Regular polygonal arrays of resonant scatterers

Abstract

Numerical results for the scattering cross section (S) of N=3–26 equally spaced monopole resonators on a circle (of radius b) indicate regularities in the values of the normalized diameter (2kb=4πb/λ=ρ with λ as the wavelength) corresponding to maximal scattering for symmetrical excitation. The peaks S∧(N) occur for ρ=ρ∧(N) between N and 2N, i.e., for circle circumference between Nλ/2 and Nλ. With increasing N, the values of ρ∧(N) in successive alternating sets (shells) of three or four values of N are close to ρm=2mπ−π/4; shell-1 consists of N=3–5, shell-2 of N=6–9, shell-3 of N=10–12, etc. The basis for the shell structure is delineated by a simple asymptotic approximation (for large ρ and N in the range ρ<2N) or a cylindrical wave representation for a sum of spherical waves. A simple approximation is also derived fo the shift in resonance frequency that occurs for ρ small enough for the array to respond as a collective monopole.