Abstract

Backscattering enhancement of electromagnetic wave scattering from a perfectly conducting two-dimensional random rough surface (three-dimensional scattering problem) is studied with Monte Carlo simulations. The magnetic-field integral equation formulation is used with the method of moments. The solution of the matrix equation is calculated exactly with an efficient method known as the sparse-matrix flat-surface iterative approach. Numerical examples are illustrated with 32,768 surface unknowns, surface areas between 256 and 1024 square wavelengths, rms heights of 0.5 and 1 wavelength, and as many as 1000 realizations. The bistatic scattering simulations show backscattering enhancement for both copolarized and cross-polarized components. Comparisons are made with controlled laboratory experimental data for which the random rough surfaces are fabricated with prescribed properties of a rms height of 1 wavelength and a correlation length equal to 2 wavelengths. Comparisons are made between simulations and experimental data for the absolute value of the bistatic scattering coefficient. The copolarized scattering coefficient is in good agreement, and the cross-polarized scattering coefficient is in excellent agreement.

© 1995 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. II.
  2. L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley-Interscience, New York, 1985).
  3. Y. Kuga, J. S. Colburn, P. Phu, “Millimeter-wave scattering from one-dimensional surface of different surface correlation functions,” Waves Random Media 3, 101–110 (1993).
    [Crossref]
  4. R. M. Axline, A. K. Fung, “Numerical computation of scattering from a perfectly conducting random surface,” IEEE Trans. Antennas Propag. AP-26, 482–488 (1978).
    [Crossref]
  5. 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]
  6. M. Nieto-Vesperinas, J. A. Sanchez-Gil, “Intensity angular correlation of light multiply scattered from random rough surfaces,” J. Opt. Soc. Am. A 10, 150–157 (1993).
    [Crossref]
  7. P. Tran, V. Celli, A. A. Maradudin, “Electromagnetic scattering from a two-dimensional, randomly rough, perfectly conducting surface: iterative methods,” J. Opt. Soc. Am. A 11, 1686–1689 (1994).
    [Crossref]
  8. D. J. Wingham, R. H. Devayya, “A note on the use of the Neumann expansion in calculating the scatter from rough surfaces,” IEEE Trans. Antennas Propag. 40, 560–563 (1992).
    [Crossref]
  9. L. Tsang, C. H. Chan, K. Pak, “Monte Carlo simulations of a two-dimensional random rough surface using the sparse-matrix flat-surface iterative approach,” Electron. Lett. 29, 1153–1154 (1993).
    [Crossref]
  10. L. Tsang, C. H. Chan, K. Pak, “Backscattering enhancement of a two-dimensional random rough surface (three-dimensional scattering) based on Monte Carlo simulations,” J. Opt. Soc. Am. A 11, 711–715 (1994).
    [Crossref]
  11. E. I. Thorsos, D. Jackson, “Studies of scattering theory using numerical methods,” Waves Random Media 1, 165–190 (1991).
    [Crossref]
  12. C. H. Chan, R. Mittra, “Some recent developments in iterative techniques for solving electromagnetic boundary value problems,” Radio Sci. 22, 929–934 (1987).
    [Crossref]
  13. Y. Kuga, P. Phu, “Experimental techniques in random media and rough surfaces,” in Progress in Electromagnetics Research, L. Tsang, M. Tateiba, eds. (Elsevier, Cambridge, Mass., to be published).
  14. A. Ishimaru, C. Le, Y. Kuga, L. Ailes-Sengers, T. K. Chan, “Polarimetric scattering theory for large slope rough surfaces,” in Progress in Electromagnetics Research, L. Tsang, M. Tateiba, eds. (Elsevier, Cambridge, Mass., to be published).
  15. L. Tsang, C. H. Chan, K. Pak, H. Sangani, A. Ishimaru, P. Phu, “Monte Carlo simulations of large-scale composite random rough surface scattering based on the banded matrix iterative approach,” J. Opt. Soc. Am. A 11, 691–696 (1994).
    [Crossref]
  16. P. Tran, A. A. Maradudin, “The scattering of electromagnetic waves from a 2D metallic surface,” Opt. Commun. 110, 269–273 (1994).
    [Crossref]

1994 (4)

1993 (3)

L. Tsang, C. H. Chan, K. Pak, “Monte Carlo simulations of a two-dimensional random rough surface using the sparse-matrix flat-surface iterative approach,” Electron. Lett. 29, 1153–1154 (1993).
[Crossref]

Y. Kuga, J. S. Colburn, P. Phu, “Millimeter-wave scattering from one-dimensional surface of different surface correlation functions,” Waves Random Media 3, 101–110 (1993).
[Crossref]

M. Nieto-Vesperinas, J. A. Sanchez-Gil, “Intensity angular correlation of light multiply scattered from random rough surfaces,” J. Opt. Soc. Am. A 10, 150–157 (1993).
[Crossref]

1992 (1)

D. J. Wingham, R. H. Devayya, “A note on the use of the Neumann expansion in calculating the scatter from rough surfaces,” IEEE Trans. Antennas Propag. 40, 560–563 (1992).
[Crossref]

1991 (1)

E. I. Thorsos, D. Jackson, “Studies of scattering theory using numerical methods,” Waves Random Media 1, 165–190 (1991).
[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 (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]

1978 (1)

R. M. Axline, A. K. Fung, “Numerical computation of scattering from a perfectly conducting random surface,” IEEE Trans. Antennas Propag. AP-26, 482–488 (1978).
[Crossref]

Ailes-Sengers, L.

A. Ishimaru, C. Le, Y. Kuga, L. Ailes-Sengers, T. K. Chan, “Polarimetric scattering theory for large slope rough surfaces,” in Progress in Electromagnetics Research, L. Tsang, M. Tateiba, eds. (Elsevier, Cambridge, Mass., to be published).

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, 482–488 (1978).
[Crossref]

Celli, V.

Chan, C. H.

L. Tsang, C. H. Chan, K. Pak, H. Sangani, A. Ishimaru, P. Phu, “Monte Carlo simulations of large-scale composite random rough surface scattering based on the banded matrix iterative approach,” J. Opt. Soc. Am. A 11, 691–696 (1994).
[Crossref]

L. Tsang, C. H. Chan, K. Pak, “Backscattering enhancement of a two-dimensional random rough surface (three-dimensional scattering) based on Monte Carlo simulations,” J. Opt. Soc. Am. A 11, 711–715 (1994).
[Crossref]

L. Tsang, C. H. Chan, K. Pak, “Monte Carlo simulations of a two-dimensional random rough surface using the sparse-matrix flat-surface iterative approach,” Electron. Lett. 29, 1153–1154 (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]

Chan, T. K.

A. Ishimaru, C. Le, Y. Kuga, L. Ailes-Sengers, T. K. Chan, “Polarimetric scattering theory for large slope rough surfaces,” in Progress in Electromagnetics Research, L. Tsang, M. Tateiba, eds. (Elsevier, Cambridge, Mass., to be published).

Colburn, J. S.

Y. Kuga, J. S. Colburn, P. Phu, “Millimeter-wave scattering from one-dimensional surface of different surface correlation functions,” Waves Random Media 3, 101–110 (1993).
[Crossref]

Devayya, R. H.

D. J. Wingham, R. H. Devayya, “A note on the use of the Neumann expansion in calculating the scatter from rough surfaces,” IEEE Trans. Antennas Propag. 40, 560–563 (1992).
[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, 482–488 (1978).
[Crossref]

Ishimaru, A.

L. Tsang, C. H. Chan, K. Pak, H. Sangani, A. Ishimaru, P. Phu, “Monte Carlo simulations of large-scale composite random rough surface scattering based on the banded matrix iterative approach,” J. Opt. Soc. Am. A 11, 691–696 (1994).
[Crossref]

A. Ishimaru, C. Le, Y. Kuga, L. Ailes-Sengers, T. K. Chan, “Polarimetric scattering theory for large slope rough surfaces,” in Progress in Electromagnetics Research, L. Tsang, M. Tateiba, eds. (Elsevier, Cambridge, Mass., to be published).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. II.

Jackson, D.

E. I. Thorsos, D. Jackson, “Studies of scattering theory using numerical methods,” Waves Random Media 1, 165–190 (1991).
[Crossref]

Kong, J. A.

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

Kuga, Y.

Y. Kuga, J. S. Colburn, P. Phu, “Millimeter-wave scattering from one-dimensional surface of different surface correlation functions,” Waves Random Media 3, 101–110 (1993).
[Crossref]

Y. Kuga, P. Phu, “Experimental techniques in random media and rough surfaces,” in Progress in Electromagnetics Research, L. Tsang, M. Tateiba, eds. (Elsevier, Cambridge, Mass., to be published).

A. Ishimaru, C. Le, Y. Kuga, L. Ailes-Sengers, T. K. Chan, “Polarimetric scattering theory for large slope rough surfaces,” in Progress in Electromagnetics Research, L. Tsang, M. Tateiba, eds. (Elsevier, Cambridge, Mass., to be published).

Le, C.

A. Ishimaru, C. Le, Y. Kuga, L. Ailes-Sengers, T. K. Chan, “Polarimetric scattering theory for large slope rough surfaces,” in Progress in Electromagnetics Research, L. Tsang, M. Tateiba, eds. (Elsevier, Cambridge, Mass., to be published).

Maradudin, A. A.

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.

Pak, K.

Phu, P.

L. Tsang, C. H. Chan, K. Pak, H. Sangani, A. Ishimaru, P. Phu, “Monte Carlo simulations of large-scale composite random rough surface scattering based on the banded matrix iterative approach,” J. Opt. Soc. Am. A 11, 691–696 (1994).
[Crossref]

Y. Kuga, J. S. Colburn, P. Phu, “Millimeter-wave scattering from one-dimensional surface of different surface correlation functions,” Waves Random Media 3, 101–110 (1993).
[Crossref]

Y. Kuga, P. Phu, “Experimental techniques in random media and rough surfaces,” in Progress in Electromagnetics Research, L. Tsang, M. Tateiba, eds. (Elsevier, Cambridge, Mass., to be published).

Sanchez-Gil, J. A.

Sangani, H.

Shin, R. T.

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

Thorsos, E. I.

E. I. Thorsos, D. Jackson, “Studies of scattering theory using numerical methods,” Waves Random Media 1, 165–190 (1991).
[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]

Tran, P.

Tsang, L.

Wingham, D. J.

D. J. Wingham, R. H. Devayya, “A note on the use of the Neumann expansion in calculating the scatter from rough surfaces,” IEEE Trans. Antennas Propag. 40, 560–563 (1992).
[Crossref]

Electron. Lett. (1)

L. Tsang, C. H. Chan, K. Pak, “Monte Carlo simulations of a two-dimensional random rough surface using the sparse-matrix flat-surface iterative approach,” Electron. Lett. 29, 1153–1154 (1993).
[Crossref]

IEEE Trans. Antennas Propag. (2)

R. M. Axline, A. K. Fung, “Numerical computation of scattering from a perfectly conducting random surface,” IEEE Trans. Antennas Propag. AP-26, 482–488 (1978).
[Crossref]

D. J. Wingham, R. H. Devayya, “A note on the use of the Neumann expansion in calculating the scatter from rough surfaces,” IEEE Trans. Antennas Propag. 40, 560–563 (1992).
[Crossref]

J. Acoust. Soc. Am. (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]

J. Opt. Soc. Am. A (4)

Opt. Commun. (1)

P. Tran, A. A. Maradudin, “The scattering of electromagnetic waves from a 2D metallic surface,” Opt. Commun. 110, 269–273 (1994).
[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]

Waves Random Media (2)

E. I. Thorsos, D. Jackson, “Studies of scattering theory using numerical methods,” Waves Random Media 1, 165–190 (1991).
[Crossref]

Y. Kuga, J. S. Colburn, P. Phu, “Millimeter-wave scattering from one-dimensional surface of different surface correlation functions,” Waves Random Media 3, 101–110 (1993).
[Crossref]

Other (4)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. II.

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

Y. Kuga, P. Phu, “Experimental techniques in random media and rough surfaces,” in Progress in Electromagnetics Research, L. Tsang, M. Tateiba, eds. (Elsevier, Cambridge, Mass., to be published).

A. Ishimaru, C. Le, Y. Kuga, L. Ailes-Sengers, T. K. Chan, “Polarimetric scattering theory for large slope rough surfaces,” in Progress in Electromagnetics Research, L. Tsang, M. Tateiba, eds. (Elsevier, Cambridge, Mass., to be published).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Comparison between the SMFSIA/CAG and the EMI results. In this and subsequent figures, hh and vh are, respectively, copolarized and cross-polarized components.

Fig. 2
Fig. 2

(a) Comparison between the original SMFSIA and the SMFSIA/CAG for one realization. Rough-surface parameters Lx = Ly = 16λ, h = 0.2λ, and lx = ly = 0.6λ; incident angles θi = 10° and ϕi = 0°. (b) A Monte Carlo simulation result with SMFSIA/CAG and 280 realizations. The parameters are those of Fig. 2(a).

Fig. 3
Fig. 3

Backscattering enhancement of cases with h = 0.5λ and lx = ly = 1.0λ. Surface parameters are Lx = Ly = 32λ (Lx × Ly = 1024λ2), and the surface is sampled at 16 points/λ2 (rd = 2.6λ, θi = 10°, ϕi = 0°, and g = Lx/2). Convergence with respect to the number of realizations is shown.

Fig. 4
Fig. 4

Convergence with respect to area (L1x = L1y = 16λ, L1x × L1y = 256λ2 and L2x = L2y = 32λ, L2x × L2y = 1024λ2). Other parameters are those of Fig. 3. The surfaces are sampled at 16 points/λ2.

Fig. 5
Fig. 5

Convergence with respect to the surface sampling rate of 64 points per square wavelength (L1x = L1y = 16λ, L1x × L1y = 256λ2) and 16 points per square wavelength (L2x = L2y = 16λ, L2x × L2y = 256λ2). Other parameters are those of Fig. 3.

Fig. 6
Fig. 6

Convergence with respect to the tapering parameter g for 64 points per square wavelength sampling. For g = Lx/2, the Lx = Ly = 16λ result of Fig. 5 is used. The g = ∞ case is for 495 realizations, with Lx = Ly = 16λ. Other parameters are those of Fig. 3.

Fig. 7
Fig. 7

Backscattering enhancement of three cases: one-dimensional (1d) and two-dimensional (2d) scalar-wave incidence and two-dimensional electromagnetic wave incidence. Comparison of normalized bistatic scattering coefficients for a rms height of 0.5 wavelength and a correlation length of 1.0 wavelength with incident angle θi = 10° and ϕi = 0°.

Fig. 8
Fig. 8

Monte Carlo simulation comparison of the SMFSIA and the experimental data of backscattering enhancement. For the SMFSIA there is an area of 1024 square wavelengths with tapering g = Lx/2. The surface has a rms height of 1 wavelength and correlation length of 2 wavelengths.

Tables (3)

Tables Icon

Table 1 CPU Comparisona

Tables Icon

Table 2 Parameters of Simulations

Tables Icon

Table 3 Average Error with Respect to the 1000-Realization Case of Fig. 3

Equations (36)

Equations on this page are rendered with MathJax. Learn more.

W ( K x , K y ) = l x l y h 2 4 π exp ( K x 2 l x 2 4 K y 2 l y 2 4 ) ,
H i ( x , y , z ) = 1 η + d k x + d k y × exp ( i k x x + i k y y i k z z ) E TE ( k x , k y ) h ̂ ( k z ) ,
h ̂ ( k z ) = k z k k ρ ( x ̂ k x + ŷ k y ) + k ρ k ,
E TE ( k x , k y ) = 1 4 π 2 + d x + d y exp ( i k x x i k y y ) × exp [ i ( k i x x + k i y y ) ( 1 + w ) ] exp ( t ) ,
t x = ( cos θ i cos ϕ i x + cos θ i sin ϕ i y ) 2 g 2 cos 2 θ i ,
t y = ( sin ϕ i x + cos ϕ i y ) 2 g 2 ,
w = 1 k 2 [ ( 2 t x 1 ) g 2 cos 2 θ i + ( 2 t y 1 ) g 2 ] .
n ̂ × H s ( r ) = 2 n ̂ × H i ( r ) + 2 n ̂ × S g × n ̂ × H s ( r ) d s ,
g = ( r r ) G ( R ) ,
G ( R ) = ( ikR 1 ) exp ( ikR ) 4 π R 3 ,
F x ( r ) 2 + f ( x , y ) y d x d y G ( R ) × [ ( x x ) F y ( r ) ( y y ) F x ( r ) ] + d x d y G ( R ) × ( { ( x x ) f ( x , y ) x + [ f ( x , y ) f ( x , y ) ] } × F x ( r ) ( x x ) f ( x , y ) y F y ( r ) ) = f ( x , y ) y H i z ( r ) H i y ( r ) ,
F y ( r ) 2 f ( x , y ) x d x d y G ( R ) × [ ( x x ) F y ( r ) ( y y ) F x ( r ) ] d x d y G ( R ) × ( ( y y ) f ( x , y ) x F x ( r ) + { ( y y ) f ( x , y ) y [ f ( x , y ) f ( x , y ) ] } F y ( r ) ) = f ( x , y ) x H i z ( r ) + H i x ( r ) ,
F x ( r ) = { 1 + [ f ( x , y ) x ] 2 + [ f ( x , y ) y ] 2 } 1 / 2 n ̂ × H s ( r ) · x ̂ ,
F y ( r ) = { 1 + [ f ( x , y ) x ] 2 + [ f ( x , y ) y ] 2 } 1 / 2 n ̂ × H s ( r ) · ŷ .
ρ R = [ ( x x ) 2 + ( y y ) 2 ] 1 / 2
G FS ( ρ R ) = ( i k ρ R 1 ) exp ( i k ρ R ) 4 π ρ R 3 .
G ( R ) = G FS ( ρ R ) + [ G ( R ) G FS ( ρ R ) ] .
F x ( r ) 2 + f ( x , y ) y ρ R < r d d x d y G ( R ) [ ( x x ) F y ( r ) ( y y ) F x ( r ) ] + ρ R < r d d x d y G ( R ) × ( { ( x x ) f ( x , y ) x + [ f ( x , y ) f ( x , y ) ] } F x ( r ) ( x x ) f ( x , y ) y F y ( r ) ) + f ( x , y ) y ρ R r d d x d y G FS ( ρ R ) [ ( x x ) F y ( r ) ( y y ) F x ( r ) ] + ρ R r d d x d y G FS ( ρ R ) × ( { ( x x ) f ( x , y ) x + [ f ( x , y ) f ( x , y ) ] } F x ( r ) ( x x ) f ( x , y ) y F y ( r ) ) = f ( x , y ) y H i z ( r ) H i y ( r ) f ( x , y ) y ρ R r d d x d y [ G ( R ) G FS ( ρ R ) ] [ ( x x ) F y ( r ) ( y y ) F x ( r ) ] ρ R r d d x d y [ G ( R ) G FS ( ρ R ) ] ( { ( x x ) f ( x , y ) x + [ f ( x , y ) f ( x , y ) ] } F x ( r ) ( x x ) f ( x , y ) y F y ( r ) ) ,
F y ( r ) 2 + f ( x , y ) x ρ R < r d d x d y G ( R ) [ ( x x ) F y ( r ) ( y y ) F x ( r ) ] ρ R < r d d x d y G ( R ) × ( ( y y ) f ( x , y ) x F x ( r ) + { ( y y ) f ( x , y ) y [ f ( x , y ) f ( x , y ) ] } F y ( r ) ) f ( x , y ) x ρ R r d d x d y G FS ( ρ R ) [ ( x x ) F y ( r ) ( y y ) F x ( r ) ] ρ R r d d x d y G FS ( ρ R ) × ( ( y y ) f ( x , y ) x F x ( r ) + { ( y y ) f ( x , y ) y [ f ( x , y ) f ( x , y ) ] } F y ( r ) ) = f ( x , y ) x H i z ( r ) H i x ( r ) f ( x , y ) y ρ R r d d x d y [ G ( R ) G FS ( ρ R ) ] [ ( x x ) F y ( r ) ( y y ) F x ( r ) ] + ρ R r d d x d y [ G ( R ) G FS ( ρ R ) ] ( ( y y ) f ( x , y ) x F x ( r ) + { ( y y ) f ( x , y ) y [ f ( x , y ) f ( x , y ) ] } F x ( r ) ) .
Z ¯ ¯ x ¯ = b ¯ .
[ Z ¯ ¯ ( s ) + Z ¯ ¯ ( FS ) + Z ¯ ¯ ( w ) ] x ¯ = b ¯ .
[ Z ¯ ¯ ( s ) + Z ¯ ¯ ( FS ) ] x ¯ ( 1 ) = b ¯ ,
[ Z ¯ ¯ ( s ) + Z ¯ ¯ ( FS ) ] x ¯ ( n + 1 ) = b ¯ ( n + 1 ) ,
b ¯ ( n + 1 ) = b ¯ Z ¯ ¯ ( w ) x ¯ ( n ) .
[ Z ¯ ¯ x ¯ ( n ) b ¯ b ¯ ] 1 / 2 < 1 %
G ( R ) G FS ( ρ R ) = ( ikR 1 ) exp ( ikR ) 4 π R 3 ( i k ρ R 1 ) exp ( i k ρ R ) 4 π ρ r 3 = m = 1 M a m ( ρ R ) ( z d 3 ρ R 2 ) m ,
( Z ¯ ¯ ( s ) + Z ¯ ¯ ( FS ) ) x ¯ ( n + 1 ) = b ¯ ( n + 1 ) ,
b ¯ ( n + 1 ) = b ¯ m = 1 6 Z ¯ ¯ m ( w ) x ¯ ( n ) ,
γ α h ( θ s , ϕ s ) = | α s | 2 2 η P h inc .
P h inc = 2 π 2 η k ρ < k d k x d k y | E TE ( k x , k y ) | 2 ( k z / k ) ,
h s = η i k 4 π d S d x d y exp ( i k β ) × [ F x ( x , y ) sin ϕ s F y ( x , y ) cos ϕ s ] ,
υ s = η i k 4 π d S d x d y exp ( i k β ) { F x ( x , y ) × [ f ( x , y ) x sin θ s cos θ s cos ϕ s ] + F y ( x , y ) [ f ( x , y ) y sin θ s cos θ s sin ϕ s ] } ,
average error ( % ) = 1 N θ s θ s | γ ( θ s ) γ ( θ s ) 1000 γ ( θ s ) | × 100 ,
a 1 ( ρ R ) = k 2 exp ( i k ρ R ) 4 π ρ R 3 i k exp ( i k ρ R ) 4 π ρ R 2 + 3 exp ( i k ρ R ) 4 π ρ R 3 ,
a 2 ( ρ R ) = i k 3 exp ( i k ρ R ) 32 π + 6 k 2 exp ( i k ρ R ) 32 π ρ R + 15 i k exp ( i k ρ R ) 32 π ρ R 2 15 exp ( i k ρ R ) 32 π ρ R 3 ,
a 3 ( ρ R ) = k 4 ρ R exp ( i k ρ R ) 192 π + 10 i k 3 exp ( i k ρ R ) 192 π 42 k 2 exp ( i k ρ R ) 192 π ρ R 96 i k exp ( i k ρ R ) 196 π ρ R 2 + 96 exp ( i k ρ R ) 196 π ρ R 3 .

Metrics