Abstract

Using a recently discovered short-coupling-range phenomenon, we have completed a computer code that is able to compute rigorously the speckle pattern diffracted by a random rough surface with an arbitrary number of asperities, illuminated by a laser beam. This computer code is used to determine the condition under which the mean intensity at infinity of the speckle pattern can be precisely obtained.

© 1986 Optical Society of America

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References

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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. D. Maystre, “Rigorous theory of light scattering from rough surfaces,” J. Opt. (Paris) 15, 43–51 (1984).
    [CrossRef]
  3. D. Maystre, O. Mata Mendez, A. Roger, “A new electromagnetic theory for scattering from shallow rough surfaces,” Opt. Acta 30, 1707–1723 (1983).
    [CrossRef]
  4. J. P. Rossi, D. Maystre, “Speckle: a rigorous electromagnetic theory for shallow surfaces,” Opt. Acta 32, 1427–1444 (1985).
    [CrossRef]
  5. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).
  6. P. Beckmann, “Scattering of light by rough surfaces,” in Progress in Optics VI, E. Wolf, ed. (North-Holland, Amsterdam, 1967).
    [CrossRef]
  7. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics (Springer-Verlag, New York, 1975).
    [CrossRef]
  8. J. Shen, A. A. Maradudin, “Multiple scattering of waves from random rough surfaces,” Phys. Rev. B. 22, 4234–4240 (1980).
    [CrossRef]
  9. N. Nieto-Vesperinas, “Depolarization of electromagnetic waves scattered from slightly rough random surfaces: a study by means of the extinction theorem,” J. Opt. Soc. Am. 72, 539–547 (1982).
    [CrossRef]
  10. O. Mata Mendez, A. Roger, D. Maystre, “Numerical solution for an inverse scattering problem of non-periodic rough surfaces,” Appl. Phys. B 32, 199–206 (1984).
    [CrossRef]
  11. D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics XXI, E. Wolf, ed. (North-Holland, Amsterdam, 1984).
    [CrossRef]
  12. Y. Wang, L. Wolfe, “Scattering from microrough surfaces: comparison of theory and experiment,” J. Opt. Soc. Am. 73, 1596–1602 (1983).
    [CrossRef]

1985 (1)

J. P. Rossi, D. Maystre, “Speckle: a rigorous electromagnetic theory for shallow surfaces,” Opt. Acta 32, 1427–1444 (1985).
[CrossRef]

1984 (2)

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

O. Mata Mendez, A. Roger, D. Maystre, “Numerical solution for an inverse scattering problem of non-periodic rough surfaces,” Appl. Phys. B 32, 199–206 (1984).
[CrossRef]

1983 (3)

Y. Wang, L. Wolfe, “Scattering from microrough surfaces: comparison of theory and experiment,” J. Opt. Soc. Am. 73, 1596–1602 (1983).
[CrossRef]

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

D. Maystre, O. Mata Mendez, A. Roger, “A new electromagnetic theory for scattering from shallow rough surfaces,” Opt. Acta 30, 1707–1723 (1983).
[CrossRef]

1982 (1)

1980 (1)

J. Shen, A. A. Maradudin, “Multiple scattering of waves from random rough surfaces,” Phys. Rev. B. 22, 4234–4240 (1980).
[CrossRef]

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

P. Beckmann, “Scattering of light by rough surfaces,” in Progress in Optics VI, E. Wolf, ed. (North-Holland, Amsterdam, 1967).
[CrossRef]

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics (Springer-Verlag, New York, 1975).
[CrossRef]

Maradudin, A. A.

J. Shen, A. A. Maradudin, “Multiple scattering of waves from random rough surfaces,” Phys. Rev. B. 22, 4234–4240 (1980).
[CrossRef]

Mata Mendez, O.

O. Mata Mendez, A. Roger, D. Maystre, “Numerical solution for an inverse scattering problem of non-periodic rough surfaces,” Appl. Phys. B 32, 199–206 (1984).
[CrossRef]

D. Maystre, O. Mata Mendez, A. Roger, “A new electromagnetic theory for scattering from shallow rough surfaces,” Opt. Acta 30, 1707–1723 (1983).
[CrossRef]

Maystre, D.

J. P. Rossi, D. Maystre, “Speckle: a rigorous electromagnetic theory for shallow surfaces,” Opt. Acta 32, 1427–1444 (1985).
[CrossRef]

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

O. Mata Mendez, A. Roger, D. Maystre, “Numerical solution for an inverse scattering problem of non-periodic rough surfaces,” Appl. Phys. B 32, 199–206 (1984).
[CrossRef]

D. Maystre, O. Mata Mendez, A. Roger, “A new electromagnetic theory for scattering from shallow rough surfaces,” Opt. Acta 30, 1707–1723 (1983).
[CrossRef]

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

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics XXI, E. Wolf, ed. (North-Holland, Amsterdam, 1984).
[CrossRef]

Nieto-Vesperinas, N.

Roger, A.

O. Mata Mendez, A. Roger, D. Maystre, “Numerical solution for an inverse scattering problem of non-periodic rough surfaces,” Appl. Phys. B 32, 199–206 (1984).
[CrossRef]

D. Maystre, O. Mata Mendez, A. Roger, “A new electromagnetic theory for scattering from shallow rough surfaces,” Opt. Acta 30, 1707–1723 (1983).
[CrossRef]

Rossi, J. P.

J. P. Rossi, D. Maystre, “Speckle: a rigorous electromagnetic theory for shallow surfaces,” Opt. Acta 32, 1427–1444 (1985).
[CrossRef]

Shen, J.

J. Shen, A. A. Maradudin, “Multiple scattering of waves from random rough surfaces,” Phys. Rev. B. 22, 4234–4240 (1980).
[CrossRef]

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

Wang, Y.

Wolfe, L.

Appl. Phys. B (1)

O. Mata Mendez, A. Roger, D. Maystre, “Numerical solution for an inverse scattering problem of non-periodic rough surfaces,” Appl. Phys. B 32, 199–206 (1984).
[CrossRef]

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. Opt. (Paris) (1)

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

J. Opt. Soc. Am. (2)

Opt. Acta (2)

D. Maystre, O. Mata Mendez, A. Roger, “A new electromagnetic theory for scattering from shallow rough surfaces,” Opt. Acta 30, 1707–1723 (1983).
[CrossRef]

J. P. Rossi, D. Maystre, “Speckle: a rigorous electromagnetic theory for shallow surfaces,” Opt. Acta 32, 1427–1444 (1985).
[CrossRef]

Phys. Rev. B. (1)

J. Shen, A. A. Maradudin, “Multiple scattering of waves from random rough surfaces,” Phys. Rev. B. 22, 4234–4240 (1980).
[CrossRef]

Other (4)

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics XXI, E. Wolf, ed. (North-Holland, Amsterdam, 1984).
[CrossRef]

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

P. Beckmann, “Scattering of light by rough surfaces,” in Progress in Optics VI, E. Wolf, ed. (North-Holland, Amsterdam, 1967).
[CrossRef]

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics (Springer-Verlag, New York, 1975).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Our theoretical model. An incident (TE- or TM-polarized) electromagnetic field illuminates a perfectly reflecting mirror that has been modulated on a finite length L.

Fig. 2
Fig. 2

Validity of the Kirchhoff approximation. Case 1, in the circle one wavelength in radius, the difference between the actual surface and Γ is small compared with the difference between the actual surface and Γ; the Kirchhoff approximation can be used. Case 2, the difference between the actual surface and Γ has, at certain points, the same order of magnitude as the difference between the actual surface and Γ, the Kirchhoff approximation cannot be used to find the scattered light.

Fig. 3
Fig. 3

The short-coupling-range phenomenon. The random rough surfaces S and S are identical between the two vertical solid lines. As a consequence, the computed surface-current densities j and j are identical between the two vertical dashed lines. Re(j) and Re(j) are the real part of the current densities on S and S.

Fig. 4
Fig. 4

Generalization of the first theory: If δ and δ are greater than one wavelength (TE polarization), the surface-current density j(M) is nearly equal to the surface-current density j(M) at M when S is illuminated under the same conditions as S.

Fig. 5
Fig. 5

Mean intensity at infinity diffracted by 2D, perfectly conducting random microrough surfaces with Gaussian correlations, illuminated under normal incidence by a Gaussian beam of wavelength λ = 1 μm. The Gaussian beam width is such that only the modulated part of the random microrough surface is illuminated; the surface-current density is negligible on the two plateaus, which have no influence on the diffracted field. The mean width d of the asperities is equal to 1.25 μm and the mean-square height equal to 0.212 μm. Q denotes the number of asperities illuminated by the Gaussian beam, and L the length of the illuminated part of the surface. A1–A3, P = 1; B1–B3, P = 10; C1–C3, P = 100. A1–C1, Q = 3; A2–C2, Q = 10; A3–C3, Q = 20.

Fig. 6
Fig. 6

Same as Fig. 5, but d = 2.5 μm.

Fig. 7
Fig. 7

Same as Fig. 5, but d = 5 μm.

Equations (12)

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y = f ( x ) ,
E i = exp [ i k ( α x β y ) ] ,
α = sin θ , β = cos θ .
E i = k + k p ( α ) exp [ i k ( α x i β y ) ] d α ,
p ( α ) = exp [ ( α α ) 2 / C 2 ] , β = ( 1 α 2 ) 1 / 2 ,
E d = + q ( α ) exp [ i k ( α x + β y ) ] d α ,
β = ( 1 α 2 ) 1 / 2 , if | α | < 1 , = i [ ( α 2 1 ) 1 / 2 ] , if | α | > 1.
I ( α ) = β | q ( α ) | 2 / I i ,
I i = 1 1 β | p ( α ) | 2 d α .
1 1 I ( α ) d α = 1 ;
S ( ν ) d ν = h 2 , ν 2 S ( ν ) d ν = h 2 / d 2 ,
y = h 2 cos ( 2 π d x ) ,

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