Integral equation analysis of propagation in a dielectric waveguide coupled to an infinite periodic sequence of ring resonators

A rigorous integral equation approach is proposed to analyse propagation in a dielectric waveguide coupled to a periodically spaced infinite sequence of ring resonators. The method utilises Green's function concept to formulate an integral equation for the field inside resonators. Floquet's theorem and Galerkin's technique are used to obtain a homogeneous linear system, involving the expansion coefficients of the unknown field in terms of cylindrical wave functions. The nontrivial solutions of the latter system provide a highly accurate estimation of the structure's Bloch modes, serving an in-depth analysis of the propagation and dispersion characteristics. The integral equation formulation allows the interaction between resonators to be expressed analytically in terms of free-space lattice sums and Sommerfeld-type series; this is an advantage for the analysis of a closely spaced sequence of resonators, which essentially acts as a coupled-resonator waveguide. The results show that coupling between the waveguide and the periodic sequence has a complicated impact on the undisturbed modes of both the coupled structures.