirchhoff scatter theory, circles indicate experimental measurements (reprinted with permission from Ref. [21]).
Fig. 20.
Fig. 20. (a) Predicted ASF (radiance) for normal incidence and $\lambda = {0.6328}\;\unicode{x00B5}{\rm m}$; (b) truncated and renormalized ASF for $\theta_i = 70^\circ$ and $\lambda = {10.6}\;\unicode{x00B5}{\rm m}$; (c) scattered intensity for $\theta_i = 70^\circ$ and 10.6 µm (adapted from Ref. [27]).
Fig. 21.
Fig. 21. (a) Comparison of MHS prediction of scattered radiance and intensity for the O’Donnell–Mendez surface with ${\theta _i} = {0}$ and $\lambda = {10.6}\;\unicode{x00B5}{\rm m}$; (b) predicted radiance and intensity for $\theta_i = 70^\circ$ and $\lambda = {10.6}\;\unicode{x00B5}{\rm m}$; (c) O’Donnell–Mendez measurements compared to MHS prediction of scattered intensity (adapted from Ref. [27]).
Fig. 22.
Fig. 22. Predictions from the empirically Modified Beckmann–Kirchhoff model are compared to the previously non-intuitive surface scatter effects illustrated in Fig. 17 (adapted from Ref. [35]).
Fig. 23.
Fig. 23. Predictions from the empirically Modified Beckmann–Kirchhoff model are compared to the Classical Beckmann–Kirchhoff theory and the Rayleigh–Rice theory (adapted from Ref. [35]).
Fig. 24.
Fig. 24. Illustration of both forward (transmitted) and backward (reflected) scattering from a moderately rough interface between two media with arbitrary refractive indices. Courtesy of SPIE, this figure previously published as Fig. 7 in Ref. [36].
Fig. 25.
Fig. 25. Comparison of scattered intensity predictions from the OHS, MHS, and GHS theories for different incident angles. Experimental data is 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. This figure previously published as Fig. 6 in Ref. [37].
Fig. 26.
Fig. 26. Comparison of the MHS and GHS surface scatter theories with the classical Beckmann–Kirchhoff scatter theory and the O’Donnell–Mendez experimental data for a wavelength of 0.6328 µm and an incident angle of 70°. Courtesy of OSA, this figure previously published as Fig 7 in Ref. [37].

Tables (1)

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Table 1. Fraunhofer Criterion for Different Tolerances upon Defocus

Equations (60)

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E ( x 2 , y 2 ) = 1 λ 2 z 2 | F { U o + ( x 1 , y 1 ) } | ξ = x 2 λ z , η = y 2 λ z | 2 .
F { U o + ( x 1 , y 1 ) } = U o + ( x 1 , y 1 ) × exp [ i 2 π ( x 1 ξ + y 1 η ) ] d x 1 d y 1 .
x ^ = x / λ , y ^ = y / λ , z ^ = z / λ , e t c .
α = x ^ / r ^ , β = y ^ / r ^ , a n d γ = z ^ / r ^ .
A o ( α , β ; 0 ) = U o ( x ^ , y ^ ; 0 ) exp [ i 2 π ( α x ^ + β y ^ ) ] d x ^ d y ^ , U o ( x ^ , y ^ ; 0 ) = A o ( α , β ; 0 ) exp [ i 2 π ( α x ^ + β y ^ ) ] d α d β ,
A ( α , β ; z ^ ) = U o ( x ^ , y ^ ; z ^ ) exp [ i 2 π ( α x ^ + β y ^ ) ] d x ^ d y ^ , U ( x ^ , y ^ ; z ^ ) = A ( α , β ; z ^ ) exp [ i 2 π ( α x ^ + β y ^ ) ] d α d β .
[ ^ 2 + ( 2 π ) 2 = λ 2 2 ] U ( x ^ , y ^ ; z ^ ) = 0 .
A ( α , β ; γ ) = A o ( α , β ; 0 ) e i 2 π γ z ^ , w h e r e γ = 1 α 2 β 2 .
H ( α , β ; z ^ ) A ( α , β ; z ^ ) / A o ( α , β ; z ^ ) = e i 2 π γ z ^ .
γ = 1 α 2 β 2 { f o r ( α 2 + β 2 1 γ is real f o r ( α 2 + β 2 1 γ is imaginary .
h ( x ^ , y ^ ; z ^ ) = F 1 { e x p ( i 2 π γ z ^ ) } = ( 1 2 π r ^ i ) z ^ r ^ exp ( i 2 π r ^ ) r ^ .
U ( x ^ , y ^ ; z ^ ) = U o ( x ^ , y ^ ; 0 ) ( 1 2 π ^ i ) × z ^ ^ exp ( i 2 π ^ ) ^ d x ^ d y ^ ,
U ( α , β ; r ^ ) = γ [ exp ( i 2 π r ^ ) / ( i r ^ ) ] F { U o ( x ^ , y ^ ; 0 ) } .
U ( α , β β o ; r ^ ) = γ [ exp ( i 2 π r ^ ) / ( i r ^ ) ] F { U o ( x ^ , y ^ ; 0 ) × exp ( i 2 π β o y ^ ) } ,
H s ( x ^ , y ^ ) = exp { ( 4 π σ ^ s ) 2 [ 1 C s ( x ^ , y ^ ) / σ s 2 ] } ,
O P D = ( γ i + γ o ) h ( x ^ , y ^ ) = 2 γ i h ( x ^ , y ^ ) ,
ϕ ( x ^ , y ^ ) = 2 π λ O P D = 4 π γ i h ^ ( x ^ , y ^ ) .
H s ( x ^ , y ^ ; γ i ) = exp { ( 4 π γ i σ ^ r e l ) 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 ^ s ) 2 ] , B ( γ i ) = 1 exp [ ( 4 π γ i σ ^ s ) 2 ] ,
G ( x ^ , y ^ ; γ i ) = exp [ ( 4 π γ i ) 2 C s ( x ^ , y ^ ) ] 1 exp [ ( 4 π γ i σ ^ s ) 2 ] 1 .
Irradiance E P A c ( watts / area ) .
Radiant Intensity I = P ω c ( watts / steradian ) .
Radiance L = 2 P ω c A s cos θ s ( watts / steradian projected area ) .
2 P = L ( θ s , ϕ s , x , y ) A s cos θ s ω c = L ( θ s , ϕ s , x , y ) A s cos θ s A c cos θ c / r 2 = L ( θ s , ϕ s , x , y ) ω s A c cos θ c .
L ( α , β ) = λ 2 A s | F { U o ( x ^ , y ^ ; 0 ) } | 2 .
L ( α , β β o ) = γ o λ 2 A s | F { U o ( x ^ , y ^ ; 0 ) exp ( i 2 π β o y ^ ) } | 2 .
L ( α , β β o ) = K γ o λ 2 A s | F { U o ( x ^ , y ^ ; 0 ) exp ( i 2 π β o y ^ ) } | 2 f o r α 2 + β 2 1 , L ( α , β β o ) = 0 f o r α 2 + β 2 > 1 ,
K = α = β = L ( α , β β o ) d α d β α = 1 1 β = 1 α 2 1 α 2 L ( α , β β o ) d α d β Normalization Factor .
z ( k / 2 ) ( x 1 2 + y 1 2 ) m a x .
W ^ = W ^ 000 ( Piston Error ) } Zero-order + W ^ 200 ρ 2 Piston + W ^ 020 a ^ 2 Defocus + W ^ 111 ρ a ^ cos ( ϕ ϕ ) Lateral Magnification Error } 2nd-order + W ^ 400 ρ 4 Piston + W ^ 040 a ^ 4 Spherical Aberration + W ^ 131 ρ a ^ 3 cos ( ϕ ϕ ) Coma + W ^ 222 ρ 2 a ^ 2 cos 2 ( ϕ ϕ ) Astigmatism + W ^ 220 ρ 2 a ^ 2 Field Curvature + W ^ 311 ρ 3 a ^ cos ( ϕ ϕ ) Distortion } 4th-order + Higher-order Terms ,
W ^ 020 = z ^ 2 ( d ^ 2 z ^ ) 2 .
z = d 2 8 W 020 .
ϕ = π 4 λ z d 2 ( r a d i a n s ) .
D { ρ } = π c 2 F 2 exp ( g ) A s m = 1 g m m ! m exp [ ( v xy 2 c 2 / 4 m ) ] ,
F = 1 + ( cos θ i cos θ sin θ i sin θ cos ϕ ) cos θ i ( cos θ i + cos θ ) .
g = [ ( 2 π σ s / λ ) ( cos θ i + cos θ ) ] 2 ,
v xy = k sin 2 θ i 2 sin θ i sin θ cos ϕ + sin 2 θ .
L ( θ , ϕ ) = K π c 2 exp ( g ) A s m = 1 g m m ! m exp [ ( v xy 2 c 2 / 4 m ) ] .
I ( θ , ϕ ) = A s L ( θ , ϕ ) cos θ A s = L ( θ , ϕ ) A s cos θ
I ( θ , ϕ ) = K π c 2 cos θ exp ( g ) m = 1 g m m ! m exp [ ( v xy 2 c 2 / 4 m ) ] .
D { ρ } = π c 2 F 2 A s v z 2 σ s 2 exp [ v xy 2 c 2 4 v z 2 σ s 2 ] , where v z = k ( cos θ i + cos θ ) .
L ( θ , φ ) = K π c 2 A s v z 2 σ s 2 exp [ v xy 2 c 2 4 v z 2 σ s 2 ] .
I ( θ , ϕ ) = K π c 2 cos θ v z 2 σ s 2 exp [ v xy 2 c 2 4 v z 2 σ s 2 ] .
O P D ( 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 O P D = 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 ] } ,
H s ( x ^ , y ^ ; γ i , γ s ) = exp { [ 2 π σ ^ r e l ( γ i + γ s ) ] 2 [ 1 C s ( x ^ , y ^ ) / σ s 2 ] } .
S ( 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 ^ r e l ] 2 } ,
B ( γ i , γ s ) = 1 exp { [ 2 π ( γ i + γ s ) σ ^ r e l ] 2 } ,
G ( x ^ , y ^ ; γ i , γ s ) = exp { [ 2 π ( γ i + γ s ) ] 2 σ r e l 2 σ s 2 C s ( x ^ , y ^ ) } 1 exp [ 2 π ( γ i + γ s ) σ ^ r e l ] 2 1 .
S ( α s , β s ; γ i , γ s ) = [ A ( γ i , γ s ) δ ( α , β β o ) + S ( α , β β o ; γ i , γ s ) ] | α = α s , β = β s ,
S ( α , β β o ; 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 jk ( α , β ; γ i ) = K ( γ i ) j k S ( α j , β k ; γ i , γ jk ) ,
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 ,
S ( α j , β k ; γ i , γ jk ) = B ( γ i , γ s ) F { G ( x ^ , y ^ ; γ i , γ jk ) exp ( i 2 π β o y ^ ) } | α s = α j , β s = β k ,
γ jk = 1 α j 2 β k 2 .