Abstract

The C method is known to be one of the most efficient and versatile tools established for modeling diffraction gratings. Its main advantage is the use of a coordinate system in which the boundary conditions apply naturally and are, ipso facto, greatly simplified. In the context of scattering from random rough surfaces, we propose an extension of this method in order to treat the problem of diffraction of an arbitrary incident beam from a perfectly conducting (PEC) rough surface. For that, we were led to revisit some numerical aspects that simplify the implementation and improve the resulting codes.

© 2008 Optical Society of America

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References

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  1. J. A. DeSanto and G. S. Brown, “Analytical techniques for multiple scattering from rough surfaces,” in Progress in Optics, E.Wolf, ed. (North-Holland, 1986), Vol. 23, pp. 3-62.
    [CrossRef]
  2. E. I. Thoros, “The validity of the Kirchoff approximation for rough surfaces scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83, 78-92 (1987).
    [CrossRef]
  3. H. Faure-Geors and D. Maystre, “Improvement of the Kirchoff approximation for scattering from rough surfaces,” J. Opt. Soc. Am. A 6, 532-542 (1989).
    [CrossRef]
  4. J. A. Desanto, “Exact spectral formalism for rough surfaces scattering,” J. Opt. Soc. Am. A 2, 2202-2207 (1985).
    [CrossRef]
  5. J. M. Soto-Crespo and M. Nieto-Vesperinas, “Electromagnetic scattering from very rough random surfaces and deep reflection gratings,” J. Opt. Soc. Am. A 6, 367-384 (1989).
    [CrossRef]
  6. D. Maystre, “Electromagnetic scattering from perfectly conducting rough surfaces in the resonance region,” IEEE Trans. Antennas Propag. 31, 885-895 (1983).
    [CrossRef]
  7. M. Sailllard and D. Maystre, “Scattering from metallic and dielectric rough surfaces,” J. Opt. Soc. Am. A 3, 982-990 (1990).
    [CrossRef]
  8. H. Giovannini, M. Saillard, and A. Sentenac, “Numerical study of scattering from rough inhomogeneous films,” J. Opt. Soc. Am. A 15, 1182-1191 (1998).
    [CrossRef]
  9. B. Guizal, D. Barchiesi, and D. Felbacq, “Electromagnetic beam diffraction by a finite lamellar structure,” J. Opt. Soc. Am. A 20, 2277-2280 (2003).
    [CrossRef]
  10. J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235-241 (1980).
    [CrossRef]
  11. L. Li and J. Chandezon, “Improvement of the coordinate transformation method for surface-relief gratings with sharp edges,” J. Opt. Soc. Am. A 13, 2247-2255 (1995).
    [CrossRef]
  12. R. Dusséaux and R. Oliveira, “Scattering of plane wave by 1-dimensional rough surface study in a nonorthogonal coordinate system,” Prog. Electromagn. Res. 34, 63-88 (2001).
    [CrossRef]
  13. G. Granet, K. Edee, and D. Felbacq, “A new curvilinear coordinate system based approach,” Prog. Electromagn. Res. 41, 235-250 (2003).
  14. K. Edee, G. Granet, R. Dusséaux, and C. Baudier, “A curvilinear coordinate based hybrid method for scattering of plane wave from rough surfaces,” J. Electromagn. Waves Appl. 15, 1001-1015 (2004).
    [CrossRef]
  15. C. Baudier, R. Dusséaux, K. Edee, and G. Granet, “Scattering of plane wave by one-dimensional dielectric random rough surfaces: study with the curvilinear coordinate method,” Waves Random Media 14, 61-74, (2004).
    [CrossRef]
  16. R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
    [CrossRef]
  17. L. Li and J. Chandezon, “Improvement of the coordinate transformation method for surface-relief gratings with sharp edges,” J. Opt. Soc. Am. A 13, 2247-2255 (1996).
    [CrossRef]
  18. J. P. Plumey, B. Guizal, and J. Chandezon, “Coordinate transformation method as applied to asymmetric gratings with vertical facets,” J. Opt. Soc. Am. A 14, 610-617 (1997).
    [CrossRef]
  19. G. Granet, J. Chandezon, J.-P. Plumey, and K. Raniriharinosy, “Reformulation of the coordinate transformation method through the concept of adaptive spatial resolution. Application to trapezoidal gratings,” J. Opt. Soc. Am. A 18, 2102-2108 (2001).
    [CrossRef]
  20. L. Tsang, J. A. Kong, K. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations, Wiley Series in Remote Sensing, J.A.Kong, ed. (Wiley, 2001), pp. 113-161.
  21. F. D. Hastings, J. B. Schneider, and S. L. Broschat, “A Monte-Carlo FDTD technique for rough surface scattering,” IEEE Trans. Antennas Propag. 43, 1183-1191 (1995).
    [CrossRef]
  22. A. Ishimaru, “Experimental and theoretical studies on enhanced backscattering from scatterers and rough surfaces,” in Scattering in Volumes and Surfaces, M.Nieto-Vesperinas and J.C.Dainty, eds. (North-Holland, 1990).
  23. A. A. Maradudin, J. Q. Lu, P. Tran, R. F. Wallis, V. Celli, Z. H. Gu, A. R. McGurn, E. R. Méndez, T. Michel, M. Nieto-Vesperinas, J. C. Dainty, and A. J. Sant, “Enhanced backscattering from one- and two-dimensional random surfaces,” Rev. Mex. Fis. 38, 343-397 (1992).

2004 (2)

K. Edee, G. Granet, R. Dusséaux, and C. Baudier, “A curvilinear coordinate based hybrid method for scattering of plane wave from rough surfaces,” J. Electromagn. Waves Appl. 15, 1001-1015 (2004).
[CrossRef]

C. Baudier, R. Dusséaux, K. Edee, and G. Granet, “Scattering of plane wave by one-dimensional dielectric random rough surfaces: study with the curvilinear coordinate method,” Waves Random Media 14, 61-74, (2004).
[CrossRef]

2003 (2)

B. Guizal, D. Barchiesi, and D. Felbacq, “Electromagnetic beam diffraction by a finite lamellar structure,” J. Opt. Soc. Am. A 20, 2277-2280 (2003).
[CrossRef]

G. Granet, K. Edee, and D. Felbacq, “A new curvilinear coordinate system based approach,” Prog. Electromagn. Res. 41, 235-250 (2003).

2001 (2)

1998 (1)

1997 (1)

1996 (1)

1995 (2)

F. D. Hastings, J. B. Schneider, and S. L. Broschat, “A Monte-Carlo FDTD technique for rough surface scattering,” IEEE Trans. Antennas Propag. 43, 1183-1191 (1995).
[CrossRef]

L. Li and J. Chandezon, “Improvement of the coordinate transformation method for surface-relief gratings with sharp edges,” J. Opt. Soc. Am. A 13, 2247-2255 (1995).
[CrossRef]

1992 (1)

A. A. Maradudin, J. Q. Lu, P. Tran, R. F. Wallis, V. Celli, Z. H. Gu, A. R. McGurn, E. R. Méndez, T. Michel, M. Nieto-Vesperinas, J. C. Dainty, and A. J. Sant, “Enhanced backscattering from one- and two-dimensional random surfaces,” Rev. Mex. Fis. 38, 343-397 (1992).

1990 (1)

1989 (2)

1987 (1)

E. I. Thoros, “The validity of the Kirchoff approximation for rough surfaces scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83, 78-92 (1987).
[CrossRef]

1985 (1)

1983 (1)

D. Maystre, “Electromagnetic scattering from perfectly conducting rough surfaces in the resonance region,” IEEE Trans. Antennas Propag. 31, 885-895 (1983).
[CrossRef]

1980 (1)

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235-241 (1980).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

D. Maystre, “Electromagnetic scattering from perfectly conducting rough surfaces in the resonance region,” IEEE Trans. Antennas Propag. 31, 885-895 (1983).
[CrossRef]

F. D. Hastings, J. B. Schneider, and S. L. Broschat, “A Monte-Carlo FDTD technique for rough surface scattering,” IEEE Trans. Antennas Propag. 43, 1183-1191 (1995).
[CrossRef]

J. Acoust. Soc. Am. (1)

E. I. Thoros, “The validity of the Kirchoff approximation for rough surfaces scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83, 78-92 (1987).
[CrossRef]

J. Electromagn. Waves Appl. (1)

K. Edee, G. Granet, R. Dusséaux, and C. Baudier, “A curvilinear coordinate based hybrid method for scattering of plane wave from rough surfaces,” J. Electromagn. Waves Appl. 15, 1001-1015 (2004).
[CrossRef]

J. Opt. (1)

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235-241 (1980).
[CrossRef]

J. Opt. Soc. Am. A (10)

B. Guizal, D. Barchiesi, and D. Felbacq, “Electromagnetic beam diffraction by a finite lamellar structure,” J. Opt. Soc. Am. A 20, 2277-2280 (2003).
[CrossRef]

H. Giovannini, M. Saillard, and A. Sentenac, “Numerical study of scattering from rough inhomogeneous films,” J. Opt. Soc. Am. A 15, 1182-1191 (1998).
[CrossRef]

J. P. Plumey, B. Guizal, and J. Chandezon, “Coordinate transformation method as applied to asymmetric gratings with vertical facets,” J. Opt. Soc. Am. A 14, 610-617 (1997).
[CrossRef]

J. A. Desanto, “Exact spectral formalism for rough surfaces scattering,” J. Opt. Soc. Am. A 2, 2202-2207 (1985).
[CrossRef]

M. Sailllard and D. Maystre, “Scattering from metallic and dielectric rough surfaces,” J. Opt. Soc. Am. A 3, 982-990 (1990).
[CrossRef]

J. M. Soto-Crespo and M. Nieto-Vesperinas, “Electromagnetic scattering from very rough random surfaces and deep reflection gratings,” J. Opt. Soc. Am. A 6, 367-384 (1989).
[CrossRef]

L. Li and J. Chandezon, “Improvement of the coordinate transformation method for surface-relief gratings with sharp edges,” J. Opt. Soc. Am. A 13, 2247-2255 (1995).
[CrossRef]

L. Li and J. Chandezon, “Improvement of the coordinate transformation method for surface-relief gratings with sharp edges,” J. Opt. Soc. Am. A 13, 2247-2255 (1996).
[CrossRef]

H. Faure-Geors and D. Maystre, “Improvement of the Kirchoff approximation for scattering from rough surfaces,” J. Opt. Soc. Am. A 6, 532-542 (1989).
[CrossRef]

G. Granet, J. Chandezon, J.-P. Plumey, and K. Raniriharinosy, “Reformulation of the coordinate transformation method through the concept of adaptive spatial resolution. Application to trapezoidal gratings,” J. Opt. Soc. Am. A 18, 2102-2108 (2001).
[CrossRef]

Prog. Electromagn. Res. (2)

R. Dusséaux and R. Oliveira, “Scattering of plane wave by 1-dimensional rough surface study in a nonorthogonal coordinate system,” Prog. Electromagn. Res. 34, 63-88 (2001).
[CrossRef]

G. Granet, K. Edee, and D. Felbacq, “A new curvilinear coordinate system based approach,” Prog. Electromagn. Res. 41, 235-250 (2003).

Rev. Mex. Fis. (1)

A. A. Maradudin, J. Q. Lu, P. Tran, R. F. Wallis, V. Celli, Z. H. Gu, A. R. McGurn, E. R. Méndez, T. Michel, M. Nieto-Vesperinas, J. C. Dainty, and A. J. Sant, “Enhanced backscattering from one- and two-dimensional random surfaces,” Rev. Mex. Fis. 38, 343-397 (1992).

Waves Random Media (1)

C. Baudier, R. Dusséaux, K. Edee, and G. Granet, “Scattering of plane wave by one-dimensional dielectric random rough surfaces: study with the curvilinear coordinate method,” Waves Random Media 14, 61-74, (2004).
[CrossRef]

Other (4)

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
[CrossRef]

L. Tsang, J. A. Kong, K. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations, Wiley Series in Remote Sensing, J.A.Kong, ed. (Wiley, 2001), pp. 113-161.

J. A. DeSanto and G. S. Brown, “Analytical techniques for multiple scattering from rough surfaces,” in Progress in Optics, E.Wolf, ed. (North-Holland, 1986), Vol. 23, pp. 3-62.
[CrossRef]

A. Ishimaru, “Experimental and theoretical studies on enhanced backscattering from scatterers and rough surfaces,” in Scattering in Volumes and Surfaces, M.Nieto-Vesperinas and J.C.Dainty, eds. (North-Holland, 1990).

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Figures (8)

Fig. 1
Fig. 1

PEC rough surface illuminated by an incident beam under the mean incidence θ i .

Fig. 2
Fig. 2

(a) Single realization of a Gaussian rough surface profile and its derivative. h = 0.05 λ , l c = 0.35 λ , d = 25.6 λ . (b) Accuracy in reflectivity for one profile realization [shown in (a)]. θ i = 30 ° , λ = 1 , g = 0.25 d , TE and TM polarizations.

Fig. 3
Fig. 3

(a) Single realization of a Gaussian rough surface profile and its derivative. h = 0.2 λ , l c = 0.35 λ , d = 25.6 λ . (b) Accuracy in reflectivity for one profile realization [shown in (a)] θ i = 30 ° , λ = 1 , g = 0.25 d , TE and TM polarizations.

Fig. 4
Fig. 4

(a) Single realization of a Gaussian rough surface profile and its derivative. h = 0.2 λ , l c = 1.5 λ , d = 25.6 λ . (b) Accuracy in reflectivity for one profile realization [shown in (a)]. θ i = 30 ° , λ = 1 , g = 0.25 d , TE and TM polarizations.

Fig. 5
Fig. 5

(a) Angular power density averaged through 100 realizations, TE polarization. θ i = 30 ° , λ = 1 , d = 25.6 λ , g = 0.25 d , h = 0.05 λ , l c = 0.35 λ , M = 30 . (b) Angular power density averaged through 100 realizations, TM polarization. θ i = 30 ° , λ = 1 , d = 25.6 λ , g = 0.25 d , h = 0.05 λ , l c = 0.35 λ , M = 30 .

Fig. 6
Fig. 6

(a) Angular power density averaged through 50 realization, TE polarization. θ i = 45 ° , λ = 1 , d = 50.6 λ , g = 0.25 d , h = 0.16 λ , l c = 0.68 λ , M = 120 . (b) Angular power density averaged through 50 realizations, TE polarization. θ i = 70 ° , λ = 1 , d = 50.6 λ , g = 0.25 d , h = 0.16 λ , l c = 0.68 λ , M = 150 .

Fig. 7
Fig. 7

(a) Angular power density average through 200 realizations, TE polarization. θ i = 30 ° , λ = 1 , d = 25.6 λ , g = 0.25 d , h = 0.2 λ , l c = 0.2 λ , M = 300 . (b) Angular power density average through 200 realizations, TM polarization. θ i = 30 ° , λ = 1 , d = 25.6 λ , g = 0.25 d , h = 0.2 λ , l c = 0.2 λ , M = 300 .

Fig. 8
Fig. 8

Surface for Poynting’s flux computation.

Equations (50)

Equations on this page are rendered with MathJax. Learn more.

x = u , y = v + a ( u ) , z = w .
( u u + k 2 I 0 0 I ) ( Ψ v Ψ ) = ( u a ̇ + a ̇ u ( 1 + a ̇ a ̇ ) I 0 ) v ( Ψ v Ψ ) ,
Ψ ( u , v ) = exp ( i k r v ) ψ ( u ) , v Ψ ( u , v ) = exp ( i k r v ) ψ r ( u ) .
( u u + k 2 I 0 0 I ) ( ψ ψ r ) = i k r ( u a ̇ + a ̇ u ( 1 + a ̇ a ̇ ) I 0 ) ( ψ ψ r ) .
f ̂ ( α ) = + f ( u ) exp ( i k α u ) d u ,
L 1 ( α ) * Γ ̂ ( α ) = i k r L 2 ( α ) * Γ ̂ ( α ) ,
α n = n Δ α , where n is a relative integer.
f ̂ ( α ) = n = + f n sinc ( π ( α n Δ α ) Δ α ) = n = + f n b ̂ n ( α ) .
( α α I 0 0 i k I ) ( ψ ψ r ) = r ( α a ̇ + a ̇ α 1 k ( I + a ̇ a ̇ ) I 0 ) ( ψ ψ r ) ,
Ψ ̂ ( α , v ) = Ψ ̂ i n c ( α , v ) + Ψ ̂ d i f f ( α , v ) ,
Ψ ̂ ( α , v ) = q = M M I q Ψ ̂ q ( α , v ) + q = M M R q Ψ ̂ q + ( α , v ) ,
Ψ ̂ q ± ( α , v ) = exp ( i k r q ± v ) n = M M ψ n q ± b ̂ n ( α ) .
Ψ ( u , v ) = Δ α Π Δ α ( u ) q I q exp ( i k r q v ) n = M M ψ n q exp ( i k α n u ) + Δ α Π Δ α ( u ) q R q exp ( i k r q + v ) n = M M ψ n q + exp ( i k α n u ) ,
Π Δ α ( u ) = { 1 if u [ 1 2 Δ α 1 2 Δ α ] 0 otherwise .
Φ ( u , v ) = η [ v a ̇ ( u ) ( u a ̇ ( u ) v ) ] Ψ ( u , v ) ,
Φ ( u , v ) = Δ α Π Δ α ( u ) q I q exp ( i k r q v ) n = M M ϕ n q exp ( i k α n u ) + Δ α Π Δ α ( u ) q R q exp ( i k r q + v ) n = M M ϕ n q + exp ( i k α n u ) ,
Δ P ̃ s ( α q ) = R q 2 R e ( n = M M ψ n q Φ n q * ) q α ̃ q [ 1 ; 1 ] I ( α ̃ q ) 2 R e ( n = M M ψ n q Φ n q * ) ,
σ ( θ q ) = Δ P ̃ s ( α q ) Δ α cos ( θ q ) .
α ̃ q = α Ψ ̂ q ( α ) 2 d α Ψ ̂ q ( α ) 2 d α .
I ( α ) = g 2 π exp [ k 2 ( α α 0 ) 2 g 2 4 ] .
q α q [ 1 ; 1 ] Δ P ̃ s ( α q ) = 1 .
ξ ( N ) = Int ( log 10 Δ P ̃ s ( α 0 ) Δ P ̃ s * ) ,
Ψ i ( x , y ) = + I ( α ) exp ( i k α x ) exp ( i k β ( α ) y ) d α ,
α 2 + β 2 = 1 .
Ψ inc ( x , y ) = Δ α Π Δ α ( x ) n I ( α n ) exp ( i k α n x ) exp ( i k β n y ) ,
α n 2 + β n 2 = 1 ,
Ψ inc ( x , y ) = Δ α Π Δ α ( x ) q I ( α q ) exp ( i k β q y ) n δ n q exp ( i k α n x ) ,
Ψ inc ( u , v ) = Δ α Π Δ α ( u ) q I ( α q ) exp ( i k β q v ) exp ( i k β q a ( u ) ) n δ n q exp ( i k α n u ) .
Ψ inc ( u , v ) = Δ α Π Δ α ( u ) q I ( α q ) exp ( i k β q v ) n L n p q exp ( i k α n u ) ,
exp ( i k β q a ( u ) ) = p L p q exp ( i k α p u ) ,
L p q = L q ( α p ) = T F [ exp ( i k β q a ( u ) ) ] ( α p ) .
Ψ inc ( u , v ) = Δ α Π Δ α ( u ) q I ( α ̃ q ) exp ( i k r q v ) n ψ n q exp ( i k α n u ) ,
I v = 1 2 R e [ E u H z * E z H u * ] .
I v = { R e [ Ψ ( u , v ) Φ * ( u , v ) ] for TE polarization R e [ Ψ * ( u , v ) Φ ( u , v ) ] for TM polarization.
P s = R e [ z d z R Ψ ( u , v ) Φ * ( u , v ) d u ] .
P s = R e [ R Ψ ( u , v ) Φ * ( u , v ) d u ] ,
R Ψ ( u , v ) Φ * ( u , v ) d u = R Ψ ̂ ( α , v ) Φ ̂ * ( α , v ) d α ,
P s = η R e ( [ q r q R R q exp ( i k r q v ) p = M M ψ p q ] [ m r m R R m * exp ( i k r m * v ) n = M M Φ n m * ] ) δ n p ,
P s = η R e ( [ q r q R m r m R R m * R q exp ( i k ( r q r m ) v ) n , p = M M ψ p q Φ n m * ] ) δ n p .
P s = η R e ( q r q R R q R q * n , p = M M ψ p q Φ n q * ) ,
P s = η q r q R R q 2 R e ( n = M M ψ n q Φ n q * ) .
Δ P s ( α q ) = R q 2 R e ( n = M M ψ n q Φ n q * ) .
P s = η q α q [ 1 ; 1 ] Δ P s ( α q ) .
P inc = η q α ̃ q [ 1 ; 1 ] I ( α ̃ q ) 2 R e ( n = M M ψ n q Φ n q * ) .
Δ P ̃ s ( α q ) = Δ P s ( α q ) P inc .
P ̃ s = q α q [ 1 ; 1 ] Δ P ̃ s ( α q ) .
q α q [ 1 ; 1 ] Δ P ̃ s ( α q ) = 1 .
σ ( θ ) = d P ̃ s ( α ( θ ) ) d θ = d P ̃ s ( α ) d α d α ( θ ) d θ .
σ ( θ ) = d P ̃ s ( α ) d α cos ( θ ) .
σ ( θ q ) = Δ P ̃ s ( α q ) Δ α cos ( θ q ) .

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