Abstract

The directional distribution of radiant flux reflected from roughened surfaces is analyzed on the basis of geometrical optics. The analytical model assumes that the surface consists of small, randomly disposed, mirror-like facets. Specular reflection from these facets plus a diffuse component due to multiple reflections and/or internal scattering are postulated as the basic mechanisms of the reflection process. The effects of shadowing and masking of facets by adjacent facets are included in the analysis. The angular distributions of reflected flux predicted by the analysis are in very good agreement with experiment for both metallic and nonmetallic surfaces. Moreover, the analysis successfully predicts the off-specular maxima in the reflection distribution which are observed experimentally and which emerge as the incidence angle increases. The model thus affords a rational explanation for the off-specular peak phenomenon in terms of mutual masking and shadowing of mirror-like, specularly reflecting surface facets.

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  1. K. E. Torrance and E. M. Sparrow, J. Heat Trans. 88, Ser. C, 223 (1966).
  2. K. E. Torrance, E. M. Sparrow, and R. C. Birkebak, J. Opt. Soc. Am. 56, 916 (1966).
  3. S. Tanaka, J. Applied Physics (Japan) 25, 207 (1956); 26, 85 (1957); 27, 600, 758 (1958); 28, 508 (1959).
  4. W. M. Brandenberg and J. T. Neu, J. Opt. Soc. Am. 56, 97 (1966).
  5. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon Press, New York, 1963).
  6. The plane of incidence includes the incident beam and the surface normal.
  7. G. I. Pokrowski, Z. Physik 30, 66 (1924); 35, 34 (1925); 35, 390 (1926); 36, 472 (1926).
  8. H. Schulz, Z. Physik 31, 496 (1925).
  9. W. E. K. Middleton and A. G. Mungall, J. Opt. Soc. Am. 42, 572 (1952).
  10. Fresnel reflectance curves for a metal and nonmetal are shown in Fig. 6 of this paper.
  11. A. G. Mungall (private communication, 23 September 1965).
  12. The original calculations were performed by hand; the present re-evaluation employed an electronic computer.
  13. The reflection triplet (ψ; θ,ø) applies to quantities which depend on the angle of incidence and the angles of reflection. A single angle in parentheses is used for quantities which depend on only one angle.
  14. J. A. Clark, Ed., Theory and Fundamental Research in Heat Transfer (Pergamon Press, New York, 1963), p. 7.
  15. H. J. McNicholas, J. Res. Natl. Bur. Std. (U. S.) 1, 29 (1928).
  16. C. von Fragstein, Optik 12, 60 (1955).
  17. F. E. Nicodemus, Appl. Opt. 4, 767 (1965).
  18. K. E. Torrance, Ph.D. dissertation, University of Minnesota (March 1966).
  19. G. M. Gorodinskii, Opt. Spectry. 16, 59 (1964).
  20. V. K. Polyanskii and V. P. Rvachev, Opt. Spectry. 20, 391 (1966).
  21. A. W. Christie, J. Opt. Soc. Am. 43, 621 (1953).
  22. S. Flugge, ed., Handbuch der Physik (Springer-Verlag, Berlin, 1928), Vol. 20, pp. 240–250.
  23. W. A. Rense, J. Opt. Soc. Am. 40, 55 (1950).
  24. American Institute of Physics Handbook (McGraw-Hill Book Co., New York 1963), Second ed., pp. 6–12 and 6–107.

Beckmann, P.

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

Birkebak, R. C.

K. E. Torrance, E. M. Sparrow, and R. C. Birkebak, J. Opt. Soc. Am. 56, 916 (1966).

Brandenberg, W. M.

W. M. Brandenberg and J. T. Neu, J. Opt. Soc. Am. 56, 97 (1966).

Christie, A. W.

A. W. Christie, J. Opt. Soc. Am. 43, 621 (1953).

Gorodinskii, G. M.

G. M. Gorodinskii, Opt. Spectry. 16, 59 (1964).

McNicholas, H. J.

H. J. McNicholas, J. Res. Natl. Bur. Std. (U. S.) 1, 29 (1928).

Middleton, W. E. K.

W. E. K. Middleton and A. G. Mungall, J. Opt. Soc. Am. 42, 572 (1952).

Mungall, A. G.

W. E. K. Middleton and A. G. Mungall, J. Opt. Soc. Am. 42, 572 (1952).

A. G. Mungall (private communication, 23 September 1965).

Neu, J. T.

W. M. Brandenberg and J. T. Neu, J. Opt. Soc. Am. 56, 97 (1966).

Nicodemus, F. E.

F. E. Nicodemus, Appl. Opt. 4, 767 (1965).

Pokrowski, G. I.

G. I. Pokrowski, Z. Physik 30, 66 (1924); 35, 34 (1925); 35, 390 (1926); 36, 472 (1926).

Polyanskii, V. K.

V. K. Polyanskii and V. P. Rvachev, Opt. Spectry. 20, 391 (1966).

Rense, W. A.

W. A. Rense, J. Opt. Soc. Am. 40, 55 (1950).

Rvachev, V. P.

V. K. Polyanskii and V. P. Rvachev, Opt. Spectry. 20, 391 (1966).

Schulz, H.

H. Schulz, Z. Physik 31, 496 (1925).

Sparrow, E. M.

K. E. Torrance, E. M. Sparrow, and R. C. Birkebak, J. Opt. Soc. Am. 56, 916 (1966).

K. E. Torrance and E. M. Sparrow, J. Heat Trans. 88, Ser. C, 223 (1966).

Spizzichino, A.

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

Tanaka, S.

S. Tanaka, J. Applied Physics (Japan) 25, 207 (1956); 26, 85 (1957); 27, 600, 758 (1958); 28, 508 (1959).

Torrance, K. E.

K. E. Torrance and E. M. Sparrow, J. Heat Trans. 88, Ser. C, 223 (1966).

K. E. Torrance, E. M. Sparrow, and R. C. Birkebak, J. Opt. Soc. Am. 56, 916 (1966).

K. E. Torrance, Ph.D. dissertation, University of Minnesota (March 1966).

von Fragstein, C.

C. von Fragstein, Optik 12, 60 (1955).

Other (24)

K. E. Torrance and E. M. Sparrow, J. Heat Trans. 88, Ser. C, 223 (1966).

K. E. Torrance, E. M. Sparrow, and R. C. Birkebak, J. Opt. Soc. Am. 56, 916 (1966).

S. Tanaka, J. Applied Physics (Japan) 25, 207 (1956); 26, 85 (1957); 27, 600, 758 (1958); 28, 508 (1959).

W. M. Brandenberg and J. T. Neu, J. Opt. Soc. Am. 56, 97 (1966).

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

The plane of incidence includes the incident beam and the surface normal.

G. I. Pokrowski, Z. Physik 30, 66 (1924); 35, 34 (1925); 35, 390 (1926); 36, 472 (1926).

H. Schulz, Z. Physik 31, 496 (1925).

W. E. K. Middleton and A. G. Mungall, J. Opt. Soc. Am. 42, 572 (1952).

Fresnel reflectance curves for a metal and nonmetal are shown in Fig. 6 of this paper.

A. G. Mungall (private communication, 23 September 1965).

The original calculations were performed by hand; the present re-evaluation employed an electronic computer.

The reflection triplet (ψ; θ,ø) applies to quantities which depend on the angle of incidence and the angles of reflection. A single angle in parentheses is used for quantities which depend on only one angle.

J. A. Clark, Ed., Theory and Fundamental Research in Heat Transfer (Pergamon Press, New York, 1963), p. 7.

H. J. McNicholas, J. Res. Natl. Bur. Std. (U. S.) 1, 29 (1928).

C. von Fragstein, Optik 12, 60 (1955).

F. E. Nicodemus, Appl. Opt. 4, 767 (1965).

K. E. Torrance, Ph.D. dissertation, University of Minnesota (March 1966).

G. M. Gorodinskii, Opt. Spectry. 16, 59 (1964).

V. K. Polyanskii and V. P. Rvachev, Opt. Spectry. 20, 391 (1966).

A. W. Christie, J. Opt. Soc. Am. 43, 621 (1953).

S. Flugge, ed., Handbuch der Physik (Springer-Verlag, Berlin, 1928), Vol. 20, pp. 240–250.

W. A. Rense, J. Opt. Soc. Am. 40, 55 (1950).

American Institute of Physics Handbook (McGraw-Hill Book Co., New York 1963), Second ed., pp. 6–12 and 6–107.

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