Parametric Mie resonances and directional amplification in time-modulated scatterers

We provide a theoretical description of light scattering by a spherical particle whose permittivity is modulated in time at twice the frequency of the incident light. Such a particle acts as a finite-sized photonic time crystal and, despite its sub-wavelength spatial extent, can host optical parametric amplification. Conditions of parametric Mie resonances in the sphere are derived. We show that time-modulated materials provide a route to tailor directional light amplification, qualitatively different from that in scatterers made from a gain media. We design two characteristic time-modulated spheres that simultaneously exhibit light amplification and desired radiation patterns, including those with zero backward and/or vanishing forward scattering. The latter sphere provides an opportunity for creating shadow-free detectors of incident light.