Abstract

Using a generalization of the integral theory of metallic and dielectric gratings developed in our laboratory 15 years ago, we propose a rigorous integral theory of scattering by metallic or dielectric nonperiodic rough surfaces leading to a single integral equation. The numerical implementation has been carried out despite strong difficulties for TM polarization and metallic surfaces because of propagation of surface plasmon waves outside the illuminated region of the rough surface. Numerical results show the influence of the statistical parameters of the asperities on the absorption phenomena for metallic surfaces. Then the influence of asperities on the total transmission around Brewster incidence is studied. Finally numerical results of enhanced backscattering from perfectly conducting, metallic, and dielectric random rough surfaces are given.

© 1990 Optical Society of America

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  1. D. Maystre, “Electromagnetic scattering from perfectly conducting rough surfaces in the resonance region,” IEEE Trans. Antennas Propag. AP-31, 885–895 (1983).
    [CrossRef]
  2. M. Nieto-Vesperinas, N. Garcia, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
    [CrossRef]
  3. D. Maystre, J. P. Rossi, “Implementation of a rigorous vector theory of speckle for two-dimensional microrough surfaces,” J. Opt. Soc. Am. A 3, 1276–1282 (1986).
    [CrossRef]
  4. M. Saillard, D. Maystre, “Scattering from random rough surfaces: a beam simulation method,” J. Opt. 19, 173–176 (1988).
    [CrossRef]
  5. D. Maystre, M. Nevière, R. Petit, “Experimental ramifications and applications of the theory,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 5.
    [CrossRef]
  6. D. Maystre, “Sur la diffraction d’une onde plane par un réseau métallique de conductivité finie,” Opt. Commun. 6, 50–54 (1972).
    [CrossRef]
  7. D. Maystre, “Sur la diffraction d’une onde plane électromagnétique par un réseau métallique,” Opt. Commun. 8, 216–219 (1973).
    [CrossRef]
  8. D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics XXI, E. Wolf, ed. (Elsevier, New York, 1984).
    [CrossRef]
  9. E. Marx, “Single integral equation for wave scattering,” J. Math. Phys. 23, 1057–1065 (1982).
    [CrossRef]
  10. R. E. Kleinman, P. A. Martin, “On single integral equations for the transmission problem of acoustics,” SIAM J. Appl. Math. 48, 307–325 (1988).
    [CrossRef]
  11. D. Maystre, “Rigorous theory of light scattering from rough surfaces,” J. Opt. 15, 43–51 (1984).
    [CrossRef]
  12. R. Mittra, C. A. Klein, Stability and Convergence of Moment Method Solution, Vol. 3 of Topics in Applied Physics (Springer-Verlag, Berlin, 1975), pp. 129–163.
  13. A. Tikhonov, V. Arsénine, Méthodes de Résolution de Problémes mal Posés (Mir, Moscow, 1976).
  14. A. Roger, D. Maystre, M. Cadilhac, “On a problem of inverse scattering in optics: the dielectric inhomogeneous medium,” J. Opt. 9, 83–90 (1978).
    [CrossRef]
  15. M. Saillard, D. Maystre, “Microrough surfaces: influence of the correlation function on the speckle pattern,” Opt. Acta 33, 1193–1206 (1986).
    [CrossRef]
  16. D. Maystre, M. Saillard, “Rigorous solution of problems of scattering by large size objects,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (Elsevier, New York, 1989), pp. 191–208.
  17. M. C. Hutley, D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
    [CrossRef]
  18. K. A. O’Donnell, E. R. Mendez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
    [CrossRef]
  19. M. Nieto-Vesperinas, J. M. Soto-Crespo, “Monte Carlo simulations for scattering of electromagnetic waves from perfectly conductive random rough surfaces,” Opt. Lett. 12, 979–981 (1987).
    [CrossRef] [PubMed]
  20. J. M. Soto-Crespo, M. Nieto-Vesperinas, “Electromagnetic scattering from very rough random surfaces and deep reflection gratings,” J. Opt. Soc. Am. A 6, 367–384 (1989).
    [CrossRef]
  21. A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
    [CrossRef]
  22. A. A. Maradudin, E. R. Mendez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large-amplitude random metallic grating,” Opt. Lett. 14, 151–153 (1989).
    [CrossRef] [PubMed]

1989 (2)

1988 (2)

M. Saillard, D. Maystre, “Scattering from random rough surfaces: a beam simulation method,” J. Opt. 19, 173–176 (1988).
[CrossRef]

R. E. Kleinman, P. A. Martin, “On single integral equations for the transmission problem of acoustics,” SIAM J. Appl. Math. 48, 307–325 (1988).
[CrossRef]

1987 (2)

1986 (2)

M. Saillard, D. Maystre, “Microrough surfaces: influence of the correlation function on the speckle pattern,” Opt. Acta 33, 1193–1206 (1986).
[CrossRef]

D. Maystre, J. P. Rossi, “Implementation of a rigorous vector theory of speckle for two-dimensional microrough surfaces,” J. Opt. Soc. Am. A 3, 1276–1282 (1986).
[CrossRef]

1985 (1)

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

1984 (1)

D. Maystre, “Rigorous theory of light scattering from rough surfaces,” J. Opt. 15, 43–51 (1984).
[CrossRef]

1983 (1)

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

1982 (1)

E. Marx, “Single integral equation for wave scattering,” J. Math. Phys. 23, 1057–1065 (1982).
[CrossRef]

1981 (1)

M. Nieto-Vesperinas, N. Garcia, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
[CrossRef]

1978 (1)

A. Roger, D. Maystre, M. Cadilhac, “On a problem of inverse scattering in optics: the dielectric inhomogeneous medium,” J. Opt. 9, 83–90 (1978).
[CrossRef]

1976 (1)

M. C. Hutley, D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
[CrossRef]

1973 (1)

D. Maystre, “Sur la diffraction d’une onde plane électromagnétique par un réseau métallique,” Opt. Commun. 8, 216–219 (1973).
[CrossRef]

1972 (1)

D. Maystre, “Sur la diffraction d’une onde plane par un réseau métallique de conductivité finie,” Opt. Commun. 6, 50–54 (1972).
[CrossRef]

Arsénine, V.

A. Tikhonov, V. Arsénine, Méthodes de Résolution de Problémes mal Posés (Mir, Moscow, 1976).

Cadilhac, M.

A. Roger, D. Maystre, M. Cadilhac, “On a problem of inverse scattering in optics: the dielectric inhomogeneous medium,” J. Opt. 9, 83–90 (1978).
[CrossRef]

Celli, V.

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

Garcia, N.

M. Nieto-Vesperinas, N. Garcia, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
[CrossRef]

Hutley, M. C.

M. C. Hutley, D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
[CrossRef]

Klein, C. A.

R. Mittra, C. A. Klein, Stability and Convergence of Moment Method Solution, Vol. 3 of Topics in Applied Physics (Springer-Verlag, Berlin, 1975), pp. 129–163.

Kleinman, R. E.

R. E. Kleinman, P. A. Martin, “On single integral equations for the transmission problem of acoustics,” SIAM J. Appl. Math. 48, 307–325 (1988).
[CrossRef]

Maradudin, A. A.

A. A. Maradudin, E. R. Mendez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large-amplitude random metallic grating,” Opt. Lett. 14, 151–153 (1989).
[CrossRef] [PubMed]

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

Martin, P. A.

R. E. Kleinman, P. A. Martin, “On single integral equations for the transmission problem of acoustics,” SIAM J. Appl. Math. 48, 307–325 (1988).
[CrossRef]

Marx, E.

E. Marx, “Single integral equation for wave scattering,” J. Math. Phys. 23, 1057–1065 (1982).
[CrossRef]

Maystre, D.

M. Saillard, D. Maystre, “Scattering from random rough surfaces: a beam simulation method,” J. Opt. 19, 173–176 (1988).
[CrossRef]

D. Maystre, J. P. Rossi, “Implementation of a rigorous vector theory of speckle for two-dimensional microrough surfaces,” J. Opt. Soc. Am. A 3, 1276–1282 (1986).
[CrossRef]

M. Saillard, D. Maystre, “Microrough surfaces: influence of the correlation function on the speckle pattern,” Opt. Acta 33, 1193–1206 (1986).
[CrossRef]

D. Maystre, “Rigorous theory of light scattering from rough surfaces,” J. Opt. 15, 43–51 (1984).
[CrossRef]

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

A. Roger, D. Maystre, M. Cadilhac, “On a problem of inverse scattering in optics: the dielectric inhomogeneous medium,” J. Opt. 9, 83–90 (1978).
[CrossRef]

M. C. Hutley, D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
[CrossRef]

D. Maystre, “Sur la diffraction d’une onde plane électromagnétique par un réseau métallique,” Opt. Commun. 8, 216–219 (1973).
[CrossRef]

D. Maystre, “Sur la diffraction d’une onde plane par un réseau métallique de conductivité finie,” Opt. Commun. 6, 50–54 (1972).
[CrossRef]

D. Maystre, M. Saillard, “Rigorous solution of problems of scattering by large size objects,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (Elsevier, New York, 1989), pp. 191–208.

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics XXI, E. Wolf, ed. (Elsevier, New York, 1984).
[CrossRef]

D. Maystre, M. Nevière, R. Petit, “Experimental ramifications and applications of the theory,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 5.
[CrossRef]

McGurn, A. R.

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

Mendez, E. R.

Michel, T.

Mittra, R.

R. Mittra, C. A. Klein, Stability and Convergence of Moment Method Solution, Vol. 3 of Topics in Applied Physics (Springer-Verlag, Berlin, 1975), pp. 129–163.

Nevière, M.

D. Maystre, M. Nevière, R. Petit, “Experimental ramifications and applications of the theory,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 5.
[CrossRef]

Nieto-Vesperinas, M.

O’Donnell, K. A.

Petit, R.

D. Maystre, M. Nevière, R. Petit, “Experimental ramifications and applications of the theory,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 5.
[CrossRef]

Roger, A.

A. Roger, D. Maystre, M. Cadilhac, “On a problem of inverse scattering in optics: the dielectric inhomogeneous medium,” J. Opt. 9, 83–90 (1978).
[CrossRef]

Rossi, J. P.

Saillard, M.

M. Saillard, D. Maystre, “Scattering from random rough surfaces: a beam simulation method,” J. Opt. 19, 173–176 (1988).
[CrossRef]

M. Saillard, D. Maystre, “Microrough surfaces: influence of the correlation function on the speckle pattern,” Opt. Acta 33, 1193–1206 (1986).
[CrossRef]

D. Maystre, M. Saillard, “Rigorous solution of problems of scattering by large size objects,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (Elsevier, New York, 1989), pp. 191–208.

Soto-Crespo, J. M.

Tikhonov, A.

A. Tikhonov, V. Arsénine, Méthodes de Résolution de Problémes mal Posés (Mir, Moscow, 1976).

IEEE Trans. Antennas Propag. (1)

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

J. Math. Phys. (1)

E. Marx, “Single integral equation for wave scattering,” J. Math. Phys. 23, 1057–1065 (1982).
[CrossRef]

J. Opt. (3)

M. Saillard, D. Maystre, “Scattering from random rough surfaces: a beam simulation method,” J. Opt. 19, 173–176 (1988).
[CrossRef]

D. Maystre, “Rigorous theory of light scattering from rough surfaces,” J. Opt. 15, 43–51 (1984).
[CrossRef]

A. Roger, D. Maystre, M. Cadilhac, “On a problem of inverse scattering in optics: the dielectric inhomogeneous medium,” J. Opt. 9, 83–90 (1978).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Acta (2)

M. Saillard, D. Maystre, “Microrough surfaces: influence of the correlation function on the speckle pattern,” Opt. Acta 33, 1193–1206 (1986).
[CrossRef]

M. Nieto-Vesperinas, N. Garcia, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
[CrossRef]

Opt. Commun. (3)

D. Maystre, “Sur la diffraction d’une onde plane par un réseau métallique de conductivité finie,” Opt. Commun. 6, 50–54 (1972).
[CrossRef]

D. Maystre, “Sur la diffraction d’une onde plane électromagnétique par un réseau métallique,” Opt. Commun. 8, 216–219 (1973).
[CrossRef]

M. C. Hutley, D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. B (1)

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

SIAM J. Appl. Math. (1)

R. E. Kleinman, P. A. Martin, “On single integral equations for the transmission problem of acoustics,” SIAM J. Appl. Math. 48, 307–325 (1988).
[CrossRef]

Other (5)

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics XXI, E. Wolf, ed. (Elsevier, New York, 1984).
[CrossRef]

D. Maystre, M. Nevière, R. Petit, “Experimental ramifications and applications of the theory,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 5.
[CrossRef]

D. Maystre, M. Saillard, “Rigorous solution of problems of scattering by large size objects,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (Elsevier, New York, 1989), pp. 191–208.

R. Mittra, C. A. Klein, Stability and Convergence of Moment Method Solution, Vol. 3 of Topics in Applied Physics (Springer-Verlag, Berlin, 1975), pp. 129–163.

A. Tikhonov, V. Arsénine, Méthodes de Résolution de Problémes mal Posés (Mir, Moscow, 1976).

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Figures (18)

Fig. 1
Fig. 1

The physical problem: C is a cylindrical surface, M is a point on the C surface at abscissa x′, and P is point (x, y).

Fig. 2
Fig. 2

Domain of integration.

Fig. 3
Fig. 3

Real part (solid curves) and imaginary part (dashed curves) of Gν(x, 0) and N ν ( x , 0 ): (a) ν = 1.5, (b) ν = 0.075 + i1.93, (c) ν = 1.3+i7.11.

Fig. 4
Fig. 4

Case of a silver surface illuminated by a TM incident beam with wavelength λ = 0.4 μm: code for random rough surfaces (solid curve) and code for diffraction gratings (crosses).

Fig. 5
Fig. 5

Comparison between the rigorous method (solid curves) and the approximation for high-conducting metals (crosses or dashed curves) for aluminum surfaces: (a) λ = 0.8 μm, (b) λ = 0.65 μm, (c) λ = 0.4 μm.

Fig. 6
Fig. 6

Same comparison as in Fig. 5, for silver surface: λ = 0.4 μm, TM polarization.

Fig. 7
Fig. 7

Silver rough surface with rms = 0.06 μm and correlation distance T = 0.24 μm.

Fig. 8
Fig. 8

Unknown of the integral equation η(x) with long transition intervals.

Fig. 9
Fig. 9

Same as Fig. 8, with truncation at x = − 6 and x = +6.

Fig. 10
Fig. 10

Plasmon surface waves: S1 and S2 are incoming waves, and S3 and S4 are outgoing waves.

Fig. 11
Fig. 11

Unknown of the integral equation for a silver sinusoidal grating with d = 0.34 μm, h = 0.023 μm and illuminated by a TM Gaussian beam with λ = 0.4 μm and width 2.5 μm.

Fig. 12
Fig. 12

Numerical experiment for surfaces illuminated under normal incidence: perfectly conducting metal (solid curve) and aluminum (dashed curve).

Fig. 13
Fig. 13

Same as Fig. 12 for TM polarization.

Fig. 14
Fig. 14

Part of the incident energy diffracted in the half-space y > 0, versus the rms σ. The correlation distance is 0.25 μm, the wavelength is 0.4 μm, the mean incidence is 0°, and the width of one beam is 20 μm.

Fig. 15
Fig. 15

Dielectric surface (ν = 1.5) with correlation distance T = 0.25 μm illuminated by a beam of width Wx = 6μm and wavelength λ = 0.5 μm under TM polarization and mean incidence 〈θi〉 = 50°: diffraction pattern (solid curves) and reflection by a perfectly plane interface (dashed curves). The arrows show Brewster’s angle, and the rms’s are (a) 0.025 μm, (b) 0.05 μm, (c) 0.09 μm, (d) 0.25 μm.

Fig. 16
Fig. 16

Perfectly conducting metal: 〈θi〉 = 40°, rms = λ, correlation distance = λ, width of the beam = 15λ.

Fig. 17
Fig. 17

Same as Fig. 16 for aluminum at λ = 0.65 μm.

Fig. 18
Fig. 18

Same as Fig. 16 for a lossless dielectric (ν = 1.5).

Equations (37)

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F s = F F i F r .
V = F s in the air ,
V = F in the material ,
Δ V + k 0 2 V = 0 if y > f ( x ) ,
Δ V + k 0 2 ν 2 V = 0 if y < f ( x ) ,
V V + = F i + F r if y = f ( x ) ,
1 c d V d n d V + d n = d d n ( F i + F r ) if y = f ( x ) ,
Δ U + k 0 2 U = 0 if y < f ( x ) , U = U + = V + if y = f ( x ) ,
U ( x , y ) = C A ( x , y , x ) η ( x ) d x ,
A ( x , y , x ) = i 4 H 0 ( 1 ) ( k P M ) ,
η ( x ) = [ 1 + f 2 ( x ) ] 1 / 2 ( d U + d n ( x ) d U d n ( x ) ) ,
U + ( x ) = C G 1 ( x , x ) η ( x ) d x ,
[ 1 + f 2 ( x ) ] 1 / 2 d U + d n ( x ) = η ( x ) 2 + C N 1 ( x , x ) η ( x ) d x ,
G 1 ( x , x ) = i 4 H 0 ( 1 ) ( k 0 M M ) ,
N 1 ( x , x ) = i k 0 4 [ f ( x ) f ( x ) f ( x ) ( x x ) ] H 1 ( 1 ) ( k 0 M M ) M M .
V ( x , y ) = C A ν ( x , y , x ) d V d n ( x ) d x C ν ( x , y , x ) V ( x ) d x ,
A ν ( x , y , x ) = i 4 H 0 ( 1 ) ( k 0 ν P M ) ,
ν ( x , y , x ) = i k 0 ν 4 [ y f ( x ) f ( x ) ( x x ) ] × H 1 ( 1 ) ( k 0 ν P M ) P M .
u ( x ) = C P ( x , x ) φ ( x ) d x
u = P { φ } ,
c 2 G ν { η } + 1 2 G 1 { η } + c G ν { N 1 { η } } + N ν { G 1 { η } } = 1 2 ( F i + F r ) N ν ( F i + F r ) c G ν { ( 1 + f 2 ) 1 / 2 ( d F i d n + d F r d n ) } ,
G ν ( x , x ) = A ν [ x , f ( x ) , x ] , N ν ( x , x ) = ν [ x , f ( x ) , x ] .
V ( x , y ) = + B ( α ) exp ( i α x + i β y ) d α ,
β = ( k 0 2 α 2 ) 1 / 2 if | α | < k , = i ( α 2 k 2 ) 1 / 2 otherwise .
α = k 0 sin θ , β = k 0 cos θ ,
I ( θ ) = β 2 ( α ) | B ( α ) | 2 .
F i = k 0 + k 0 p ( α ) exp ( i α x i β y ) d α ,
p ( α ) = A k 0 π exp [ A 2 ( α α 0 ) 2 / 4 k 0 2 ] ,
lim x x N 1 ( x , x ) = 1 4 π f ( x ) 1 + f 2 ( x ) ,
I = C G ν ( x , x ) α ( x ) d x ,
I = α ( x ) C G ν ( x , x ) d x .
C G ν ( x , x ) i 2 k 0 ν ( 1 + f 2 ) 1 / 2 ,
C N ν ( x , x ) i 4 k 0 ν f ( x ) ( 1 + f 2 ) 3 / 2 .
G 1 ( η ) = ( F i + F r ) for TE case ,
N 1 ( η ) + η 2 = 1 + f 2 ( d F i d n + d F r d n ) for TM case .
α p = ν ( 1 + ν 2 ) 1 / 2 .
α p = 1.030 + i 0.001 .

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