Abstract

Numerical simulations exhibiting backscattering enhancement of electromagnetic waves from two-dimensional dielectric random rough surfaces (three-dimensional scattering problem) are presented. The Stratton–Chu surface integral equation formulation is used with the method of moments to solve for the tangential and normal components of surface fields. The solution of the matrix equation is calculated efficiently by using the sparse-matrix canonical grid (SMCG) method. The accuracy of the solution is assessed by comparing the bistatic scattering coefficients obtained from the SMCG and the matrix inversion method. Also, a sufficient sampling rate is established with respect to the dielectric constant below the rough-surface boundary. Numerical simulations are illustrated for moderate rms heights of 0.2 and 0.5 electromagnetic wavelengths with rms slopes of 0.5 and 0.7. The magnitude of the relative permittivity ranges from 3 to 7. With use of the SMCG method, scattered fields from a surface area of 256 square wavelengths (98,304 surface unknowns) are found. For a rms height of 0.5 wavelength and a correlation length of 1.0 wavelength, backscattering enhancement is observed in both co-polarization and cross polarization. However, in the case in which the rms height is 0.2 wavelength and the correlation length is 0.6 wavelength, backscattering enhancement is observed in cross polarization only.

© 1997 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. J. J. Greffet, “Theoretical model of the shift of the Brewster angle on a rough surface,” Opt. Lett. 17, 238–240 (1992).
    [CrossRef] [PubMed]
  4. M. Nieto-Vesperinas, J. M. Soto-Crespo, “Monte Carlo calculations of speckle contrast from perfectly conducting rough surfaces,” Opt. Commun. 75, 215–218 (1990).
    [CrossRef]
  5. 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]
  6. C. H. Chan, S. H. Lou, L. Tsang, J. A. Kong, “Electromagnetic scattering of waves by random rough surface: a finite-difference time-domain approach,” Microwave Opt. Technol. Lett. 4, 355–359 (1991).
    [CrossRef]
  7. S. H. Lou, L. Tsang, C. H. Chan, “Application of the finite element method to Monte Carlo simulations of scattering of waves by random rough surfaces: penetrable case,” Waves Random Media 1, 287–307 (1991).
    [CrossRef]
  8. D. Maystre, M. Saillard, J. Ingers, “Scattering by one- or two-dimensional randomly rough surfaces,” Waves Random Media 1, 143–155 (1991).
    [CrossRef]
  9. E. I. Thorsos, D. Jackson, “Studies of scattering theory using numerical methods,” Waves Random Media 1, 165–190 (1991).
    [CrossRef]
  10. 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]
  11. P. Tran, V. Celli, A. A. Maradudin, “Electromagnetic scattering from a two-dimensional, randomly rough, perfectly conducting surface: iterative methods,” J. Opt. Soc. Am. 11, 1686–1689 (1994).
    [CrossRef]
  12. P. Tran, A. A. Maradudin, “The scattering of electromagnetic waves from 2D metallic surface,” Opt. Commun. 110, 269–273 (1994).
    [CrossRef]
  13. A. K. Fung, M. R. Shah, S. Tjuatja, “Numerical simulations of scattering from three-dimensional randomly rough surfaces,” IEEE Trans. Geosci. Remote Sens. 32, 986–994 (1994).
    [CrossRef]
  14. L. Tsang, S. H. Lou, C. H. Chan, “Application of the extended boundary condition method to Monte Carlo simulations of scattering of waves by two-dimensional random rough surfaces,” Microwave Opt. Technol. Lett. 4, 527–531 (1991).
    [CrossRef]
  15. L. Tsang, C. H. Chan, K. Pak, H. Sangani, “Monte Carlo simulations of large-scale problems of random rough surface scattering and applications to grazing incidence with the BMIA/canonical grid method,” IEEE Trans. Antennas Propag. 43, 851–859 (1995).
    [CrossRef]
  16. 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]
  17. 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]
  18. K. Pak, L. Tsang, C. H. Chan, J. Johnson, “Backscatter enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces based on Monte Carlo simulations,” J. Opt. Soc. Am. A 12, 2491–2499 (1995).
    [CrossRef]
  19. J. T. Johnson, L. Tsang, R. 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]
  20. 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]
  21. C. H. Chan, R. Mittra, “Some recent developments in iterative techniques for solving electromagnetic boundary value problems,” Radio Sci. 22, 929–934 (1987).
    [CrossRef]
  22. J. T. Johnson, “Applications of numerical models for rough surface scattering,” Ph.D. dissertation (Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Mass., 1996).
  23. C. H. Chan, L. Tsang, “A sparse-matrix canonical-grid method for scattering by many scatterers,” Microwave Opt. Technol. Lett. 8, 114–118 (1995).
    [CrossRef]

1996 (1)

J. T. Johnson, L. Tsang, R. 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 (3)

K. Pak, L. Tsang, C. H. Chan, J. Johnson, “Backscatter enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces based on Monte Carlo simulations,” J. Opt. Soc. Am. A 12, 2491–2499 (1995).
[CrossRef]

L. Tsang, C. H. Chan, K. Pak, H. Sangani, “Monte Carlo simulations of large-scale problems of random rough surface scattering and applications to grazing incidence with the BMIA/canonical grid method,” IEEE Trans. Antennas Propag. 43, 851–859 (1995).
[CrossRef]

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

1994 (4)

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]

P. Tran, V. Celli, A. A. Maradudin, “Electromagnetic scattering from a two-dimensional, randomly rough, perfectly conducting surface: iterative methods,” J. Opt. Soc. Am. 11, 1686–1689 (1994).
[CrossRef]

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

A. K. Fung, M. R. Shah, S. Tjuatja, “Numerical simulations of scattering from three-dimensional randomly rough surfaces,” IEEE Trans. Geosci. Remote Sens. 32, 986–994 (1994).
[CrossRef]

1993 (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]

1992 (2)

J. J. Greffet, “Theoretical model of the shift of the Brewster angle on a rough surface,” Opt. Lett. 17, 238–240 (1992).
[CrossRef] [PubMed]

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 (5)

L. Tsang, S. H. Lou, C. H. Chan, “Application of the extended boundary condition method to Monte Carlo simulations of scattering of waves by two-dimensional random rough surfaces,” Microwave Opt. Technol. Lett. 4, 527–531 (1991).
[CrossRef]

C. H. Chan, S. H. Lou, L. Tsang, J. A. Kong, “Electromagnetic scattering of waves by random rough surface: a finite-difference time-domain approach,” Microwave Opt. Technol. Lett. 4, 355–359 (1991).
[CrossRef]

S. H. Lou, L. Tsang, C. H. Chan, “Application of the finite element method to Monte Carlo simulations of scattering of waves by random rough surfaces: penetrable case,” Waves Random Media 1, 287–307 (1991).
[CrossRef]

D. Maystre, M. Saillard, J. Ingers, “Scattering by one- or two-dimensional randomly rough surfaces,” Waves Random Media 1, 143–155 (1991).
[CrossRef]

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

1990 (1)

M. Nieto-Vesperinas, J. M. Soto-Crespo, “Monte Carlo calculations of speckle contrast from perfectly conducting rough surfaces,” Opt. Commun. 75, 215–218 (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 (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]

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.

P. Tran, V. Celli, A. A. Maradudin, “Electromagnetic scattering from a two-dimensional, randomly rough, perfectly conducting surface: iterative methods,” J. Opt. Soc. Am. 11, 1686–1689 (1994).
[CrossRef]

Chan, C. H.

J. T. Johnson, L. Tsang, R. 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. H. Chan, J. Johnson, “Backscatter enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces based on Monte Carlo simulations,” J. Opt. Soc. Am. A 12, 2491–2499 (1995).
[CrossRef]

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

L. Tsang, C. H. Chan, K. Pak, H. Sangani, “Monte Carlo simulations of large-scale problems of random rough surface scattering and applications to grazing incidence with the BMIA/canonical grid method,” IEEE Trans. Antennas Propag. 43, 851–859 (1995).
[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]

L. Tsang, S. H. Lou, C. H. Chan, “Application of the extended boundary condition method to Monte Carlo simulations of scattering of waves by two-dimensional random rough surfaces,” Microwave Opt. Technol. Lett. 4, 527–531 (1991).
[CrossRef]

C. H. Chan, S. H. Lou, L. Tsang, J. A. Kong, “Electromagnetic scattering of waves by random rough surface: a finite-difference time-domain approach,” Microwave Opt. Technol. Lett. 4, 355–359 (1991).
[CrossRef]

S. H. Lou, L. Tsang, C. H. Chan, “Application of the finite element method to Monte Carlo simulations of scattering of waves by random rough surfaces: penetrable case,” Waves Random Media 1, 287–307 (1991).
[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]

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.

A. K. Fung, M. R. Shah, S. Tjuatja, “Numerical simulations of scattering from three-dimensional randomly rough surfaces,” IEEE Trans. Geosci. Remote Sens. 32, 986–994 (1994).
[CrossRef]

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]

Greffet, J. J.

Ingers, J.

D. Maystre, M. Saillard, J. Ingers, “Scattering by one- or two-dimensional randomly rough surfaces,” Waves Random Media 1, 143–155 (1991).
[CrossRef]

Ishimaru, A.

J. T. Johnson, L. Tsang, R. 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), Vol. II.

Jackson, D.

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

Johnson, J.

Johnson, J. T.

J. T. Johnson, L. Tsang, R. 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]

J. T. Johnson, “Applications of numerical models for rough surface scattering,” Ph.D. dissertation (Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Mass., 1996).

Kong, J. A.

C. H. Chan, S. H. Lou, L. Tsang, J. A. Kong, “Electromagnetic scattering of waves by random rough surface: a finite-difference time-domain approach,” Microwave Opt. Technol. Lett. 4, 355–359 (1991).
[CrossRef]

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

Kuga, Y.

J. T. Johnson, L. Tsang, R. 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]

Lou, S. H.

S. H. Lou, L. Tsang, C. H. Chan, “Application of the finite element method to Monte Carlo simulations of scattering of waves by random rough surfaces: penetrable case,” Waves Random Media 1, 287–307 (1991).
[CrossRef]

C. H. Chan, S. H. Lou, L. Tsang, J. A. Kong, “Electromagnetic scattering of waves by random rough surface: a finite-difference time-domain approach,” Microwave Opt. Technol. Lett. 4, 355–359 (1991).
[CrossRef]

L. Tsang, S. H. Lou, C. H. Chan, “Application of the extended boundary condition method to Monte Carlo simulations of scattering of waves by two-dimensional random rough surfaces,” Microwave Opt. Technol. Lett. 4, 527–531 (1991).
[CrossRef]

Maradudin, A. A.

P. Tran, A. A. Maradudin, “The scattering of electromagnetic waves from 2D metallic surface,” Opt. Commun. 110, 269–273 (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. 11, 1686–1689 (1994).
[CrossRef]

Maystre, D.

D. Maystre, M. Saillard, J. Ingers, “Scattering by one- or two-dimensional randomly rough surfaces,” Waves Random Media 1, 143–155 (1991).
[CrossRef]

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.

M. Nieto-Vesperinas, J. M. Soto-Crespo, “Monte Carlo calculations of speckle contrast from perfectly conducting rough surfaces,” Opt. Commun. 75, 215–218 (1990).
[CrossRef]

Pak, K.

J. T. Johnson, L. Tsang, R. 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. H. Chan, J. Johnson, “Backscatter enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces based on Monte Carlo simulations,” J. Opt. Soc. Am. A 12, 2491–2499 (1995).
[CrossRef]

L. Tsang, C. H. Chan, K. Pak, H. Sangani, “Monte Carlo simulations of large-scale problems of random rough surface scattering and applications to grazing incidence with the BMIA/canonical grid method,” IEEE Trans. Antennas Propag. 43, 851–859 (1995).
[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]

Saillard, M.

D. Maystre, M. Saillard, J. Ingers, “Scattering by one- or two-dimensional randomly rough surfaces,” Waves Random Media 1, 143–155 (1991).
[CrossRef]

Sangani, H.

L. Tsang, C. H. Chan, K. Pak, H. Sangani, “Monte Carlo simulations of large-scale problems of random rough surface scattering and applications to grazing incidence with the BMIA/canonical grid method,” IEEE Trans. Antennas Propag. 43, 851–859 (1995).
[CrossRef]

Shah, M. R.

A. K. Fung, M. R. Shah, S. Tjuatja, “Numerical simulations of scattering from three-dimensional randomly rough surfaces,” IEEE Trans. Geosci. Remote Sens. 32, 986–994 (1994).
[CrossRef]

Shin, R.

J. T. Johnson, L. Tsang, R. 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]

Shin, R. T.

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

Soto-Crespo, J. M.

M. Nieto-Vesperinas, J. M. Soto-Crespo, “Monte Carlo calculations of speckle contrast from perfectly conducting rough surfaces,” Opt. Commun. 75, 215–218 (1990).
[CrossRef]

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]

Tjuatja, S.

A. K. Fung, M. R. Shah, S. Tjuatja, “Numerical simulations of scattering from three-dimensional randomly rough surfaces,” IEEE Trans. Geosci. Remote Sens. 32, 986–994 (1994).
[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. 11, 1686–1689 (1994).
[CrossRef]

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

Tsang, L.

J. T. Johnson, L. Tsang, R. 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. H. Chan, J. Johnson, “Backscatter enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces based on Monte Carlo simulations,” J. Opt. Soc. Am. A 12, 2491–2499 (1995).
[CrossRef]

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

L. Tsang, C. H. Chan, K. Pak, H. Sangani, “Monte Carlo simulations of large-scale problems of random rough surface scattering and applications to grazing incidence with the BMIA/canonical grid method,” IEEE Trans. Antennas Propag. 43, 851–859 (1995).
[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]

L. Tsang, S. H. Lou, C. H. Chan, “Application of the extended boundary condition method to Monte Carlo simulations of scattering of waves by two-dimensional random rough surfaces,” Microwave Opt. Technol. Lett. 4, 527–531 (1991).
[CrossRef]

C. H. Chan, S. H. Lou, L. Tsang, J. A. Kong, “Electromagnetic scattering of waves by random rough surface: a finite-difference time-domain approach,” Microwave Opt. Technol. Lett. 4, 355–359 (1991).
[CrossRef]

S. H. Lou, L. Tsang, C. H. Chan, “Application of the finite element method to Monte Carlo simulations of scattering of waves by random rough surfaces: penetrable case,” Waves Random Media 1, 287–307 (1991).
[CrossRef]

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

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. (4)

L. Tsang, C. H. Chan, K. Pak, H. Sangani, “Monte Carlo simulations of large-scale problems of random rough surface scattering and applications to grazing incidence with the BMIA/canonical grid method,” IEEE Trans. Antennas Propag. 43, 851–859 (1995).
[CrossRef]

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]

J. T. Johnson, L. Tsang, R. 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]

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]

IEEE Trans. Geosci. Remote Sens. (1)

A. K. Fung, M. R. Shah, S. Tjuatja, “Numerical simulations of scattering from three-dimensional randomly rough surfaces,” IEEE Trans. Geosci. Remote Sens. 32, 986–994 (1994).
[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. (1)

P. Tran, V. Celli, A. A. Maradudin, “Electromagnetic scattering from a two-dimensional, randomly rough, perfectly conducting surface: iterative methods,” J. Opt. Soc. Am. 11, 1686–1689 (1994).
[CrossRef]

J. Opt. Soc. Am. A (2)

Microwave Opt. Technol. Lett. (3)

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

L. Tsang, S. H. Lou, C. H. Chan, “Application of the extended boundary condition method to Monte Carlo simulations of scattering of waves by two-dimensional random rough surfaces,” Microwave Opt. Technol. Lett. 4, 527–531 (1991).
[CrossRef]

C. H. Chan, S. H. Lou, L. Tsang, J. A. Kong, “Electromagnetic scattering of waves by random rough surface: a finite-difference time-domain approach,” Microwave Opt. Technol. Lett. 4, 355–359 (1991).
[CrossRef]

Opt. Commun. (2)

M. Nieto-Vesperinas, J. M. Soto-Crespo, “Monte Carlo calculations of speckle contrast from perfectly conducting rough surfaces,” Opt. Commun. 75, 215–218 (1990).
[CrossRef]

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

Opt. Lett. (1)

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 (3)

S. H. Lou, L. Tsang, C. H. Chan, “Application of the finite element method to Monte Carlo simulations of scattering of waves by random rough surfaces: penetrable case,” Waves Random Media 1, 287–307 (1991).
[CrossRef]

D. Maystre, M. Saillard, J. Ingers, “Scattering by one- or two-dimensional randomly rough surfaces,” Waves Random Media 1, 143–155 (1991).
[CrossRef]

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

Other (3)

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).

J. T. Johnson, “Applications of numerical models for rough surface scattering,” Ph.D. dissertation (Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Mass., 1996).

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 (13)

Fig. 1
Fig. 1

Comparison between the sparse-matrix canonical grid (SMCG) and full matrix inversion (MI): co-polarized result (hh). The simulation parameters are Lx=Ly=4.0λ, h =0.2λ, lx=ly=1.0λ, r=3.0+i0.2, and rd=2.0λ. The incidence angles are θi=10° and ϕi=0°.

Fig. 2
Fig. 2

Same as Fig. 1, but for the cross-polarized result (vh).

Fig. 3
Fig. 3

Surface field magnitude comparison |(vi-wi)/vi 100%. The result is for one iteration of Eq. (7). The simulation parameters are h=0.5λ, lx=ly=1.0λ, r=6.5+i1.0, Lx=Ly=8.0λ, 64×64 grid, θi=10°, ϕi=0°, and rd=3.5λ. The average errors are less than 0.3% for Ix(r), Iy(r), Fx(r), and Fy(r). The curves are offset by 2.0% for illustration.

Fig. 4
Fig. 4

Convergence with respect to the neighborhood distance rd for the co-polarized component. The simulation parameters are Lx=Ly=16λ, h=0.2λ, lx=ly=0.6λ, r=6.5+i1.0, θi=10°, and ϕi=0°.

Fig. 5
Fig. 5

Same as Fig. 4, but for the cross-polarized component.

Fig. 6
Fig. 6

Dependence of CPU time on neighborhood distance rd. The simulation parameters are those of Fig. 4.

Fig. 7
Fig. 7

Dependence of CPU time on r. The simulation parameters are Lx=Ly=8.0λ, h=0.5λ, lx=ly=1.0λ, rd=3.5λ, θi=10°, and ϕi=0°.

Fig. 8
Fig. 8

Dependence of power conservation error on r. The simulation parameters are those of Fig. 7.

Fig. 9
Fig. 9

Bistatic co-polarized scattering coefficients for h=0.2λ, lx=ly=0.6λ, Lx=Ly=16λ, r=6.5+i1.0, rd=3.0λ, θi=10°, and ϕi=0°. The result illustrates convergence with respect to the number of realizations.  

Fig. 10
Fig. 10

Bistatic co-polarized scattering coefficients (hh) with the simulation parameters of Fig. 9. The result illustrates both the incoherent and coherent components.

Fig. 11
Fig. 11

Bistatic cross-polarized scattering coefficients with the simulation parameters of Fig. 9. The result illustrates convergence with respect to the number of realizations.

Fig. 12
Fig. 12

Bistatic scattering coefficients for h=0.5λ, lx=ly=1.0λ, Lx=Ly=16λ, r=6.5+i1.0, rd=3.5λ, θi=10°, and ϕi=0°.

Fig. 13
Fig. 13

Backscattering enhancement of two cases: 1-D and 2-D electromagnetic wave incidence on a dielectric surface (r=6.5+i1.0). A comparison is shown of normalized bistatic scattering coefficients for h=0.5λ, l=1.0λ, θi=10°, and ϕi=0°. The number of realizations in each case is given in parentheses.

Equations (62)

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

Zx=b.
Z=ZS+ZW,
ZW=m=0MZmW,
ZFS=Z0W.
(ZS+ZFS)x(1)=b,
(ZS+ZFS)x(n+1)=b(n+1),
b(n+1)=b-m=1MZmWx(n).
Ei(x, y, z)
=-+ dkx-+ dky exp(ikxx+ikyy-ikzz)×ETE(kx, ky)eˆ(-kz),
Hi(x, ,y z)
=-1η1 -+ dkx-+dky exp(ikx+ikyy-ikzz)×ETE(kx, ky)hˆ(-kz),
eˆ(-kz)=1kρ (xˆky-yˆky),
hˆ(-kz)=kzkkρ (xˆkx+yˆky)+kρk zˆ,
ETE(kx, ky)=14π2 -+dx-+dy×exp(-ikxx-ikyy)×exp[i(kixx+kiyy)(1+w)]×exp(-t),
tx=[(cos θi cos ϕτ)x+(cos θi sin ϕτ)y]2g2 cos2 θi,
ty=[-(sin ϕi)x+(cos ϕi)y]2g2,
w=1k2 2tx-1g2 cos2 θi+2ty-1g2.
0=-nˆ×Er2-nˆ×iωnˆ×H(r)μ2×g2dS+P[n^×E(r)×g2]+n^E(r) 12 g2dS,
nˆ×Hinc(r)=nˆ×H(r)2-nˆ×-iωn^×E(r)1g1dS+P[n^×H(r)×g1+n^H(r)g1]dS,
nˆEinc(r)=nˆE(r)2-nˆn^×H(r)iωμ1g1dS+P{[n^×E(r)×g1]+g1n^E(r)}dS,
0=-nˆH(r)2-nˆ-n^E(r)iω2g2dS+P{[n^×H(r)×g2]+g2n^H(r){dS,
g1,2=exp(ik1,2R)4πR,
g1,2=(r-r)G1,2(R),
G1,2(R)=(1-ik1,2R)exp(ik1,2R)4πR3
0=-Ix(r)2+PdxdyIx(r)G2(R)f(x, y)y (y-y)+f(x, y)x (x-x)-(z-z)+Iy(r)G2(R)-f(x, y)y (x-x)+f(x, y)y (x-x)+In(r) 12 G2(R)f(x, y)y (z-z)+(y-y)+ dxdyik1η1g2Fx(r) f(x, y)y f(x, y)x+ik1η1g2Fy(r)f(x, y)y f(x, y)y+1,
0=-Iy(r)2+PdxdyIx(r)G2(R)f(x, y)x (y-y)-f(x, y)x (y-y)+Iy(r)G2(R)-(z-z)+f(x, y)y (y-y)+f(x, y)x (x-x)+In(r) 12 G2(R)-f(x, y)x (z-z)-(x-x)+dxdyik1η1g2Fx(r)-1-f(x, y)x f(x, y)x+iFy(r)k1η1g2-f(x, y)x f(x, y)y,
Ininc(r)=In(r)2+PdxdyG1(R)Ix(r)-f(x, y)x f(x, y)x (y-y)+f(x, y)y f(x, y)x (x-x)-(z-z)-(y-y)+G1(R)Iy(r)f(x, y)x (z-z)-f(x, y)y (y-y)+f(x, y)y f(x, y)y (x-x)+(x-x)+G1(R)In(r)f(x, y)x (x-x)+f(x, y)y (y-y)-(z-z)+dxdyik1η1g1Fx(r)f(x, y)x-f(x, y)x+ik1η1g1Fy(r)f(x, y)y-f(x, y)y,
Fxinc(r)=Fx(r)2+dxdy-i k1η1 g1Ix(r)f(x, y)y f(x, y)x-i k1η1 g1Iy(r)f(x, y)y f(x, y)y+1+PdxdyG1(R)Fx(r)f(x, y)y (y-y)+f(x, y)x (x-x)-(z-z)+G1(R)Fy(r)-f(x, y)y (x-x)+f(x, y)y (x-x)+G1(R)Fn(r)f(x, y)y (z-z)+(y-y),
Fyinc(r)=Fy(r)2+dxdy-i k1η1 g1Ix(r)-1-f(x, y)x f(x, y)x+i k1η1 g1Iy(r) f(x, y)y f(x, y)x+PdxdyG1(R)Fx(r)f(x, y)x (y-y)-f(x, y)x (y-y)+G1(R)Fy(r)-(z-z)+f(x, y)y (y-y)+f(x, y)x (x-x)+G1(R)Fn(r)-f(x, y)x (z-z)-(x-x),
0=-Fn(r)2+dxdy-ik1η1 21 g2Ix(r)f(x, y)x-f(x, y)x-ik1η1 21 Iy(r)g2f(x, y)y-f(x, y)y+PdxdyG2(R)Fx(r)-f(x, y)x f(x, y)x (y-y)+f(x, y)y f(x, y)x (x-x)-(z-z)-(y-y)+G2(R)Fy(r)f(x, y)x (z-z)+f(x, y)y f(x, y)y (x-x)-f(x, y)y f(x, y)x×(y-y)+(x-x)+ G2(R)Fn(r)f(x, y)x (x-x)+f(x, y)y (y-y)-(z-z),
Fx(r)=1+f(x, y)x2+f(x, y)y21/2nˆ×Hs(r)xˆ,
Fy(r)=1+f(x, y)x2+f(x, y)y21/2nˆ×Hs(r)yˆ,
Fn(r)=1+f(x, y)x2+f(x, y)y21/2nˆHs(r),
Ix(r)=1+f(x, y)x2+f(x, y)y21/2nˆ×Es(r)xˆ,
Iy(r)=1+f(x, y)x2+f(x, y)y21/2nˆ×Es(r)yˆ,
In(r)=1+f(x, y)x2+f(x, y)y21/2nˆEs(r).
ρR=[(x-x)2+(y-y)2]1/2
G1,2(R)=(1-ik1,2R)exp(ik1,2R)4πR3=m=0Mam(1, 2)(ρR)zd2ρR2m,
g1,2=exp(ik1,2R)4πR=m=0Mbm(1, 2)(ρR)zd2ρR2m,
a0(1, 2)(ρR)=(1-ik1,2ρR) exp(ik1,2ρR)4πρR3,
a1(1, 2)(ρR)=exp(ik1,2ρR)4π k1,222ρR+3ik1,22ρR2-32ρR3,
a2(1, 2)(ρR)=exp(ik1,2ρR)4π ik1,238-6k1,228ρR-15ik1,28ρR2+158ρR3,
a3(1, 2)(ρR)=exp(ik1,2ρR)4π -k1,24ρR48-10ik1,2348+105k1,22112ρR+35ik1,216ρR2-3516ρR3,
b0(1, 2)(ρR)=exp(ik1,2ρR)4πρR,
b1(1, 2)(ρR)=[exp(ik1,2ρR)]ik1,28π-18πρR,
b2(1, 2)(ρR)=[exp(ik1,2ρR)]-k1,22ρR32π-3ik1,232π+332πρR,
b3(1, 2)(ρR)=[exp(ik1,2ρR)]-i k1,23ρR2192π+k1,22ρR32π+13ik1,2192π-13192πρR.
NCGM[256rd2nN+2N(log N)mFFT]+NSMCG[72N(log N)mFFT],
ZX(n)-bb1/2<1%
γαh(θs, ϕs)=|Eαs|22ηPninc.
Phinc=2π2η kρ<k dkxdky|ETE(kx, ky)|2 kzk.
Ehs=ik4π dS dxdy exp(-ikβ)×Ix(x, y)cos θs cos ϕs+Iy(x, y)cos θs sin θs-Ix(x, y) f(x, y)x sin θs-Iy(x, y)f(x, y)y sin θs-η[Fx(x, y)sin ϕs-Fy(x, y)cos ϕs],
Evs=ik4π dS dxdy exp(-ikβ)[Ix(x, y)sin ϕs-Iy(x, y)cos ϕs]+ηFx(x, y)cos θs cos θs+Fycos θs sin ϕs-Fx(x, y) f(x, y)x sin θs-Fy(x, y) f(x, y)y sin θs,
W(Kx, Ky)=lxlyh24π exp-Kx2lx24-Ky2ly24,
γcoherent(θs, ϕs)=|Ehs|22ηPhinc,
γincoherent(θs, ϕs)=|Ehs|2-|Ehs|22ηPhinc,
-m=15ρR>rd dxdyIx(n)(r)am(2)(ρR)zd2ρRmf(x, y)y (y-y)+f(x, y)x (x-x)-(z-z)+Iy(n)(r)am(2)(ρR)zd2ρRm-f(x, y)y (x-x)+f(x, y)y (x-x)+In(n)(r) 12 am(2)zd2ρRm×f(x, y)y (z-z)+(y-y)-m=15ρR>rd dxdyik1η1bm(2)(ρR)zd2ρRmFx(n)(r)×f(x, y)y f(x, y)x+ik1η1bm(2)(ρR)zd2ρRmFy(n)(r)f(x, y)y f(x, y)y+1=-Ix(n+1)(r)2+P ρRrd dxdyIx(n+1)(r)G2(R)f(x, y)y (y-y)+f(x, y)x (x-x)-(z-z)+Iy(n+1)(r)G2(R)-f(x, y)y (x-x)+f(x, y)y (x-x)+In(n+1)(r) 12 G2(R)×f(x, y)y (z-z)+(y-y)+ρRrd dxdyik1η1g2Fx(n+1)(r) f(x, y)y f(x, y)x+ik1η1g2Fy(n+1)(r)×f(x, y)y f(x, y)y+1+ρR>rd dxdyIx(n+1)(r)a0(2)(ρR)f(x, y)y (y-y)+f(x, y)x (x-x)-(z-z)+Iy(n+1)(r)a0(2)(ρR)-f(x, y)y (x-x)+f(x, y)y (x-x)+In(n+1)(r) 12 a0(2)(ρR)f(x, y)y (z-z)+(y-y)+ dxdyik1η1b0(2)(ρR)Fx(n+1)(r)×f(x, y)y f(x, y)x+ik1η1b0(2)(ρR)Fy(n+1)(r)f(x, y)y f(x, y)y+1,
-m=15ρR>rd dxdyIx(n)(r)am(2)(ρR)zdρRmf(x, y)x (y-y)-f(x, y)x (y-y)+Iy(n)(r)am(2)(ρR)zdρRm-(z-z)+f(x, y)y (y-y)+f(x, y)x (x-x)+In(n)(r) 12 am(2)(ρR)zdρRm-f(x, y)x (z-z)+(x-x)-m=15ρR>rd dxdyik1η1bm(2)(ρR)zdρRm×Fx(n)(r)-1-f(x, y)x f(x, y)x+iFy(n)(r)k1η1bm(2)(ρR)zdρRm-f(x, y)x f(x, y)y=-Iy(n+1)(r)2+P ρR<rd dxdyIx(n+1)(r)G2(R)f(x, y)x (y-y)-f(x, y)x (y-y)+Iy(n+1)(r)G2(R)-(z-z)+f(x, y)y (y-y)+f(x, y)x (x-x)+In(n+1)(r) 12 G2(R)×-f(x, y)x (z-z)+(x-x)+ρR<rd dxdyik1η1g2Fx(n+1)(r)-1-f(x, y)x f(x, y)x+iFy(n+1)(r)k1η1g2-f(x, y)x f(x, y)y+ρR>rd dxdyIx(n+1)a0(2)(ρR)f(x, y)x (y-y)-f(x, y)x (y-y)+Iy(n+1)(r)a0(2)(ρR)-(z-z)+f(x, y)y (y-y)+f(x, y)x (x-x)+In(n+1)(r) 12 a0(2)(ρR)-f(x, y)x (z-z)+(x-x)+ρR>rd dxdyik1η1b0(2)(ρR)Fx(n+1)(r)×-1-f(x, y)x f(x, y)x+iFy(n+1)(r)k1η1b0(2)(ρR)-f(x, y)x f(x, y)y,
-m=15ρR>rd dxdyam(1)(ρR)zdρRmIx(r)-f(x, y)x f(x, y)x (y-y)+f(x, y)y f(x, y)x (x-x)-(z-z)-(y-y)+am(1)(ρR)zdρRmIy(r)f(x, y)x (z-z)-f(x, y)y (y-y)+f(x, y)y f(x, y)y (x-x)+(x-x)+am(1)(ρR)zdρRmIn(r)f(x, y)x (x-x)+f(x, y)y (y-y)-(z-z)-m=15ρR>rd dxdyik1η1bm(1)(ρR)zdρRmFx(r)f(x, y)x-f(x, y)x+ik1η1bm(1)(ρR)zdρRmFy(r)×f(x, y)y-f(x, y)y+Ininc(r)=In(r)2+P  dxdyG1(R)Ix(r)-f(x, y)x f(x, y)x (y-y)+f(x, y)y×f(x, y)x (x-x)-(z-z)-(y-y)+G1(R)Iy(r)f(x, y)x (z-z)-f(x, y)y (y-y)+f(x, y)y f(x, y)y (x-x)+(x-x)+G1(R)In(r)f(x, y)x (x-x)+f(x, y)y (y-y)-(z-z)+ dxdyik1η1g1Fx(r)f(x, y)x-f(x, y)x+ik1η1g1Fy(r)f(x, y)y-f(x, y)y+ρR>rd dxdya0(1)(ρR)Ix(r)-f(x, y)x f(x, y)x (y-y)+f(x, y)y×f(x, y)x (x-x)-(z-z)-(y-y)+a0(1)(ρR)Iy(r)f(x, y)x (z-z)-f(x, y)y (y-y)+f(x, y)y f(x, y)y (x-x)+(x-x)+a0(1)(ρR)In(r)f(x, y)x (x-x)+f(x, y)y (y-y)-(z-z)+ρR>rd dxdyik1η1b0(1)(ρR)Fx(r)f(x, y)x-f(x, y)x+ik1η1b0(1)(ρR)Fy(r)f(x, y)y-f(x, y)y,
Fxinc(r)-m=15ρR>rd dxdy-i k1η1 bm(1)(ρR)zd2ρR2mIx(n)(r)f(x, y)y f(x, y)x-i k1η1 bm(1)(ρR)zd2ρR2mIy(n)(r)×f(x, y)y f(x, y)y+1-m=15ρR>rd dxdyam(1)(ρR)zd2ρR2mFx(n)(r)f(x, y)y (y-y)+f(x, y)x (x-x)-(z-z)+am(1)(ρR)zd2ρR2mFy(n)(r)-f(x, y)y (x-x)+f(x, y)y (x-x)+am(1)(ρR)zd2ρR2mFn(n)(r)f(x, y)y (z-z)+(y-y)=Fx(n+1)(r)2+ρrd dxdy-i k1η1 g1Ix(n+1)(r)f(x, y)y f(x, y)x-i k1η1 g1Iy(n+1)(r)f(x, y)y f(x, y)y+1+P ρRrd dxdyG1(R)Fx(n+1)(r)f(x, y)y (y-y)+f(x, y)x (x-x)-(z-z)+G1(R)Fy(n+1)(r)-f(x, y)y (x-x)+f(x, y)y (x-x)+G1(R)Fy(n+1)(r)-f(x, y)y (z-z)+(y-y)+ρR>rddxdy-i k1η1 b0(1)(ρR)Ix(n+1)(r)×f(x, y)y f(x, y)x -i k1η1 b0(1)(ρR)Ix(n+1)(r)f(x, y)y f(x, y)y+1+ρR>rddxdy×a0(1)(ρR)Fx(n+1)(r)f(x, y)y (y-y)+f(x, y)x (x-x)-(z-z)+a0(1) (ρR)Fy(n+1)(r)×-f(x, y)y (x-x)+f(x, y)y (x-x)+a0(1) (ρR)Fy(n+1)(r)-f(x, y)y (z-z)-(y-y),
Fyinc(r)-m=15ρR>rd dxdy-i k1η1 bm(1)(ρR)zd2ρR2mIx(n)(r)-1-f(x, y)x-f(x, y)x+i k1η1 bm(1)(ρR)zd2ρR2mIy(n)×(r) f(x, y)y f(x, y)x-m=15ρR>rd dxdyam(1)(ρR)zd2ρR2mFx(n)(r)f(x, y)x (y-y)-f(x, y)x (y-y)+am(1)(ρR)zd2ρR2mFy(n)(r)-(z-z)+f(x, y)y (y-y)+f(x, y)x (x-x)+am(1)(ρR)zd2ρR2mFn(n)(r)-f(x, y)x (z-z)-(x-x)=Fy(n+1)(r)2+ρRrd dxdy-i k1η1 g1Ix(n+1)(r)×-1-f(x, y)x-f(x, y)x+i k1η1 g1Iy(n+1)(r) f(x, y)y f(x, y)x+P ρRrd dxdy×G1(R)Fx(n+1)(r)f(x, y)x (y-y)-f(x, y)x (y-y)+G1(R)Fy(n+1)(r)×-(z-z)+f(x, y)y (y-y)+f(x, y)x (x-x)+G1(R)Fn(n+1)(r)-f(x, y)x (z-z)-(x-x)+ρR>rd dxdy-i k1η1 b0(1)(ρR)Ix(n+1)(r)-1-f(x, y)x-f(x, y)x+i k1η1 b0(1)(ρR)[Iy(n+1)(r)]×f(x, y)y f(x, y)x+ρR>rd dxdya0(1)(ρR)Fx(n+1)(r)f(x, y)x (y-y)-f(x, y)x (y-y)+a0(1)(ρR)Fy(n+1)(r)-(z-z)+f(x, y)y (y-y)+f(x, y)x (x-x)+a0(1)(ρR)Fn(n+1)(r)-f(x, y)x (z-z)-(x-x),
m=15ρ>rd dxdy-ik1η1 21 bm(2)(ρR)zd2ρR2mIx(n)(r)f(x, y)x-f(x, y)x-ik1η1 21 Iy(n)(r)bm(2)(ρR)zd2ρR2m×f(x, y)y-f(x, y)y-m=15ρ>rd dxdyam(2)(ρR)zd2ρR2mFx(n)(r)-f(x, y)x f(x, y)x (y-y)+f(x, y)yf(x, y)x (x-x)-(z-z)-(y-y)+am(2)(ρR)zd2ρR2mFy(n)(r)f(x, y)x (z-z)+f(x, y)y f(x, y)y (x-x)-f(x, y)y f(x, y)x (y-y)+(x-x)+am(2)(ρR)zd2ρR2m[Fn(n)(r)]×f(x, y)x (x-x)+f(x, y)y (y-y)-(z-z)=-Fn(r)2+ρRrd dxdy-ik1η1 21 g2Ix(n+1)(r)f(x, y)x-f(x, y)x-ik1η1 21 Iy(n+1)(r)g2×f(x, y)y-f(x, y)y+P ρRrd dxdyG2Fx(n+1)(r)-f(x, y)x f(x, y)x (y-y)+f(x, y)y×f(x, y)x (x-x)-(z-z)-(y-y)+G2(R)Fy(n+1)(r)f(x, y)x (z-z)+f(x, y)y f(x, y)y×(x-x)-f(x, y)y f(x, y)x (y-y)+(x-x)+G2(R)Fn(n+1)(r)f(x, y)x (x-x)+f(x, y)y (y-y)-(z-z)+ρ>rd dxdy-ik1η1 21 b0(2)(ρR)Ix(n+1)(r)×f(x, y)x-f(x, y)x-ik1η1 21 Iy(n+1)(r)b0(2)(ρR)f(x, y)y-f(x, y)y+ρ>rd dxdy×a0(2)(ρR)Fx(n+1)(r)-f(x, y)x f(x, y)x (y-y)+f(x, y)y f(x, y)x (x-x)-(z-z)-(y-y)+a0(2)(ρR)Fy(n+1)(r)f(x, y)x (z-z)+f(x, y)y f(x, y)y (x-x)-f(x, y)y f(x, y)x (y-y)+(x-x)+a0(2)(ρR)Fn(n+1)(r)f(x, y)x (x-x)+f(x, y)y (y-y)-(z-z).

Metrics