Abstract

We present Monte Carlo numerical calculations for the near-field and scattered diffuse intensities of p-polarized light incident upon a random Gaussian correlated surface. We observe that the near field is dominated by two evanescent waves of momentum parallel to the surface Qs and ∼−Qs that in turn produce oscillations in the near field. This is in agreement with a theoretical diagrammatic expansion in the spatial disorder that included the so-called fan diagrams. Our calculations, within a resolution larger than 4° in the scatter angle, do not show a sharp peak or even an extra contribution in the backward diffuse scattered intensity predicted by that theoretical expansion, however. At present, we cannot make any firm conclusions about this because we have to increase resolution and decrease statistical error to elucidate this point.

© 1985 Optical Society of America

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References

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  1. N. Garcia, E. Stoll, “Monte-Carlo calculation for electromagnetic-wave scattering from random rough surfaces,” Phys. Rev. Lett. 52, 1798 (1984).We note that the angle of incidence θi, when not explicitly indicated, is 24°.
    [CrossRef]
  2. N. Garcia, “Exact calculations of p-polarized electromagnetic fields incident on grating surfaces: surface polariton resonances,” Opt. Commun. 45, 307 (1983).
    [CrossRef]
  3. J. S. Langer, T. E. Neal, “Breakdown of the concentration expansion for the impurity resistivity in metals,” Phys. Rev. Lett. 16, 984 (1966).
    [CrossRef]
  4. L. P. Gorkov, A. I. Larkin, D. E. Khemelnitskii, “Particle conductivity in a two-dimensional random potential,” Pis’ma Zh. Eksp. Teor. Fiz. 30, 248 (1979) [JEPT Lett. 30, 228 (1979)].
  5. D. Vollhardt, P. Wolfle, “Diagrammatic self-consistent treatment of the Anderson localization problem in d< 2 dimensions,” Phys. Rev. B 22, 4666 (1980).
    [CrossRef]
  6. A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866 (1985).
    [CrossRef]
  7. C. Lopez, F. J. Yndurain, N. Garcia, “Iterative series for calculating the scattering of waves from a hard corrugated surface,” Phys. Rev. B 18, 970 (1978).
    [CrossRef]
  8. S. Kirkpatrick, E. Stoll, “A very fast shift register sequence random number generator,” J. Comput. Phys. 40, 517 (1981).
    [CrossRef]
  9. N. Garcia, N. Cabrera, “New method for solving the scattering of waves from a periodic hard surface: solutions and numerical comparisons with the various formalisms,” Phys. Rev. B 18, 576 (1978).
    [CrossRef]
  10. This assumption of MMC [Eq. (22)] was pointed out by A. Baratoff, IBM Zurich Research Laboratory, 8803 Rüischlikon, Switzerland (personal communication).
  11. P. W. Anderson, “Localization redux,” Physica 117-B, 30 (1983).

1985 (1)

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

1984 (1)

N. Garcia, E. Stoll, “Monte-Carlo calculation for electromagnetic-wave scattering from random rough surfaces,” Phys. Rev. Lett. 52, 1798 (1984).We note that the angle of incidence θi, when not explicitly indicated, is 24°.
[CrossRef]

1983 (2)

N. Garcia, “Exact calculations of p-polarized electromagnetic fields incident on grating surfaces: surface polariton resonances,” Opt. Commun. 45, 307 (1983).
[CrossRef]

P. W. Anderson, “Localization redux,” Physica 117-B, 30 (1983).

1981 (1)

S. Kirkpatrick, E. Stoll, “A very fast shift register sequence random number generator,” J. Comput. Phys. 40, 517 (1981).
[CrossRef]

1980 (1)

D. Vollhardt, P. Wolfle, “Diagrammatic self-consistent treatment of the Anderson localization problem in d< 2 dimensions,” Phys. Rev. B 22, 4666 (1980).
[CrossRef]

1979 (1)

L. P. Gorkov, A. I. Larkin, D. E. Khemelnitskii, “Particle conductivity in a two-dimensional random potential,” Pis’ma Zh. Eksp. Teor. Fiz. 30, 248 (1979) [JEPT Lett. 30, 228 (1979)].

1978 (2)

N. Garcia, N. Cabrera, “New method for solving the scattering of waves from a periodic hard surface: solutions and numerical comparisons with the various formalisms,” Phys. Rev. B 18, 576 (1978).
[CrossRef]

C. Lopez, F. J. Yndurain, N. Garcia, “Iterative series for calculating the scattering of waves from a hard corrugated surface,” Phys. Rev. B 18, 970 (1978).
[CrossRef]

1966 (1)

J. S. Langer, T. E. Neal, “Breakdown of the concentration expansion for the impurity resistivity in metals,” Phys. Rev. Lett. 16, 984 (1966).
[CrossRef]

Anderson, P. W.

P. W. Anderson, “Localization redux,” Physica 117-B, 30 (1983).

Baratoff, A.

This assumption of MMC [Eq. (22)] was pointed out by A. Baratoff, IBM Zurich Research Laboratory, 8803 Rüischlikon, Switzerland (personal communication).

Cabrera, N.

N. Garcia, N. Cabrera, “New method for solving the scattering of waves from a periodic hard surface: solutions and numerical comparisons with the various formalisms,” Phys. Rev. B 18, 576 (1978).
[CrossRef]

Celli, V.

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

Garcia, N.

N. Garcia, E. Stoll, “Monte-Carlo calculation for electromagnetic-wave scattering from random rough surfaces,” Phys. Rev. Lett. 52, 1798 (1984).We note that the angle of incidence θi, when not explicitly indicated, is 24°.
[CrossRef]

N. Garcia, “Exact calculations of p-polarized electromagnetic fields incident on grating surfaces: surface polariton resonances,” Opt. Commun. 45, 307 (1983).
[CrossRef]

N. Garcia, N. Cabrera, “New method for solving the scattering of waves from a periodic hard surface: solutions and numerical comparisons with the various formalisms,” Phys. Rev. B 18, 576 (1978).
[CrossRef]

C. Lopez, F. J. Yndurain, N. Garcia, “Iterative series for calculating the scattering of waves from a hard corrugated surface,” Phys. Rev. B 18, 970 (1978).
[CrossRef]

Gorkov, L. P.

L. P. Gorkov, A. I. Larkin, D. E. Khemelnitskii, “Particle conductivity in a two-dimensional random potential,” Pis’ma Zh. Eksp. Teor. Fiz. 30, 248 (1979) [JEPT Lett. 30, 228 (1979)].

Khemelnitskii, D. E.

L. P. Gorkov, A. I. Larkin, D. E. Khemelnitskii, “Particle conductivity in a two-dimensional random potential,” Pis’ma Zh. Eksp. Teor. Fiz. 30, 248 (1979) [JEPT Lett. 30, 228 (1979)].

Kirkpatrick, S.

S. Kirkpatrick, E. Stoll, “A very fast shift register sequence random number generator,” J. Comput. Phys. 40, 517 (1981).
[CrossRef]

Langer, J. S.

J. S. Langer, T. E. Neal, “Breakdown of the concentration expansion for the impurity resistivity in metals,” Phys. Rev. Lett. 16, 984 (1966).
[CrossRef]

Larkin, A. I.

L. P. Gorkov, A. I. Larkin, D. E. Khemelnitskii, “Particle conductivity in a two-dimensional random potential,” Pis’ma Zh. Eksp. Teor. Fiz. 30, 248 (1979) [JEPT Lett. 30, 228 (1979)].

Lopez, C.

C. Lopez, F. J. Yndurain, N. Garcia, “Iterative series for calculating the scattering of waves from a hard corrugated surface,” Phys. Rev. B 18, 970 (1978).
[CrossRef]

Maradudin, A. A.

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

McGurn, A. R.

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

Neal, T. E.

J. S. Langer, T. E. Neal, “Breakdown of the concentration expansion for the impurity resistivity in metals,” Phys. Rev. Lett. 16, 984 (1966).
[CrossRef]

Stoll, E.

N. Garcia, E. Stoll, “Monte-Carlo calculation for electromagnetic-wave scattering from random rough surfaces,” Phys. Rev. Lett. 52, 1798 (1984).We note that the angle of incidence θi, when not explicitly indicated, is 24°.
[CrossRef]

S. Kirkpatrick, E. Stoll, “A very fast shift register sequence random number generator,” J. Comput. Phys. 40, 517 (1981).
[CrossRef]

Vollhardt, D.

D. Vollhardt, P. Wolfle, “Diagrammatic self-consistent treatment of the Anderson localization problem in d< 2 dimensions,” Phys. Rev. B 22, 4666 (1980).
[CrossRef]

Wolfle, P.

D. Vollhardt, P. Wolfle, “Diagrammatic self-consistent treatment of the Anderson localization problem in d< 2 dimensions,” Phys. Rev. B 22, 4666 (1980).
[CrossRef]

Yndurain, F. J.

C. Lopez, F. J. Yndurain, N. Garcia, “Iterative series for calculating the scattering of waves from a hard corrugated surface,” Phys. Rev. B 18, 970 (1978).
[CrossRef]

J. Comput. Phys. (1)

S. Kirkpatrick, E. Stoll, “A very fast shift register sequence random number generator,” J. Comput. Phys. 40, 517 (1981).
[CrossRef]

Opt. Commun. (1)

N. Garcia, “Exact calculations of p-polarized electromagnetic fields incident on grating surfaces: surface polariton resonances,” Opt. Commun. 45, 307 (1983).
[CrossRef]

Phys. Rev. B (4)

D. Vollhardt, P. Wolfle, “Diagrammatic self-consistent treatment of the Anderson localization problem in d< 2 dimensions,” Phys. Rev. B 22, 4666 (1980).
[CrossRef]

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

C. Lopez, F. J. Yndurain, N. Garcia, “Iterative series for calculating the scattering of waves from a hard corrugated surface,” Phys. Rev. B 18, 970 (1978).
[CrossRef]

N. Garcia, N. Cabrera, “New method for solving the scattering of waves from a periodic hard surface: solutions and numerical comparisons with the various formalisms,” Phys. Rev. B 18, 576 (1978).
[CrossRef]

Phys. Rev. Lett. (2)

N. Garcia, E. Stoll, “Monte-Carlo calculation for electromagnetic-wave scattering from random rough surfaces,” Phys. Rev. Lett. 52, 1798 (1984).We note that the angle of incidence θi, when not explicitly indicated, is 24°.
[CrossRef]

J. S. Langer, T. E. Neal, “Breakdown of the concentration expansion for the impurity resistivity in metals,” Phys. Rev. Lett. 16, 984 (1966).
[CrossRef]

Physica (1)

P. W. Anderson, “Localization redux,” Physica 117-B, 30 (1983).

Pis’ma Zh. Eksp. Teor. Fiz. (1)

L. P. Gorkov, A. I. Larkin, D. E. Khemelnitskii, “Particle conductivity in a two-dimensional random potential,” Pis’ma Zh. Eksp. Teor. Fiz. 30, 248 (1979) [JEPT Lett. 30, 228 (1979)].

Other (1)

This assumption of MMC [Eq. (22)] was pointed out by A. Baratoff, IBM Zurich Research Laboratory, 8803 Rüischlikon, Switzerland (personal communication).

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Figures (6)

Fig. 1
Fig. 1

(a) Values of the modulus square of the field on the boundary described by Eq. (1). The values of the parameters are = −11 + 0.33i, λ = 5154 Å, θi = 24°, CL = 355 Å, and σ = 0.00032. The average value of |Ψ2| is 172.6 with small standard deviation. (b) Same as (a) but for σ = 0.01. Note that the pronounced oscillation of the field <|Ψ2|> is now 26. To describe a situation like that in Fig. (b) in a perturbation expansion may be difficult, even if it is assumed that the series converges.

Fig. 2
Fig. 2

Same as in Fig. 1(a) for θi = 20°, a = 8000 Å, and <|Ψ2|> = 3.5. The other parameters are given in the text. Results are shown for two typical random profiles.

Fig. 3
Fig. 3

(a) Same as Fig. 2 for a = 17 000 Å (see also text). (b) Histogram of scattering amplitudes versus parallel momentum Q in units of k0. When the modulus of Q/k0 exceeds 1, the waves are evanescent. Note the two almost symmetric prominent peaks that cause the oscillations in (a). We stress that this behavior of the near field dies exponentially away from the boundary C(x).

Fig. 4
Fig. 4

Same as Fig. 3 for a = 20 082 Å.

Fig. 5
Fig. 5

(a) Same as Fig. 3 for a = 31 000 Å. (c) Same as (b) for J = 100 instead of J = 50. Note that the spectrum is practically the same so that the calculation converges.

Fig. 6
Fig. 6

Plot of the diffuse reflectivity ∂R/∂θ versus θ for different incident angles (solid curve, 20°; dashed curve, 40°; dotted curve, 50°). The coherent specular peak is 3 or 4 orders of magnitude larger and has not been plotted. The parameters of the calculation are given in the text and correspond exactly to those of MMC in Fig. 2(a) of Ref. 6. Within our resolution of 4°, we do not observe any extra intensity in the backward intensity, but the general intensity is in remarkable agreement with that of Ref. 6.

Equations (5)

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C ( x ) = a [ H cos ( 2 π x a ) + σ Z c ( x ) ] .
Z c ( x ) = 0 Z c ( x ) Z c ( y ) = exp [ ( x y ) 2 / 2 C L 2 ) .
k 0 = ( K , q 0 ) ,
k Q = ( Q , q Q ) , q Q = ( k 0 2 Q 2 ) 1 / 2 ,
I ( Q ) = 1 N ( q Q q 0 ) i = i N | A Q i | 2 ,

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