Abstract

By introducing the scattering probability of a subsurface defect (SSD) and statistical distribution functions of SSD radius, refractive index, and position, we derive an extended bidirectional reflectance distribution function (BRDF) from the Jones scattering matrix. This function is applicable to the calculation for comparison with measurement of polarized light-scattering resulting from a SSD. A numerical calculation of the extended BRDF for the case of p-polarized incident light was performed by means of the Monte Carlo method. Our numerical results indicate that the extended BRDF strongly depends on the light incidence angle, the light scattering angle, and the out-of-plane azimuth angle. We observe a 180° symmetry with respect to the azimuth angle. We further investigate the influence of the SSD density, the substrate refractive index, and the statistical distributions of the SSD radius and refractive index on the extended BRDF. For transparent substrates, we also find the dependence of the extended BRDF on the SSD positions.

© 2006 Optical Society of America

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    [Crossref]

2006 (1)

J. Shen, S. J. Liu, W. J. Kong, Z. C. Shen, J. D. Shao, and J. Yao, "Extended bidirectional reflectance distribution function for subsurface defects scattering," Microelectron. Eng. 83, 1047-1050 (2006).
[Crossref]

2005 (3)

2004 (1)

2002 (3)

2001 (1)

2000 (1)

T. A. Germer, "Measurement of roughness of two interfaces of a dielectric film by scattering ellipsometry," Phys. Rev. Lett. 85, 349-352 (2000).
[Crossref] [PubMed]

1999 (2)

D. Rats, J. V. Stebut, and F. Augereau, "High frequency scanning acoustic microscopy: a novel non-destructive surface analytical tool for assessment of coating-specific elastic moduli and tomographic study of subsurface defects," Thin Solid Films 355, 347-352 (1999).
[Crossref]

T. A. Germer and C. C. Asmail, "Polarization of light scattered by microrough surfaces and subsurface defects," J. Opt. Soc. Am. A 16, 1326-1332 (1999).
[Crossref]

1998 (4)

1997 (2)

1995 (1)

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

1994 (1)

1985 (1)

1982 (1)

G. C. Wetsel, Jr., and F. A. Mcdonald, "Subsurface-structure determination using photothermal laser-beam deflection," Appl. Phys. Lett. 41, 926-928 (1982).
[Crossref]

Alexander, C.

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

Amra, C.

Asmail, C. C.

Augereau, F.

D. Rats, J. V. Stebut, and F. Augereau, "High frequency scanning acoustic microscopy: a novel non-destructive surface analytical tool for assessment of coating-specific elastic moduli and tomographic study of subsurface defects," Thin Solid Films 355, 347-352 (1999).
[Crossref]

Bashkansky, M.

Bennett, J. M.

Booij, S. M.

Braat, J. J. M.

Camp, D. W.

D. W. Camp, M. R. Kozlowski, L. M. Sheehan, M. A. Nichols, M. Dovik, R. G. Raether, and I. M. Thomas, "Subsurface damage and polishing compound affect the 355-nm laser damage threshold of fused silica surfaces," in Laser-Induced Damage in Optical Materials, G. J. Exarhos, A. H. Guenther, M. R. Kozlowski, and M. J. Soileau, eds., Proc. SPIE 3244, 356-364 (1998).

Debruyne, S.

Deumié, C.

Dovik, M.

D. W. Camp, M. R. Kozlowski, L. M. Sheehan, M. A. Nichols, M. Dovik, R. G. Raether, and I. M. Thomas, "Subsurface damage and polishing compound affect the 355-nm laser damage threshold of fused silica surfaces," in Laser-Induced Damage in Optical Materials, G. J. Exarhos, A. H. Guenther, M. R. Kozlowski, and M. J. Soileau, eds., Proc. SPIE 3244, 356-364 (1998).

Duncan, M. D.

Duparre, A.

J. Steinert, S. Gliech, A. Wuttig, A. Duparre, and H. Truckenbrodt, "Advanced methods for surface and subsurface defect characterization of optical components," in Optical Metrology Roadmap for the Semiconductor, Optical, and Data Storage Industries, G. A. Al-Jumaily, A. Duparre, and B. Singh, eds., Proc. SPIE 4099, 290-298 (2000).

Fähnle, O. W.

Feit, M. D.

M. D. Feit and A. M. Rubenchik, "Influence of subsurface cracks on laser induced surface damage," in Laser-Induced Damage in Optical Materials, G. J. Exarhos, A. H. Guenther, N. Kaiser, K. L. Lewis, M. J. Soileau, and C. J. Stolz, eds., Proc. SPIE 5273, 264-272 (2004).

Flynn, D. S.

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

Fujita, H.

K. Yoshida, T. Hirao, T. Kamimura, K. Ochi, S. Kaku, H. Yoshida, H. Fujita, F. Tani, M. Sunagawa, and Y. Okamoto, "In-situ optical coatings on subsurface damage-removed substrate," in 27th Annual Boulder Damage Symposium: Laser-Induced Damage in Optical Materials, H. E. Bennett, A. H. Guenther, M. R. Kozlowski, B. E. Newnam, and M. J. Soileau, eds., Proc. SPIE 2714, 340-350 (1996).

Fukumoto, S.

Germer, T. A.

Gilbert, O.

Giovannini, H.

Gliech, S.

J. Steinert, S. Gliech, A. Wuttig, A. Duparre, and H. Truckenbrodt, "Advanced methods for surface and subsurface defect characterization of optical components," in Optical Metrology Roadmap for the Semiconductor, Optical, and Data Storage Industries, G. A. Al-Jumaily, A. Duparre, and B. Singh, eds., Proc. SPIE 4099, 290-298 (2000).

Golini, D.

J. A. Menapace, B. Penetrante, D. Golini, A. F. Slomba, P. E. Miller, T. G. Parham, M. Nichols, and J. Peterson, "Combined advanced finishing and UV-laser conditioning for producing UV-damage-resistant fused-silica optics," in Laser-Induced Damage in Optical Materials, G. J. Exarhos, A. H. Guenther, K. L. Lewis, M. J. Soileau, and C. J. Stolz, eds., Proc. SPIE 4679, 56-68 (2002).

Hirao, T.

K. Yoshida, T. Hirao, T. Kamimura, K. Ochi, S. Kaku, H. Yoshida, H. Fujita, F. Tani, M. Sunagawa, and Y. Okamoto, "In-situ optical coatings on subsurface damage-removed substrate," in 27th Annual Boulder Damage Symposium: Laser-Induced Damage in Optical Materials, H. E. Bennett, A. H. Guenther, M. R. Kozlowski, B. E. Newnam, and M. J. Soileau, eds., Proc. SPIE 2714, 340-350 (1996).

Jabr, S. N.

Jacobs, S. D.

Jungling, K. C.

Kaku, S.

K. Yoshida, T. Hirao, T. Kamimura, K. Ochi, S. Kaku, H. Yoshida, H. Fujita, F. Tani, M. Sunagawa, and Y. Okamoto, "In-situ optical coatings on subsurface damage-removed substrate," in 27th Annual Boulder Damage Symposium: Laser-Induced Damage in Optical Materials, H. E. Bennett, A. H. Guenther, M. R. Kozlowski, B. E. Newnam, and M. J. Soileau, eds., Proc. SPIE 2714, 340-350 (1996).

Kamimura, T.

T. Kamimura, S. Fukumoto, R. Ono, Y. K. Yap, M. Yoshimura, Y. Mori, T. Sasaki, and K. Yoshida, "Enhancement of CsLiB6O10 surface-damage resistance by improved crystallinity and ion-beam etching," Opt. Lett. 27, 616-618 (2002).
[Crossref]

T. Kamimura, Y. Mori, T. Sasaki, H. Yoshida, and T. Okamoto, "Ion etching of fused silica glasses for high-power lasers," Jpn. J. Appl. Phys., Part 1 37, 4840-4841 (1998).
[Crossref]

K. Yoshida, T. Hirao, T. Kamimura, K. Ochi, S. Kaku, H. Yoshida, H. Fujita, F. Tani, M. Sunagawa, and Y. Okamoto, "In-situ optical coatings on subsurface damage-removed substrate," in 27th Annual Boulder Damage Symposium: Laser-Induced Damage in Optical Materials, H. E. Bennett, A. H. Guenther, M. R. Kozlowski, B. E. Newnam, and M. J. Soileau, eds., Proc. SPIE 2714, 340-350 (1996).

Koch, E.

Kong, W. J.

J. Shen, S. J. Liu, W. J. Kong, Z. C. Shen, J. D. Shao, and J. Yao, "Extended bidirectional reflectance distribution function for subsurface defects scattering," Microelectron. Eng. 83, 1047-1050 (2006).
[Crossref]

Kozlowski, M. R.

Z. L. Wu, L. Sheehan, and M. R. Kozlowski, "Laser modulated scattering as a nondestructive evaluation tool for defect inspection in optical materials for high power laser applications," Opt. Express 3, 376-383 (1998).
[Crossref] [PubMed]

D. W. Camp, M. R. Kozlowski, L. M. Sheehan, M. A. Nichols, M. Dovik, R. G. Raether, and I. M. Thomas, "Subsurface damage and polishing compound affect the 355-nm laser damage threshold of fused silica surfaces," in Laser-Induced Damage in Optical Materials, G. J. Exarhos, A. H. Guenther, M. R. Kozlowski, and M. J. Soileau, eds., Proc. SPIE 3244, 356-364 (1998).

Kranenberg, C. F.

Kunieda, H.

Lambropoulos, J. C.

Liu, S. J.

J. Shen, S. J. Liu, W. J. Kong, Z. C. Shen, J. D. Shao, and J. Yao, "Extended bidirectional reflectance distribution function for subsurface defects scattering," Microelectron. Eng. 83, 1047-1050 (2006).
[Crossref]

Lodha, G. S.

Marx, E.

Mcdonald, F. A.

G. C. Wetsel, Jr., and F. A. Mcdonald, "Subsurface-structure determination using photothermal laser-beam deflection," Appl. Phys. Lett. 41, 926-928 (1982).
[Crossref]

Meeder, M.

Menapace, J. A.

J. A. Menapace, B. Penetrante, D. Golini, A. F. Slomba, P. E. Miller, T. G. Parham, M. Nichols, and J. Peterson, "Combined advanced finishing and UV-laser conditioning for producing UV-damage-resistant fused-silica optics," in Laser-Induced Damage in Optical Materials, G. J. Exarhos, A. H. Guenther, K. L. Lewis, M. J. Soileau, and C. J. Stolz, eds., Proc. SPIE 4679, 56-68 (2002).

Miller, P. E.

J. A. Menapace, B. Penetrante, D. Golini, A. F. Slomba, P. E. Miller, T. G. Parham, M. Nichols, and J. Peterson, "Combined advanced finishing and UV-laser conditioning for producing UV-damage-resistant fused-silica optics," in Laser-Induced Damage in Optical Materials, G. J. Exarhos, A. H. Guenther, K. L. Lewis, M. J. Soileau, and C. J. Stolz, eds., Proc. SPIE 4679, 56-68 (2002).

Mori, Y.

T. Kamimura, S. Fukumoto, R. Ono, Y. K. Yap, M. Yoshimura, Y. Mori, T. Sasaki, and K. Yoshida, "Enhancement of CsLiB6O10 surface-damage resistance by improved crystallinity and ion-beam etching," Opt. Lett. 27, 616-618 (2002).
[Crossref]

T. Kamimura, Y. Mori, T. Sasaki, H. Yoshida, and T. Okamoto, "Ion etching of fused silica glasses for high-power lasers," Jpn. J. Appl. Phys., Part 1 37, 4840-4841 (1998).
[Crossref]

Namba, Y.

Nichols, M.

J. A. Menapace, B. Penetrante, D. Golini, A. F. Slomba, P. E. Miller, T. G. Parham, M. Nichols, and J. Peterson, "Combined advanced finishing and UV-laser conditioning for producing UV-damage-resistant fused-silica optics," in Laser-Induced Damage in Optical Materials, G. J. Exarhos, A. H. Guenther, K. L. Lewis, M. J. Soileau, and C. J. Stolz, eds., Proc. SPIE 4679, 56-68 (2002).

Nichols, M. A.

D. W. Camp, M. R. Kozlowski, L. M. Sheehan, M. A. Nichols, M. Dovik, R. G. Raether, and I. M. Thomas, "Subsurface damage and polishing compound affect the 355-nm laser damage threshold of fused silica surfaces," in Laser-Induced Damage in Optical Materials, G. J. Exarhos, A. H. Guenther, M. R. Kozlowski, and M. J. Soileau, eds., Proc. SPIE 3244, 356-364 (1998).

Ochi, K.

K. Yoshida, T. Hirao, T. Kamimura, K. Ochi, S. Kaku, H. Yoshida, H. Fujita, F. Tani, M. Sunagawa, and Y. Okamoto, "In-situ optical coatings on subsurface damage-removed substrate," in 27th Annual Boulder Damage Symposium: Laser-Induced Damage in Optical Materials, H. E. Bennett, A. H. Guenther, M. R. Kozlowski, B. E. Newnam, and M. J. Soileau, eds., Proc. SPIE 2714, 340-350 (1996).

Okamoto, T.

T. Kamimura, Y. Mori, T. Sasaki, H. Yoshida, and T. Okamoto, "Ion etching of fused silica glasses for high-power lasers," Jpn. J. Appl. Phys., Part 1 37, 4840-4841 (1998).
[Crossref]

Okamoto, Y.

K. Yoshida, T. Hirao, T. Kamimura, K. Ochi, S. Kaku, H. Yoshida, H. Fujita, F. Tani, M. Sunagawa, and Y. Okamoto, "In-situ optical coatings on subsurface damage-removed substrate," in 27th Annual Boulder Damage Symposium: Laser-Induced Damage in Optical Materials, H. E. Bennett, A. H. Guenther, M. R. Kozlowski, B. E. Newnam, and M. J. Soileau, eds., Proc. SPIE 2714, 340-350 (1996).

Ono, R.

Parham, T. G.

J. A. Menapace, B. Penetrante, D. Golini, A. F. Slomba, P. E. Miller, T. G. Parham, M. Nichols, and J. Peterson, "Combined advanced finishing and UV-laser conditioning for producing UV-damage-resistant fused-silica optics," in Laser-Induced Damage in Optical Materials, G. J. Exarhos, A. H. Guenther, K. L. Lewis, M. J. Soileau, and C. J. Stolz, eds., Proc. SPIE 4679, 56-68 (2002).

Penetrante, B.

J. A. Menapace, B. Penetrante, D. Golini, A. F. Slomba, P. E. Miller, T. G. Parham, M. Nichols, and J. Peterson, "Combined advanced finishing and UV-laser conditioning for producing UV-damage-resistant fused-silica optics," in Laser-Induced Damage in Optical Materials, G. J. Exarhos, A. H. Guenther, K. L. Lewis, M. J. Soileau, and C. J. Stolz, eds., Proc. SPIE 4679, 56-68 (2002).

Peterson, J.

J. A. Menapace, B. Penetrante, D. Golini, A. F. Slomba, P. E. Miller, T. G. Parham, M. Nichols, and J. Peterson, "Combined advanced finishing and UV-laser conditioning for producing UV-damage-resistant fused-silica optics," in Laser-Induced Damage in Optical Materials, G. J. Exarhos, A. H. Guenther, K. L. Lewis, M. J. Soileau, and C. J. Stolz, eds., Proc. SPIE 4679, 56-68 (2002).

Raether, R. G.

D. W. Camp, M. R. Kozlowski, L. M. Sheehan, M. A. Nichols, M. Dovik, R. G. Raether, and I. M. Thomas, "Subsurface damage and polishing compound affect the 355-nm laser damage threshold of fused silica surfaces," in Laser-Induced Damage in Optical Materials, G. J. Exarhos, A. H. Guenther, M. R. Kozlowski, and M. J. Soileau, eds., Proc. SPIE 3244, 356-364 (1998).

Randi, J. A.

Rats, D.

D. Rats, J. V. Stebut, and F. Augereau, "High frequency scanning acoustic microscopy: a novel non-destructive surface analytical tool for assessment of coating-specific elastic moduli and tomographic study of subsurface defects," Thin Solid Films 355, 347-352 (1999).
[Crossref]

Rubenchik, A. M.

M. D. Feit and A. M. Rubenchik, "Influence of subsurface cracks on laser induced surface damage," in Laser-Induced Damage in Optical Materials, G. J. Exarhos, A. H. Guenther, N. Kaiser, K. L. Lewis, M. J. Soileau, and C. J. Stolz, eds., Proc. SPIE 5273, 264-272 (2004).

Sasaki, T.

T. Kamimura, S. Fukumoto, R. Ono, Y. K. Yap, M. Yoshimura, Y. Mori, T. Sasaki, and K. Yoshida, "Enhancement of CsLiB6O10 surface-damage resistance by improved crystallinity and ion-beam etching," Opt. Lett. 27, 616-618 (2002).
[Crossref]

T. Kamimura, Y. Mori, T. Sasaki, H. Yoshida, and T. Okamoto, "Ion etching of fused silica glasses for high-power lasers," Jpn. J. Appl. Phys., Part 1 37, 4840-4841 (1998).
[Crossref]

Savvides, N.

N. Savvides, "Surface microroughness of ion-beam etched optical surfaces," J. Appl. Phys. 97, 053517 (2005).
[Crossref]

Shao, J. D.

J. Shen, S. J. Liu, W. J. Kong, Z. C. Shen, J. D. Shao, and J. Yao, "Extended bidirectional reflectance distribution function for subsurface defects scattering," Microelectron. Eng. 83, 1047-1050 (2006).
[Crossref]

Sheehan, L.

Sheehan, L. M.

D. W. Camp, M. R. Kozlowski, L. M. Sheehan, M. A. Nichols, M. Dovik, R. G. Raether, and I. M. Thomas, "Subsurface damage and polishing compound affect the 355-nm laser damage threshold of fused silica surfaces," in Laser-Induced Damage in Optical Materials, G. J. Exarhos, A. H. Guenther, M. R. Kozlowski, and M. J. Soileau, eds., Proc. SPIE 3244, 356-364 (1998).

Shen, J.

J. Shen, S. J. Liu, W. J. Kong, Z. C. Shen, J. D. Shao, and J. Yao, "Extended bidirectional reflectance distribution function for subsurface defects scattering," Microelectron. Eng. 83, 1047-1050 (2006).
[Crossref]

Shen, Z. C.

J. Shen, S. J. Liu, W. J. Kong, Z. C. Shen, J. D. Shao, and J. Yao, "Extended bidirectional reflectance distribution function for subsurface defects scattering," Microelectron. Eng. 83, 1047-1050 (2006).
[Crossref]

Slomba, A. F.

J. A. Menapace, B. Penetrante, D. Golini, A. F. Slomba, P. E. Miller, T. G. Parham, M. Nichols, and J. Peterson, "Combined advanced finishing and UV-laser conditioning for producing UV-damage-resistant fused-silica optics," in Laser-Induced Damage in Optical Materials, G. J. Exarhos, A. H. Guenther, K. L. Lewis, M. J. Soileau, and C. J. Stolz, eds., Proc. SPIE 4679, 56-68 (2002).

Stebut, J. V.

D. Rats, J. V. Stebut, and F. Augereau, "High frequency scanning acoustic microscopy: a novel non-destructive surface analytical tool for assessment of coating-specific elastic moduli and tomographic study of subsurface defects," Thin Solid Films 355, 347-352 (1999).
[Crossref]

Steinert, J.

J. Steinert, S. Gliech, A. Wuttig, A. Duparre, and H. Truckenbrodt, "Advanced methods for surface and subsurface defect characterization of optical components," in Optical Metrology Roadmap for the Semiconductor, Optical, and Data Storage Industries, G. A. Al-Jumaily, A. Duparre, and B. Singh, eds., Proc. SPIE 4099, 290-298 (2000).

Stover, J. C.

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

Sunagawa, M.

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Tani, F.

K. Yoshida, T. Hirao, T. Kamimura, K. Ochi, S. Kaku, H. Yoshida, H. Fujita, F. Tani, M. Sunagawa, and Y. Okamoto, "In-situ optical coatings on subsurface damage-removed substrate," in 27th Annual Boulder Damage Symposium: Laser-Induced Damage in Optical Materials, H. E. Bennett, A. H. Guenther, M. R. Kozlowski, B. E. Newnam, and M. J. Soileau, eds., Proc. SPIE 2714, 340-350 (1996).

Tawara, Y.

Thomas, I. M.

D. W. Camp, M. R. Kozlowski, L. M. Sheehan, M. A. Nichols, M. Dovik, R. G. Raether, and I. M. Thomas, "Subsurface damage and polishing compound affect the 355-nm laser damage threshold of fused silica surfaces," in Laser-Induced Damage in Optical Materials, G. J. Exarhos, A. H. Guenther, M. R. Kozlowski, and M. J. Soileau, eds., Proc. SPIE 3244, 356-364 (1998).

Truckenbrodt, H.

J. Steinert, S. Gliech, A. Wuttig, A. Duparre, and H. Truckenbrodt, "Advanced methods for surface and subsurface defect characterization of optical components," in Optical Metrology Roadmap for the Semiconductor, Optical, and Data Storage Industries, G. A. Al-Jumaily, A. Duparre, and B. Singh, eds., Proc. SPIE 4099, 290-298 (2000).

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[Crossref]

Wons, T.

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J. Steinert, S. Gliech, A. Wuttig, A. Duparre, and H. Truckenbrodt, "Advanced methods for surface and subsurface defect characterization of optical components," in Optical Metrology Roadmap for the Semiconductor, Optical, and Data Storage Industries, G. A. Al-Jumaily, A. Duparre, and B. Singh, eds., Proc. SPIE 4099, 290-298 (2000).

Yamashita, K.

Yao, J.

J. Shen, S. J. Liu, W. J. Kong, Z. C. Shen, J. D. Shao, and J. Yao, "Extended bidirectional reflectance distribution function for subsurface defects scattering," Microelectron. Eng. 83, 1047-1050 (2006).
[Crossref]

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T. Kamimura, Y. Mori, T. Sasaki, H. Yoshida, and T. Okamoto, "Ion etching of fused silica glasses for high-power lasers," Jpn. J. Appl. Phys., Part 1 37, 4840-4841 (1998).
[Crossref]

K. Yoshida, T. Hirao, T. Kamimura, K. Ochi, S. Kaku, H. Yoshida, H. Fujita, F. Tani, M. Sunagawa, and Y. Okamoto, "In-situ optical coatings on subsurface damage-removed substrate," in 27th Annual Boulder Damage Symposium: Laser-Induced Damage in Optical Materials, H. E. Bennett, A. H. Guenther, M. R. Kozlowski, B. E. Newnam, and M. J. Soileau, eds., Proc. SPIE 2714, 340-350 (1996).

Yoshida, K.

T. Kamimura, S. Fukumoto, R. Ono, Y. K. Yap, M. Yoshimura, Y. Mori, T. Sasaki, and K. Yoshida, "Enhancement of CsLiB6O10 surface-damage resistance by improved crystallinity and ion-beam etching," Opt. Lett. 27, 616-618 (2002).
[Crossref]

K. Yoshida, T. Hirao, T. Kamimura, K. Ochi, S. Kaku, H. Yoshida, H. Fujita, F. Tani, M. Sunagawa, and Y. Okamoto, "In-situ optical coatings on subsurface damage-removed substrate," in 27th Annual Boulder Damage Symposium: Laser-Induced Damage in Optical Materials, H. E. Bennett, A. H. Guenther, M. R. Kozlowski, B. E. Newnam, and M. J. Soileau, eds., Proc. SPIE 2714, 340-350 (1996).

Yoshimura, M.

Yu, J.

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[Crossref]

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[Crossref] [PubMed]

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[Crossref] [PubMed]

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[Crossref]

T. A. Germer and E. Marx, "Ray model of light scattering by flake pigments or rough surfaces with smooth transparent coatings," Appl. Opt. 43, 1266-1274 (2004).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

G. C. Wetsel, Jr., and F. A. Mcdonald, "Subsurface-structure determination using photothermal laser-beam deflection," Appl. Phys. Lett. 41, 926-928 (1982).
[Crossref]

J. Appl. Phys. (1)

N. Savvides, "Surface microroughness of ion-beam etched optical surfaces," J. Appl. Phys. 97, 053517 (2005).
[Crossref]

J. Opt. Soc. Am. A (2)

Jpn. J. Appl. Phys., Part 1 (1)

T. Kamimura, Y. Mori, T. Sasaki, H. Yoshida, and T. Okamoto, "Ion etching of fused silica glasses for high-power lasers," Jpn. J. Appl. Phys., Part 1 37, 4840-4841 (1998).
[Crossref]

Microelectron. Eng. (1)

J. Shen, S. J. Liu, W. J. Kong, Z. C. Shen, J. D. Shao, and J. Yao, "Extended bidirectional reflectance distribution function for subsurface defects scattering," Microelectron. Eng. 83, 1047-1050 (2006).
[Crossref]

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Other (7)

J. Steinert, S. Gliech, A. Wuttig, A. Duparre, and H. Truckenbrodt, "Advanced methods for surface and subsurface defect characterization of optical components," in Optical Metrology Roadmap for the Semiconductor, Optical, and Data Storage Industries, G. A. Al-Jumaily, A. Duparre, and B. Singh, eds., Proc. SPIE 4099, 290-298 (2000).

D. W. Camp, M. R. Kozlowski, L. M. Sheehan, M. A. Nichols, M. Dovik, R. G. Raether, and I. M. Thomas, "Subsurface damage and polishing compound affect the 355-nm laser damage threshold of fused silica surfaces," in Laser-Induced Damage in Optical Materials, G. J. Exarhos, A. H. Guenther, M. R. Kozlowski, and M. J. Soileau, eds., Proc. SPIE 3244, 356-364 (1998).

M. D. Feit and A. M. Rubenchik, "Influence of subsurface cracks on laser induced surface damage," in Laser-Induced Damage in Optical Materials, G. J. Exarhos, A. H. Guenther, N. Kaiser, K. L. Lewis, M. J. Soileau, and C. J. Stolz, eds., Proc. SPIE 5273, 264-272 (2004).

J. A. Menapace, B. Penetrante, D. Golini, A. F. Slomba, P. E. Miller, T. G. Parham, M. Nichols, and J. Peterson, "Combined advanced finishing and UV-laser conditioning for producing UV-damage-resistant fused-silica optics," in Laser-Induced Damage in Optical Materials, G. J. Exarhos, A. H. Guenther, K. L. Lewis, M. J. Soileau, and C. J. Stolz, eds., Proc. SPIE 4679, 56-68 (2002).

K. Yoshida, T. Hirao, T. Kamimura, K. Ochi, S. Kaku, H. Yoshida, H. Fujita, F. Tani, M. Sunagawa, and Y. Okamoto, "In-situ optical coatings on subsurface damage-removed substrate," in 27th Annual Boulder Damage Symposium: Laser-Induced Damage in Optical Materials, H. E. Bennett, A. H. Guenther, M. R. Kozlowski, B. E. Newnam, and M. J. Soileau, eds., Proc. SPIE 2714, 340-350 (1996).

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

T. A. Germer, "Polarized light diffusely scattered under smooth and rough interfaces," in Polarization Science and Remote Sensing, J. A. Shaw and J. S. Tyo, eds., Proc. SPIE 5158, 193-204 (2003).

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Figures (6)

Fig. 1
Fig. 1

Coordinate system for the SSD scattering model.

Fig. 2
Fig. 2

Schematic of a single SSD scattering.

Fig. 3
Fig. 3

BRDF dependence on the incident, the scattering, and the azimuth angles. n 2 = 1.52 , N A = 10 4 , a = 1 10 nm , d = 10 100 nm , and n sub = 1 1.52 ; a, n sub and d uniform; (a) θ i = 0 ° , (b) θ i = 30 ° , (c) θ i = 45 ° , (d) θ i = 56.66 ° , (e) θ i = 75 ° , (f) θ i = 85 ° .

Fig. 4
Fig. 4

BRDF dependence on statistical distributions of defect sizes and refractive indices. θ i = 45 ° , n 2 = 1.52 , N A = 10 4 , a = 1 10 nm , d = 10 100 nm , and n sub = 1 1.52 ; (a) a Gaussian, n sub uniform, (b) a uniform, n sub Gaussian, (c) a, n sub Gaussian.

Fig. 5
Fig. 5

BRDF data for various SSD densities versus azimuth angle at fixed incident and scattering angles. θ i = θ s = 45 ° , n 2 = 1.52 , a = 1 10 nm , d = 1 100 nm , and n sub = 1 1.52 ; a, n sub , and d uniform.

Fig. 6
Fig. 6

BRDF data for various substrate refractive index values at fixed incident and scattering angles. θ i = θ s = 45 ° , N A = 10 4 , a = 1 10 nm , d = 10 100 nm , and n sub = 1 1.52 ; a, n sub , and d uniform.

Equations (21)

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( E p scat E s scat ) = exp ( i k R ) R [ S p p sub S s p sub S p s sub S s s sub ] ( E p inc E s inc ) ,
p 1 ( ξ ) = 1 l 1 exp ( ξ l 1 ) d ξ ,
p 2 ( ξ ) = 1 l 2 exp ( ξ l 2 ) d ξ .
p 1 ( ξ ) = ξ p 1 ( ξ ) d ξ = exp ( ξ l 1 ) ,
p 2 ( ξ ) = ξ p 2 ( ξ ) d ξ = exp ( ξ l 2 ) .
p 3 ( h ) d h = p 1 ( h ) p 2 ( h ) d h = 1 l 1 cos θ i exp ( h l 1 cos θ i ) 1 l 2 cos θ i exp ( h l 2 cos θ i ) d h .
p 4 = 0 d p 3 ( h ) d h = 0 d 1 d 2 exp ( h d ) exp ( h d ) d h = 1 2 d ( 1 1 e 2 ) ,
p 5 = 0 2 π 0 θ c sin θ d θ d ϕ 0 2 π 0 π 2 sin θ d θ d ϕ = 1 cos θ c = 1 1 1 n 2 2 .
p = p 4 p 5 = 1 2 d ( 1 1 e 2 ) ( 1 1 1 n 2 2 ) .
q s s sub = cos ϕ s ( cos θ i + n 2 2 sin 2 θ i ) ( cos θ s + n 2 2 sin 2 θ s ) ,
q s p sub = sin ϕ s n 2 2 sin 2 θ s ( cos θ i + n 2 2 sin 2 θ i ) ( n 2 2 cos θ s + n 2 2 sin 2 θ s ) ,
q p s sub = sin ϕ s n 2 2 sin 2 θ i ( n 2 2 cos θ i + n 2 2 sin 2 θ i ) ( cos θ s + n 2 2 sin 2 θ s ) ,
q p p sub = sin θ i sin θ s n 2 2 sin 2 θ i n 2 2 sin 2 θ s cos ϕ s ( n 2 2 cos θ i + n 2 2 sin 2 θ i ) ( n 2 2 cos θ s + n 2 2 sin 2 θ s ) .
S 0 sub = 4 p δ γ ( n sub 2 n 2 2 n sub 2 + 2 n 2 2 ) cos θ s cos θ i a 3 k 2 n 2 1.5 [ ( n 2 2 sin 2 θ s ) 1 2 cos θ s ] 1 2 ,
S 0 sub = 4 p exp [ i n 2 k F 3 ( d ) cos θ s ] exp [ i n 2 k F 3 ( d ) cos θ i ] × { [ F 1 ( n sub ) ] 2 n 2 2 [ F 1 ( n sub ) ] 2 + 2 n 2 2 } ( cos θ s ) 1 2 cos θ i [ F 2 ( a ) ] 3 k 2 n 2 1.5 ( n 2 2 sin 2 θ s ) 1 4 .
BRDF = N F A S e ̂ 2 cos θ s cos θ i ,
BRDF = N F S 0 sub 2 A cos θ s cos θ i [ q p p sub 2 q s p sub 2 q p s sub 2 q s s sub 2 ] , = 256 π 4 N F p 2 λ 4 A cos θ i n 2 3 ( n 2 2 sin 2 θ s ) 1 2 [ q p p sub 2 q s p sub 2 q p s sub 2 q s s sub 2 ] exp [ i n 2 k F 3 ( d ) cos θ s ] exp [ i n 2 k F 3 ( d ) cos θ i ] 2 [ F 2 ( a ) ] 6 { [ F 1 ( n sub ) ] 2 n 2 2 [ F 1 ( n sub ) ] 2 + 2 n 2 2 } 2 .
BRDF sub = 256 π 4 N p 2 λ 4 A M cos θ i n 2 3 ( n 2 2 sin 2 θ s ) 1 2 [ q p p sub 2 q s p sub 2 q p s sub 2 q s s sub 2 ] × exp [ i n 2 k F 3 ( d ) cos θ s ] exp [ i n 2 k F 3 ( d ) cos θ i ] 2 m = 1 M l = 1 N F l [ F 2 ( a ) ] 6 { [ F 1 ( n sub ) ] 2 n 2 2 [ F 1 ( n sub ) ] 2 + 2 n 2 2 } 2 ,
BRDF sub p = 256 π 4 N p 2 λ 4 A M cos θ i n 2 3 ( n 2 2 sin 2 θ s ) 1 2 [ q p p sub 2 0 q p s sub 2 0 ] × exp [ i n 2 k F 3 ( d ) cos θ s ] exp [ i n 2 k F 3 ( d ) cos θ i ] 2 m = 1 M l = 1 N F l [ F 2 ( a ) ] 6 { [ F 1 ( n sub ) ] 2 n 2 2 [ F 1 ( n sub ) ] 2 + 2 n 2 2 } 2 .
BRDF sub s = 256 π 4 N p 2 λ 4 A M cos θ i n 2 3 ( n 2 2 sin 2 θ s ) 1 2 [ 0 q s p sub 2 0 q s s sub 2 ] × exp [ i n 2 k F 3 ( d ) cos θ s ] exp [ i n 2 k F 3 ( d ) cos θ i ] 2 m = 1 M l = 1 N F l [ F 2 ( a ) ] 6 { [ F 1 ( n sub ) ] 2 n 2 2 [ F 1 ( n sub ) ] 2 + 2 n 2 2 } 2 .
BRDF sub p = 256 π 4 N p 2 λ 4 A M cos θ i n 2 3 ( n 2 2 sin 2 θ s ) 1 2 ( q p p sub 2 + q p s sub 2 ) × m = 1 M l = 1 N F l [ F 2 ( a ) ] 6 { [ F 1 ( n sub ) ] 2 n 2 2 [ F 1 ( n sub ) ] 2 + 2 n 2 2 } 2 ,

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