Abstract

Energy conservation of the scattering from one-dimensional strongly rough dielectric surfaces is investigated using the Kirchhoff approximation with single reflection and by taking the shadowing phenomenon into account, both in reflection and transmission. In addition, because no shadowing function in transmission exists in the literature, this function is presented here in detail. The model is reduced to the high-frequency limit (or geometric optics). The energy conservation criterion is investigated versus the incidence angle, the permittivity of the lower medium, and the surface rms slope.

© 2005 Optical Society of America

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References

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  1. P. Beckman and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces. Part I. Theory (Pergamon, 1963).
  2. J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Institute of Physics, 1991).
  3. A. K. Fung, Microwave Scattering and Emission Models and Their Applications (Artech House, 1994).
  4. J. A. Kong, Electromagnetic Wave Theory (Wiley, 1975).
  5. A. K. Fung, IEEE Trans. Antennas Propag. 29, 463 (1981).
    [CrossRef]
  6. J. Caron, J. Lafait, and C. Andraud, Opt. Commun. 207, 17 (2002).
    [CrossRef]
  7. C. Bourlier and G. Berginc, Waves Random Media 14, 229 (2004).
    [CrossRef]
  8. C. Bourlier, G. Berginc, and J. Saillard, Waves Random Media 12, 145 (2002).
    [CrossRef]
  9. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964).

2004 (1)

C. Bourlier and G. Berginc, Waves Random Media 14, 229 (2004).
[CrossRef]

2002 (2)

C. Bourlier, G. Berginc, and J. Saillard, Waves Random Media 12, 145 (2002).
[CrossRef]

J. Caron, J. Lafait, and C. Andraud, Opt. Commun. 207, 17 (2002).
[CrossRef]

1981 (1)

A. K. Fung, IEEE Trans. Antennas Propag. 29, 463 (1981).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964).

Andraud, C.

J. Caron, J. Lafait, and C. Andraud, Opt. Commun. 207, 17 (2002).
[CrossRef]

Beckman, P.

P. Beckman and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces. Part I. Theory (Pergamon, 1963).

Berginc, G.

C. Bourlier and G. Berginc, Waves Random Media 14, 229 (2004).
[CrossRef]

C. Bourlier, G. Berginc, and J. Saillard, Waves Random Media 12, 145 (2002).
[CrossRef]

Bourlier, C.

C. Bourlier and G. Berginc, Waves Random Media 14, 229 (2004).
[CrossRef]

C. Bourlier, G. Berginc, and J. Saillard, Waves Random Media 12, 145 (2002).
[CrossRef]

Caron, J.

J. Caron, J. Lafait, and C. Andraud, Opt. Commun. 207, 17 (2002).
[CrossRef]

Fung, A. K.

A. K. Fung, IEEE Trans. Antennas Propag. 29, 463 (1981).
[CrossRef]

A. K. Fung, Microwave Scattering and Emission Models and Their Applications (Artech House, 1994).

Kong, J. A.

J. A. Kong, Electromagnetic Wave Theory (Wiley, 1975).

Lafait, J.

J. Caron, J. Lafait, and C. Andraud, Opt. Commun. 207, 17 (2002).
[CrossRef]

Ogilvy, J. A.

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Institute of Physics, 1991).

Saillard, J.

C. Bourlier, G. Berginc, and J. Saillard, Waves Random Media 12, 145 (2002).
[CrossRef]

Spizzichino, A.

P. Beckman and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces. Part I. Theory (Pergamon, 1963).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964).

IEEE Trans. Antennas Propag. (1)

A. K. Fung, IEEE Trans. Antennas Propag. 29, 463 (1981).
[CrossRef]

Opt. Commun. (1)

J. Caron, J. Lafait, and C. Andraud, Opt. Commun. 207, 17 (2002).
[CrossRef]

Waves Random Media (2)

C. Bourlier and G. Berginc, Waves Random Media 14, 229 (2004).
[CrossRef]

C. Bourlier, G. Berginc, and J. Saillard, Waves Random Media 12, 145 (2002).
[CrossRef]

Other (5)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964).

P. Beckman and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces. Part I. Theory (Pergamon, 1963).

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Institute of Physics, 1991).

A. K. Fung, Microwave Scattering and Emission Models and Their Applications (Artech House, 1994).

J. A. Kong, Electromagnetic Wave Theory (Wiley, 1975).

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Figures (4)

Fig. 1
Fig. 1

Illustration of the studied problem.

Fig. 2
Fig. 2

Comparison between S 11 and S 12 , with θ i = 80 ° and slope rms σ s = 0.3 .

Fig. 3
Fig. 3

Simulations for σ s = 0.2 and ϵ r 2 = i (case of a perfectly conducting surface: V polar H polar).

Fig. 4
Fig. 4

Simulations for σ s = 0.2 and ϵ r 2 = 4 .

Equations (17)

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E r ( r ) = + [ E ( r 0 ) G 1 n 0 E ( r 0 ) n 0 G 1 ] d S 0 ,
E t ( r ) = [ E t ( r 0 ) G 2 n 0 E t ( r 0 ) n 0 G 2 ] d S 0 .
G 1 , 2 ( r 0 , r ) i 4 2 π k 1 , 2 r exp ( i π 4 ) exp [ i ( k 1 , 2 r k r , t r 0 ) ] .
E ( r 0 ) = [ 1 + R ( θ ) ] E i ( r 0 ) ,
E ( r 0 ) , n 0 = i ( k i n ̂ 0 ) [ 1 R ( θ ) ] E i ( r 0 ) ,
E t ( r 0 ) = T ( θ ) E i ( r 0 ) ,
E t ( r 0 ) n 0 = i ( k t s p n ̂ 0 ) T ( θ ) E i ( r 0 ) ,
γ r 0 = k ̂ r k ̂ i q ̂ r q ̂ i , γ t 0 = k 2 k ̂ t k 1 k ̂ i k 2 q ̂ t k 1 q ̂ i .
E r ( r ) E 0 = exp [ i ( k 1 r + π 4 ) ] 2 π k 1 r R ( θ ) f r ( θ i , θ r ) × L 0 + L 0 Ξ ( x 0 ) exp [ i ( k i k r ) r 0 ] d x 0 ,
E t ( r ) E 0 = exp [ i ( k 2 r π 4 ) ] 2 π k 2 r T ( χ ) f t ( θ i , θ t ) × L 0 + L 0 Ξ ( x 0 ) exp [ i ( k i k t ) r 0 ] d x 0 ,
σ r = R 2 ( θ ) cos θ i f r 2 ( θ i , θ r ) p s ( γ r 0 ) q ̂ r q ̂ i S 11 ( θ i , θ r γ r 0 ) ,
σ t = T 2 ( χ ) cos θ i f t 2 ( θ i , θ t ) p s ( γ t 0 ) q ̂ t k 1 k 2 q ̂ i S 12 ( θ i , θ t γ t 0 ) , }
S 1 ( θ 1 ζ 0 , γ 0 ) = Υ ( μ 1 γ 0 ) [ P h ( ζ 0 ) P h ( ) ] Λ ( μ 1 ) ,
S 2 ( θ 2 ζ 0 , γ 0 ) = Υ ( μ 2 γ 0 ) { 1 [ P h ( ζ 0 ) P h ( ) ] } Λ ( μ 2 ) ,
S 11 ( θ i , θ r γ r 0 ) = { [ 1 + Λ ( μ r ) ] 1 ( θ r [ π 2 ; θ i [ ) [ 1 + Λ ( μ i ) ] 1 ( θ r [ θ i ; 0 [ ) , [ 1 + Λ ( μ i ) + Λ ( μ r ) ] 1 ( θ r [ 0 ; π 2 ] ) }
S 12 ( θ i , θ t γ t 0 ) = B [ 1 + Λ ( μ i ) , 1 + Λ ( μ t ) ] ,
P r , t P i = π 2 + π 2 σ r , t ( θ i , θ r , t ) d θ r , t .

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