Coupled electric and magnetic dipole formulation for planar arrays of particles: Resonances and bound states in the continuum for all-dielectric metasurfaces

The optical properties of infinite planar array of scattering particles, metasurfaces and metagratings, are attracting special attention lately for their rich phenomenology, including both plasmonic and high-refractive-index dielectric meta-atoms with a variety of electric and magnetic resonant responses. Herein we derive a coupled electric and magnetic dipole analytical formulation to describe the reflection and transmission of such periodic arrays, including specular and diffractive orders, valid in the spectral regimes where only dipolar multipoles are needed. The two-dimensional lattice Green function is rewritten in terms of a one-dimensional (chain) version that fully converges in the complex frequency plane and can be easily calculated. Modes emerging as poles of such lattice Green function can be extracted, as evidenced by calculating resonances and bound states in the continuum for an array of Si spheres. This formulation can be applied to investigate a wealth of plasmonic, all-dielectric, and hybrid metasurfaces and metagratings of interest throughout the electromagnetic spectrum.