Second harmonic generation and spontaneous parametric down-conversion in Mie nanoresonators

The recent active research in the field of nonlinear Mie nanoresonators has led to significant increase of harmonic generation efficiency from subwavelength structures. The progress in the second order nonlinear processes became evident after switching to nanoresonators made of materials that possess a bulk nonlinearity such as semiconductor and perovskite-type materials. Despite of that, the theory of second-harmonic generation from spherical Mie nanoparticles with a bulk nonlinearity has not been developed yet. In this work, we present a theoretical approach based on Mie theory for describing the second order nonlinear processes in all-dielectric nanoresonators. We focus on the second harmonic generation (SHG) process as well as on the inverse one, the process of spontaneous parametric down-conversion (SPDC). We demonstrate the SHG-SPDC correspondence in terms of Mie resonances and identify the selection rules governing these second order nonlinear processes. We show how one can control the spatial correlations of photons in SPDC from a single Mie-nanoparticle by properly choosing the parameters of the system, in particular, allowing for Kerker-type unidirectional emission of photons. (paper)