Abstract

A novel multilevel algorithm to analyze scattering from dielectric random rough surfaces is presented. This technique, termed the steepest-descent fast-multipole method, exploits the quasi-planar nature of dielectric rough surfaces to expedite the iterative solution of the pertinent integral equation. A combination of the fast-multipole method and Sommerfeld steepest-descent-path integral representations is used to efficiently compute electric and magnetic fields that are due to source distributions residing on the rough surface. The CPU time and memory requirements of the technique scale linearly with problem size, thereby permitting the rapid analysis of scattering by large dielectric surfaces and permitting Monte Carlo simulations with realistic computing resources. Numerical results are presented to demonstrate the efficacy of the steepest-decent fast-multipole method.

© 1998 Optical Society of America

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  1. See Proceedings of the Workshop on Rough Surface Scattering and Related Phenomena (Napa Valley Lodge, Yountville, Calif., 1996).
  2. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  3. V. Rokhlin, “Rapid solution of integral equations of scattering theory in two dimensions,” J. Comput. Phys. 36, 414–439 (1990).
    [CrossRef]
  4. N. Engheta, W. D. Murphy, V. Rokhlin, M. S. Vassiliou, “The fast multipole method (FMM) for electromagnetic scattering problems,” IEEE Trans. Antennas Propag. 40, 634–641 (1992).
    [CrossRef]
  5. R. Coifman, V. Rokhlin, S. Wandzura, “The fast multipole method for the wave equation: a pedestrian description,” IEEE Antennas Propag. Mag. 35, 7–12 (1993).
    [CrossRef]
  6. C. C. Lu, W. C. Chew, “A multilevel algorithm for solving a boundary integral equation of wave scattering,” Microwave Opt. Technol. Lett. 7, 466–470 (1994).
    [CrossRef]
  7. J. M. Song, W. C. Chew, “Multilevel fast-multipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microwave Opt. Technol. Lett. 10, 14–19 (1995).
    [CrossRef]
  8. R. L. Wagner, W. C. Chew, “A ray-propagation fast multipole algorithm,” Microwave Opt. Technol. Lett. 7, 435–438 (1994).
    [CrossRef]
  9. M. A. Epton, B. Dembart, “Multipole translation theory for the three-dimensional Laplace and Helmholtz equations,” SIAM J. Sci. Comput. 16, 865–897 (1995).
    [CrossRef]
  10. C. C. Lu, W. C. Chew, “Fast algorithm for solving hybrid integral equations,” IEE Proc. H 140, 455–460 (1993).
  11. E. Michielssen, A. Boag, “A multilevel matrix decomposition algorithm for analyzing scattering from large structures,” IEEE Trans. Antennas Propag. 44, 1086–1093 (1996).
    [CrossRef]
  12. C. H. Chan, L. S. Tsang, “A sparse-matrix canonical-grid method for scattering by many scatterers,” Microwave Opt. Technol. Lett. 8, 114–118 (1995).
    [CrossRef]
  13. E. Bleszynski, M. Bleszynski, T. Jaroszewicz, “A fast integral-equation solver for electromagnetic scattering problems,” in Proceedings of the IEEE/APS International Symposium, Seattle, Washington (Institute of Electrical and Electronics Engineers, Piscataway, N. J., 1994), pp. 416–419.
  14. J. T. Johnson, L. Tsang, R. T. Shin, K. Pak, C. H. Chan, A. Ishimaru, Y. Kuga, “Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces—a comparison of Monte Carlo simulations with experimental data,” IEEE Trans. Antennas Propag. 44, 748–756 (1996).
    [CrossRef]
  15. K. Pak, L. Tsang, C. Chan, J. Johnson, Q. Li, “Scattering of electromagnetic waves in large-scale rough surface problems based on the sparse-matrix canonical-grid method,” in Proceedings of the 13th Annual Review of Progress in Applied Computational Electromagnetics (Applied Computational Electromagnetics Society, Monterey, Calif., 1997).
  16. R. L. Wagner, J. Song, W. Chew, “Monte Carlo simulation of electromagnetic scattering from two-dimensional random rough surfaces,” IEEE Trans. Antennas Propag. 45, 235–245 (1997).
    [CrossRef]
  17. D. A. Kapp, G. Brown, “A new numerical method for rough-surface scattering calculations,” IEEE Trans. Antennas Propag. 44, 711–721 (1996).
    [CrossRef]
  18. E. Michielssen, A. Boag, W. C. Chew, “Scattering from elongated objects: direct solution in O(N log2 N) operations,” IEE Proc. Microwaves Antennas Propag. 143, 277–283 (1996).
    [CrossRef]
  19. V. Jandhyala, E. Michielssen, B. Shanker, W. Chew, “A combined steepest descent–fast multipole algorithm for the fast analysis of three-dimensional scattering by rough surfaces,” (Center for Computational Electromagnetics, University of Illinois, Urbana, 1997).
  20. L. Medgyesi-Mitschang, J. Putnam, M. Gedera, “Generalized method of moments for three-dimensional penetrable scatterers,” J. Opt. Soc. Am. A 12, 1383–1398 (1994).
    [CrossRef]
  21. P. Tran, A. A. Maradudin, “The scattering of electromagnetic waves from a randomly rough 2-D metallic surface,” Opt. Commun. 110, 269–273 (1994).
    [CrossRef]
  22. K. Umashankar, A. Taflove, S. Rao, “Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric object,” IEEE Trans. Antennas Propag. 34, 758–765 (1986).
    [CrossRef]
  23. 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]
  24. S. M. Rao, D. R. Wilton, A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
    [CrossRef]
  25. W. C. Chew, Waves and Fields in Inhomogeneous Media (Institute of Electrical and Electronics Engineers, Piscataway, N. J., 1995).
  26. J. Rahola, “Efficient solution of linear equations in electromagnetic scattering calculations,” (Center for Scientific Computing, Espoo, Finland, 1996).
  27. O. Bucci, C. Gennarelli, C. Savarese, “Optimal interpolation of radiated fields over a sphere,” IEEE Trans. Antennas Propag. 39, 1633–1643 (1991).
    [CrossRef]
  28. J. Volakis, “EM programmer’s notebook,” IEEE Antennas Propag. Mag. 37, 94–100 (1995).

1997

R. L. Wagner, J. Song, W. Chew, “Monte Carlo simulation of electromagnetic scattering from two-dimensional random rough surfaces,” IEEE Trans. Antennas Propag. 45, 235–245 (1997).
[CrossRef]

1996

D. A. Kapp, G. Brown, “A new numerical method for rough-surface scattering calculations,” IEEE Trans. Antennas Propag. 44, 711–721 (1996).
[CrossRef]

E. Michielssen, A. Boag, W. C. Chew, “Scattering from elongated objects: direct solution in O(N log2 N) operations,” IEE Proc. Microwaves Antennas Propag. 143, 277–283 (1996).
[CrossRef]

E. Michielssen, A. Boag, “A multilevel matrix decomposition algorithm for analyzing scattering from large structures,” IEEE Trans. Antennas Propag. 44, 1086–1093 (1996).
[CrossRef]

J. T. Johnson, L. Tsang, R. T. Shin, K. Pak, C. H. Chan, A. Ishimaru, Y. Kuga, “Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces—a comparison of Monte Carlo simulations with experimental data,” IEEE Trans. Antennas Propag. 44, 748–756 (1996).
[CrossRef]

1995

J. Volakis, “EM programmer’s notebook,” IEEE Antennas Propag. Mag. 37, 94–100 (1995).

C. H. Chan, L. S. Tsang, “A sparse-matrix canonical-grid method for scattering by many scatterers,” Microwave Opt. Technol. Lett. 8, 114–118 (1995).
[CrossRef]

J. M. Song, W. C. Chew, “Multilevel fast-multipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microwave Opt. Technol. Lett. 10, 14–19 (1995).
[CrossRef]

M. A. Epton, B. Dembart, “Multipole translation theory for the three-dimensional Laplace and Helmholtz equations,” SIAM J. Sci. Comput. 16, 865–897 (1995).
[CrossRef]

1994

R. L. Wagner, W. C. Chew, “A ray-propagation fast multipole algorithm,” Microwave Opt. Technol. Lett. 7, 435–438 (1994).
[CrossRef]

C. C. Lu, W. C. Chew, “A multilevel algorithm for solving a boundary integral equation of wave scattering,” Microwave Opt. Technol. Lett. 7, 466–470 (1994).
[CrossRef]

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]

L. Medgyesi-Mitschang, J. Putnam, M. Gedera, “Generalized method of moments for three-dimensional penetrable scatterers,” J. Opt. Soc. Am. A 12, 1383–1398 (1994).
[CrossRef]

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

1993

R. Coifman, V. Rokhlin, S. Wandzura, “The fast multipole method for the wave equation: a pedestrian description,” IEEE Antennas Propag. Mag. 35, 7–12 (1993).
[CrossRef]

C. C. Lu, W. C. Chew, “Fast algorithm for solving hybrid integral equations,” IEE Proc. H 140, 455–460 (1993).

1992

N. Engheta, W. D. Murphy, V. Rokhlin, M. S. Vassiliou, “The fast multipole method (FMM) for electromagnetic scattering problems,” IEEE Trans. Antennas Propag. 40, 634–641 (1992).
[CrossRef]

1991

O. Bucci, C. Gennarelli, C. Savarese, “Optimal interpolation of radiated fields over a sphere,” IEEE Trans. Antennas Propag. 39, 1633–1643 (1991).
[CrossRef]

1990

V. Rokhlin, “Rapid solution of integral equations of scattering theory in two dimensions,” J. Comput. Phys. 36, 414–439 (1990).
[CrossRef]

1986

K. Umashankar, A. Taflove, S. Rao, “Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric object,” IEEE Trans. Antennas Propag. 34, 758–765 (1986).
[CrossRef]

1982

S. M. Rao, D. R. Wilton, A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
[CrossRef]

Bleszynski, E.

E. Bleszynski, M. Bleszynski, T. Jaroszewicz, “A fast integral-equation solver for electromagnetic scattering problems,” in Proceedings of the IEEE/APS International Symposium, Seattle, Washington (Institute of Electrical and Electronics Engineers, Piscataway, N. J., 1994), pp. 416–419.

Bleszynski, M.

E. Bleszynski, M. Bleszynski, T. Jaroszewicz, “A fast integral-equation solver for electromagnetic scattering problems,” in Proceedings of the IEEE/APS International Symposium, Seattle, Washington (Institute of Electrical and Electronics Engineers, Piscataway, N. J., 1994), pp. 416–419.

Boag, A.

E. Michielssen, A. Boag, “A multilevel matrix decomposition algorithm for analyzing scattering from large structures,” IEEE Trans. Antennas Propag. 44, 1086–1093 (1996).
[CrossRef]

E. Michielssen, A. Boag, W. C. Chew, “Scattering from elongated objects: direct solution in O(N log2 N) operations,” IEE Proc. Microwaves Antennas Propag. 143, 277–283 (1996).
[CrossRef]

Brown, G.

D. A. Kapp, G. Brown, “A new numerical method for rough-surface scattering calculations,” IEEE Trans. Antennas Propag. 44, 711–721 (1996).
[CrossRef]

Bucci, O.

O. Bucci, C. Gennarelli, C. Savarese, “Optimal interpolation of radiated fields over a sphere,” IEEE Trans. Antennas Propag. 39, 1633–1643 (1991).
[CrossRef]

Celli, V.

Chan, C.

K. Pak, L. Tsang, C. Chan, J. Johnson, Q. Li, “Scattering of electromagnetic waves in large-scale rough surface problems based on the sparse-matrix canonical-grid method,” in Proceedings of the 13th Annual Review of Progress in Applied Computational Electromagnetics (Applied Computational Electromagnetics Society, Monterey, Calif., 1997).

Chan, C. H.

J. T. Johnson, L. Tsang, R. T. Shin, K. Pak, C. H. Chan, A. Ishimaru, Y. Kuga, “Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces—a comparison of Monte Carlo simulations with experimental data,” IEEE Trans. Antennas Propag. 44, 748–756 (1996).
[CrossRef]

C. H. Chan, L. S. Tsang, “A sparse-matrix canonical-grid method for scattering by many scatterers,” Microwave Opt. Technol. Lett. 8, 114–118 (1995).
[CrossRef]

Chew, W.

R. L. Wagner, J. Song, W. Chew, “Monte Carlo simulation of electromagnetic scattering from two-dimensional random rough surfaces,” IEEE Trans. Antennas Propag. 45, 235–245 (1997).
[CrossRef]

V. Jandhyala, E. Michielssen, B. Shanker, W. Chew, “A combined steepest descent–fast multipole algorithm for the fast analysis of three-dimensional scattering by rough surfaces,” (Center for Computational Electromagnetics, University of Illinois, Urbana, 1997).

Chew, W. C.

E. Michielssen, A. Boag, W. C. Chew, “Scattering from elongated objects: direct solution in O(N log2 N) operations,” IEE Proc. Microwaves Antennas Propag. 143, 277–283 (1996).
[CrossRef]

J. M. Song, W. C. Chew, “Multilevel fast-multipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microwave Opt. Technol. Lett. 10, 14–19 (1995).
[CrossRef]

C. C. Lu, W. C. Chew, “A multilevel algorithm for solving a boundary integral equation of wave scattering,” Microwave Opt. Technol. Lett. 7, 466–470 (1994).
[CrossRef]

R. L. Wagner, W. C. Chew, “A ray-propagation fast multipole algorithm,” Microwave Opt. Technol. Lett. 7, 435–438 (1994).
[CrossRef]

C. C. Lu, W. C. Chew, “Fast algorithm for solving hybrid integral equations,” IEE Proc. H 140, 455–460 (1993).

W. C. Chew, Waves and Fields in Inhomogeneous Media (Institute of Electrical and Electronics Engineers, Piscataway, N. J., 1995).

Coifman, R.

R. Coifman, V. Rokhlin, S. Wandzura, “The fast multipole method for the wave equation: a pedestrian description,” IEEE Antennas Propag. Mag. 35, 7–12 (1993).
[CrossRef]

Dembart, B.

M. A. Epton, B. Dembart, “Multipole translation theory for the three-dimensional Laplace and Helmholtz equations,” SIAM J. Sci. Comput. 16, 865–897 (1995).
[CrossRef]

Engheta, N.

N. Engheta, W. D. Murphy, V. Rokhlin, M. S. Vassiliou, “The fast multipole method (FMM) for electromagnetic scattering problems,” IEEE Trans. Antennas Propag. 40, 634–641 (1992).
[CrossRef]

Epton, M. A.

M. A. Epton, B. Dembart, “Multipole translation theory for the three-dimensional Laplace and Helmholtz equations,” SIAM J. Sci. Comput. 16, 865–897 (1995).
[CrossRef]

Gedera, M.

L. Medgyesi-Mitschang, J. Putnam, M. Gedera, “Generalized method of moments for three-dimensional penetrable scatterers,” J. Opt. Soc. Am. A 12, 1383–1398 (1994).
[CrossRef]

Gennarelli, C.

O. Bucci, C. Gennarelli, C. Savarese, “Optimal interpolation of radiated fields over a sphere,” IEEE Trans. Antennas Propag. 39, 1633–1643 (1991).
[CrossRef]

Glisson, A.

S. M. Rao, D. R. Wilton, A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
[CrossRef]

Ishimaru, A.

J. T. Johnson, L. Tsang, R. T. Shin, K. Pak, C. H. Chan, A. Ishimaru, Y. Kuga, “Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces—a comparison of Monte Carlo simulations with experimental data,” IEEE Trans. Antennas Propag. 44, 748–756 (1996).
[CrossRef]

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

Jandhyala, V.

V. Jandhyala, E. Michielssen, B. Shanker, W. Chew, “A combined steepest descent–fast multipole algorithm for the fast analysis of three-dimensional scattering by rough surfaces,” (Center for Computational Electromagnetics, University of Illinois, Urbana, 1997).

Jaroszewicz, T.

E. Bleszynski, M. Bleszynski, T. Jaroszewicz, “A fast integral-equation solver for electromagnetic scattering problems,” in Proceedings of the IEEE/APS International Symposium, Seattle, Washington (Institute of Electrical and Electronics Engineers, Piscataway, N. J., 1994), pp. 416–419.

Johnson, J.

K. Pak, L. Tsang, C. Chan, J. Johnson, Q. Li, “Scattering of electromagnetic waves in large-scale rough surface problems based on the sparse-matrix canonical-grid method,” in Proceedings of the 13th Annual Review of Progress in Applied Computational Electromagnetics (Applied Computational Electromagnetics Society, Monterey, Calif., 1997).

Johnson, J. T.

J. T. Johnson, L. Tsang, R. T. Shin, K. Pak, C. H. Chan, A. Ishimaru, Y. Kuga, “Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces—a comparison of Monte Carlo simulations with experimental data,” IEEE Trans. Antennas Propag. 44, 748–756 (1996).
[CrossRef]

Kapp, D. A.

D. A. Kapp, G. Brown, “A new numerical method for rough-surface scattering calculations,” IEEE Trans. Antennas Propag. 44, 711–721 (1996).
[CrossRef]

Kuga, Y.

J. T. Johnson, L. Tsang, R. T. Shin, K. Pak, C. H. Chan, A. Ishimaru, Y. Kuga, “Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces—a comparison of Monte Carlo simulations with experimental data,” IEEE Trans. Antennas Propag. 44, 748–756 (1996).
[CrossRef]

Li, Q.

K. Pak, L. Tsang, C. Chan, J. Johnson, Q. Li, “Scattering of electromagnetic waves in large-scale rough surface problems based on the sparse-matrix canonical-grid method,” in Proceedings of the 13th Annual Review of Progress in Applied Computational Electromagnetics (Applied Computational Electromagnetics Society, Monterey, Calif., 1997).

Lu, C. C.

C. C. Lu, W. C. Chew, “A multilevel algorithm for solving a boundary integral equation of wave scattering,” Microwave Opt. Technol. Lett. 7, 466–470 (1994).
[CrossRef]

C. C. Lu, W. C. Chew, “Fast algorithm for solving hybrid integral equations,” IEE Proc. H 140, 455–460 (1993).

Maradudin, A. A.

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]

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

Medgyesi-Mitschang, L.

L. Medgyesi-Mitschang, J. Putnam, M. Gedera, “Generalized method of moments for three-dimensional penetrable scatterers,” J. Opt. Soc. Am. A 12, 1383–1398 (1994).
[CrossRef]

Michielssen, E.

E. Michielssen, A. Boag, W. C. Chew, “Scattering from elongated objects: direct solution in O(N log2 N) operations,” IEE Proc. Microwaves Antennas Propag. 143, 277–283 (1996).
[CrossRef]

E. Michielssen, A. Boag, “A multilevel matrix decomposition algorithm for analyzing scattering from large structures,” IEEE Trans. Antennas Propag. 44, 1086–1093 (1996).
[CrossRef]

V. Jandhyala, E. Michielssen, B. Shanker, W. Chew, “A combined steepest descent–fast multipole algorithm for the fast analysis of three-dimensional scattering by rough surfaces,” (Center for Computational Electromagnetics, University of Illinois, Urbana, 1997).

Murphy, W. D.

N. Engheta, W. D. Murphy, V. Rokhlin, M. S. Vassiliou, “The fast multipole method (FMM) for electromagnetic scattering problems,” IEEE Trans. Antennas Propag. 40, 634–641 (1992).
[CrossRef]

Pak, K.

J. T. Johnson, L. Tsang, R. T. Shin, K. Pak, C. H. Chan, A. Ishimaru, Y. Kuga, “Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces—a comparison of Monte Carlo simulations with experimental data,” IEEE Trans. Antennas Propag. 44, 748–756 (1996).
[CrossRef]

K. Pak, L. Tsang, C. Chan, J. Johnson, Q. Li, “Scattering of electromagnetic waves in large-scale rough surface problems based on the sparse-matrix canonical-grid method,” in Proceedings of the 13th Annual Review of Progress in Applied Computational Electromagnetics (Applied Computational Electromagnetics Society, Monterey, Calif., 1997).

Putnam, J.

L. Medgyesi-Mitschang, J. Putnam, M. Gedera, “Generalized method of moments for three-dimensional penetrable scatterers,” J. Opt. Soc. Am. A 12, 1383–1398 (1994).
[CrossRef]

Rahola, J.

J. Rahola, “Efficient solution of linear equations in electromagnetic scattering calculations,” (Center for Scientific Computing, Espoo, Finland, 1996).

Rao, S.

K. Umashankar, A. Taflove, S. Rao, “Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric object,” IEEE Trans. Antennas Propag. 34, 758–765 (1986).
[CrossRef]

Rao, S. M.

S. M. Rao, D. R. Wilton, A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
[CrossRef]

Rokhlin, V.

R. Coifman, V. Rokhlin, S. Wandzura, “The fast multipole method for the wave equation: a pedestrian description,” IEEE Antennas Propag. Mag. 35, 7–12 (1993).
[CrossRef]

N. Engheta, W. D. Murphy, V. Rokhlin, M. S. Vassiliou, “The fast multipole method (FMM) for electromagnetic scattering problems,” IEEE Trans. Antennas Propag. 40, 634–641 (1992).
[CrossRef]

V. Rokhlin, “Rapid solution of integral equations of scattering theory in two dimensions,” J. Comput. Phys. 36, 414–439 (1990).
[CrossRef]

Savarese, C.

O. Bucci, C. Gennarelli, C. Savarese, “Optimal interpolation of radiated fields over a sphere,” IEEE Trans. Antennas Propag. 39, 1633–1643 (1991).
[CrossRef]

Shanker, B.

V. Jandhyala, E. Michielssen, B. Shanker, W. Chew, “A combined steepest descent–fast multipole algorithm for the fast analysis of three-dimensional scattering by rough surfaces,” (Center for Computational Electromagnetics, University of Illinois, Urbana, 1997).

Shin, R. T.

J. T. Johnson, L. Tsang, R. T. Shin, K. Pak, C. H. Chan, A. Ishimaru, Y. Kuga, “Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces—a comparison of Monte Carlo simulations with experimental data,” IEEE Trans. Antennas Propag. 44, 748–756 (1996).
[CrossRef]

Song, J.

R. L. Wagner, J. Song, W. Chew, “Monte Carlo simulation of electromagnetic scattering from two-dimensional random rough surfaces,” IEEE Trans. Antennas Propag. 45, 235–245 (1997).
[CrossRef]

Song, J. M.

J. M. Song, W. C. Chew, “Multilevel fast-multipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microwave Opt. Technol. Lett. 10, 14–19 (1995).
[CrossRef]

Taflove, A.

K. Umashankar, A. Taflove, S. Rao, “Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric object,” IEEE Trans. Antennas Propag. 34, 758–765 (1986).
[CrossRef]

Tran, P.

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]

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

Tsang, L.

J. T. Johnson, L. Tsang, R. T. Shin, K. Pak, C. H. Chan, A. Ishimaru, Y. Kuga, “Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces—a comparison of Monte Carlo simulations with experimental data,” IEEE Trans. Antennas Propag. 44, 748–756 (1996).
[CrossRef]

K. Pak, L. Tsang, C. Chan, J. Johnson, Q. Li, “Scattering of electromagnetic waves in large-scale rough surface problems based on the sparse-matrix canonical-grid method,” in Proceedings of the 13th Annual Review of Progress in Applied Computational Electromagnetics (Applied Computational Electromagnetics Society, Monterey, Calif., 1997).

Tsang, L. S.

C. H. Chan, L. S. Tsang, “A sparse-matrix canonical-grid method for scattering by many scatterers,” Microwave Opt. Technol. Lett. 8, 114–118 (1995).
[CrossRef]

Umashankar, K.

K. Umashankar, A. Taflove, S. Rao, “Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric object,” IEEE Trans. Antennas Propag. 34, 758–765 (1986).
[CrossRef]

Vassiliou, M. S.

N. Engheta, W. D. Murphy, V. Rokhlin, M. S. Vassiliou, “The fast multipole method (FMM) for electromagnetic scattering problems,” IEEE Trans. Antennas Propag. 40, 634–641 (1992).
[CrossRef]

Volakis, J.

J. Volakis, “EM programmer’s notebook,” IEEE Antennas Propag. Mag. 37, 94–100 (1995).

Wagner, R. L.

R. L. Wagner, J. Song, W. Chew, “Monte Carlo simulation of electromagnetic scattering from two-dimensional random rough surfaces,” IEEE Trans. Antennas Propag. 45, 235–245 (1997).
[CrossRef]

R. L. Wagner, W. C. Chew, “A ray-propagation fast multipole algorithm,” Microwave Opt. Technol. Lett. 7, 435–438 (1994).
[CrossRef]

Wandzura, S.

R. Coifman, V. Rokhlin, S. Wandzura, “The fast multipole method for the wave equation: a pedestrian description,” IEEE Antennas Propag. Mag. 35, 7–12 (1993).
[CrossRef]

Wilton, D. R.

S. M. Rao, D. R. Wilton, A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
[CrossRef]

IEE Proc. H

C. C. Lu, W. C. Chew, “Fast algorithm for solving hybrid integral equations,” IEE Proc. H 140, 455–460 (1993).

IEE Proc. Microwaves Antennas Propag.

E. Michielssen, A. Boag, W. C. Chew, “Scattering from elongated objects: direct solution in O(N log2 N) operations,” IEE Proc. Microwaves Antennas Propag. 143, 277–283 (1996).
[CrossRef]

IEEE Antennas Propag. Mag.

J. Volakis, “EM programmer’s notebook,” IEEE Antennas Propag. Mag. 37, 94–100 (1995).

R. Coifman, V. Rokhlin, S. Wandzura, “The fast multipole method for the wave equation: a pedestrian description,” IEEE Antennas Propag. Mag. 35, 7–12 (1993).
[CrossRef]

IEEE Trans. Antennas Propag.

N. Engheta, W. D. Murphy, V. Rokhlin, M. S. Vassiliou, “The fast multipole method (FMM) for electromagnetic scattering problems,” IEEE Trans. Antennas Propag. 40, 634–641 (1992).
[CrossRef]

E. Michielssen, A. Boag, “A multilevel matrix decomposition algorithm for analyzing scattering from large structures,” IEEE Trans. Antennas Propag. 44, 1086–1093 (1996).
[CrossRef]

J. T. Johnson, L. Tsang, R. T. Shin, K. Pak, C. H. Chan, A. Ishimaru, Y. Kuga, “Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces—a comparison of Monte Carlo simulations with experimental data,” IEEE Trans. Antennas Propag. 44, 748–756 (1996).
[CrossRef]

R. L. Wagner, J. Song, W. Chew, “Monte Carlo simulation of electromagnetic scattering from two-dimensional random rough surfaces,” IEEE Trans. Antennas Propag. 45, 235–245 (1997).
[CrossRef]

D. A. Kapp, G. Brown, “A new numerical method for rough-surface scattering calculations,” IEEE Trans. Antennas Propag. 44, 711–721 (1996).
[CrossRef]

O. Bucci, C. Gennarelli, C. Savarese, “Optimal interpolation of radiated fields over a sphere,” IEEE Trans. Antennas Propag. 39, 1633–1643 (1991).
[CrossRef]

K. Umashankar, A. Taflove, S. Rao, “Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric object,” IEEE Trans. Antennas Propag. 34, 758–765 (1986).
[CrossRef]

S. M. Rao, D. R. Wilton, A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
[CrossRef]

J. Comput. Phys.

V. Rokhlin, “Rapid solution of integral equations of scattering theory in two dimensions,” J. Comput. Phys. 36, 414–439 (1990).
[CrossRef]

J. Opt. Soc. Am. A

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]

L. Medgyesi-Mitschang, J. Putnam, M. Gedera, “Generalized method of moments for three-dimensional penetrable scatterers,” J. Opt. Soc. Am. A 12, 1383–1398 (1994).
[CrossRef]

Microwave Opt. Technol. Lett.

C. C. Lu, W. C. Chew, “A multilevel algorithm for solving a boundary integral equation of wave scattering,” Microwave Opt. Technol. Lett. 7, 466–470 (1994).
[CrossRef]

J. M. Song, W. C. Chew, “Multilevel fast-multipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microwave Opt. Technol. Lett. 10, 14–19 (1995).
[CrossRef]

R. L. Wagner, W. C. Chew, “A ray-propagation fast multipole algorithm,” Microwave Opt. Technol. Lett. 7, 435–438 (1994).
[CrossRef]

C. H. Chan, L. S. Tsang, “A sparse-matrix canonical-grid method for scattering by many scatterers,” Microwave Opt. Technol. Lett. 8, 114–118 (1995).
[CrossRef]

Opt. Commun.

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

SIAM J. Sci. Comput.

M. A. Epton, B. Dembart, “Multipole translation theory for the three-dimensional Laplace and Helmholtz equations,” SIAM J. Sci. Comput. 16, 865–897 (1995).
[CrossRef]

Other

See Proceedings of the Workshop on Rough Surface Scattering and Related Phenomena (Napa Valley Lodge, Yountville, Calif., 1996).

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

E. Bleszynski, M. Bleszynski, T. Jaroszewicz, “A fast integral-equation solver for electromagnetic scattering problems,” in Proceedings of the IEEE/APS International Symposium, Seattle, Washington (Institute of Electrical and Electronics Engineers, Piscataway, N. J., 1994), pp. 416–419.

K. Pak, L. Tsang, C. Chan, J. Johnson, Q. Li, “Scattering of electromagnetic waves in large-scale rough surface problems based on the sparse-matrix canonical-grid method,” in Proceedings of the 13th Annual Review of Progress in Applied Computational Electromagnetics (Applied Computational Electromagnetics Society, Monterey, Calif., 1997).

V. Jandhyala, E. Michielssen, B. Shanker, W. Chew, “A combined steepest descent–fast multipole algorithm for the fast analysis of three-dimensional scattering by rough surfaces,” (Center for Computational Electromagnetics, University of Illinois, Urbana, 1997).

W. C. Chew, Waves and Fields in Inhomogeneous Media (Institute of Electrical and Electronics Engineers, Piscataway, N. J., 1995).

J. Rahola, “Efficient solution of linear equations in electromagnetic scattering calculations,” (Center for Scientific Computing, Espoo, Finland, 1996).

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Figures (12)

Fig. 1
Fig. 1

Rough surface formed at the interface of two regions. Regions 1 and 2 have distinct permittivities. Typically, Region 1 is free space.

Fig. 2
Fig. 2

SDFMM far-field interactions in a single-level implementation. The dielectric rough-surface of size L×L has been divided into blocks of surface dimensions l×l. These blocks have the z coordinate of their center (small dark circles) determined by the mean height of the portion of the rough surface they enclose. These circles represent plane-wave expansions. The block with its expansion shown as a black circle is in the far field of the blocks with expansions represented by gray circles. The maximum peak-to-peak height of the rough surface is H.

Fig. 3
Fig. 3

Schematic description of a single-level and a two-level SDFMM: (a) SDP integration (short bold arrows), azimuthal FMM spectral integration (thinner arrows), and diagonal translation operators (long bold arrow). At an observation block, another combined SDFMM integration is used to find field values in the block. (b) In a two-level SDFMM, inhomogeneous plane-wave expansions are shifted to centers of parent blocks, and incoming plane-wave spectra are translated (thin arrows) to centers of child blocks. Two kinds of interaction are possible, one at the fine level and another at the coarser level, as shown. Distinct SDP rules, FMM integration points, and translation operators (bold arrows) are used at each level. In a multilevel SDFMM, such an approach would be continued across several levels.

Fig. 4
Fig. 4

SDFMM blocks at (a) the finest level (b) a coarse level. The effective angle between a source and an observation block is reduced as one goes to coarser levels.

Fig. 5
Fig. 5

Comparison of RCS results obtained by MoM (solid curve) and SDFMM (*). The dielectric rough surface is of dimensions 5.5λ2, with σ=0.4λ and lc=1.0λ, where λ is the free-space wavelength. The dielectric constant is 2=3+0.1i, thus making the surface dimensions equal to 16.4λ2 in the dielectric. A finest-level box size of 0.15λ×0.15λ is used in the SDFMM.

Fig. 6
Fig. 6

Matrix–vector product times for direct multiplication and the SDFMM, as a function of problem size.

Fig. 7
Fig. 7

Memory requirements for the standard MoM and the SDFMM, as a function of problem size.

Fig. 8
Fig. 8

Comparison of matrix-fill time for the standard MoM and setup time for the SDFMM, as a function of problem size.

Fig. 9
Fig. 9

Copolarized bistatic RCS for a dielectric rough surface of area 162λ2, with σ=0.2λ and lc=1.5λ, where λ is the free-space wavelength.

Fig. 10
Fig. 10

Copolarized bistatic RCS for a dielectric rough surface of area 205λ2, with σ=0.05λ and lc=1.0λ, where λ is the free-space wavelength.

Fig. 11
Fig. 11

Copolarized bistatic RCS for a dielectric rough surface of area 205λ2, with σ=0.4λ and lc=1.5λ, where λ is the free-space wavelength.

Fig. 12
Fig. 12

Copolarized bistatic scattering coefficient obtained through a Monte Carlo simulation involving 50 realizations of rough surfaces of size 22.28λ2, with σ=0.4λ and lc=1.5λ, where λ is the free-space wavelength.

Equations (37)

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z(x, y)=0,
z(x, y)z(x, y)=σ2 exp{-[(x-x)2+(y-y)2]/lc2},
Einc(r)=2πW2L2|K|k1eˆinc(K, kz)×exp(iK·ρ)exp(-ikzz)×exp(-|K-K0|2 W2/2),
eˆinc(Kx, Ky, kz)=1(Kx2+kz2)1/2(kz, 0, Kx),
Einc(r)|tan=(L1+L2)J(r)|tan-(K1+K2)M(r)|tan
Hinc(r)|tan=(K1+K2)J(r)|tan+1η12L1+1η22L2M(r)|tan
LqX(r)=Sds-iωμqX(r)+-iωq·X(r)gq(r, r),
KqX(r)=SdsX(r)×gq(r, r),
gq(r, r)=exp(ikq|r-r|)4π|r-r|.
σγδ(θ, ϕ)=limr4πr2Sδ(θ, ϕ, r)Pγinc.
J(r)n=1NIn1jn(r),
M(r)n=1NIn2jn(r).
Z¯·I=V.
Z¯=Z¯11Z¯12Z¯21Z¯22,
Zmn11=fm, (L1+L2)jn,
Zmn12=fm, -(K1+K2)jn,
Zmn21=fm, (K1+K2)jn,
Zmn22=fm,1η12L1+1η22L2jn.
I=I1I2,
V=V1V2,
Vm1=fm, Einc(r),
Vm2=fm, Hinc(r).
Z¯=Z¯+Z¯,
exp(ikq|r-r|)4π|r-r|=i8π-dkzq exp[ikzq(z-z)]H0(1)×(kρq|ρ-ρ|),
H0(1)(kρq|ρ-ρ|)=Hˆ0(1)(kρq|ρ-ρ|)exp{ikρq|ρ-ρ|},
gq(r, r)=-i8π-dkzq exp[ikzq(δz-δz)]×Hˆ0(1)(kρq|ρ-ρ|)exp[ikρq|ρ-ρ|+ikzq(zc-zc)],
gq(r, r)=-ikq8πΓdα sin α exp[ikzq(δz-δz)]×Hˆ0(1)(kq sin α|ρ-ρ|)×exp[ikq|r1-r1|cos(α-θ1)],
ikq|r1-r1|cos(α-θc)=ikq|r1-r1|-s2,
gq(r, r)=-ikq8πexp(ikq|r1-r1|)×-dsdαdssin α exp{ikzq(δz-δz)}×Hˆ0(1)(kq sin α|ρ-ρ|)×exp{ikq|r1-r1|[cos(α-θ1)-cos(α-θc)]}exp{-s2}.
gq(r, r)=i8πexp(ikq|r1-r1|)×j=1n(sd),qwjq(sd)kpq(j) exp[ikzq(j)(δz-δz)]×Hˆ0(1)(kρq(j)|ρ-ρ|)exp{ikq|r1-r1|×[cos(αj-θ1)-cos(αj-θc)]}exp(-sjq2),
gq(r, r)i8πj=1n(sd),qwjq(sd)kρq(j)H0(1)(kρq(j)|ρ-ρ|)×exp[ikzq(j)(z-z)].
H0(1)(kρq(j)|ρ-ρ|)=12π02πdϕ exp[ikρq(j)(ρ-ρb)·sˆq]×T(kρq(j), sˆq, ρb-ρa)exp[ikρq(j)(ρa-ρ)·sˆq]=12πj=1Pqwjq(fmm) exp{ikρq(j)(ρ-ρb)·sˆjq}×T(kρq(j), sˆjq, ρb-ρa)exp[ikρq(j)(ρa-ρ)·sˆjq]
kq(j)=kρq(j)sˆjq+kzq(j)zˆ,
T(kρq(j), sˆq, ρb-ρa)
=p=-PqPqHp(1)(kρq(j)|ρb-ρa|)exp{-ip[θ-ϕq-π/2]},
fm, Lqjn=i16π2j=1n(sd),qj=1Pqwjq(sd)wjq(fmm)kρq(j)×Sdrfm exp[ikq(j)·(r-rb)]×T(kρq(j), sˆjq, ρb-ρa)I¯-kq(j)kq(j)kq2×Sdrjn exp[ikq(j)·(ra-r)],
fm, Kqjn=-116π2j=1n(sd),qj=1Pqwjq(sd)wjq(fmm)kρq(j)×Sdr[fm×kq(j)]exp[ikq(j)·(r-rb)]×T(kρq(j), sˆjq, ρb-ρa)×Sdrjn exp[ikq(j)·(ra-r)].

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