Magnetoinductive Waves and Complex Modes in Two-Dimensional Periodic Arrays of Split Ring Resonators

We investigate magnetoinductive waves in two-dimensional periodic arrays of split ring resonators or capacitively loaded loops and characterize the modes with real and complex wavenumber excitable in such arrays. Each resonator is modeled as a single magnetic dipole, and the computation of the modal wavenumbers is performed by searching for the zeroes of the homogeneous scalar equation characterizing the field in the array. We provide original developments for the Ewald method applied to the required dyadic periodic Green's function for the array of magnetic dipoles, including the quasi-static case. The Ewald representation is analytically continued into the complex wavenumber space and also provides series with Gaussian convergence rate. In particular, we analyze and classify proper, improper, forward, backward, bound, and leaky magnetoinductive waves varying frequency and compare the fully retarded solution to the quasi-static one. We highlight the importance of accounting for field retardation effects for the prediction of the physical waves excitable in the array when the dimensions of its unit cell are approximately greater than a tenth of the free-space wavelength. The proposed method complements previous investigations and is a powerful tool for the design of waveguiding or radiating structures based on magnetoinductive waves.