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2007-03-22
Fast Multipole Accelerated Scattering Matrix Method for Multiple Scattering of a Large Number of Cylinders
By
Progress In Electromagnetics Research, Vol. 72, 105-126, 2007
Abstract
The lowering and raising operators of cylindrical harmonics are used to derive the general fast multipole expressions of arbitrary order Hankel functions. These expressions are then employed to transform the dense matrix in the scattering matrix method (SMM) into a combination of sparse matrices (aggregation, translation and disaggregation matrices). The novel method is referred to as fast multipole accelerated scattering matrix method (FMA-SMM). Theoretical study shows FMA-SMM has lower complexity O(N1.5) instead of SMM's O(N2), where N stands for total harmonics number used. An empirical formula is derived to relate the minimum group size in FMA-SMM to the highest order Hankel functions involved. The various implementation parameters are carefully investigated to guarantee the algorithm's accuracy and efficiency. The impact of the cylinders density on convergence rate of iterative solvers (BiCGStab(2) here), memory cost as well as CPU time is also investigated. Up to thousands of cylinders can be easily simulated and potential applications in photonic crystal devices are illustrated.
Citation
Yao Jiang Zhang, and Er Ping Li, "Fast Multipole Accelerated Scattering Matrix Method for Multiple Scattering of a Large Number of Cylinders," Progress In Electromagnetics Research, Vol. 72, 105-126, 2007.
doi:10.2528/PIER07030503
References

1. Foldy, L. L., "The multiple scattering of waves I. General theory of isotropic scattering by randomly distributed scatterers," Phys. Rev., Vol. 67, No. 2, 107-119, 1945.
doi:10.1103/PhysRev.67.107

2. Lax, M., "Multiple scattering of waves," Rev. Mod. Phys., Vol. 23, No. 10, 287-310, 1951.
doi:10.1103/RevModPhys.23.287

3. Twersky, V., "Multiple scattering of radiation by an arbitrary configuration of parallel cylinders," J. Acoust. Soc. Am, Vol. 24, No. 1, 42-46, 1952.
doi:10.1121/1.1906845

4. Waterman, P. C., "New formulation of acoustic scattering," J. Acoust. Soc. Am., Vol. 45, No. 6, 1417-1429, 1969.
doi:10.1121/1.1911619

5. Waterman, P. C., "Symmetry, unitarity, and geometry in electromagnetic scattering," Phys. Rev. D, Vol. 3, No. 2, 825-829, 1971.
doi:10.1103/PhysRevD.3.825

6. Peterson, B. and S. Ström, "T-matrix for electromagnetic scattering from an arbitrary number of scatterers and representation of E(3)," Phys. Rev. D, Vol. 8, No. 11, 3661-3678, 1973.
doi:10.1103/PhysRevD.8.3661

7. Chew, W. C., C. C. Lu, and Y. M. Wang, "Efficient computation of three-dimensional scattering of vector electromagnetic waves," J. Opt. Soc. Am. A, Vol. 11, No. 4, 1528-1537, 1994.

8. Tayeb, G. and D. Maystre, "Rigorous theoretical study of finitesize two-dimensional photonic crystals doped by microcavities," J. Opt. Soc. Am. A, Vol. 14, No. 12, 3323-3332, 1997.

9. Yonekura, J., M. Ikeda, and T. Baba, "Analysis of finite 2- D photonic crystals of columns and lightwave devices using the scattering matrix method," J. Lightwave Tech., Vol. 17, No. 8, 1500-1508, 1999.
doi:10.1109/50.779177

10. Li, E. P., Q. X. Wang, Y. J. Zhang, and B. L. Ooi, "Analysis of finite-size coated electromagnetic bandgap structure by an efficient scattering matrix method," IEEE J. Selected Topics Quantum Elect., Vol. 11, No. 4, 485-492, 2005.
doi:10.1109/JSTQE.2005.845619

11. Kuo, C.-H. and Z. Ye, "Negative-refraction like behavior revealed by arrays of dielectric cylinders," Phys. Rev. E, Vol. 70, 026608, 2004.
doi:10.1103/PhysRevE.70.026608

12. Shooshtari, A. and A. R. Sebak, "Electromagnetic scattering by parallel metamaterial cylinders," Progress In Electromagnetics Research, Vol. 57, 165-177, 2006.
doi:10.2528/PIER05071103

13. Boscolo, S. and M. Midrio, "Three-dimensional multiplescattering technique for the analysis of photonic-crystal slabs," J. Lightwave Tech., Vol. 22, No. 12, 2778-2786, 2004.
doi:10.1109/JLT.2004.833276

14. Talebi, N., M. Shahabadi, and C. Hafner, "Analysis of a lossy microring using the generalized multipole technique," Progress In Electromagnetics Research, Vol. 66, 287-299, 2006.
doi:10.2528/PIER06112801

15. Koc, S. and W. C. Chew, "Calculation of acoustical scattering from a cluster of scatterers," J. Acoust. Soc. Am, Vol. 103, No. 2, 721-734, 1998.
doi:10.1121/1.421231

16. Gumerov, N. A. and R. Duraiswami, "Computation of scattering from clusters of spheres using the fast multipole method," J. Acoust. Soc. Am, Vol. 117, No. 4, 1744-1761, 2005.
doi:10.1121/1.1853017

17. Cheng, H., W. Y. Crutchfield, Z. Gimbutas, L. F. Greengard, J. F. Ethridge, J. Huang, V. Rokhlin, N. Yarvin, and J. Zhao, "A wideband fast multipole method for the Helmholtz equation in three dimensions," J. Comput. Phys., Vol. 216, No. 7, 300-325, 2006.
doi:10.1016/j.jcp.2005.12.001

18. Gumerov, N. A. and R. Duraiswami, Fast Multipol Methods for the Helmholtz Equation in Three Dimensions, Elsevier Ltd., 2004.

19. Rokhlin, V., "Rapid solution of integral equations of scattering theory in two dimensions," J. Comput. Phys., Vol. 86, No. 2, 414-439, 1990.
doi:10.1016/0021-9991(90)90107-C

20. Engheta, N., W. D. Murphy, V. Rokhlin, and M. S. Vassiliou, "The fast multipole method (FMM) for electomagnetic scattering problems," IEEE Trans. Antennas Propagat., Vol. 40, No. 6, 634-641, 1992.
doi:10.1109/8.144597

21. Lu, C. C. and W. C. Chew, "Fast algorithm for solving hybrid integral equations," IEE Proc.-H, Vol. 140, No. 12, 455-460, 1993.

22. Chew, W. C., J. M. Jin, E. Michielssen, and J. M. Song, Fast and Efficient Algorithms in Computational Electromagnetics, Artech House, 2001.

23. Chew, W. C., Waves and Fields in Inhomogeneous Media, Van Nostrand Reinhold, New York, U.S.A, 1990.

24. Van der Vorst, H. A., "Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems," SIAM J. Sci. Stat. Comput., Vol. 13, 631-644, 1992.
doi:10.1137/0913035

25. Ohnuki, S. and W. C. Chew, "Numerical accuracy of multipole expansion for 2-D MLFMA," IEEE Trans. Antennas Propagat., Vol. 51, No. 8, 1883-1890, 2003.
doi:10.1109/TAP.2003.815425

26. Felbacq, D., G. Tayeb, and D. Maystre, "Scattering by a random set of parallel cylinders," J. Opt. Soc. Am. A, Vol. 11, No. 9, 2526-2538, 1994.

27. Elsherbeni, A. Z. and M. Hamid, "Scattering by parallel conducting circular cylinders," IEEE Trans. Antennas Propagat., Vol. AP-35, No. 3, 355-358, 1987.
doi:10.1109/TAP.1987.1144098

28. Ragheb, H. A. and M. Hamid, "Simulation of a cylindrical reflector by conducting circular cylinders," IEEE Trans. Antennas Propagat., Vol. AP-35, No. 3, 349-353, 1987.
doi:10.1109/TAP.1987.1144096

29. Liu, T., A. R. Zakharian, M. Fallahi, V. Moloney, and M. Mansuripur, "Multimode Interference-based photonic crystal waveguide power splitter," J. Lightwave Tech., Vol. 22, No. 12, 2842-2846, 2004.
doi:10.1109/JLT.2004.834479