Abstract

An expression is derived for the mutual coherence function of laser radiation scattered from a rough surface. The rough surface is described by the superposition of two random-height distributions. The expression descibes the correlation of the radiation amplitude at spatially separated far-field points caused by the scattering from spatially separate surface elements. Numerical predictions of this mutual interaction or coherent contribution to the mutual coherence function are computed for a square surface area and typical values of the surface statistical parameters. These results are compared with the scattering cross section of a flat surface area and the degree of coherence of a perfectly diffuse surface. The functional form of the incoherent contribution to the mutual coherence function, resulting from the integration of the radiation that is diffracted by the surface irregularities, is deduced from an analysis of the single scale of roughness scattering problem. The derived mutual coherence function is shown to reduce to the proper functional form for the ideal limiting cases of a perfectly flat surface and a perfectly diffuse scattering area.

© 1976 Optical Society of America

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References

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  1. J. W. Goodman, Proc. IEEE 53, 1688 (1965).
    [CrossRef]
  2. H. T. Yura, Appl. Opt. 13, 150 (1974).
    [CrossRef] [PubMed]
  3. R. B. Crane, J. Opt. Soc. Am. 60, 1658 (1970).
    [CrossRef]
  4. H. H. Arsenault, J. Opt. Soc. Am. 61, 1425 (1971).
    [CrossRef]
  5. B. I. Semyonov, Radio Eng. Electron. Phys. (USSR) 11, 1179 (1966).
  6. P. Beckmann, Proc. IEEE 53, 1012 (1965).
    [CrossRef]
  7. A. K. Fung and H-L Chan, IEEE Trans. Antennas Propag. AP-17, 590 (1969).
    [CrossRef]
  8. J. W. Wright, IEEE Trans. Antennas Propag. AP-16, 217 (1968).
    [CrossRef]
  9. D. E. Barrick and W. H. Peake, Rad. Sci. 3, 865 (1968).
  10. Portions of this section’s development were presented by the author at the 1973 International IEEE Antennas and Propagation Symposium and USNC/URSI Meeting, Boulder, Colo.22–24 Aug. 1973. These portions are outlined in McDonnell Aircraft Co. Paper No. MCAIR 73-020.
  11. J. C. Leader, J. Appl. Phys. 42, 4808 (1971).
    [CrossRef]
  12. J. C. Leader, J. Opt. Soc. Am. 62, 1356 (1972).
  13. M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice Hall, Englewood Cliffs, N.J., 1964) p. 40.
  14. J. C. Leader, “Polarization Discrimination in Remote Sensing,” AGARD Electromagnetic Propagation Panel Symposium of Electromagnetic Wave Propagation Involving Irregular Surfaces and Inhomogeneous Media, The Hague, Netherlands (March 1974), AGARD Conference Proceedings No. 144.
  15. L. I. Goldfischer, J. Opt. Soc. Am. 55, 247 (1965).
    [CrossRef]
  16. Goldfischer calculated the irradiance correlation function R12 = 〈I1I2〉, which is related to the degree of coherence by R12 = 〈I1〉 〈I2〉 [1 + 2 |γ12|2] for Gaussian fluctuation statistics.
  17. J. E. Seltzer, IEEE Trans. Antennas Propag. AP-20, 726 (1972).
  18. D. E. Barrick, IEEE Trans. Antennas Propag. AP-16, 449 (1968).
    [CrossRef]

1974 (1)

1972 (2)

J. C. Leader, J. Opt. Soc. Am. 62, 1356 (1972).

J. E. Seltzer, IEEE Trans. Antennas Propag. AP-20, 726 (1972).

1971 (2)

J. C. Leader, J. Appl. Phys. 42, 4808 (1971).
[CrossRef]

H. H. Arsenault, J. Opt. Soc. Am. 61, 1425 (1971).
[CrossRef]

1970 (1)

1969 (1)

A. K. Fung and H-L Chan, IEEE Trans. Antennas Propag. AP-17, 590 (1969).
[CrossRef]

1968 (3)

J. W. Wright, IEEE Trans. Antennas Propag. AP-16, 217 (1968).
[CrossRef]

D. E. Barrick and W. H. Peake, Rad. Sci. 3, 865 (1968).

D. E. Barrick, IEEE Trans. Antennas Propag. AP-16, 449 (1968).
[CrossRef]

1966 (1)

B. I. Semyonov, Radio Eng. Electron. Phys. (USSR) 11, 1179 (1966).

1965 (3)

P. Beckmann, Proc. IEEE 53, 1012 (1965).
[CrossRef]

J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[CrossRef]

L. I. Goldfischer, J. Opt. Soc. Am. 55, 247 (1965).
[CrossRef]

Arsenault, H. H.

Barrick, D. E.

D. E. Barrick and W. H. Peake, Rad. Sci. 3, 865 (1968).

D. E. Barrick, IEEE Trans. Antennas Propag. AP-16, 449 (1968).
[CrossRef]

Beckmann, P.

P. Beckmann, Proc. IEEE 53, 1012 (1965).
[CrossRef]

Beran, M. J.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice Hall, Englewood Cliffs, N.J., 1964) p. 40.

Chan, H-L

A. K. Fung and H-L Chan, IEEE Trans. Antennas Propag. AP-17, 590 (1969).
[CrossRef]

Crane, R. B.

Fung, A. K.

A. K. Fung and H-L Chan, IEEE Trans. Antennas Propag. AP-17, 590 (1969).
[CrossRef]

Goldfischer, L. I.

Goodman, J. W.

J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[CrossRef]

Leader, J. C.

J. C. Leader, J. Opt. Soc. Am. 62, 1356 (1972).

J. C. Leader, J. Appl. Phys. 42, 4808 (1971).
[CrossRef]

J. C. Leader, “Polarization Discrimination in Remote Sensing,” AGARD Electromagnetic Propagation Panel Symposium of Electromagnetic Wave Propagation Involving Irregular Surfaces and Inhomogeneous Media, The Hague, Netherlands (March 1974), AGARD Conference Proceedings No. 144.

Parrent, G. B.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice Hall, Englewood Cliffs, N.J., 1964) p. 40.

Peake, W. H.

D. E. Barrick and W. H. Peake, Rad. Sci. 3, 865 (1968).

Seltzer, J. E.

J. E. Seltzer, IEEE Trans. Antennas Propag. AP-20, 726 (1972).

Semyonov, B. I.

B. I. Semyonov, Radio Eng. Electron. Phys. (USSR) 11, 1179 (1966).

Wright, J. W.

J. W. Wright, IEEE Trans. Antennas Propag. AP-16, 217 (1968).
[CrossRef]

Yura, H. T.

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (4)

A. K. Fung and H-L Chan, IEEE Trans. Antennas Propag. AP-17, 590 (1969).
[CrossRef]

J. W. Wright, IEEE Trans. Antennas Propag. AP-16, 217 (1968).
[CrossRef]

J. E. Seltzer, IEEE Trans. Antennas Propag. AP-20, 726 (1972).

D. E. Barrick, IEEE Trans. Antennas Propag. AP-16, 449 (1968).
[CrossRef]

J. Appl. Phys. (1)

J. C. Leader, J. Appl. Phys. 42, 4808 (1971).
[CrossRef]

J. Opt. Soc. Am. (4)

Proc. IEEE (2)

P. Beckmann, Proc. IEEE 53, 1012 (1965).
[CrossRef]

J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[CrossRef]

Rad. Sci. (1)

D. E. Barrick and W. H. Peake, Rad. Sci. 3, 865 (1968).

Radio Eng. Electron. Phys. (USSR) (1)

B. I. Semyonov, Radio Eng. Electron. Phys. (USSR) 11, 1179 (1966).

Other (4)

Goldfischer calculated the irradiance correlation function R12 = 〈I1I2〉, which is related to the degree of coherence by R12 = 〈I1〉 〈I2〉 [1 + 2 |γ12|2] for Gaussian fluctuation statistics.

Portions of this section’s development were presented by the author at the 1973 International IEEE Antennas and Propagation Symposium and USNC/URSI Meeting, Boulder, Colo.22–24 Aug. 1973. These portions are outlined in McDonnell Aircraft Co. Paper No. MCAIR 73-020.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice Hall, Englewood Cliffs, N.J., 1964) p. 40.

J. C. Leader, “Polarization Discrimination in Remote Sensing,” AGARD Electromagnetic Propagation Panel Symposium of Electromagnetic Wave Propagation Involving Irregular Surfaces and Inhomogeneous Media, The Hague, Netherlands (March 1974), AGARD Conference Proceedings No. 144.

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Figures (8)

FIG. 1
FIG. 1

Coordinate system used to describe the scattered field interaction produced by scattering from spatially-separate, rough-surface elements.

FIG. 2
FIG. 2

Coordinate system relating the relative stationary-phase point displacements (ũ, v ˜ , h ˜) to displacements (X, Y, Z) about the mean plane of the undulating surface.

FIG. 3
FIG. 3

Calculated backscatter cross section of a perfectly smooth (a) and a rough (b) square surface area. Data are normalized to unity at normal incidence. A normalized length of kL = 2π × 105 is assumed for both rough and smooth surfaces. An rms slope s = 0.25 is assumed for the rough surface.

FIG. 4
FIG. 4

Calculated angular distribution of Cosc[C, L, a] for 0 < Θ < 0.1 mrad as a function of the normalized height deviation, kH. An rms slope of s = 0.25 and a normalized length, kL = 2π × 105 are assumed.

FIG. 5
FIG. 5

Calculated angular distribution of Cosc[C, L, a] for 0° < Θ< 10° as a function of the normalized height deviation, kH. An rms slope of s = 0.25 and a normalized length, kL = 2π × 105 are assumed.

FIG. 6
FIG. 6

Calculated real part of the mutual-interaction degree of coherence of a rough square surface area with rms slopes of (a) 0.25, (b) 0.25 × 10−3, (c) 1.25 × 10−4, and (d) 6.25 × 10−5. The corresponding degree of coherence of a perfectly diffuse surface area is illustrated in Figs. (a)–(d) for comparison. A normalized length of kL = 2π × 105 is assumed.

FIG. 7
FIG. 7

Calculated real part of the mutual-interaction degree of coherence of a rough square surface area as a function of the backscatter angle, Θ. An rms slope of s = 0.25 and a normalized length, kL = 2π × 105 are assumed.

FIG. 8
FIG. 8

Calculated real part of the mutual-interaction degree of coherence of a rough square surface area as a function of the normalized length, kL. An rms slope of s = 0.25 and a normalized height deviation, kH = 1.0 are assumed for the calculations.

Equations (77)

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E j ( P ) = K [ α j - ( n y / n z ) α y j ] S E 0 exp [ i k ( n y y - n z z ) ] d x d y .
sin θ 0 R 1 , 2 - + sin θ s R 1 , 2 + - ( cos θ s + cos θ 0 ) × ( R 1 , 2 / Z y ) z x = 0 , z y = 0 , n y = sin θ s - sin θ 0 ,             v y = k n y , n z = cos θ s + cos θ 0 ,             v z = k n z ,
j k ( P , P , τ ) = E j ( P , t ) E k * ( P , t + τ ) = 1 ( 4 π ) 2 R R 0 k 2 C j C k * S 1 S 2 exp [ i k ( n y y 1 - n y y 2 - n z z 1 + n z z 2 ) ] × d x 1 d y 1 d x 2 d y 2 Î ( ν ) exp { - 2 π i ν [ τ - ( R - R ) / c ] } d ν ,
Γ ( P , P , τ ) = 0 S Γ ˆ ( S 1 , S 2 , ν ) G 1 ( P , P 1 , ν ) n s 1 | P 1 = S 1 × G 2 ( P , P 2 , ν ) n s 2 | P 2 = S 2 d S 1 d S 2 e - 2 π i ν τ d ν .
[ i 2 + k 2 ( ν ) ] G ( P , P i , ν ) = - δ ( P - P i )
G ( P , P i , ν ) P i = S i = 0.
θ s ( P ) = θ s ( P ) + δ ,             where δ 1.
u = x 1 - x 2 , v = y 1 - y 2 ,
n y - n y cos θ s sin δ , n z - n z - sin θ s sin δ ,
j j ( δ , τ ) = 1 ( 4 π ) 2 R R 0 · k 2 C j 2 S S exp ( i k n y v - i k cos θ s sin δ y 2 ) exp [ i k n z ( z 1 - z 2 ) ] × exp ( i k sin θ s sin δ z 2 ) d x 2 d y 2 d u d v Î ( ν ) exp { - 2 π i ν [ τ - ( R - R ) / c ] } d ν .
exp [ i k n z ( z 1 - z 2 ) ] exp [ i k sin θ s sin δ z 2 ] = exp { - v z 2 σ 2 [ 1 - sin θ s sin δ / n z ] [ 1 - C ( ρ ) ] } exp ( - k 2 sin 2 θ s sin 2 δ σ 2 / 2 ) = exp { - v z 2 σ 2 [ 1 - C ( ρ ) ] } exp ( - k 2 sin 2 θ s sin 2 δ σ 2 / 2 ) ,
u = ρ cos ξ , v = ρ sin ξ , x 2 = ρ 2 cos ζ , y 2 = ρ 2 sin ζ ,
j j ( δ , τ ) = 1 ( 4 π ) 2 R R 0 k 2 C j 2 0 2 π 0 0 2 π 0 exp ( i k n y ρ sin ξ ) exp ( - i k cos θ s sin δ ρ 2 sin ζ ) exp { - v z 2 σ 2 [ 1 - C ( ρ ) ] } × exp ( - k 2 sin 2 θ s sin 2 δ σ 2 / 2 ) A ( ρ ) A ( ρ 2 ) ρ d ρ d ξ ρ 2 d ρ 2 d ζ × Î ( ν ) exp { - 2 π i ν [ τ - ( R - R ) / c ] } d ν ,
k 2 ( 4 π R ) 2 C j 2 0 2 π 0 exp ( i k n y ρ sin ξ ) exp { - v z 2 σ 2 [ 1 - C ( ρ ) ] } A ( ρ ) ρ d ρ d ξ = 2 π k 2 ( 4 π R ) 2 C j 2 0 J 0 ( v y ρ ) exp { - v z 2 σ 2 [ 1 - C ( ρ ) ] } A ( ρ ) ρ d ρ = σ j j 0 ( θ 0 , θ s ; ν ) 1 4 π R 2 ,
j j ( δ , τ ) = 1 2 R R 0 σ j j 0 ( θ 0 , θ s , ν ) Î ( ν ) exp { - 2 π i ν [ τ - ( R - R ) / c ] } exp ( - k 2 sin 2 θ s sin 2 δ σ 2 / 2 ) × 0 A ( ρ 2 ) J 0 ( k ρ 2 cos θ s sin δ ) ρ 2 d ρ 2 d ν .
j j ( δ , τ ) = exp ( - k 2 σ 2 sin 2 θ s sin 2 δ / 2 ) exp [ 2 π i ν ¯ ( R - R ) / c ] 1 2 R R σ j j 0 ( θ 0 , θ s , ν ¯ ) 0 exp ( - 2 π i ν τ ) Î ( ν ) d ν × 0 A ( ρ 2 ) J 0 ( k ρ 2 cos θ s sin δ ) ρ 2 d ρ 2 ,
exp ( - i k n z z ) = exp [ - ( k n z σ ) 2 / 2 ] ;
E j ( P ) 2 = | K E 0 C j exp [ - ( k n z σ ) 2 / 2 ] × S exp ( i k n y y ) d x d y | 2 .
σ j j ( θ 0 , θ s ) = k 2 4 π C j ( θ 0 , θ s ) 2 A exp [ - ( k n z σ ) 2 ] × ( 2 sin ( k n y L / 2 ) k n y L ) 2 ,
j j ( P , P ) = E j ( P ) E j * ( P ) = E j ( P , S 1 ) E j * ( P , S 1 ) + E j ( P , S 2 ) E j * ( P , S 2 ) + E j ( P , S 1 ) E j * ( P , S 2 ) + E j ( P , S 2 ) E j * ( P , S 1 ) .
y 1 = y 2 cos ξ + z 2 sin ξ + v , z 1 = z 2 cos ξ + y 2 sin ξ + h ( v ) ,
j j , s 1 x s 2 ( P , P ) = K E 0 2 C j 1 ( P ) S 1 S 2 C j 2 * ( P ) × exp [ i k ( n y 1 y 1 - n y 2 y 2 - n z 1 z 1 + n z 2 z 2 ) ] d x 1 d y 1 d x 2 d y 2 ,
n y 1 = sin θ s - sin θ 0 , n y 2 = sin ( θ s + ξ + δ ) - sin ( θ 0 - ξ ) , n z 1 = cos θ s + cos θ 0 , n z 2 = cos ( θ s + ξ + δ ) + cos ( θ 0 - ξ ) .
j j , S 1 x S 2 ( P , P ) = K E 0 2 C j 1 ( P ) × S 1 S 2 C j 2 * ( P ) exp [ i k L ( x 2 , y 2 , u , v ) ] × d u d v d x 2 d y 2 ,
L = n y 1 v + n z 1 h - y 2 sin δ cos ( θ s + ξ - 2 y 2 n z 1 sin ξ ) - z 2 sin δ sin ( θ s + ξ ) + ( sin ξ n y 1 - cos ξ n z 1 ) ( z 2 e - z 2 )
exp i k [ B ( z 2 e - z 2 ) - z 2 sin δ sin ( θ s + ξ ) ] = exp { - ( k σ ) 2 [ ( B + sin δ sin ( θ + ξ ) ) 2 / 2 + B 2 / 2 ] } ,
B = sin ξ n y 1 - cos ξ n z 1 = - [ cos ( θ s + ξ ) + cos ( θ 0 - ξ ) ]
z z e z 2 = 0 ,
p = L y 2 = - sin δ cos ( θ s + ξ ) - 2 n z 1 sin ξ = 0 ,
q = L v = n y 1 + n z 1 h v - y 2 [ sin δ ( - cos θ s sin ξ - sin θ s cos ξ ) ξ v + 2 n z 1 cos ξ ξ v ] = 0.
y 2 = 0 ,             h v = - n y 1 n z 1
h v = n · ĵ n · k ˆ ,
θ 0 = θ ˜ + ξ and θ s = θ ˜ - ξ .
sin δ cos ( θ s + ξ ) = sin δ cos θ ˜ = - 2 sin ξ ( cos θ 0 + cos θ s ) = - 4 cos θ ˜ sin ξ cos ξ
sin δ = - 2 sin 2 ξ .
ξ - δ / 4 ,
j j , s 1 x s 2 ( P , P ) = i ( K E 0 2 C j 1 [ θ ˜ + ( δ / 4 ) , θ ˜ - ( δ / 4 ) ] C j 2 ( θ ˜ , θ ˜ + δ ) 2 π i ( M ) 2 exp ( i k n y z 1 · r ) exp [ - ( 2 k σ cos θ ˜ ) 2 ] k { - [ sin 2 δ sin 2 θ ˜ - 4 n z 1 sin δ sin θ ˜ cos ( δ / 4 ) + 4 n z 1 2 cos 2 ( δ / 4 ) ] ( 2 h / v 2 ) 2 } 1 / 2 ) i ,
n y z 1 = n y 1 ĵ + n z 1 k ˆ ,             r = v ĵ + h k ˆ .
n y 1 = sin θ 0 - sin θ s = 2 cos θ ˜ sin ( δ / 4 )
n z 1 = 2 cos θ ˜ cos ( δ / 4 )
n y z 1 · r = 2 cos θ ˜ [ sin ( δ / 4 ) v + cos ( δ / 4 ) h ] .
( 2 h v 2 ) - 1 = ρ v [ 1 + ( h v ) 2 ] - 3 / 2 ,
1 + ( h v ) 2 = 1 + ( n y 1 n z 1 ) 2 = 1 / cos 2 ( δ / 4 ) 1.
C j 1 [ θ ˜ + ( δ / 4 ) , θ ˜ - ( δ / 4 ) ] C j 2 ( θ ˜ , θ ˜ + δ ) C j ( θ ˜ , θ ˜ ) ,
j j , T x ( P , P ) = K E 0 2 C j ( θ ˜ , θ ˜ ) 2 ( M ) 2 ( 2 k cos θ ˜ ) - 1 exp [ - ( 2 k σ cos θ ˜ ) 2 ] × i 1 2 ( ρ v 2 exp { i k 2 cos θ ˜ [ sin ( δ / 4 ) v + cos ( δ / 4 ) h ] } + ρ v 1 exp { - i k 2 cos θ ˜ [ - sin ( δ / 4 ) v + cos ( δ / 4 ) h ] } ) i ,
cos θ s 2 = cos ( θ s + ξ + δ ) cos η
cos θ 02 = cos ( θ 0 - ξ ) cos η .
x 1 = cos η x 2 + sin ξ sin η y 2 - cos ξ sin η z 2 + u , y 1 = cos ξ y 2 + sin ξ z 2 + v , z 1 = cos ξ cos η z 2 - sin ξ cos η y 2 + sin η x 2 + h ( u , v ) ,
L ( x 2 , y 2 , z 2 , u , v , h ) = n y 1 v + n z 1 h - y 2 sin δ cos ( θ s + ξ ) - y 2 sin ξ n z 1 - y 2 sin ξ n z 1 cos η - sin η x 2 n z 1
- 2 cos θ ˜ sin η cos δ 4 = 0 sin η = 0
2 cos θ ˜ cos δ 4 h u = 0 h u = 0
j j , T x ( P , P ) E 0 2 R j ( θ ˜ ) 2 8 R 2 exp [ - ( 2 k σ cos θ ˜ ) 2 ] × i [ ρ v 2 ρ u 2 exp ( i k 2 cos θ ˜ ) ( sin δ 4 v + cos δ 4 h ) + ρ v 1 ρ u 1 exp ( - i k 2 cos θ ˜ ) ( - sin δ 4 v + cos δ 4 h ) ] i ,
j j , T x ( P , P ) E 0 2 R j ( θ ˜ ) 2 4 R 2 N ( N - 1 ) 2 ρ v ( θ ˜ ) ρ u ( 0 ) × exp ( i k 2 cos θ ˜ ) sin δ 4 v ˜ cos [ 2 k cos θ ˜ cos δ 4 h ˜ ] × exp [ - ( 2 k σ cos θ ˜ ) 2 ] ,
v ˜ = Y ˜ cos Θ + Z ˜ sin Θ
h ˜ = Z ˜ cos Θ - Y ˜ sin Θ ,
cos [ 2 k cos θ ˜ cos δ 4 h ˜ ] = cos ( 2 k cos θ ˜ cos δ 4 Z ˜ cos Θ ) cos ( 2 k cos θ ˜ cos δ Y Y ˜ sin Θ ) + sin ( 2 k cos θ ˜ cos δ 4 Z ˜ cos Θ ) × sin ( 2 k cos θ ˜ cos δ 4 Y ˜ sin Θ )
exp ( i k 2 cos θ ˜ sin δ 4 v ˜ ) = exp ( 2 i k cos θ ˜ sin δ 4 Y ˜ cos Θ ) × exp ( 2 i k cos θ ˜ sin δ 4 Z ˜ sin Θ )
cos C Y ˜ = n = 1 N ( N + 1 - n ) cos C n a / n = 1 N ( N + 1 - n ) ,
cos C Y ˜ = y n = a L ( L + a - y n ) cos ( C y n ) Δ y / y n = a L y n Δ y .
cos C Y ˜ = Cosc [ C , L , a ] = 2 ( C a sin C L - C L sin C a + cos C a - cos C L ) × [ ( C L ) 2 - ( C a ) 2 ] - 1 ,
exp ( i F Y ˜ ) = Eosc [ F , L , a ] = 2 [ e i F a - e i F L - i ( F a e i F L - F L e i F a ) ] × [ ( F L ) 2 - ( F a ) 2 ] - 1 .
cos C Z ˜ = 1 l - cos C Z exp ( - Z 2 / h ¯ 2 ) d Z = exp [ - ( C 2 h ¯ 2 / 4 ) ] l ( π h ¯ 2 ) 1 / 2 , sin C Z ˜ = 1 l - sin C Z exp ( - Z 2 / h ¯ 2 ) d Z = 0 ,
exp ( i F Z ˜ ) = 1 l - exp ( i F Z ) exp ( - Z 2 / h ¯ 2 ) d Z = exp [ - ( F 2 h ¯ 2 / 4 ) ] l ( π h ¯ 2 ) 1 / 2 ,
N d = ( 7.255 / π 2 Λ 2 ) exp ( - tan 2 Θ / s 2 ) = specular - point density
ρ v ρ u = 0.827 ( π / 2 ) ( sec 4 Θ / γ 2 ) ,
s 2 = 4 H 2 / Λ 2 = surface - slope variance , γ 2 = 12 ( H 2 / Λ 4 ) ,
N ( N - 1 ) N ( θ ) N ( θ + δ / 4 ) .
N ( θ ) N ( θ + δ / 4 ) N ( θ ) N ( θ + δ / 4 )
tan ( θ + δ ) = tan θ + tan δ 1 - tan θ tan δ tan θ + tan δ ,
4 π R 2 A E 0 2 j j = R j ( θ ˜ ) 2 7.255 16 A π 2 H 2 sec 4 Θ Eosc [ F , L , a ] Cosc [ C , L , a ] exp [ - ( 2 k σ cos θ ˜ ) 2 ] × exp - { δ 2 / 16 [ ( 1.24 k H cos θ ˜ sin Θ ) 2 + s - 2 ] } exp ( - δ tan Θ / 4 s 2 ) exp { - [ 2 tan 2 Θ / s 2 + ( 1.24 k H cos θ ˜ cos Θ ) 2 ] } ,
Cosc [ C , L , 0 ] = 2 ( 1 - cos C L ) ( C L ) 2 = 4 sin 2 ( C L / 2 ) ( C L ) 2 = [ sin ( k L sin θ ) k L sin θ ] 2 ,
Eosc [ F , x , a ] = 2 e i F a { 1 - e i F x - i [ F a e i F x - F ( x + a ) ] } F 2 ( x 2 + 2 a x ) ,
lim x 0 Eosc [ F , x , a ] = 1 ,
I = c d a b F ( x , y ) e i k g ( x , y ) d x d y ,
p 0 = q 0 = 0             and             r 0 t 0 - s 0 2 0 ,
R = ( x , y ) x 0 - δ x x 0 + δ , y 0 - y y 0 + ) ,
I = ± 2 π i exp [ i k g ( x 0 , y 0 ) ] F ( x 0 , y 0 ) k [ r 0 t 0 - s 0 2 ] 1 / 2 + 0 ( 1 k 3 / 2 ) as k .