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)

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

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

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]

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