Abstract

The angular distributions of light scattered by gold-coated and aluminum-coated gratings with amplitudes of ∼90 nm and periods of 6.67 µm were measured and calculated for light incident from a He–Ne laser at an angle of 6°. Experimental results are compared with predictions of Beckmann’s scalar theory and Rayleigh’s vector theory. The measured scattering pattern has a background of scattered light due mainly to residual surface roughness. Also the power in the higher-order peaks is larger by several orders of magnitude than the computed one, which can be attributed mainly to the low-order contributions of the harmonics in the profile.

© 2000 Optical Society of America

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  1. E. Marx, T. R. Lettieri, T. V. Vorburger, M. McIntosh, “Sinusoidal surfaces as standards for BRDF instruments,” in Optical Scattering: Applications, Measurement, and Theory, J. C. Stover, ed., Proc. SPIE1530, 15–21 (1991).
  2. E. Marx, T. V. Vorburger, “Windowing effects on light scattered by sinusoidal surfaces,” in Optical Scattering: Applications, Measurement, and Theory II, J. C. Stover, ed., Proc. SPIE1995, 2–14 (1993).
  3. E. Marx, T. R. Lettieri, T. V. Vorburger, “Light scattering by sinusoidal surfaces: illumination windows and harmonics in standards,” Appl. Opt. 34, 1269–1277 (1995).
    [CrossRef] [PubMed]
  4. B. C. Park, T. V. Vorburger, T. A. Germer, E. Marx, “Scattering from sinusoidal gratings,” in Scattering and Surface Roughness, Z.-H. Gu, A. Maradudin, eds., Proc. SPIE3141, 65–77 (1997).
  5. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), Chaps. 4 and 5.
  6. J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Institute of Physics, Bristol, 1991), Chap. 4, pp. 80–84.
  7. Ref. 6, p. 42.
  8. J. C. Stover, Optical Scattering: Measurement and Analysis (McGraw-Hill, New York, 1990), p. 219.
  9. R. Petit, “A tutorial introduction,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 16–17.
  10. Ref. 5, p. 81 or Ref. 6, p. 89.
  11. Ref. 8, p. 60.
  12. D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 76–81.
  13. G. R. Jiracek, “Numerical comparisons of a modified Rayleigh approach with other rough surface EM scattering solutions,” IEEE Trans. Antennas Propag. 21, 393–396 (1973).
    [CrossRef]
  14. Certain commercial equipment is identified to specify adequately the experimental procedure. In no case does such identification imply recommendation or endorsement by NIST or does it imply that the equipment is necessarily the best available for the purpose.
  15. C. C. Asmail, C. L. Cromer, J. E. Proctor, J. J. Hsia, “Instrumentation at the National Institute of Standards and Technology for bidirectional reflectance distribution function (BRDF) measurements,” in Stray Radiation in Optical Systems III, R. P. Breault, ed., Proc. SPIE2260, 52–61 (1994).
  16. T. A. Germer, C. C. Asmail, “Goniometric optical scatter instrument for out-of-plane ellipsometry measurements,” Rev. Sci. Instrum. 70, 3688–3695 (1999).
    [CrossRef]
  17. J. E. Harvey, C. L. Vernold, A. Krywonos, P. L. Thompson, “Diffracted radiance: a fundamental quantity of nonparaxial scalar diffraction theory,” Appl. Opt. 38, 6469–6481 (1999).
    [CrossRef]
  18. E. D. Palik, Handbook of Optical Constants (Academic, San Diego, Calif., 1985), pp. 286–295, 369–406.
  19. Ref. 8, pp. 112–115.
  20. A. Wirgin, “Scattering from sinusoidal gratings: an evaluation of the Kirchhoff approximation,” J. Opt. Soc. Am. 73, 1028–1041 (1983).
    [CrossRef]
  21. D. A. Content, “Diffraction grating groove analysis used to predict efficiency and scatter performance,” in Gradient Index, Miniature, and Diffractive Optical Systems, A. D. Kathman, ed., Proc. SPIE3778, 19–30 (1999).

1999 (2)

T. A. Germer, C. C. Asmail, “Goniometric optical scatter instrument for out-of-plane ellipsometry measurements,” Rev. Sci. Instrum. 70, 3688–3695 (1999).
[CrossRef]

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

1995 (1)

1983 (1)

1973 (1)

G. R. Jiracek, “Numerical comparisons of a modified Rayleigh approach with other rough surface EM scattering solutions,” IEEE Trans. Antennas Propag. 21, 393–396 (1973).
[CrossRef]

Asmail, C. C.

T. A. Germer, C. C. Asmail, “Goniometric optical scatter instrument for out-of-plane ellipsometry measurements,” Rev. Sci. Instrum. 70, 3688–3695 (1999).
[CrossRef]

C. C. Asmail, C. L. Cromer, J. E. Proctor, J. J. Hsia, “Instrumentation at the National Institute of Standards and Technology for bidirectional reflectance distribution function (BRDF) measurements,” in Stray Radiation in Optical Systems III, R. P. Breault, ed., Proc. SPIE2260, 52–61 (1994).

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), Chaps. 4 and 5.

Content, D. A.

D. A. Content, “Diffraction grating groove analysis used to predict efficiency and scatter performance,” in Gradient Index, Miniature, and Diffractive Optical Systems, A. D. Kathman, ed., Proc. SPIE3778, 19–30 (1999).

Cromer, C. L.

C. C. Asmail, C. L. Cromer, J. E. Proctor, J. J. Hsia, “Instrumentation at the National Institute of Standards and Technology for bidirectional reflectance distribution function (BRDF) measurements,” in Stray Radiation in Optical Systems III, R. P. Breault, ed., Proc. SPIE2260, 52–61 (1994).

Germer, T. A.

T. A. Germer, C. C. Asmail, “Goniometric optical scatter instrument for out-of-plane ellipsometry measurements,” Rev. Sci. Instrum. 70, 3688–3695 (1999).
[CrossRef]

B. C. Park, T. V. Vorburger, T. A. Germer, E. Marx, “Scattering from sinusoidal gratings,” in Scattering and Surface Roughness, Z.-H. Gu, A. Maradudin, eds., Proc. SPIE3141, 65–77 (1997).

Harvey, J. E.

Hsia, J. J.

C. C. Asmail, C. L. Cromer, J. E. Proctor, J. J. Hsia, “Instrumentation at the National Institute of Standards and Technology for bidirectional reflectance distribution function (BRDF) measurements,” in Stray Radiation in Optical Systems III, R. P. Breault, ed., Proc. SPIE2260, 52–61 (1994).

Jiracek, G. R.

G. R. Jiracek, “Numerical comparisons of a modified Rayleigh approach with other rough surface EM scattering solutions,” IEEE Trans. Antennas Propag. 21, 393–396 (1973).
[CrossRef]

Krywonos, A.

Lettieri, T. R.

E. Marx, T. R. Lettieri, T. V. Vorburger, “Light scattering by sinusoidal surfaces: illumination windows and harmonics in standards,” Appl. Opt. 34, 1269–1277 (1995).
[CrossRef] [PubMed]

E. Marx, T. R. Lettieri, T. V. Vorburger, M. McIntosh, “Sinusoidal surfaces as standards for BRDF instruments,” in Optical Scattering: Applications, Measurement, and Theory, J. C. Stover, ed., Proc. SPIE1530, 15–21 (1991).

Marx, E.

E. Marx, T. R. Lettieri, T. V. Vorburger, “Light scattering by sinusoidal surfaces: illumination windows and harmonics in standards,” Appl. Opt. 34, 1269–1277 (1995).
[CrossRef] [PubMed]

E. Marx, T. R. Lettieri, T. V. Vorburger, M. McIntosh, “Sinusoidal surfaces as standards for BRDF instruments,” in Optical Scattering: Applications, Measurement, and Theory, J. C. Stover, ed., Proc. SPIE1530, 15–21 (1991).

E. Marx, T. V. Vorburger, “Windowing effects on light scattered by sinusoidal surfaces,” in Optical Scattering: Applications, Measurement, and Theory II, J. C. Stover, ed., Proc. SPIE1995, 2–14 (1993).

B. C. Park, T. V. Vorburger, T. A. Germer, E. Marx, “Scattering from sinusoidal gratings,” in Scattering and Surface Roughness, Z.-H. Gu, A. Maradudin, eds., Proc. SPIE3141, 65–77 (1997).

Maystre, D.

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 76–81.

McIntosh, M.

E. Marx, T. R. Lettieri, T. V. Vorburger, M. McIntosh, “Sinusoidal surfaces as standards for BRDF instruments,” in Optical Scattering: Applications, Measurement, and Theory, J. C. Stover, ed., Proc. SPIE1530, 15–21 (1991).

Ogilvy, J. A.

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Institute of Physics, Bristol, 1991), Chap. 4, pp. 80–84.

Palik, E. D.

E. D. Palik, Handbook of Optical Constants (Academic, San Diego, Calif., 1985), pp. 286–295, 369–406.

Park, B. C.

B. C. Park, T. V. Vorburger, T. A. Germer, E. Marx, “Scattering from sinusoidal gratings,” in Scattering and Surface Roughness, Z.-H. Gu, A. Maradudin, eds., Proc. SPIE3141, 65–77 (1997).

Petit, R.

R. Petit, “A tutorial introduction,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 16–17.

Proctor, J. E.

C. C. Asmail, C. L. Cromer, J. E. Proctor, J. J. Hsia, “Instrumentation at the National Institute of Standards and Technology for bidirectional reflectance distribution function (BRDF) measurements,” in Stray Radiation in Optical Systems III, R. P. Breault, ed., Proc. SPIE2260, 52–61 (1994).

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), Chaps. 4 and 5.

Stover, J. C.

J. C. Stover, Optical Scattering: Measurement and Analysis (McGraw-Hill, New York, 1990), p. 219.

Thompson, P. L.

Vernold, C. L.

Vorburger, T. V.

E. Marx, T. R. Lettieri, T. V. Vorburger, “Light scattering by sinusoidal surfaces: illumination windows and harmonics in standards,” Appl. Opt. 34, 1269–1277 (1995).
[CrossRef] [PubMed]

E. Marx, T. V. Vorburger, “Windowing effects on light scattered by sinusoidal surfaces,” in Optical Scattering: Applications, Measurement, and Theory II, J. C. Stover, ed., Proc. SPIE1995, 2–14 (1993).

E. Marx, T. R. Lettieri, T. V. Vorburger, M. McIntosh, “Sinusoidal surfaces as standards for BRDF instruments,” in Optical Scattering: Applications, Measurement, and Theory, J. C. Stover, ed., Proc. SPIE1530, 15–21 (1991).

B. C. Park, T. V. Vorburger, T. A. Germer, E. Marx, “Scattering from sinusoidal gratings,” in Scattering and Surface Roughness, Z.-H. Gu, A. Maradudin, eds., Proc. SPIE3141, 65–77 (1997).

Wirgin, A.

Appl. Opt. (2)

IEEE Trans. Antennas Propag. (1)

G. R. Jiracek, “Numerical comparisons of a modified Rayleigh approach with other rough surface EM scattering solutions,” IEEE Trans. Antennas Propag. 21, 393–396 (1973).
[CrossRef]

J. Opt. Soc. Am. (1)

Rev. Sci. Instrum. (1)

T. A. Germer, C. C. Asmail, “Goniometric optical scatter instrument for out-of-plane ellipsometry measurements,” Rev. Sci. Instrum. 70, 3688–3695 (1999).
[CrossRef]

Other (16)

D. A. Content, “Diffraction grating groove analysis used to predict efficiency and scatter performance,” in Gradient Index, Miniature, and Diffractive Optical Systems, A. D. Kathman, ed., Proc. SPIE3778, 19–30 (1999).

E. D. Palik, Handbook of Optical Constants (Academic, San Diego, Calif., 1985), pp. 286–295, 369–406.

Ref. 8, pp. 112–115.

Certain commercial equipment is identified to specify adequately the experimental procedure. In no case does such identification imply recommendation or endorsement by NIST or does it imply that the equipment is necessarily the best available for the purpose.

C. C. Asmail, C. L. Cromer, J. E. Proctor, J. J. Hsia, “Instrumentation at the National Institute of Standards and Technology for bidirectional reflectance distribution function (BRDF) measurements,” in Stray Radiation in Optical Systems III, R. P. Breault, ed., Proc. SPIE2260, 52–61 (1994).

E. Marx, T. R. Lettieri, T. V. Vorburger, M. McIntosh, “Sinusoidal surfaces as standards for BRDF instruments,” in Optical Scattering: Applications, Measurement, and Theory, J. C. Stover, ed., Proc. SPIE1530, 15–21 (1991).

E. Marx, T. V. Vorburger, “Windowing effects on light scattered by sinusoidal surfaces,” in Optical Scattering: Applications, Measurement, and Theory II, J. C. Stover, ed., Proc. SPIE1995, 2–14 (1993).

B. C. Park, T. V. Vorburger, T. A. Germer, E. Marx, “Scattering from sinusoidal gratings,” in Scattering and Surface Roughness, Z.-H. Gu, A. Maradudin, eds., Proc. SPIE3141, 65–77 (1997).

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), Chaps. 4 and 5.

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Institute of Physics, Bristol, 1991), Chap. 4, pp. 80–84.

Ref. 6, p. 42.

J. C. Stover, Optical Scattering: Measurement and Analysis (McGraw-Hill, New York, 1990), p. 219.

R. Petit, “A tutorial introduction,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 16–17.

Ref. 5, p. 81 or Ref. 6, p. 89.

Ref. 8, p. 60.

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 76–81.

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