Parametric Mie Resonances and Directional Amplification in Time-Modulated Scatterers

We provide a theoretical description of light scattering by a spherical particle the permittivity of which 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 subwavelength 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.

Figure 1

A spherical particle with a time-modulated bulk carrier density illuminated by incident light. Temporal modulation leads to parametric Mie resonances with simultaneous scattered-field amplification and the possibility of far-field pattern manipulation.

Figure 2

The band-structure diagram of a time-modulated material plotted for the case with modulation strength $M=0.1$ and damping factor $\gamma =0$ Hz. The thick blue and thin red lines correspond, respectively, to low- and high-frequency bulk plasmon-polariton bands in the Lorentzian dispersion.

Figure 3

The colored curves depict threshold values of the modulation strength $M$ that provide parametric oscillations at fixed frequency ${\omega }_m/2$ for different multipolar modes in a time-modulated sphere versus its normalized radius. The gray dots depict values of the normalized imaginary part of permittivity that support lasing at fixed frequency ${\omega }_{\mathrm{las}}={\omega }_m/2$ for different modes in a time-invariant sphere with optical gain. While the horizontal coordinates of these points match to those of the minima of the colored curves for the corresponding multipolar modes, their vertical coordinates do differ and the difference depends on the chosen value of the stationary permittivity ${\epsilon }_{\mathrm{st},0}=\mathrm{Re}({\epsilon }_{\mathrm{inv}})$ (see Sec. 4 of the Supplemental Material [38]).

Figure 4

The scattered far-field patterns of time-modulated spheres with parameters (a) $M=0.68$ and $R=1.048\frac{2c}{{\omega }_m}$ and (b) $M=0.093$ and $R=1.481\frac{2c}{{\omega }_m}$. The patterns are calculated at the frequency of incident wave ${\omega }_{\mathrm{inc}}$. The red contours depict cross sections of the patterns parallel to the $x$-$y$, $y$-$z$, and $x$-$z$ planes, calculated at the center of the coordinate system. The colors of the patterns denote the scattering amplitude (the dark red and dark blue colors denote the maximum and minimum values, respectively).