Abstract

A novel approach to reducing the matrix size associated with the method-of-moments solution of the problem of electromagnetic scattering from arbitrarily shaped closed bodies is presented. The key step in this approach is to represent the scattered field in terms of a series of beams that are produced by multipole sources located in a complex space. On the scatterer boundary the fields that are generated by these multipole sources resemble the Gabor basis functions. By utilizing the properties of the Gabor series, guidelines for selecting the orders as well as the locations of the multipole sources are developed. The present approach not only reduces the number of unknowns but also generates a generalized impedance matrix with a banded structure. We verify the accuracy of the proposed method by using internal accuracy checks and by comparing the numerical results with the analytic solution for a spherical scatterer.

© 1994 Optical Society of America

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References

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  1. R. F. Harrington, Field Computation by Moment Methods (Macmillan, New York, 1968).
  2. F. X. Canning, “The impedance matrix localization (IML) method for moment-method calculations,” IEEE Antennas Propag. Mag. 32(5), 18–30 (1990).
    [CrossRef]
  3. D. Gabor, “Theory of Communication,” Proc. Inst. Electr. Eng. 93, 429–457 (1946).
  4. F. X. Canning, “A new combined field integral equation for impedance matrix localization (IML),” in Proceedings of the IEEE Antennas and Propagation Society International Symposium (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 1140–1143.
    [CrossRef]
  5. Y. Leviatan, E. Hudis, P. D. Einziger, “A method of moments analysis of electromagnetic coupling through slots using a Gaussian beam expansion,” IEEE Trans. Antennas Propag. 37, 1537–1544 (1989).
    [CrossRef]
  6. C. Hafner, The Generalized Multipole Technique for Computational Electromagnetics (Artech House, Norwood, Mass., 1990).
  7. Y. Leviatan, A. Boag, A. Boag, “Analysis of electromagnetic scattering using a current model method,” Comput. Phys. Commun. 68, 331–345 (1991).
    [CrossRef]
  8. A. Boag, Y. Leviatan, A. Boag, “On the use of SVD-improved point matching in the current-model method,” IEEE Trans. Antennas Propag. 41, 926–933 (1993).
    [CrossRef]
  9. R. J. Pogorzelski, “Improved computational efficiency via near-field localization,” presented at the Union Radio-Scientifique Internationale Radio Science Meeting, Chicago, III., July 1992.
  10. E. Erez, Y. Leviatan, “Analysis of scattering from structures containing a variety of length-scales using a source-model technique,” J. Acoust. Soc. Am. 93, 3027–3031 (1993).
    [CrossRef]
  11. G. A. Deschamps, “Gaussian beams as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
    [CrossRef]
  12. P. D. Einziger, S. Raz, “Beam-series representation and the parabolic approximation: the frequency domain,” J. Opt. Soc. Am. A 5, 1883–1892 (1988).
    [CrossRef]
  13. J. M. Klosner, L. B. Felsen, I. T. Lu, H. Grossfeld, “Three-dimensional source field modeling by self-consistent Gaussian beam superposition,” J. Acoust. Soc. Am. 91, 1809–1822 (1992).
    [CrossRef]
  14. J. Wexler, S. Raz, “Discrete Gabor expansions,” Signal Process. 21, 207–220 (1990).
    [CrossRef]
  15. Y. Y. Zeevi, M. Porat, “The generalized Gabor scheme of image representation in biological and machine vision,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 452–468 (1988).
    [CrossRef]
  16. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  17. S. Y. Shin, L. B. Felsen, “Gaussian beam modes by multipoles with complex source points,” J. Opt. Soc. Am. 67, 699–700 (1977).
    [CrossRef]
  18. A. Boag, R. Mittra, “Complex multipole beam approach to electromagnetic scattering problems,” IEEE Trans. Antennas Propag. (to be published).
  19. J. G. Daugman, “Complete discrete 2-D Gabor transforms by neural networks for image analysis and compression,” IEEE Trans. Acoust. Speech Signal Process. 36, 1169–1179 (1988).
    [CrossRef]
  20. A. Boag, R. Mittra, “Hybrid multipole beam approach to electromagnetic scattering problems,” presented at the Second International Conference on Approximations and Numerical Solution of the Maxwell Equations, Washington, D.C., October 1993.

1993 (2)

A. Boag, Y. Leviatan, A. Boag, “On the use of SVD-improved point matching in the current-model method,” IEEE Trans. Antennas Propag. 41, 926–933 (1993).
[CrossRef]

E. Erez, Y. Leviatan, “Analysis of scattering from structures containing a variety of length-scales using a source-model technique,” J. Acoust. Soc. Am. 93, 3027–3031 (1993).
[CrossRef]

1992 (1)

J. M. Klosner, L. B. Felsen, I. T. Lu, H. Grossfeld, “Three-dimensional source field modeling by self-consistent Gaussian beam superposition,” J. Acoust. Soc. Am. 91, 1809–1822 (1992).
[CrossRef]

1991 (1)

Y. Leviatan, A. Boag, A. Boag, “Analysis of electromagnetic scattering using a current model method,” Comput. Phys. Commun. 68, 331–345 (1991).
[CrossRef]

1990 (2)

F. X. Canning, “The impedance matrix localization (IML) method for moment-method calculations,” IEEE Antennas Propag. Mag. 32(5), 18–30 (1990).
[CrossRef]

J. Wexler, S. Raz, “Discrete Gabor expansions,” Signal Process. 21, 207–220 (1990).
[CrossRef]

1989 (1)

Y. Leviatan, E. Hudis, P. D. Einziger, “A method of moments analysis of electromagnetic coupling through slots using a Gaussian beam expansion,” IEEE Trans. Antennas Propag. 37, 1537–1544 (1989).
[CrossRef]

1988 (3)

Y. Y. Zeevi, M. Porat, “The generalized Gabor scheme of image representation in biological and machine vision,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 452–468 (1988).
[CrossRef]

P. D. Einziger, S. Raz, “Beam-series representation and the parabolic approximation: the frequency domain,” J. Opt. Soc. Am. A 5, 1883–1892 (1988).
[CrossRef]

J. G. Daugman, “Complete discrete 2-D Gabor transforms by neural networks for image analysis and compression,” IEEE Trans. Acoust. Speech Signal Process. 36, 1169–1179 (1988).
[CrossRef]

1977 (1)

1971 (1)

G. A. Deschamps, “Gaussian beams as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

1946 (1)

D. Gabor, “Theory of Communication,” Proc. Inst. Electr. Eng. 93, 429–457 (1946).

Boag, A.

A. Boag, Y. Leviatan, A. Boag, “On the use of SVD-improved point matching in the current-model method,” IEEE Trans. Antennas Propag. 41, 926–933 (1993).
[CrossRef]

A. Boag, Y. Leviatan, A. Boag, “On the use of SVD-improved point matching in the current-model method,” IEEE Trans. Antennas Propag. 41, 926–933 (1993).
[CrossRef]

Y. Leviatan, A. Boag, A. Boag, “Analysis of electromagnetic scattering using a current model method,” Comput. Phys. Commun. 68, 331–345 (1991).
[CrossRef]

Y. Leviatan, A. Boag, A. Boag, “Analysis of electromagnetic scattering using a current model method,” Comput. Phys. Commun. 68, 331–345 (1991).
[CrossRef]

A. Boag, R. Mittra, “Complex multipole beam approach to electromagnetic scattering problems,” IEEE Trans. Antennas Propag. (to be published).

A. Boag, R. Mittra, “Hybrid multipole beam approach to electromagnetic scattering problems,” presented at the Second International Conference on Approximations and Numerical Solution of the Maxwell Equations, Washington, D.C., October 1993.

Canning, F. X.

F. X. Canning, “The impedance matrix localization (IML) method for moment-method calculations,” IEEE Antennas Propag. Mag. 32(5), 18–30 (1990).
[CrossRef]

F. X. Canning, “A new combined field integral equation for impedance matrix localization (IML),” in Proceedings of the IEEE Antennas and Propagation Society International Symposium (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 1140–1143.
[CrossRef]

Daugman, J. G.

J. G. Daugman, “Complete discrete 2-D Gabor transforms by neural networks for image analysis and compression,” IEEE Trans. Acoust. Speech Signal Process. 36, 1169–1179 (1988).
[CrossRef]

Deschamps, G. A.

G. A. Deschamps, “Gaussian beams as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

Einziger, P. D.

Y. Leviatan, E. Hudis, P. D. Einziger, “A method of moments analysis of electromagnetic coupling through slots using a Gaussian beam expansion,” IEEE Trans. Antennas Propag. 37, 1537–1544 (1989).
[CrossRef]

P. D. Einziger, S. Raz, “Beam-series representation and the parabolic approximation: the frequency domain,” J. Opt. Soc. Am. A 5, 1883–1892 (1988).
[CrossRef]

Erez, E.

E. Erez, Y. Leviatan, “Analysis of scattering from structures containing a variety of length-scales using a source-model technique,” J. Acoust. Soc. Am. 93, 3027–3031 (1993).
[CrossRef]

Felsen, L. B.

J. M. Klosner, L. B. Felsen, I. T. Lu, H. Grossfeld, “Three-dimensional source field modeling by self-consistent Gaussian beam superposition,” J. Acoust. Soc. Am. 91, 1809–1822 (1992).
[CrossRef]

S. Y. Shin, L. B. Felsen, “Gaussian beam modes by multipoles with complex source points,” J. Opt. Soc. Am. 67, 699–700 (1977).
[CrossRef]

Gabor, D.

D. Gabor, “Theory of Communication,” Proc. Inst. Electr. Eng. 93, 429–457 (1946).

Grossfeld, H.

J. M. Klosner, L. B. Felsen, I. T. Lu, H. Grossfeld, “Three-dimensional source field modeling by self-consistent Gaussian beam superposition,” J. Acoust. Soc. Am. 91, 1809–1822 (1992).
[CrossRef]

Hafner, C.

C. Hafner, The Generalized Multipole Technique for Computational Electromagnetics (Artech House, Norwood, Mass., 1990).

Harrington, R. F.

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

Hudis, E.

Y. Leviatan, E. Hudis, P. D. Einziger, “A method of moments analysis of electromagnetic coupling through slots using a Gaussian beam expansion,” IEEE Trans. Antennas Propag. 37, 1537–1544 (1989).
[CrossRef]

Klosner, J. M.

J. M. Klosner, L. B. Felsen, I. T. Lu, H. Grossfeld, “Three-dimensional source field modeling by self-consistent Gaussian beam superposition,” J. Acoust. Soc. Am. 91, 1809–1822 (1992).
[CrossRef]

Leviatan, Y.

E. Erez, Y. Leviatan, “Analysis of scattering from structures containing a variety of length-scales using a source-model technique,” J. Acoust. Soc. Am. 93, 3027–3031 (1993).
[CrossRef]

A. Boag, Y. Leviatan, A. Boag, “On the use of SVD-improved point matching in the current-model method,” IEEE Trans. Antennas Propag. 41, 926–933 (1993).
[CrossRef]

Y. Leviatan, A. Boag, A. Boag, “Analysis of electromagnetic scattering using a current model method,” Comput. Phys. Commun. 68, 331–345 (1991).
[CrossRef]

Y. Leviatan, E. Hudis, P. D. Einziger, “A method of moments analysis of electromagnetic coupling through slots using a Gaussian beam expansion,” IEEE Trans. Antennas Propag. 37, 1537–1544 (1989).
[CrossRef]

Lu, I. T.

J. M. Klosner, L. B. Felsen, I. T. Lu, H. Grossfeld, “Three-dimensional source field modeling by self-consistent Gaussian beam superposition,” J. Acoust. Soc. Am. 91, 1809–1822 (1992).
[CrossRef]

Mittra, R.

A. Boag, R. Mittra, “Hybrid multipole beam approach to electromagnetic scattering problems,” presented at the Second International Conference on Approximations and Numerical Solution of the Maxwell Equations, Washington, D.C., October 1993.

A. Boag, R. Mittra, “Complex multipole beam approach to electromagnetic scattering problems,” IEEE Trans. Antennas Propag. (to be published).

Pogorzelski, R. J.

R. J. Pogorzelski, “Improved computational efficiency via near-field localization,” presented at the Union Radio-Scientifique Internationale Radio Science Meeting, Chicago, III., July 1992.

Porat, M.

Y. Y. Zeevi, M. Porat, “The generalized Gabor scheme of image representation in biological and machine vision,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 452–468 (1988).
[CrossRef]

Raz, S.

Shin, S. Y.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Wexler, J.

J. Wexler, S. Raz, “Discrete Gabor expansions,” Signal Process. 21, 207–220 (1990).
[CrossRef]

Zeevi, Y. Y.

Y. Y. Zeevi, M. Porat, “The generalized Gabor scheme of image representation in biological and machine vision,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 452–468 (1988).
[CrossRef]

Comput. Phys. Commun. (1)

Y. Leviatan, A. Boag, A. Boag, “Analysis of electromagnetic scattering using a current model method,” Comput. Phys. Commun. 68, 331–345 (1991).
[CrossRef]

Electron. Lett. (1)

G. A. Deschamps, “Gaussian beams as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

IEEE Antennas Propag. Mag. (1)

F. X. Canning, “The impedance matrix localization (IML) method for moment-method calculations,” IEEE Antennas Propag. Mag. 32(5), 18–30 (1990).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (1)

J. G. Daugman, “Complete discrete 2-D Gabor transforms by neural networks for image analysis and compression,” IEEE Trans. Acoust. Speech Signal Process. 36, 1169–1179 (1988).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

Y. Leviatan, E. Hudis, P. D. Einziger, “A method of moments analysis of electromagnetic coupling through slots using a Gaussian beam expansion,” IEEE Trans. Antennas Propag. 37, 1537–1544 (1989).
[CrossRef]

A. Boag, Y. Leviatan, A. Boag, “On the use of SVD-improved point matching in the current-model method,” IEEE Trans. Antennas Propag. 41, 926–933 (1993).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

Y. Y. Zeevi, M. Porat, “The generalized Gabor scheme of image representation in biological and machine vision,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 452–468 (1988).
[CrossRef]

J. Acoust. Soc. Am. (2)

J. M. Klosner, L. B. Felsen, I. T. Lu, H. Grossfeld, “Three-dimensional source field modeling by self-consistent Gaussian beam superposition,” J. Acoust. Soc. Am. 91, 1809–1822 (1992).
[CrossRef]

E. Erez, Y. Leviatan, “Analysis of scattering from structures containing a variety of length-scales using a source-model technique,” J. Acoust. Soc. Am. 93, 3027–3031 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Proc. Inst. Electr. Eng. (1)

D. Gabor, “Theory of Communication,” Proc. Inst. Electr. Eng. 93, 429–457 (1946).

Signal Process. (1)

J. Wexler, S. Raz, “Discrete Gabor expansions,” Signal Process. 21, 207–220 (1990).
[CrossRef]

Other (7)

A. Boag, R. Mittra, “Complex multipole beam approach to electromagnetic scattering problems,” IEEE Trans. Antennas Propag. (to be published).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

F. X. Canning, “A new combined field integral equation for impedance matrix localization (IML),” in Proceedings of the IEEE Antennas and Propagation Society International Symposium (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 1140–1143.
[CrossRef]

C. Hafner, The Generalized Multipole Technique for Computational Electromagnetics (Artech House, Norwood, Mass., 1990).

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

R. J. Pogorzelski, “Improved computational efficiency via near-field localization,” presented at the Union Radio-Scientifique Internationale Radio Science Meeting, Chicago, III., July 1992.

A. Boag, R. Mittra, “Hybrid multipole beam approach to electromagnetic scattering problems,” presented at the Second International Conference on Approximations and Numerical Solution of the Maxwell Equations, Washington, D.C., October 1993.

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Figures (9)

Fig. 1
Fig. 1

Distribution of the y component of the equivalent magnetic current for the M11 multipole located at r 0 = j a x ̂ over a disk of radius a in the yz plane.

Fig. 2
Fig. 2

Contour plots of the absolute value for the θ component of the multipole field Mnn(rr0) computed on a sphere | r | = r = 3 λ, with r 0 = j x ̂ for various values of n.

Fig. 3
Fig. 3

Contour plots of the absolute value of the 2D discrete Fourier transform of the field distributions presented in Fig. 2.

Fig. 4
Fig. 4

Example of a perfectly conducting body of arbitrarily smooth cross section illuminated by a plane wave.

Fig. 5
Fig. 5

Local k-space polar grid of an approximate 2D Gabor expansion based on the multipole beams with K = 5 and L = 4. The areas covered by the multipole beams are schematically shown as shaded ellipses. Only few out of N(l) beams of order nl are shown for l = 3,4.

Fig. 6
Fig. 6

Scattering cross section of the perfectly conducting sphere of radius R = 4.2λ illuminated by an x-polarized plane wave incident at θinc = 0° in the plane in which (a)φ = 0° and (b) φ = 90°.

Fig. 7
Fig. 7

Scattering cross section versus θ in planes in which (a) φ = 0° and (b) φ = 90° for the case of Fig. 4 with R1 = R2 = 4.2λ, D = 2λ, E 0 inc = x ̂, and θinc = 0°.

Fig. 8
Fig. 8

Normalized absolute value of the elements in the first column of the generalized impedance matrix versus row index for the case of Fig. 4 with R1 = 4.3λ, R2 = 4.1λ, and D = 2λ.

Fig. 9
Fig. 9

Scattering cross section versus θ in planes in which (a) φ = 0° and (b) φ = 90° for the case of Fig. 4 with R1 = 4.3λ, R2 = 4.1λ, D = 2λ, E 0 inc = x ̂, and θinc = 0°.

Tables (1)

Tables Icon

Table 1 Multipole Parameters Optimized for W ≈ 3λ on a Spherical Surface 4.5λ in Radius with r = 3 λ and K = 5

Equations (11)

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

M m n ( r ) = h n ( 2 ) ( k r ) exp ( j m φ ) × [ j m sin θ P n m ( cos θ ) θ ̂ θ P n m ( cos θ ) φ ̂ ] ,
N m n ( r ) = 1 k r h n ( 2 ) ( k r ) n ( n + 1 ) P n m ( cos θ ) exp ( j m φ ) r ̂ + 1 k r r h n ( 2 ) ( k r ) exp ( j m φ ) × [ j m sin θ P n m ( cos θ ) θ ̂ θ P n m ( cos θ ) φ ̂ ]
r = [ ( x x 0 ) 2 + ( y y 0 ) 2 + ( z z 0 ) 2 ] 1 / 2 , Re { r } 0 ,
cos θ = z z 0 r , sin θ = ρ r , cos φ = x x 0 ρ , sin φ = y y 0 ρ ,
exp ( j n φ ) = ( cos φ + j sin φ ) n ,
ρ = [ ( x x 0 ) 2 + ( y y 0 ) 2 ] 1 / 2 , Re { ρ } 0 .
M n n = j n ( 1 ) n ( 2 n 1 ) ! ! h n ( 2 ) ( k r ) [ ( x ̂ + j ŷ ) r z E n 1 E n ] ,
N n n = n ( 1 ) n ( 2 n 1 ) ! ! k r × ( ( x ̂ + j ŷ ) E n 1 r [ r h n ( 2 ) ( k r ) ] + ( r x x ̂ + r y ŷ + r y ) E n × { ( n + 1 ) h n ( 2 ) ( k r ) r [ r h n ( 2 ) ( k r ) ] } ) ,
W l p = [ S | E ( r ) | d 2 s max r S | E ( r ) | ] 1 / 2
E S ( r ) = i = l P l = 0 L ν = 1 N ( l ) p = 1 2 I i l ν p E i l ν p ( r ) ,
E i l ν p ( r ) = { R i l ν T M n l , n l [ R i l ν ( r r i ) + j a l M ] p = 1 R i l ν T M n l , n l [ R i l ν ( r r i ) + j a l N ] p = 2 ,

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