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.