A study of electromagnetic scattering from conducting targets above and below the dielectric rough surface |
Optics Express, Vol. 19, Issue 7, pp. 5785-5801 (2011)
http://dx.doi.org/10.1364/OE.19.005785
Acrobat PDF (1495 KB)
Abstract
The composite scattering from the conducting targets above and below the dielectric rough surface using the extended Propagation-Inside-Layer Expansion (EPILE) combined with the Forward-Backward method (FBM) is studied. The established integral equations are approved by comparing with the related theory. The accuracy and efficiency of the EPILE + FBM are compared with the method of moments (MOM). The influences of target size, target height/depth, target position, and the rms height, the correlation length, as well as the incident angle on the bistatic scattering coefficient (BSC) for different polarizations are also investigated. The presented algorithm is of generality for the target-rough surface composite scattering problems.
© 2011 OSA
1. Introduction
2. D. Holliday, “Resolution of a controversy surrounding the Kirchhoff approach and the small perturbation method in rough surface scattering theory,” IEEE Trans. Antenn. Propag. 35(1), 120–122 (1987). [CrossRef]
3. L. X. Guo and Z. S. Wu, “Application of the extended boundary condition method to electromagnetic scattering from rough dielectric fractal sea surface,” J. Electromagn. Waves Appl. 18(9), 1219–1234 (2004). [CrossRef]
4. S. L. Durden and J. F. Vesecky, “A numerical study of the separation wavenumber in the two-scale scattering approximation,” IEEE Trans. Geosci. Rem. Sens. 28(2), 271–272 (1990). [CrossRef]
5. D. Winebrenner and A. Ishimaru, “Investigation of a surface field phase perturbation technique for scattering from rough surfaces,” Radio Sci. 20(2), 161–170 (1985). [CrossRef]
6. A. Voronovich, “Small-slope approximation for electromagnetic wave scattering at a rough interface of two dielectric half- spaces,” Waves Random Media 4(3), 337–367 (1994). [CrossRef]
7. R. R. Lentz, “A numerical study of electromagnetic scattering from ocean-like surfaces,” Radio Sci. 9(12), 1139–1146 (1974). [CrossRef]
8. R. M. Axline and A. K. Fung, “Numerical computations of scattering from a perfectly conducting random surface,” IEEE Trans. Antenn. Propag. 26(3), 482–488 (1978). [CrossRef]
9. C. H. Chan, S. H. Lou, L. Tsang, and J. A. Kong, “Electromagnetic scattering of waves by random rough surface: A finite- difference time-domain approach,” Microw. Opt. Technol. Lett. 4(9), 355–359 (1991). [CrossRef]
10. S. H. Lou, L. Tsang, and C. H. Chan, “Application of finite element method to Monte Carlo simulations of scattering of waves by random rough surfaces: penetrable case,” Waves Random Media 1(4), 287–307 (1991). [CrossRef]
11. D. A. Kapp and G. S. Brown, “A new numerical method for rough surface scattering calculations,” IEEE Trans. Antenn. Propag. 44(5), 711–721 (1996). [CrossRef]
12. V. Jandhyala, E. Michielssen, S. Balasubramaniam, and W. C. Chew, “A combined steepest descent-fast multipole algorithm for the fast analysis of three-dimensional scattering by rough surfaces,” IEEE Trans. Geosci. Rem. Sens. 36(3), 738–748 (1998). [CrossRef]
13. L. Tsang, C. H. Chan, H. Sangani, A. Ishimaru, and P. Phu, “A Banded Matrix Iterative Approach to Monte Carlo simulations of large scale random rough surface scattering: TE case,” J. Electromagn. Waves Appl. 7(9), 1185–1200 (1993). [CrossRef]
14. A. Iodice, “Forward–backward method for scattering from dielectric rough surfaces,” IEEE Trans. Antenn. Propag. 50(7), 901–911 (2002). [CrossRef]
15. X. Wang, C. F. Wang, Y. B. Gan, and L. W. Li, “Electromagnetic scattering from a circular target above or below rough surface,” Prog. Electromagn. Res. 40, 207–227 (2003). [CrossRef]
16. Y. Q. Jin and G. Li, “Detection of a scatter target over a randomly rough surface by using the angular correlation function in a finite-element approach,” Waves Random Media 10(2), 273–280 (2000). [CrossRef]
18. Y. Zhang, Y. E. Yang, H. Braunisch, and J. A. Kong, “Electromagnetic wave interaction of conducting object with rough surface by hybrid SPM/MOM technique,” Prog. Electromagn. Res. 22, 315–335 (1999). [CrossRef]
19. H. Ye and Y. Q. Jin, “A hybrid KA-MOM algorithm for computation of scattering from a 3-D PEC target above a dielectric rough surface,” Radio Sci. 43(3), RS3005 (2008). [CrossRef]
20. S. Y. He and G. Q. Zhu, “A hybrid MM-PO method combining UV technique for scattering from two-dimensional target above a rough surface,” Microw. Opt. Technol. Lett. 49(12), 2957–2960 (2007). [CrossRef]
21. T. Chiu and K. Sarabandi, “Electromagnetic scattering interaction between a dielectric cylinder and a slightly rough surface,” IEEE Trans. Antenn. Propag. 47(5), 902–913 (1999). [CrossRef]
22. G. Lixin and K. Cheyoung, “Light scattering models for a spherical particle above a slightly dielectric rough surface,” Microw. Opt. Technol. Lett. 33(2), 142–146 (2002). [CrossRef]
24. Z. Li and Y. Q. Jin, “Bistatic scattering from a fractal dynamic rough sea surface with a ship presence at low grazing-angle incidence using the FBM/SAA,” Microw. Opt. Technol. Lett. 31(2), 146–151 (2001). [CrossRef]
25. N. Déchamps, N. de Beaucoudrey, C. Bourlier, and S. Toutain, “Fast numerical method for electromagnetic scattering by rough layered interfaces: propagation-inside-layer expansion method,” J. Opt. Soc. Am. A 23(2), 359–369 (2006). [CrossRef]
25. N. Déchamps, N. de Beaucoudrey, C. Bourlier, and S. Toutain, “Fast numerical method for electromagnetic scattering by rough layered interfaces: propagation-inside-layer expansion method,” J. Opt. Soc. Am. A 23(2), 359–369 (2006). [CrossRef]
26. G. Kubické, C. Bourlier, and J. Saillard, “Scattering by an object above a randomly rough surface from a fast numerical method: Extended PILE method combined with FB-SA,” Waves Random Complex Media 18(3), 495–519 (2008). [CrossRef]
27. C. Bourlier, G. Kubické, and N. Déchamps, “Fast method to compute scattering by a buried object under a randomly rough surface: PILE combined with FB-SA,” J. Opt. Soc. Am. A 25(4), 891–902 (2008). [CrossRef]
25. N. Déchamps, N. de Beaucoudrey, C. Bourlier, and S. Toutain, “Fast numerical method for electromagnetic scattering by rough layered interfaces: propagation-inside-layer expansion method,” J. Opt. Soc. Am. A 23(2), 359–369 (2006). [CrossRef]
27. C. Bourlier, G. Kubické, and N. Déchamps, “Fast method to compute scattering by a buried object under a randomly rough surface: PILE combined with FB-SA,” J. Opt. Soc. Am. A 25(4), 891–902 (2008). [CrossRef]
26. G. Kubické, C. Bourlier, and J. Saillard, “Scattering by an object above a randomly rough surface from a fast numerical method: Extended PILE method combined with FB-SA,” Waves Random Complex Media 18(3), 495–519 (2008). [CrossRef]
27. C. Bourlier, G. Kubické, and N. Déchamps, “Fast method to compute scattering by a buried object under a randomly rough surface: PILE combined with FB-SA,” J. Opt. Soc. Am. A 25(4), 891–902 (2008). [CrossRef]
2. Formulation of the composite problem
29. E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83(1), 78–92 (1988). [CrossRef]
15. X. Wang, C. F. Wang, Y. B. Gan, and L. W. Li, “Electromagnetic scattering from a circular target above or below rough surface,” Prog. Electromagn. Res. 40, 207–227 (2003). [CrossRef]
15. X. Wang, C. F. Wang, Y. B. Gan, and L. W. Li, “Electromagnetic scattering from a circular target above or below rough surface,” Prog. Electromagn. Res. 40, 207–227 (2003). [CrossRef]
15. X. Wang, C. F. Wang, Y. B. Gan, and L. W. Li, “Electromagnetic scattering from a circular target above or below rough surface,” Prog. Electromagn. Res. 40, 207–227 (2003). [CrossRef]
3. The EPILE combined with the FBM for conducting targets both above and below the dielectric rough surface
25. N. Déchamps, N. de Beaucoudrey, C. Bourlier, and S. Toutain, “Fast numerical method for electromagnetic scattering by rough layered interfaces: propagation-inside-layer expansion method,” J. Opt. Soc. Am. A 23(2), 359–369 (2006). [CrossRef]
26. G. Kubické, C. Bourlier, and J. Saillard, “Scattering by an object above a randomly rough surface from a fast numerical method: Extended PILE method combined with FB-SA,” Waves Random Complex Media 18(3), 495–519 (2008). [CrossRef]
27. C. Bourlier, G. Kubické, and N. Déchamps, “Fast method to compute scattering by a buried object under a randomly rough surface: PILE combined with FB-SA,” J. Opt. Soc. Am. A 25(4), 891–902 (2008). [CrossRef]
25. N. Déchamps, N. de Beaucoudrey, C. Bourlier, and S. Toutain, “Fast numerical method for electromagnetic scattering by rough layered interfaces: propagation-inside-layer expansion method,” J. Opt. Soc. Am. A 23(2), 359–369 (2006). [CrossRef]
27. C. Bourlier, G. Kubické, and N. Déchamps, “Fast method to compute scattering by a buried object under a randomly rough surface: PILE combined with FB-SA,” J. Opt. Soc. Am. A 25(4), 891–902 (2008). [CrossRef]
25. N. Déchamps, N. de Beaucoudrey, C. Bourlier, and S. Toutain, “Fast numerical method for electromagnetic scattering by rough layered interfaces: propagation-inside-layer expansion method,” J. Opt. Soc. Am. A 23(2), 359–369 (2006). [CrossRef]
27. C. Bourlier, G. Kubické, and N. Déchamps, “Fast method to compute scattering by a buried object under a randomly rough surface: PILE combined with FB-SA,” J. Opt. Soc. Am. A 25(4), 891–902 (2008). [CrossRef]
26. G. Kubické, C. Bourlier, and J. Saillard, “Scattering by an object above a randomly rough surface from a fast numerical method: Extended PILE method combined with FB-SA,” Waves Random Complex Media 18(3), 495–519 (2008). [CrossRef]
27. C. Bourlier, G. Kubické, and N. Déchamps, “Fast method to compute scattering by a buried object under a randomly rough surface: PILE combined with FB-SA,” J. Opt. Soc. Am. A 25(4), 891–902 (2008). [CrossRef]
14. A. Iodice, “Forward–backward method for scattering from dielectric rough surfaces,” IEEE Trans. Antenn. Propag. 50(7), 901–911 (2002). [CrossRef]
14. A. Iodice, “Forward–backward method for scattering from dielectric rough surfaces,” IEEE Trans. Antenn. Propag. 50(7), 901–911 (2002). [CrossRef]
4. Numerical results and discussions
15. X. Wang, C. F. Wang, Y. B. Gan, and L. W. Li, “Electromagnetic scattering from a circular target above or below rough surface,” Prog. Electromagn. Res. 40, 207–227 (2003). [CrossRef]
5. Conclusions
Acknowledgments
References and links
1. | S. O. Rice, “Reflection of Electromagnetic Waves from Slightly Rough Surfaces,” in Theory of Electromagnetic Waves, M. Kline, ed. (Wiley, 1951). |
2. | D. Holliday, “Resolution of a controversy surrounding the Kirchhoff approach and the small perturbation method in rough surface scattering theory,” IEEE Trans. Antenn. Propag. 35(1), 120–122 (1987). [CrossRef] |
3. | L. X. Guo and Z. S. Wu, “Application of the extended boundary condition method to electromagnetic scattering from rough dielectric fractal sea surface,” J. Electromagn. Waves Appl. 18(9), 1219–1234 (2004). [CrossRef] |
4. | S. L. Durden and J. F. Vesecky, “A numerical study of the separation wavenumber in the two-scale scattering approximation,” IEEE Trans. Geosci. Rem. Sens. 28(2), 271–272 (1990). [CrossRef] |
5. | D. Winebrenner and A. Ishimaru, “Investigation of a surface field phase perturbation technique for scattering from rough surfaces,” Radio Sci. 20(2), 161–170 (1985). [CrossRef] |
6. | A. Voronovich, “Small-slope approximation for electromagnetic wave scattering at a rough interface of two dielectric half- spaces,” Waves Random Media 4(3), 337–367 (1994). [CrossRef] |
7. | R. R. Lentz, “A numerical study of electromagnetic scattering from ocean-like surfaces,” Radio Sci. 9(12), 1139–1146 (1974). [CrossRef] |
8. | R. M. Axline and A. K. Fung, “Numerical computations of scattering from a perfectly conducting random surface,” IEEE Trans. Antenn. Propag. 26(3), 482–488 (1978). [CrossRef] |
9. | C. H. Chan, S. H. Lou, L. Tsang, and J. A. Kong, “Electromagnetic scattering of waves by random rough surface: A finite- difference time-domain approach,” Microw. Opt. Technol. Lett. 4(9), 355–359 (1991). [CrossRef] |
10. | S. H. Lou, L. Tsang, and C. H. Chan, “Application of finite element method to Monte Carlo simulations of scattering of waves by random rough surfaces: penetrable case,” Waves Random Media 1(4), 287–307 (1991). [CrossRef] |
11. | D. A. Kapp and G. S. Brown, “A new numerical method for rough surface scattering calculations,” IEEE Trans. Antenn. Propag. 44(5), 711–721 (1996). [CrossRef] |
12. | V. Jandhyala, E. Michielssen, S. Balasubramaniam, and W. C. Chew, “A combined steepest descent-fast multipole algorithm for the fast analysis of three-dimensional scattering by rough surfaces,” IEEE Trans. Geosci. Rem. Sens. 36(3), 738–748 (1998). [CrossRef] |
13. | L. Tsang, C. H. Chan, H. Sangani, A. Ishimaru, and P. Phu, “A Banded Matrix Iterative Approach to Monte Carlo simulations of large scale random rough surface scattering: TE case,” J. Electromagn. Waves Appl. 7(9), 1185–1200 (1993). [CrossRef] |
14. | A. Iodice, “Forward–backward method for scattering from dielectric rough surfaces,” IEEE Trans. Antenn. Propag. 50(7), 901–911 (2002). [CrossRef] |
15. | X. Wang, C. F. Wang, Y. B. Gan, and L. W. Li, “Electromagnetic scattering from a circular target above or below rough surface,” Prog. Electromagn. Res. 40, 207–227 (2003). [CrossRef] |
16. | Y. Q. Jin and G. Li, “Detection of a scatter target over a randomly rough surface by using the angular correlation function in a finite-element approach,” Waves Random Media 10(2), 273–280 (2000). [CrossRef] |
17. | J. Li, L. X. Guo, and H. Zeng, “FDTD investigation on the electromagnetic scattering from a target above a randomly rough sea surface,” Waves Random Media 10, 273–280 (2008). |
18. | Y. Zhang, Y. E. Yang, H. Braunisch, and J. A. Kong, “Electromagnetic wave interaction of conducting object with rough surface by hybrid SPM/MOM technique,” Prog. Electromagn. Res. 22, 315–335 (1999). [CrossRef] |
19. | H. Ye and Y. Q. Jin, “A hybrid KA-MOM algorithm for computation of scattering from a 3-D PEC target above a dielectric rough surface,” Radio Sci. 43(3), RS3005 (2008). [CrossRef] |
20. | S. Y. He and G. Q. Zhu, “A hybrid MM-PO method combining UV technique for scattering from two-dimensional target above a rough surface,” Microw. Opt. Technol. Lett. 49(12), 2957–2960 (2007). [CrossRef] |
21. | T. Chiu and K. Sarabandi, “Electromagnetic scattering interaction between a dielectric cylinder and a slightly rough surface,” IEEE Trans. Antenn. Propag. 47(5), 902–913 (1999). [CrossRef] |
22. | G. Lixin and K. Cheyoung, “Light scattering models for a spherical particle above a slightly dielectric rough surface,” Microw. Opt. Technol. Lett. 33(2), 142–146 (2002). [CrossRef] |
23. | M. R. Pino, L. Landesa, J. L. Rodriguez, F. Obelleiro, and R. J. Burkholder, “The generalized forward-backward method for analyzing the scattering from targets on ocean-like rough surfaces,” IEEE Trans. Antenn. Propag. 3, 961–968 (1998). |
24. | Z. Li and Y. Q. Jin, “Bistatic scattering from a fractal dynamic rough sea surface with a ship presence at low grazing-angle incidence using the FBM/SAA,” Microw. Opt. Technol. Lett. 31(2), 146–151 (2001). [CrossRef] |
25. | N. Déchamps, N. de Beaucoudrey, C. Bourlier, and S. Toutain, “Fast numerical method for electromagnetic scattering by rough layered interfaces: propagation-inside-layer expansion method,” J. Opt. Soc. Am. A 23(2), 359–369 (2006). [CrossRef] |
26. | G. Kubické, C. Bourlier, and J. Saillard, “Scattering by an object above a randomly rough surface from a fast numerical method: Extended PILE method combined with FB-SA,” Waves Random Complex Media 18(3), 495–519 (2008). [CrossRef] |
27. | C. Bourlier, G. Kubické, and N. Déchamps, “Fast method to compute scattering by a buried object under a randomly rough surface: PILE combined with FB-SA,” J. Opt. Soc. Am. A 25(4), 891–902 (2008). [CrossRef] |
28. | L. Tsang, and J. A. Kong, Scattering of Electromagnetic Waves- Numerical Simulations, (Wiley, 2000). |
29. | E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83(1), 78–92 (1988). [CrossRef] |
30. | M. R. Hestenes and E. Stiefel, “Method of conjugate gradients for solving linear systems,” J. Res. Natl. Bur. Stand. 49, 409–436 (1952). |
31. | G. L. G. Sleijpeny and D. R. Fokkema, “Bicgstab(L) for linear equations involving unsymmetric matrices with complex spectrum,” Electron. Trans. Numer. Anal. 1, 11–32 (1993). |
OCIS Codes
(290.5880) Scattering : Scattering, rough surfaces
ToC Category:
Scattering
History
Original Manuscript: January 4, 2011
Revised Manuscript: March 3, 2011
Manuscript Accepted: March 7, 2011
Published: March 14, 2011
Citation
Lixin Guo, Yu Liang, and Zhensen Wu, "A study of electromagnetic scattering from conducting targets above and below the dielectric rough surface," Opt. Express 19, 5785-5801 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-7-5785
Sort: Year | Journal | Reset
References
- S. O. Rice, “Reflection of Electromagnetic Waves from Slightly Rough Surfaces,” in Theory of Electromagnetic Waves, M. Kline, ed. (Wiley, 1951).
- D. Holliday, “Resolution of a controversy surrounding the Kirchhoff approach and the small perturbation method in rough surface scattering theory,” IEEE Trans. Antenn. Propag. 35(1), 120–122 (1987). [CrossRef]
- L. X. Guo and Z. S. Wu, “Application of the extended boundary condition method to electromagnetic scattering from rough dielectric fractal sea surface,” J. Electromagn. Waves Appl. 18(9), 1219–1234 (2004). [CrossRef]
- S. L. Durden and J. F. Vesecky, “A numerical study of the separation wavenumber in the two-scale scattering approximation,” IEEE Trans. Geosci. Rem. Sens. 28(2), 271–272 (1990). [CrossRef]
- D. Winebrenner and A. Ishimaru, “Investigation of a surface field phase perturbation technique for scattering from rough surfaces,” Radio Sci. 20(2), 161–170 (1985). [CrossRef]
- A. Voronovich, “Small-slope approximation for electromagnetic wave scattering at a rough interface of two dielectric half- spaces,” Waves Random Media 4(3), 337–367 (1994). [CrossRef]
- R. R. Lentz, “A numerical study of electromagnetic scattering from ocean-like surfaces,” Radio Sci. 9(12), 1139–1146 (1974). [CrossRef]
- R. M. Axline and A. K. Fung, “Numerical computations of scattering from a perfectly conducting random surface,” IEEE Trans. Antenn. Propag. 26(3), 482–488 (1978). [CrossRef]
- C. H. Chan, S. H. Lou, L. Tsang, and J. A. Kong, “Electromagnetic scattering of waves by random rough surface: A finite- difference time-domain approach,” Microw. Opt. Technol. Lett. 4(9), 355–359 (1991). [CrossRef]
- S. H. Lou, L. Tsang, and C. H. Chan, “Application of finite element method to Monte Carlo simulations of scattering of waves by random rough surfaces: penetrable case,” Waves Random Media 1(4), 287–307 (1991). [CrossRef]
- D. A. Kapp and G. S. Brown, “A new numerical method for rough surface scattering calculations,” IEEE Trans. Antenn. Propag. 44(5), 711–721 (1996). [CrossRef]
- V. Jandhyala, E. Michielssen, S. Balasubramaniam, and W. C. Chew, “A combined steepest descent-fast multipole algorithm for the fast analysis of three-dimensional scattering by rough surfaces,” IEEE Trans. Geosci. Rem. Sens. 36(3), 738–748 (1998). [CrossRef]
- L. Tsang, C. H. Chan, H. Sangani, A. Ishimaru, and P. Phu, “A Banded Matrix Iterative Approach to Monte Carlo simulations of large scale random rough surface scattering: TE case,” J. Electromagn. Waves Appl. 7(9), 1185–1200 (1993). [CrossRef]
- A. Iodice, “Forward–backward method for scattering from dielectric rough surfaces,” IEEE Trans. Antenn. Propag. 50(7), 901–911 (2002). [CrossRef]
- X. Wang, C. F. Wang, Y. B. Gan, and L. W. Li, “Electromagnetic scattering from a circular target above or below rough surface,” Prog. Electromagn. Res. 40, 207–227 (2003). [CrossRef]
- Y. Q. Jin and G. Li, “Detection of a scatter target over a randomly rough surface by using the angular correlation function in a finite-element approach,” Waves Random Media 10(2), 273–280 (2000). [CrossRef]
- J. Li, L. X. Guo, and H. Zeng, “FDTD investigation on the electromagnetic scattering from a target above a randomly rough sea surface,” Waves Random Media 10, 273–280 (2008).
- Y. Zhang, Y. E. Yang, H. Braunisch, and J. A. Kong, “Electromagnetic wave interaction of conducting object with rough surface by hybrid SPM/MOM technique,” Prog. Electromagn. Res. 22, 315–335 (1999). [CrossRef]
- H. Ye and Y. Q. Jin, “A hybrid KA-MOM algorithm for computation of scattering from a 3-D PEC target above a dielectric rough surface,” Radio Sci. 43(3), RS3005 (2008). [CrossRef]
- S. Y. He and G. Q. Zhu, “A hybrid MM-PO method combining UV technique for scattering from two-dimensional target above a rough surface,” Microw. Opt. Technol. Lett. 49(12), 2957–2960 (2007). [CrossRef]
- T. Chiu and K. Sarabandi, “Electromagnetic scattering interaction between a dielectric cylinder and a slightly rough surface,” IEEE Trans. Antenn. Propag. 47(5), 902–913 (1999). [CrossRef]
- G. Lixin and K. Cheyoung, “Light scattering models for a spherical particle above a slightly dielectric rough surface,” Microw. Opt. Technol. Lett. 33(2), 142–146 (2002). [CrossRef]
- M. R. Pino, L. Landesa, J. L. Rodriguez, F. Obelleiro, and R. J. Burkholder, “The generalized forward-backward method for analyzing the scattering from targets on ocean-like rough surfaces,” IEEE Trans. Antenn. Propag. 3, 961–968 (1998).
- Z. Li and Y. Q. Jin, “Bistatic scattering from a fractal dynamic rough sea surface with a ship presence at low grazing-angle incidence using the FBM/SAA,” Microw. Opt. Technol. Lett. 31(2), 146–151 (2001). [CrossRef]
- N. Déchamps, N. de Beaucoudrey, C. Bourlier, and S. Toutain, “Fast numerical method for electromagnetic scattering by rough layered interfaces: propagation-inside-layer expansion method,” J. Opt. Soc. Am. A 23(2), 359–369 (2006). [CrossRef]
- G. Kubické, C. Bourlier, and J. Saillard, “Scattering by an object above a randomly rough surface from a fast numerical method: Extended PILE method combined with FB-SA,” Waves Random Complex Media 18(3), 495–519 (2008). [CrossRef]
- C. Bourlier, G. Kubické, and N. Déchamps, “Fast method to compute scattering by a buried object under a randomly rough surface: PILE combined with FB-SA,” J. Opt. Soc. Am. A 25(4), 891–902 (2008). [CrossRef]
- L. Tsang, and J. A. Kong, Scattering of Electromagnetic Waves- Numerical Simulations, (Wiley, 2000).
- E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83(1), 78–92 (1988). [CrossRef]
- M. R. Hestenes and E. Stiefel, “Method of conjugate gradients for solving linear systems,” J. Res. Natl. Bur. Stand. 49, 409–436 (1952).
- G. L. G. Sleijpeny and D. R. Fokkema, “Bicgstab(L) for linear equations involving unsymmetric matrices with complex spectrum,” Electron. Trans. Numer. Anal. 1, 11–32 (1993).
Cited By |
Alert me when this paper is cited |
OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.
Figures
Fig. 1 | Fig. 2 | Fig. 4 |
Fig. 5 | Fig. 3 | Fig. 6 |
Fig. 7 | Fig. 8 | Fig. 9 |
Fig. 11 | Fig. 10 | Fig. 12 |
Fig. 13 | Fig. 14 | |
« Previous Article | Next Article »
OSA is a member of CrossRef.