Fast Multipole Accelerated Scattering Matrix Method for Multiple Scattering of a Large Number of Cylinders

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.

[1]  M. Midrio,et al.  Three-dimensional multiple-scattering technique for the analysis of photonic-Crystal slabs , 2004, Journal of Lightwave Technology.

[2]  Jian-Ming Jin,et al.  Fast and Efficient Algorithms in Computational Electromagnetics , 2001 .

[3]  M. Mansuripur,et al.  Multimode interference-based photonic crystal waveguide power splitter , 2004, Journal of Lightwave Technology.

[4]  Zydrunas Gimbutas,et al.  A wideband fast multipole method for the Helmholtz equation in three dimensions , 2006, J. Comput. Phys..

[5]  Atef Z. Elsherbeni,et al.  Scattering by parallel conducting circular cylinders , 1987 .

[6]  Zhen Ye,et al.  Negative-refraction-like behavior revealed by arrays of dielectric cylinders. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  Weng Cho Chew,et al.  Numerical accuracy of multipole expansion for 2D MLFMA , 2003 .

[8]  ANALYSIS OF A LOSSY MICRORING USING THE GENERALIZED MULTIPOLE TECHNIQUE , 2006 .

[9]  B. Peterson,et al.  T matrix for electromagnetic scattering from an arbitrary number of scatterers and representations of E(3) , 1973 .

[10]  Weng Cho Chew,et al.  Calculation of acoustical scattering from a cluster of scatterers , 1998 .

[11]  D. Felbacq,et al.  Scattering by a random set of parallel cylinders , 1994 .

[12]  L. Foldy,et al.  The Multiple Scattering of Waves. I. General Theory of Isotropic Scattering by Randomly Distributed Scatterers , 1945 .

[13]  Weng Cho Chew,et al.  Fast algorithm for solving hybrid integral equations , 1993 .

[14]  P. Waterman,et al.  New Formulation of Acoustic Scattering , 1969 .

[15]  T. Baba,et al.  Analysis of finite 2D photonic crystals of columns and lightwave devices using the scattering matrix method , 1999 .

[16]  M. Lax.  Multiple Scattering of Waves , 1951 .

[17]  V. Rokhlin.  Rapid Solution of Integral Equations of Scattering Theory , 1990 .

[18]  Henk A. van der Vorst,et al.  Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems , 1992, SIAM J. Sci. Comput..

[19]  W. Chew.  Waves and Fields in Inhomogeneous Media , 1990 .

[20]  M. Hamid,et al.  Simulation of a cylindrical reflector by conducting circular cylinders , 1987 .

[21]  Gérard Tayeb,et al.  Rigorous theoretical study of finite-size two-dimensional photonic crystals doped by microcavities , 1997 .

[22]  V. Twersky,et al.  Multiple Scattering of Radiation by an Arbitrary Configuration of Parallel Cylinders , 1952 .

[23]  V. Rokhlin,et al.  The fast multipole method (FMM) for electromagnetic scattering problems , 1992 .

[24]  P. Waterman,et al.  SYMMETRY, UNITARITY, AND GEOMETRY IN ELECTROMAGNETIC SCATTERING. , 1971 .

[25]  R. Duraiswami,et al.  Fast Multipole Methods for the Helmholtz Equation in Three Dimensions , 2005 .

[26]  Yaojiang Zhang,et al.  Analysis of finite-size coated electromagnetic bandgap structure by an efficient scattering matrix method , 2005 .

[27]  Ramani Duraiswami,et al.  Computation of scattering from clusters of spheres using the fast multipole method. , 2005, The Journal of the Acoustical Society of America.

[28]  ELECTROMAGNETIC SCATTERING BY PARALLEL METAMATERIAL CYLINDERS , 2006 .

[29]  Weng Cho Chew,et al.  Efficient computation of three-dimensional scattering of vector electromagnetic waves , 1994 .