Fano resonances in all-dielectric electromagnetic metasurfaces

We are interested in the resonant electromagnetic scattering by all -dielectric metasurfaces made of a two-dimensional lattice of nanoparticles with high refractive indices. In [Ammari, Li, and Zou, Trans. Amer. Math. Soc., 376 (2023), pp. 39--90], it has been shown that a single high -index nanoresonator can couple with the incident wave and exhibit a strong magnetic dipole response. Recent physics experiments reveal that when the particles are arranged in certain periodic configurations, they may have different anomalous scattering effects in the macroscopic scale, compared to the single -particle case. In this work, we shall develop a rigorous mathematical framework for analyzing the resonant behaviors of all -dielectric metasurfaces. We start with the characterization of subwavelength scattering resonances in such a periodic setting and their asymptotic expansions in terms of the refractive index of the nanoparticles. Then we show that real resonances always exist below the essential spectrum of the periodic Maxwell operator and that they are the simple poles of the scattering resolvent with the exponentially decaying resonant modes. By using group theory, we discuss the implications of the symmetry of the metasurface on the subwavelength band functions and their associated eigenfunctions. For the symmetric metasurfaces with the normal incidence, we use a variational method to show the existence of embedded eigenvalues (i.e., real subwavelength resonances embedded in the continuous radiation spectrum). Furthermore, we break the configuration symmetry either by introducing a small deformation of particles or by slightly deviating from the normal incidence and prove that Fano-type reflection and transmission anomalies can arise in both of these scenarios.