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(N-1.5) instead of SMM's O(N-2), 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.