E-polarized plane-wave scattering from a PEC strip grating on a dielectric substrate: analytical regularization and lattice-mode resonances

We consider a plane E-polarized wave scattering from an infinite flat grating of slots cut in a perfect electrically conducting plane, backed with a dielectric slab. At first, we reduce this problem to a dual series equation for the complex amplitudes of the Floquet spatial harmonics. Then we perform its analytical regularization, based on the inversion of the main part with the aid of the Riemann-Hilbert Problem. This yields a Fredholm second-kind infinite matrix equation, numerical solution of which has a guaranteed convergence. We perform numerical experiments demonstrating how the rate of convergence depends on the thickness and dielectric permittivity of the slab. Our computations demonstrate the Fano-shape resonances on the so-called lattice modes, responsible for the phased array blindness. Their frequencies lay near to the Rayleigh anomalies and, if the slots or strips are narrow and the slab is thin, may have ultrahigh Q-factors.