Abstract

A new ray model is presented for the reflection of electromagnetic radiation from the rough air-material interface of a randomly rough surface. Unlike previous derivations that modeled the rough interface as consisting of microareas randomly oriented but flat (facets), this derivation models it as consisting of microareas not only randomly oriented but also randomly curved. Physically, the models are the same, but this new derivation leads to some new results. (1) For any given rough surface, there exists a single, optically smooth, curved surface of revolution of very restricted shape that will reflect radiation in the same distribution as that reflected by the rough interface. (2) Modeling that surface as an ellipsoid of revolution gives a surface-structure function that appears more accurate and useful than existing ones. (3) Unlike the facet derivations, this derivation lends itself to a normalization that gives the absolute, instead of just a comparative, reflectance-distribution function.

© 1975 Optical Society of America

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References

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  1. P. Bouguer, Histoire de l’Academie Royale des sciences, Paris, 1757, (1762); and Traite d’optique sur la gradation de la muniere (Ouυrage posthume de M. Bouguer), (l’Abbe de Lacaille, Paris, 1760).
  2. G. I. Pokrowski, Z. Physik 30, 66 (1924).
    [Crossref]
  3. W. E. K. Middleton and A. G. Mungall, J. Opt. Soc. Am. 42, 572 (1952).
    [Crossref]
  4. H. Schulz, Z. Physik 31, 496 (1925).
    [Crossref]
  5. J. C. Richmond, J. Opt. Soc. Am. 56, 253 (1966).
    [Crossref]
  6. N. A. Umov, Selected Works (State Publishers of Technical–Theoretical Literature, Moscow, 1950).
  7. E. M. Berry, J. Opt. Soc. Am. 7, 627 (1923).
    [Crossref]
  8. G. I. Pokrowski, Z. Physik 36, 472 (1926).
    [Crossref]
  9. W. W. Barkas, Proc. Phys. Soc. (Lond.) 51, 274 (1939).
    [Crossref]
  10. A. W. Christie, J. Opt. Soc. Am. 43, 621 (1953).
    [Crossref]
  11. K. E. Torrance and E. M. Sparrow, J. Opt. Soc. Am. 57, 1105 (1967).
    [Crossref]
  12. W. A. Rense, J. Opt. Soc. Am. 40, 55 (1950).
    [Crossref]
  13. R. S. Sirohi, J. Phys. D 3, 1407 (1970).
    [Crossref]
  14. F. E. Nicodemus, private communication.
  15. F. E. Nicodemus, Appl. Opt. 9, 1474 (1970).
    [Crossref] [PubMed]
  16. Max Planck, The Theory of Heat Radiation (Dover, New York, 1959).
  17. R. C. Jones, in Appendix to Spiro, Jones, and Wark, Infrared Phys. 5, 11 (1965).
    [Crossref]
  18. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, New York, 1963), p. 88.
  19. D. B. Judd, J. Opt. Soc. Am. 57, 445 (1967).
    [Crossref] [PubMed]

1970 (2)

1967 (2)

1966 (1)

1965 (1)

R. C. Jones, in Appendix to Spiro, Jones, and Wark, Infrared Phys. 5, 11 (1965).
[Crossref]

R. C. Jones, in Appendix to Spiro, Jones, and Wark, Infrared Phys. 5, 11 (1965).
[Crossref]

1953 (1)

1952 (1)

1950 (1)

1939 (1)

W. W. Barkas, Proc. Phys. Soc. (Lond.) 51, 274 (1939).
[Crossref]

1926 (1)

G. I. Pokrowski, Z. Physik 36, 472 (1926).
[Crossref]

1925 (1)

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

1924 (1)

G. I. Pokrowski, Z. Physik 30, 66 (1924).
[Crossref]

1923 (1)

Barkas, W. W.

W. W. Barkas, Proc. Phys. Soc. (Lond.) 51, 274 (1939).
[Crossref]

Beckmann, P.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, New York, 1963), p. 88.

Berry, E. M.

Bouguer, P.

P. Bouguer, Histoire de l’Academie Royale des sciences, Paris, 1757, (1762); and Traite d’optique sur la gradation de la muniere (Ouυrage posthume de M. Bouguer), (l’Abbe de Lacaille, Paris, 1760).

Christie, A. W.

Jones,

R. C. Jones, in Appendix to Spiro, Jones, and Wark, Infrared Phys. 5, 11 (1965).
[Crossref]

Jones, R. C.

R. C. Jones, in Appendix to Spiro, Jones, and Wark, Infrared Phys. 5, 11 (1965).
[Crossref]

Judd, D. B.

Middleton, W. E. K.

Mungall, A. G.

Nicodemus, F. E.

Planck, Max

Max Planck, The Theory of Heat Radiation (Dover, New York, 1959).

Pokrowski, G. I.

G. I. Pokrowski, Z. Physik 36, 472 (1926).
[Crossref]

G. I. Pokrowski, Z. Physik 30, 66 (1924).
[Crossref]

Rense, W. A.

Richmond, J. C.

Schulz, H.

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

Sirohi, R. S.

R. S. Sirohi, J. Phys. D 3, 1407 (1970).
[Crossref]

Sparrow, E. M.

Spiro,

R. C. Jones, in Appendix to Spiro, Jones, and Wark, Infrared Phys. 5, 11 (1965).
[Crossref]

Spizzichino, A.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, New York, 1963), p. 88.

Torrance, K. E.

Umov, N. A.

N. A. Umov, Selected Works (State Publishers of Technical–Theoretical Literature, Moscow, 1950).

Wark,

R. C. Jones, in Appendix to Spiro, Jones, and Wark, Infrared Phys. 5, 11 (1965).
[Crossref]

Appl. Opt. (1)

Infrared Phys. (1)

R. C. Jones, in Appendix to Spiro, Jones, and Wark, Infrared Phys. 5, 11 (1965).
[Crossref]

J. Opt. Soc. Am. (7)

J. Phys. D (1)

R. S. Sirohi, J. Phys. D 3, 1407 (1970).
[Crossref]

Proc. Phys. Soc. (Lond.) (1)

W. W. Barkas, Proc. Phys. Soc. (Lond.) 51, 274 (1939).
[Crossref]

Z. Physik (3)

G. I. Pokrowski, Z. Physik 36, 472 (1926).
[Crossref]

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

G. I. Pokrowski, Z. Physik 30, 66 (1924).
[Crossref]

Other (5)

P. Bouguer, Histoire de l’Academie Royale des sciences, Paris, 1757, (1762); and Traite d’optique sur la gradation de la muniere (Ouυrage posthume de M. Bouguer), (l’Abbe de Lacaille, Paris, 1760).

N. A. Umov, Selected Works (State Publishers of Technical–Theoretical Literature, Moscow, 1950).

F. E. Nicodemus, private communication.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, New York, 1963), p. 88.

Max Planck, The Theory of Heat Radiation (Dover, New York, 1959).

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Figures (4)

FIG. 1
FIG. 1

Reflection from a curved microarea of the rough-surface air–material interface. A small optically smooth, slightly curved area ΔAs with a normal n ˆ intercepts the light from an area ΔAI of the incident beam diverges it into Δω = ΔAR/R2 steradians in a direction r ˆ such that r ˆ × n ˆ = n ˆ × l ˆ.

FIG. 2
FIG. 2

The shape of a curved microarea in terms of two orthogonal curved lines.

FIG. 3
FIG. 3

Cross section of an average surface irregularity. The shape of h(x) may be restricted to the general shape illustrated and still be general enough to represent any microstructure. This general shape consists of the slope being 0 at x = 0 and infinity at h = 0 and the curve between having no inflection points or straight-line sections.

FIG. 4
FIG. 4

Comparison of best fits of some surface-structure functions (equivalent to the microarea or facet distribution function), ○ Data taken from Rense (Ref. 12). —— Plot of our function e4/(e2cos2α + sin2α)2, e = 0.40. ······· Plot of function e2/(e2cos2α + sin2α), e = 0.25, originated by Berry (Ref. 7). –··· Plot of function (cos−4α)exp(−A2tan2α), A2 = 7.62, derived from Beckmann’s (Ref. 18) result for a surface characterized by a gaussian distribution of surface heights and an autocorrelation length, in the ray approximation. ---- Plot of the function (cos−4α)exp(−A2tan2α), A2 = 6.93, used by Sirohi (Ref. 13), originated by Berry (Ref. 7). ——Plot of an approximation exp(−A2α2), A2 = 0.0021/deg2) of – – – used by Rense (Ref. 12) for small values of α.

Equations (20)

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I = R ( s , n , k ) E n cos s cos θ | Δ α Δ z Δ θ Δ ψ | σ α σ z .
I T = i = 1 N I i = i = 1 N R ( s , n , k ) E n i cos s cos θ | J ( α , z θ , ψ ) | σ α i σ z i .
I T = R ( s , n , k ) Φ I cos s cos θ | J ( α , z θ , ψ ) | 1 A I i = 1 N σ α i σ z i .
I T = R ( s , n , k ) Φ I cos s cos β cos θ | J ( α , z θ , ψ ) | 1 A I i = 1 N n σ α i σ z i ,
D ( α ) = 1 A I i = 1 N n σ α i σ z i ( dimensionless )
f r I ( β ; θ , ψ ) = R ( s , n , k ) cos s cos β cos θ | J ( α , z θ , ψ ) | D ( α ) .
D ( α ) = C ρ α ( α ) ρ z ( α )
D [ tan 1 ( h ) ] = C | x | [ 1 + ( h ) 2 ] 3 / 2 / | h | .
tan z = sin ψ cos θ / ( sin β + cos ψ cos θ ) ,
tan α = sin ψ cos θ / [ sin z ( sin θ + cos β ) ] ,
cos s = cos z sin α sin β + cos α cos β ,
J ( α , z θ , ψ ) = cos θ 4 cos s sin α .
f r I ( β ; θ , ψ ) = R ( s ) 1 4 cos β C ρ α ( α ) ρ z ( α ) sin α .
f r I ( 0 ; 90 ° , 0 ) = R ( 0 ) ρ α 2 ( 0 ) / [ 4 π ρ z 2 ( π / 2 ) ] .
f r I ( 0 ; 90 ° , 0 ) = 1 4 C R ( 0 ) ρ α ( 0 ) lim α 0 [ ρ z ( α ) / sin α ] .
f r I ( β ; θ , ψ ) = R ( s ) 1 4 π cos β ρ α ( 0 ) ρ α ( α ) ρ z ( α ) ρ z 2 ( π / 2 ) lim α 0 [ ρ z ( α ) / sin α ] sin α .
h = e ( 1 x 2 ) 1 / 2 ,
h = tan α , x = tan α ( tan 2 α + e 2 ) 1 / 2 = ρ z ( α ) , ρ α ( α ) = [ 1 + ( h ) 2 ] 3 / 2 / | h | ,
f r I ( β ; θ , ψ ) = R ( s , n , k ) 4 π cos β e 2 ( e 2 cos 2 α + sin 2 α ) 2 .
e 4 / ( e 2 cos 2 α + sin 2 α ) 2 .