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.

© 1967 Optical Society of America

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References

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  1. K. E. Torrance and E. M. Sparrow, J. Heat Trans. 88, Ser. C, 223 (1966).
    [Crossref]
  2. K. E. Torrance, E. M. Sparrow, and R. C. Birkebak, J. Opt. Soc. Am. 56, 916 (1966).
    [Crossref]
  3. S. Tanaka, J. Applied Physics (Japan) 25, 207 (1956);J. Applied Physics (Japan) 26, 85 (1957);J. Applied Physics (Japan) 27, 600, 758 (1958);J. Applied Physics (Japan) 28, 508 (1959).
  4. W. M. Brandenberg and J. T. Neu, J. Opt. Soc. Am. 56, 97 (1966).
    [Crossref]
  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);Z. Physik 35, 34 (1925);Z. Physik 35, 390 (1926);Z. Physik 36, 472 (1926).
    [Crossref]
  8. H. Schulz, Z. Physik 31, 496 (1925).
    [Crossref]
  9. W. E. K. Middleton and A. G. Mungall, J. Opt. Soc. Am. 42, 572 (1952).
    [Crossref]
  10. Fresnel reflectance curves for a metal and nonmetal are shown in Fig. 6 of this paper.
  11. A. G. Mungall (private communication, 23September1965).
  12. The original calculations were performed by hand; the present re-evaluation employed an electronic computer.
  13. The reflection triplet (ψ;θ,ϕ) 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).
    [Crossref]
  18. K. E. Torrance, Ph.D. dissertation, University of Minnesota (March1966).
  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).
    [Crossref]
  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).
    [Crossref]
  24. American Institute of Physics Handbook (McGraw-Hill Book Co., New York1963), Second ed., pp. 6–12 and 6–107.

1966 (4)

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

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

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

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

1965 (1)

1964 (1)

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

1956 (1)

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

1955 (1)

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

1953 (1)

1952 (1)

1950 (1)

1928 (1)

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

1925 (1)

H. Schulz, Z. Physik 31, 496 (1925).
[Crossref]

1924 (1)

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

Beckmann, P.

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

Birkebak, R. C.

Brandenberg, W. M.

Christie, A. W.

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.

Mungall, A. G.

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

A. G. Mungall (private communication, 23September1965).

Neu, J. T.

Nicodemus, F. E.

Pokrowski, G. I.

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

Polyanskii, V. K.

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

Rense, W. A.

Rvachev, V. P.

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

Schulz, H.

H. Schulz, Z. Physik 31, 496 (1925).
[Crossref]

Sparrow, E. M.

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

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

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);J. Applied Physics (Japan) 26, 85 (1957);J. Applied Physics (Japan) 27, 600, 758 (1958);J. Applied Physics (Japan) 28, 508 (1959).

Torrance, K. E.

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

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

K. E. Torrance, Ph.D. dissertation, University of Minnesota (March1966).

von Fragstein, C.

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

Appl. Opt. (1)

J. Applied Physics (Japan) (1)

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

J. Heat Trans. (1)

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

J. Opt. Soc. Am. (5)

J. Res. Natl. Bur. Std. (U. S.) (1)

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

Opt. Spectry. (2)

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

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

Optik (1)

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

Z. Physik (2)

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

H. Schulz, Z. Physik 31, 496 (1925).
[Crossref]

Other (10)

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.

K. E. Torrance, Ph.D. dissertation, University of Minnesota (March1966).

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

A. G. Mungall (private communication, 23September1965).

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

The reflection triplet (ψ;θ,ϕ) 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.

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

American Institute of Physics Handbook (McGraw-Hill Book Co., New York1963), Second ed., pp. 6–12 and 6–107.

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Figures (11)

F. 1
F. 1

Spatial angles of incident and reflected flux.

F. 2
F. 2

Bidirectional reflectance distributions in the plane of incidence for various angles of incidence ψ, λ = 0.5 μ. (a) Aluminum (2024-T4), aluminum coated, σm =1.3 μ. (b) Magnesium oxide ceramic, σm = 1.9 μ.

F. 3
F. 3

(a) Reflection at a mirror-like facet. (b) Spherical triangle of reflection.

F. 4
F. 4

(a) Geometry of the V-groove cavity. (b) Simultaneous masking–shadowing.

F. 5
F. 5

(a) and (b) Illustration of the reflection angles θp* and θp**, respectively, (c) Reflection triangle for V-groove cavity.

F. 6
F. 6

Fresnel reflectance.

F. 7
F. 7

The factor G(ψ,θ)/cosθ in the plane of incidence for various incidence angles ψ.

F. 8
F. 8

Bidirectional reflectance distributions in the plane of incidence for various incidence angles ψ as calculated from Eq. (23). F ( ψ , n ̂ ) = 1, c = 0.05, and g = 2 3.

F. 9
F. 9

Predicted bidirectional reflectance distributions corresponding to the experimental distributions in Fig. 2. Calculated from Eq. (23) with c = 0.05 and F ( ψ , n ̂ ) evaluated at λ = 0.5 μ. (a) Aluminum g = 2 3. (b) Magnesium oxide, g = 2.

F. 10
F. 10

The factor G(ψpp)/cosθ for various azimuthal angles ϕ. (a)ψ = 30°. (b) ψ = 60°. (c) ψ = 75°.

F. 11
F. 11

Bidirectional reflectance distributions for various azimuthal angles ϕ as predicted by Eq. (23). F ( ψ , n ̂ ) = 1, c = 0.05, and g = 2 3. (a) ψ = 30°. (b) ψ = 60°. (c) ψ = 75°.

Tables (1)

Tables Icon

Table I Calculation formulas for the geometrical attenuation factor G(ψpp).

Equations (36)

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d Φ i ( ψ ) = N i ( ψ ) cos ψ dAd ω i ,
d Φ r ( ψ ; θ , ϕ ) = d N r ( ψ ; θ , ϕ ) cos θ dAd ω r .
ρ ( ψ ; θ , ϕ ) d N r ( ψ ; θ , ϕ ) d Φ i ( ψ ) / d A = d N r ( ψ ; θ , ϕ ) N i ( ψ ) cos ψ d ω i .
d N r ( ψ ; θ , ϕ ) = d N r , s ( ψ ; θ , ϕ ) + d N r , d ( ψ ) .
d N r , d ( ψ ) = a N i cos ψ ,
P = P ( α ) = b exp ( c 2 α 2 ) ,
P ( α ) d ω d A .
f P ( α ) d ω d A
f cos ψ P ( α ) d ω d A .
d Φ i = f N i cos ψ P ( α ) d ω dAd ω i .
F ( ψ , n ̂ ) d Φ i .
d Φ r = G ( ψ p , θ p ) F ( ψ , n ̂ ) d Φ i = f N i G ( ψ p , θ p ) F ( ψ , n ̂ ) cos ψ P ( α ) d ω dAd ω i .
d Φ r = d N r , s ( ψ ; θ , ϕ ) cos θ dAd ω r .
d ω = d ω r / 4 cos ψ ,
d N r , s ( ψ ; θ , ϕ ) = ( f N i d ω i / 4 ) F ( ψ , n ̂ ) [ G ( ψ p , θ p ) / cos θ ] P ( α ) .
d N r ( ψ ; θ , ϕ ) = ( b f N i d ω i / 4 ) F ( ψ , n ̂ ) [ G ( ψ p , θ p ) / cos θ ] × exp ( c 2 α 2 ) + a N i cos ψ .
ρ ( ψ ; θ , ϕ ) ρ ( ψ ; ψ , 0 ° ) = d N r ( ψ ; θ , ϕ ) d N r ( ψ ; ψ , 0 ° ) = g F ( ψ , n ̂ ) [ G ( ψ p , θ p ) / cos θ ] exp ( c 2 α 2 ) + cos ψ g F ( ψ , n ̂ ) [ G ( ψ , ψ ) / cos ψ ] + cos ψ ,
ψ = 1 2 cos 1 [ cos θ cos ψ sin θ sin ψ cos ϕ ] , α = cos 1 [ cos ψ cos ψ + sin ψ sin ψ cos β 1 ] , ψ p = tan 1 [ cos β 2 tan ψ ] , θ p = ψ p + 2 α ,
β 1 = sin 1 [ sin ϕ sin θ / sin 2 ψ ] , β 2 = π sin 1 [ sin β 1 sin ψ / sin α ] .
ψ = ( ψ + θ ) / 2 , α = ( θ ψ ) / 2 , ψ p = ψ , θ p = θ .
G ( ψ p , θ p ) = 1 ( m / l ) .
θ p * = ( ψ p + π ) / 3.
θ p * * = ( ψ p π ) / 3 .
θ p * * = 3 ψ p π .
n 2 = m 2 + l 2 2 m l cos [ π + ψ p θ p ] .
n cos [ ( π / 2 ) θ p ] = ( l + m ) cos [ ( θ p ψ p ) / 2 ] .
G ( ψ p , θ p ) = 1 ( m / l ) = 1 [ 1 ( 1 A 2 ) 1 2 ] / A ,
A = sin 2 θ p cos 2 [ ( θ p ψ p ) / 2 ] cos 2 [ ( θ p ψ p ) / 2 ] cos [ θ p ψ p ] sin 2 θ p .
lim θ π / 2 [ G ( ψ p , θ p ) / cos θ ] = 2 cot α / cos ( β 2 ϕ ) ,
ρ ( ψ ; θ , ϕ ) ρ ( ψ ; ψ , 0 ° ) = g F ( ψ , n ̂ ) [ G ( ψ p , θ p ) / cos θ ] exp ( c 2 α 2 ) + cos ψ g [ F ( ψ , n ̂ ) / cos ψ ] + cos ψ .
F ( ψ , n ̂ ) [ G ( ψ p , θ p ) / cos θ ] exp ( c 2 α 2 ) .
π / 2 θ p ψ p π 3
ψ p π 3 θ p ψ p + π 3
3 ψ p π θ p ψ p + π 3
ψ p + π 3 θ p π / 2
ψ p + π 3 θ p π / 2