Abstract

We have calculated reflected and refracted electromagnetic waves from a rough surface with the boundary conditions exactly satisfied. The surface model consists of three parameters: the dimension of the independent cells, h; the parameter in the slope distribution function of the tangent plane t0; and the average radius of curvature of the surface, R. Within the reasonable range of these parameters, we have calculated the polarized component P and the depolarized component D measured by Renau et al. and have found the following features of the scattering of the reflected wave from a rough surface that have not been previously explained theoretically. (i) P and D have very different angular dependences. (ii) Whereas P is a sensitive function of t0, D is almost independent of t0. (iii) D does not vanish at all angles. A qualitative comparison between the data and the calculation is given.

© 1978 Optical Society of America

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References

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  1. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975). We follow the scattering geometry in Jackson.
  2. J. Renau, R. K. Cheo, and H. G. Cooper, J. Opt. Soc. Am. 57, 459 (1967).
    [Crossref] [PubMed]
  3. G. J. Wilhelmi, J. W. Rouse, and A. J. Blanchard, J. Opt. Soc. Am. 65, 1036 (1975).
    [Crossref]
  4. P. Beckmann, The Depolarization of Electromagnetic Waves (Golem, Boulder, Colo., 1968); R. D. Kodis, IEEE Trans. Antennas and Propag.,  AP-14, 77 (1963); A. Stogryn, Radio Sci. 2, 415 (1967).
  5. A. K. Fung, Planet. Space Sci. 14, 563 (1966).
    [Crossref]
  6. A. K. Fung and H. L. Chan, IEEE Trans. Antennas Propag. 17, 590 (1969).
    [Crossref]
  7. J. M. Elson and R. H. Ritchie, Phys. Rev. B 4, 4129 (1972).
    [Crossref]
  8. A. A. Maradudin and D. C. Mills, Phys. Rev. B 11, 1392 (1975).
    [Crossref]
  9. V. Celli, A. Marvin, and F. Toigo, Phys. Rev. B 11, 1779 (1975).
    [Crossref]
  10. E. Kröger and E. Kretschmann, Z. Phys. 237, 1 (1970).
    [Crossref]
  11. S. O. Rice, Commun. Pure Appl. Math. 4, 351 (1951).
    [Crossref]
  12. C. C. Sung and J. A. Holzer, Appl. Phys. Lett. 28, 429 (1976); ibid., J. Appl. Phys.48, 1739 (1977).
    [Crossref]
  13. Ref. 6 is intended to fit the experimental data on Ref. 2 as a function of the incident angle θ, but the solution does not satisfy the boundary conditions. It should be pointed out that since the data are simple functions of θ, it is easy to fit the data with several parameters even if the calculations are based on poor approximations.
  14. C. C. Sung and W. D. Eberhardt, J. Appl. Phys. (to be published).

1976 (1)

C. C. Sung and J. A. Holzer, Appl. Phys. Lett. 28, 429 (1976); ibid., J. Appl. Phys.48, 1739 (1977).
[Crossref]

1975 (3)

G. J. Wilhelmi, J. W. Rouse, and A. J. Blanchard, J. Opt. Soc. Am. 65, 1036 (1975).
[Crossref]

A. A. Maradudin and D. C. Mills, Phys. Rev. B 11, 1392 (1975).
[Crossref]

V. Celli, A. Marvin, and F. Toigo, Phys. Rev. B 11, 1779 (1975).
[Crossref]

1972 (1)

J. M. Elson and R. H. Ritchie, Phys. Rev. B 4, 4129 (1972).
[Crossref]

1970 (1)

E. Kröger and E. Kretschmann, Z. Phys. 237, 1 (1970).
[Crossref]

1969 (1)

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

1967 (1)

1966 (1)

A. K. Fung, Planet. Space Sci. 14, 563 (1966).
[Crossref]

1951 (1)

S. O. Rice, Commun. Pure Appl. Math. 4, 351 (1951).
[Crossref]

Beckmann, P.

P. Beckmann, The Depolarization of Electromagnetic Waves (Golem, Boulder, Colo., 1968); R. D. Kodis, IEEE Trans. Antennas and Propag.,  AP-14, 77 (1963); A. Stogryn, Radio Sci. 2, 415 (1967).

Blanchard, A. J.

Celli, V.

V. Celli, A. Marvin, and F. Toigo, Phys. Rev. B 11, 1779 (1975).
[Crossref]

Chan, H. L.

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

Cheo, R. K.

Cooper, H. G.

Eberhardt, W. D.

C. C. Sung and W. D. Eberhardt, J. Appl. Phys. (to be published).

Elson, J. M.

J. M. Elson and R. H. Ritchie, Phys. Rev. B 4, 4129 (1972).
[Crossref]

Fung, A. K.

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

A. K. Fung, Planet. Space Sci. 14, 563 (1966).
[Crossref]

Holzer, J. A.

C. C. Sung and J. A. Holzer, Appl. Phys. Lett. 28, 429 (1976); ibid., J. Appl. Phys.48, 1739 (1977).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975). We follow the scattering geometry in Jackson.

Kretschmann, E.

E. Kröger and E. Kretschmann, Z. Phys. 237, 1 (1970).
[Crossref]

Kröger, E.

E. Kröger and E. Kretschmann, Z. Phys. 237, 1 (1970).
[Crossref]

Maradudin, A. A.

A. A. Maradudin and D. C. Mills, Phys. Rev. B 11, 1392 (1975).
[Crossref]

Marvin, A.

V. Celli, A. Marvin, and F. Toigo, Phys. Rev. B 11, 1779 (1975).
[Crossref]

Mills, D. C.

A. A. Maradudin and D. C. Mills, Phys. Rev. B 11, 1392 (1975).
[Crossref]

Renau, J.

Rice, S. O.

S. O. Rice, Commun. Pure Appl. Math. 4, 351 (1951).
[Crossref]

Ritchie, R. H.

J. M. Elson and R. H. Ritchie, Phys. Rev. B 4, 4129 (1972).
[Crossref]

Rouse, J. W.

Sung, C. C.

C. C. Sung and J. A. Holzer, Appl. Phys. Lett. 28, 429 (1976); ibid., J. Appl. Phys.48, 1739 (1977).
[Crossref]

C. C. Sung and W. D. Eberhardt, J. Appl. Phys. (to be published).

Toigo, F.

V. Celli, A. Marvin, and F. Toigo, Phys. Rev. B 11, 1779 (1975).
[Crossref]

Wilhelmi, G. J.

Appl. Phys. Lett. (1)

C. C. Sung and J. A. Holzer, Appl. Phys. Lett. 28, 429 (1976); ibid., J. Appl. Phys.48, 1739 (1977).
[Crossref]

Commun. Pure Appl. Math. (1)

S. O. Rice, Commun. Pure Appl. Math. 4, 351 (1951).
[Crossref]

IEEE Trans. Antennas Propag. (1)

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

J. Opt. Soc. Am. (2)

Phys. Rev. B (3)

J. M. Elson and R. H. Ritchie, Phys. Rev. B 4, 4129 (1972).
[Crossref]

A. A. Maradudin and D. C. Mills, Phys. Rev. B 11, 1392 (1975).
[Crossref]

V. Celli, A. Marvin, and F. Toigo, Phys. Rev. B 11, 1779 (1975).
[Crossref]

Planet. Space Sci. (1)

A. K. Fung, Planet. Space Sci. 14, 563 (1966).
[Crossref]

Z. Phys. (1)

E. Kröger and E. Kretschmann, Z. Phys. 237, 1 (1970).
[Crossref]

Other (4)

Ref. 6 is intended to fit the experimental data on Ref. 2 as a function of the incident angle θ, but the solution does not satisfy the boundary conditions. It should be pointed out that since the data are simple functions of θ, it is easy to fit the data with several parameters even if the calculations are based on poor approximations.

C. C. Sung and W. D. Eberhardt, J. Appl. Phys. (to be published).

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975). We follow the scattering geometry in Jackson.

P. Beckmann, The Depolarization of Electromagnetic Waves (Golem, Boulder, Colo., 1968); R. D. Kodis, IEEE Trans. Antennas and Propag.,  AP-14, 77 (1963); A. Stogryn, Radio Sci. 2, 415 (1967).

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Figures (4)

FIG. 1
FIG. 1

Angular dependence of the normalized backscattered unperturbed polarized component Pp(θ) for different incident angles θ and t0. h = 0.1, = 2.5.

FIG. 2
FIG. 2

Angular dependence of the normalized backscattered perturbed polarized component for different incident angles θ and t0. h = 0.1, = 2.5.

FIG. 3
FIG. 3

Angular dependence of the backscattered depolarized component for different incident angles θ and t0. h = 0.1, = 2.5.

FIG. 4
FIG. 4

The ratio of the backscattered depolarized components to the sum of the backscattered depolarized and perturbed polarized component for different incident angles θ. The parameters used in A are h = 0.2, = 1.3; B are h = 0.2, = 2.5; C are h = 0.1, = 2.5; and D are h = 0.2, = 100. For all curves t0 = 0.2.

Equations (50)

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z ( x , y ) = α x + β y + ( x 2 / R x ) + ( x y / R 0 ) + ( y 2 / R y ) ,
f ( α , β ) = ( 1 / π t 0 2 ) exp [ - ( α 2 + β 2 ) / t 0 2 ] .
R x - 2 = R y - 2 = R 0 - 2 = R - 2 .
Δ z = m , n z m , n exp ( i S m , n · x ) ,
h 2 / λ R 1.
A = A 0 exp [ i k · x ] ,
D = D 0 exp [ i d · x ]
R = R 0 exp [ i r · x ] .
d x = k cos ϕ sin θ s , d y = k sin ϕ sin θ s ,
d 2 = k 2
d · x = k · x
d z = ( k 2 - d x 2 - d y 2 ) 1 / 2 , α 0 = ( d x - k x ) / ( d z + k z ) , β 0 = ( d y - k y ) / ( d z + k z ) .
r 2 = k 2
r · x = k · x
r z = ( k 2 - r x 2 - r y 2 ) 1 / 2 , r x = k x + α 0 ( k z - r z ) ,
r y = k y + β 0 ( k z - r z ) .
k · n > 0 and d · n < 0
n = ( - α , - β , 1 ) / ( 1 + α 2 + β 2 ) 1 / 2 .
t i · ( A + D ) = t i · R t i · ( k × A + d × D ) = t i · ( r × R ) } i = 1 , 2 ,
t 1 = ( 1 , 0 , α ) / ( 1 + α 2 ) 1 / 2
t 2 = ( 0 , 1 , β ) / ( 1 + β 2 ) 1 / 2
div D = d · D 0 = 0
div R = r · R 0 = 0.
R x = F 1 A x + F 2 A y + F 3 A z + F 4 D x + F 5 D y ,
R y = f 1 A x + f 2 A y + f 3 A z + f 4 D x + f 5 D y ,
B = 1 - α ( r x / r z ) - β ( r y / r z ) , F 1 = [ 1 - β ( r y / r z ) ] / B , F 2 = α ( r y / r z ) / B , F 3 = α / B , F 4 = [ 1 + α ( d x / d z ) - β ( r y / r z ) ] / B , F 5 = α [ ( r y / r z ) + ( d y / d z ) ] / B , f 1 = β ( r x / r z ) / B , f 2 = ( 1 - α ( r x / r z ) ] / B , f 3 = β / B , f 4 = β [ ( r x / r z ) + ( d x / d z ) ] / B , f 5 = ( 1 + β ( d y / d z ) - α ( r x / r z ) ] / B ,
a 11 D x + a 12 D y = b 1 ,
a 21 D x + a 22 D y = b 2 ,
g 1 = r z - β r y + ( r x 2 / r z ) , g 2 = β r x + ( r x r y / r z ) , G 1 = - α r y - ( r x r y / r z ) , G 2 = α r x - r z - ( r y 2 / r z ) ,
a 11 = - d z - β d y - ( d x 2 / d z ) - g 1 F 4 - g 2 f 4 , a 12 = β d x - ( d x d y / d z ) - g 1 F 5 - g 2 f 5 , a 21 = - α d y + ( d x d y / d z ) - G 1 F 4 - G 2 f 4 , a 22 = α d x + d z + ( d y 2 / d z ) - G 1 F 5 - G 2 f 5 , b 1 = ( F 1 g 1 + f 1 g 2 - k z ) A x + ( F 2 g 1 + f 2 g 2 - β k x ) A y + ( F 3 g 1 + f 3 g 2 + k x ) A z , b 2 = ( F 1 G 1 + f 1 G 2 ) A x + ( F 2 G 1 + f 2 G 2 - α k x + k z ) A y + ( F 3 G 1 + f 3 G 2 ) A z .
P s = D x sin ϕ - D y cos ϕ 2 f ( α 0 , β 0 )
D s = D x cos ϕ + D y sin ϕ 2 f ( α 0 , β 0 ) / cos 2 θ s ,
A = i m , n z m , n exp [ i S m , n · x ] × ( k z A exp [ i k · x ] - d z D exp [ i d · x ] - r z R exp [ i r · x ] )
D = m , n D m , n exp [ i ( d + S m , n ) · x ] ,
R = m , n R m , n exp [ i ( r + S m , n ) · x ] .
( d + S m , n ) · D m , n = 0 ,
( r + S m , n ) · R m , n = 0 ,
r + S m , n 2 / = d + S m , n 2 = k 2 .
k · x = d · x = r · x | z = α x + β y
t i · ( A + D ) = t i · R .
- h h d x d y exp [ - i S m , n · x ]
i z m , n t i · ( k z A 0 - d z D 0 - r z R 0 ) + t i · D m , n = t i R m , n .
i z m , n t i · ( k z k × A 0 - d z d × D 0 - r z r × R 0 ) + t i · ( d ˜ m , n × D m , n ) = t i · ( r ˜ m , n × R m , n ) ,
d ˜ m , n = d + S m , n , r ˜ m , n = r + S m , n .
α m = [ sin θ s cos ϕ - sin θ - ( m λ / h ) ] / ( cos θ + cos θ s )
β n = [ sin θ s sin ϕ - ( n λ / h ) ] / ( cos θ + cos θ s ) ,
b 1 = z m , n { A x k z [ - k z + F 1 g 1 + f 1 g 2 ] + A y k z [ - β k x + F 2 g 1 + f 2 g 2 ] + A z k z [ k x + F 3 g 1 + f 3 g 2 ] + D x d z [ - d z - β d y - F 1 g 1 - f 1 g 2 ] + D y d z [ β d x - F 2 g 1 - f 2 g 2 ] + D z d z [ - d x - F 3 g 1 - f 3 g 2 ] + R x r z ( r z - β r y - F 1 g 1 - f 1 g 2 ) + R y r z [ β r x - F 2 g 1 - f 2 g 2 ] + R z r z [ - r x - F 3 g 1 - f 3 g 2 ] }
b 2 = z m , n { A x k z [ G 1 F 1 + G 2 f 1 ] + A y k z [ k z - α k x + G 1 F 2 + G 2 f 2 ] + A z k z [ G 1 F 3 + G 2 f 3 ] + D x d z [ - α d y - G 1 F 1 - G 2 f 1 ] + D y d z [ α d x + d z - G 1 F 2 - G 2 f 2 ] + D z d z [ d y - G 1 F 3 - G 2 f 3 ] + R x r z [ - α r y - G 1 F 1 - G 2 f 1 ] + R y r z [ α r x - r z - G 1 F 2 - G 2 f 2 ] + R z r z [ r y - G 1 F 3 - G 2 f 3 ] } .
P s = m , n D x ( m , n ) sin ϕ - D y ( m , n ) cos ϕ 2 f ( α m , β n )
D s = m , n D x ( m , n ) cos ϕ + D y ( m , n ) × sin ϕ 2 f ( α m , β n ) / cos 2 θ s .