Abstract

The exact solution of scattering by a two-dimensional random rough surface (three-dimensional scattering problem) of an area of 80 square wavelengths with 4096 surface unknowns is computed, and the results show backscattering enhancement. The computation is based on a new numerical method called the sparse-matrix flat-surface iterative approach. The approach decomposes the matrix of the integral equation as a sum of a sparse matrix, a flat-surface block Toeplitz matrix, and a weak remainder that is followed by an iterative solution until convergence is achieved.

© 1994 Optical Society of America

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References

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  1. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).
  2. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, San Diego, Calif., 1978).
  3. G. S. Brown, “Simplifications in the stochastic Fourier transform approach to random surface scattering,”IEEE Trans. Antennas Propag. AP-33, 48–55 (1985).
    [CrossRef]
  4. L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).
  5. S. John, “Localization of light,” Phys. Today 44(5), 32–40 (1991).
    [CrossRef]
  6. A. Ishimaru, J. S. Chen, “Scattering from very rough surfaces based on the modified second-order Kirchhoff approximation with angular and propagation shadowing,”J. Acoust. Soc. Am. 88, 1877–1883 (1990).
    [CrossRef]
  7. R. M. Axline, A. K. Fung, “Numerical computation of scattering from a perfectly conducting random surface,”IEEE Trans. Antennas Propag. AP-26, 483–488 (1978).
  8. N. Garcia, E. Stoll, “Monte Carlo calculation for electromagnetic-wave scattering from random rough surfaces,” Phys. Rev. Lett. 52, 1798–1801 (1984).
    [CrossRef]
  9. E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,”J. Acoust. Soc. Am. 82, 78–92 (1988).
    [CrossRef]
  10. J. S. Chen, A. Ishimaru, “Numerical simulation of the second order Kirchhoff approximation from very rough surfaces and study of backscattering enhancement,”J. Acoust. Soc. Am. 88, 1846–1850 (1990).
    [CrossRef]
  11. M. Nieto-Vesperinas, J. M. Soto-Crespo, “Monte Carlo simulations for scattering of electromagnetic waves from perfectly conducting random rough surfaces,” Opt. Lett. 12, 979–981 (1987).
    [CrossRef] [PubMed]
  12. P. Phu, A. Ishimaru, Y. Kuga, “Controlled millimeter wave experiments and numerical simulations on the enhanced backscattering from one-dimensional very rough surfaces,” Radio Sci. (to be published).
  13. R. Harrington, Field Computation by Moment Methods (Macmillan, New York, 1968).
  14. L. Tsang, C. H. Chan, H. Sangani, “Application of a banded matrix iterative approach to Monte Carlo simulations of scattering of waves by random rough surface problems: TM case,” Microwave Opt. Tech. Lett. 6, 148–151 (1993).
    [CrossRef]
  15. C. H. Chan, R. Mittra, “Some recent developments in iterative techniques for solving electromagnetic boundary value problems,” Radio Sci. 22, 929–934 (1987).
    [CrossRef]
  16. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975).

1993 (1)

L. Tsang, C. H. Chan, H. Sangani, “Application of a banded matrix iterative approach to Monte Carlo simulations of scattering of waves by random rough surface problems: TM case,” Microwave Opt. Tech. Lett. 6, 148–151 (1993).
[CrossRef]

1991 (1)

S. John, “Localization of light,” Phys. Today 44(5), 32–40 (1991).
[CrossRef]

1990 (2)

A. Ishimaru, J. S. Chen, “Scattering from very rough surfaces based on the modified second-order Kirchhoff approximation with angular and propagation shadowing,”J. Acoust. Soc. Am. 88, 1877–1883 (1990).
[CrossRef]

J. S. Chen, A. Ishimaru, “Numerical simulation of the second order Kirchhoff approximation from very rough surfaces and study of backscattering enhancement,”J. Acoust. Soc. Am. 88, 1846–1850 (1990).
[CrossRef]

1988 (1)

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,”J. Acoust. Soc. Am. 82, 78–92 (1988).
[CrossRef]

1987 (2)

M. Nieto-Vesperinas, J. M. Soto-Crespo, “Monte Carlo simulations for scattering of electromagnetic waves from perfectly conducting random rough surfaces,” Opt. Lett. 12, 979–981 (1987).
[CrossRef] [PubMed]

C. H. Chan, R. Mittra, “Some recent developments in iterative techniques for solving electromagnetic boundary value problems,” Radio Sci. 22, 929–934 (1987).
[CrossRef]

1985 (1)

G. S. Brown, “Simplifications in the stochastic Fourier transform approach to random surface scattering,”IEEE Trans. Antennas Propag. AP-33, 48–55 (1985).
[CrossRef]

1984 (1)

N. Garcia, E. Stoll, “Monte Carlo calculation for electromagnetic-wave scattering from random rough surfaces,” Phys. Rev. Lett. 52, 1798–1801 (1984).
[CrossRef]

1978 (1)

R. M. Axline, A. K. Fung, “Numerical computation of scattering from a perfectly conducting random surface,”IEEE Trans. Antennas Propag. AP-26, 483–488 (1978).

Axline, R. M.

R. M. Axline, A. K. Fung, “Numerical computation of scattering from a perfectly conducting random surface,”IEEE Trans. Antennas Propag. AP-26, 483–488 (1978).

Beckmann, P.

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

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975).

Brown, G. S.

G. S. Brown, “Simplifications in the stochastic Fourier transform approach to random surface scattering,”IEEE Trans. Antennas Propag. AP-33, 48–55 (1985).
[CrossRef]

Chan, C. H.

L. Tsang, C. H. Chan, H. Sangani, “Application of a banded matrix iterative approach to Monte Carlo simulations of scattering of waves by random rough surface problems: TM case,” Microwave Opt. Tech. Lett. 6, 148–151 (1993).
[CrossRef]

C. H. Chan, R. Mittra, “Some recent developments in iterative techniques for solving electromagnetic boundary value problems,” Radio Sci. 22, 929–934 (1987).
[CrossRef]

Chen, J. S.

A. Ishimaru, J. S. Chen, “Scattering from very rough surfaces based on the modified second-order Kirchhoff approximation with angular and propagation shadowing,”J. Acoust. Soc. Am. 88, 1877–1883 (1990).
[CrossRef]

J. S. Chen, A. Ishimaru, “Numerical simulation of the second order Kirchhoff approximation from very rough surfaces and study of backscattering enhancement,”J. Acoust. Soc. Am. 88, 1846–1850 (1990).
[CrossRef]

Fung, A. K.

R. M. Axline, A. K. Fung, “Numerical computation of scattering from a perfectly conducting random surface,”IEEE Trans. Antennas Propag. AP-26, 483–488 (1978).

Garcia, N.

N. Garcia, E. Stoll, “Monte Carlo calculation for electromagnetic-wave scattering from random rough surfaces,” Phys. Rev. Lett. 52, 1798–1801 (1984).
[CrossRef]

Harrington, R.

R. Harrington, Field Computation by Moment Methods (Macmillan, New York, 1968).

Ishimaru, A.

J. S. Chen, A. Ishimaru, “Numerical simulation of the second order Kirchhoff approximation from very rough surfaces and study of backscattering enhancement,”J. Acoust. Soc. Am. 88, 1846–1850 (1990).
[CrossRef]

A. Ishimaru, J. S. Chen, “Scattering from very rough surfaces based on the modified second-order Kirchhoff approximation with angular and propagation shadowing,”J. Acoust. Soc. Am. 88, 1877–1883 (1990).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, San Diego, Calif., 1978).

P. Phu, A. Ishimaru, Y. Kuga, “Controlled millimeter wave experiments and numerical simulations on the enhanced backscattering from one-dimensional very rough surfaces,” Radio Sci. (to be published).

John, S.

S. John, “Localization of light,” Phys. Today 44(5), 32–40 (1991).
[CrossRef]

Kong, J. A.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

Kuga, Y.

P. Phu, A. Ishimaru, Y. Kuga, “Controlled millimeter wave experiments and numerical simulations on the enhanced backscattering from one-dimensional very rough surfaces,” Radio Sci. (to be published).

Mittra, R.

C. H. Chan, R. Mittra, “Some recent developments in iterative techniques for solving electromagnetic boundary value problems,” Radio Sci. 22, 929–934 (1987).
[CrossRef]

Nieto-Vesperinas, M.

Phu, P.

P. Phu, A. Ishimaru, Y. Kuga, “Controlled millimeter wave experiments and numerical simulations on the enhanced backscattering from one-dimensional very rough surfaces,” Radio Sci. (to be published).

Sangani, H.

L. Tsang, C. H. Chan, H. Sangani, “Application of a banded matrix iterative approach to Monte Carlo simulations of scattering of waves by random rough surface problems: TM case,” Microwave Opt. Tech. Lett. 6, 148–151 (1993).
[CrossRef]

Shin, R. T.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

Soto-Crespo, J. M.

Spizzichino, A.

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

Stoll, E.

N. Garcia, E. Stoll, “Monte Carlo calculation for electromagnetic-wave scattering from random rough surfaces,” Phys. Rev. Lett. 52, 1798–1801 (1984).
[CrossRef]

Thorsos, E. I.

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,”J. Acoust. Soc. Am. 82, 78–92 (1988).
[CrossRef]

Tsang, L.

L. Tsang, C. H. Chan, H. Sangani, “Application of a banded matrix iterative approach to Monte Carlo simulations of scattering of waves by random rough surface problems: TM case,” Microwave Opt. Tech. Lett. 6, 148–151 (1993).
[CrossRef]

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975).

IEEE Trans. Antennas Propag. (2)

G. S. Brown, “Simplifications in the stochastic Fourier transform approach to random surface scattering,”IEEE Trans. Antennas Propag. AP-33, 48–55 (1985).
[CrossRef]

R. M. Axline, A. K. Fung, “Numerical computation of scattering from a perfectly conducting random surface,”IEEE Trans. Antennas Propag. AP-26, 483–488 (1978).

J. Acoust. Soc. Am. (3)

A. Ishimaru, J. S. Chen, “Scattering from very rough surfaces based on the modified second-order Kirchhoff approximation with angular and propagation shadowing,”J. Acoust. Soc. Am. 88, 1877–1883 (1990).
[CrossRef]

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,”J. Acoust. Soc. Am. 82, 78–92 (1988).
[CrossRef]

J. S. Chen, A. Ishimaru, “Numerical simulation of the second order Kirchhoff approximation from very rough surfaces and study of backscattering enhancement,”J. Acoust. Soc. Am. 88, 1846–1850 (1990).
[CrossRef]

Microwave Opt. Tech. Lett. (1)

L. Tsang, C. H. Chan, H. Sangani, “Application of a banded matrix iterative approach to Monte Carlo simulations of scattering of waves by random rough surface problems: TM case,” Microwave Opt. Tech. Lett. 6, 148–151 (1993).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

N. Garcia, E. Stoll, “Monte Carlo calculation for electromagnetic-wave scattering from random rough surfaces,” Phys. Rev. Lett. 52, 1798–1801 (1984).
[CrossRef]

Phys. Today (1)

S. John, “Localization of light,” Phys. Today 44(5), 32–40 (1991).
[CrossRef]

Radio Sci. (1)

C. H. Chan, R. Mittra, “Some recent developments in iterative techniques for solving electromagnetic boundary value problems,” Radio Sci. 22, 929–934 (1987).
[CrossRef]

Other (6)

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975).

P. Phu, A. Ishimaru, Y. Kuga, “Controlled millimeter wave experiments and numerical simulations on the enhanced backscattering from one-dimensional very rough surfaces,” Radio Sci. (to be published).

R. Harrington, Field Computation by Moment Methods (Macmillan, New York, 1968).

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

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

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, San Diego, Calif., 1978).

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Figures (3)

Fig. 1
Fig. 1

Comparison of normalized bistatic scattering coefficients of 2-D and 1-D random rough surfaces (310 and 4000 realizations, respectively) for a rms height of 0.5 wavelength and correlation length of 0.707 wavelength with incident angle θ0 = 20°.

Fig. 2
Fig. 2

Convergence of bistatic scattering coefficient with the number of realizations for a 2-D rough surface. Four cases are shown: 155, 225, 275, and 310 realizations. The cases of 275 and 310 realizations overlap each other.

Fig. 3
Fig. 3

Comparison between the SMFSIA and the second-order Kirchhoff method. The second-order Kirchhoff method result is based on 360 realizations.

Tables (2)

Tables Icon

Table 1 Bistatic Coefficients of the 2-D Rough Surface of Fig. 1

Tables Icon

Table 2 Convergence of the SMFSIA of 2-D Surface of a Single Realization

Equations (20)

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ψ inc ( x , y , z ) = exp [ - i k ( cos θ 0 z - x sin θ 0 cos ϕ 0 - y sin θ 0 sin ϕ 0 ) ( 1 + ω ) ] exp ( - t ) ,
t x = ( cos θ 0 cos ϕ 0 x + cos θ 0 sin ϕ 0 y + sin θ 0 z ) 2 g 2 cos 2 θ 0 ,
t y = ( - sin ϕ 0 x + cos ϕ 0 y ) 2 g 2 ,
ω = 1 k 2 [ ( 2 t x - 1 ) g 2 cos 2 θ 0 + ( 2 t y - 1 ) g 2 ] ,
0 = ψ inc ( r ) - d x d y G 0 [ x , y , f ( x , y ) ; x , y , f ( x , y ) ] U ( x , y ) ,
U ( x , y ) = ψ ( r ) n [ 1 + ( f x ) 2 + ( f y ) 2 ] 1 / 2 .
G FS ( x - x , y - y ) = G 0 ( x , y , z = 0 ; x , y , z = 0 ) = exp ( i k ρ ) 4 π ρ .
ρ < r d d x d y G 0 ( r , r ) U ( x , y ) + ρ r d d x d y G FS ( x - x , y - y ) U ( x , y ) = ψ inc ( r ) - ρ r d d x d y [ G 0 ( r , r ) - G FS ( x - x , y - y ) ] U ( x , y ) .
L x = b ,
L = L ( s ) + L ( FS ) + L ( w ) ,
[ L ( s ) + L ( FS ) ] x ( 1 ) = b ,
[ L ( s ) + L ( FS ) ] x ( n + 1 ) = b ( n + 1 ) ,
[ L ( s ) + L ( FS ) ] x ( u ) = b ( u ) ,
E ( n ) = [ L x ( n ) - b b ] 1 / 2 × 100 % .
σ ( k ^ s ) = F ( k ^ s ) 2 8 π 3 g 2 cos θ 0 [ 1 - ( 1 + cos 2 θ 0 + 2 tan 2 θ 0 ) 2 k 2 g 2 cos 2 θ 0 ] ,
F ( k ^ s ) = - d x d y U ( x , y ) exp [ - i k x sin θ s cos ϕ s - i k y sin θ s sin ϕ s - i k f ( x , y ) cos θ s ] .
L x ( n ) - b = [ L ( s ) + L ( F S ) ] x ( n ) - [ b - L ( w ) x ( n ) ] = b ( n ) - b ( n + 1 ) .
O ( L ( w ) x ) = O { ρ > r d d x d y × [ exp [ i k ( x d 2 + y d 2 + z d 2 ) 1 / 2 ] 4 π ( x d 2 + y d 2 + z d 2 ) 1 / 2 - exp ( i k ρ ) 4 π ρ ] U ( x , y ) } ,
O [ L ( w ) x ] = O [ ρ > r d d x d y exp ( i k ρ ) 4 π ρ ( i k z d 2 2 ρ ) U ( x , y ) ] = O [ r d d ρ ρ 0 2 π d ϕ exp ( i k ρ ) 4 π ρ ( i k z d 2 2 ρ ) U ( x , y ) ] = O [ r d d ρ ρ exp ( i k ρ ) 4 ρ 2 i k z d 2 U ( x , y ) ] .
O [ L ( w ) x ] = O ( k h 2 r d ) ,

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