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 control of the temporal modulation strength provides a qualitatively new route to tailor the far-field pattern of a scatterer. Two characteristic spheres are designed, providing amplification of scattered light and satisfying nearly ideally the first and second Kerker scattering conditions.