Fano resonances in photonic crystal slabs near optical bound states in the continuum

Photonic crystal slabs are able to support optical bound states in the continuum. The latter are eigenmodes of the structure that are truly guided (no radiation leakage) despite the fact that they lie above the light cone within the continuum of radiation modes. Such peculiar states can be viewed as modes with an infinite quality factor $Q$. Therefore, the question of the behavior of Fano resonances, as optogeometrical parameters are tuned close to optical bound states in the continuum, is of importance for applications of photonic crystal slabs with ultrahigh $Q$ factors. We study theoretically the reflection and transmission of a photonic crystal slab close to an optical bound state in the continuum with a phenomenological approach involving the poles and zeros of the scattering matrix. In particular, we derive a general relation valid for asymmetric structures that gives the position of a pole in the complex plane as a function of the positions of the zeros. We provide closed-form expressions for the reflection and transmission. The proposed phenomenological approach is validated through rigorous numerical calculations.