Perturbative Born theory for light scattering by time-modulated scatterers

TL;DR

The paper develops a perturbative first-order Born theory for electromagnetic scattering by time-periodically modulated scatterers, deriving an explicit expression for the first-order T-matrix and showing that inelastic scattering amplitudes are governed by overlaps between static modes at the input and output frequencies. It validates the approach by comparing with time-Floquet calculations and dynamic EBCM for a dielectric sphere and a high-permittivity cylinder, highlighting resonance-to-resonance transitions and the role of symmetry in suppressing or enhancing frequency conversion. The work provides physical intuition for frequency conversion in time-dependent photonics and offers practical guidance for designing dynamically tunable resonators with tailored inelastic channels. Potential extensions include higher-order perturbative corrections, nonuniform or anisotropic time variations, and integration into multiple-scatterer photonic architectures.

Abstract

We present a theoretical framework for electromagnetic scattering by particles with a permittivity that is periodically varying in time, based on a perturbative approach. Within this framework, we derive explicit expressions for the scattering matrix of the dynamic system in a first-order Born approximation, relating it directly to the corresponding static problem. We show that inelastic scattering amplitudes are governed by overlap integrals between static modes at the input and output frequencies. Using this insight, we analyze scattering from a time-modulated, isotropic, dielectric sphere and a high-permittivity dielectric cylinder, and demonstrate how modal orthogonality can suppress inelastic channels, while appropriate tuning of geometric parameters can significantly enhance them. In particular, we show that cylindrical resonators support strong inelastic scattering when resonance-to-resonance optical transitions, induced by the temporal variation, involve a high-Q supercavity mode. Comparison with full time-Floquet calculations confirms that the first-order Born approximation remains quantitatively accurate for modest modulation amplitudes and provides clear physical intuition for frequency conversion and resonance-mediated scattering processes in time-modulated photonic resonators.

Figures (4)

• Figure 1: Light scattering by a sphere of radius $R$ with static permittivity $ε = 12$ and permeability $μ = 1$. Spherical waves up to $\ell_{\rm{max}} = 5$ have been retained. (a): Static scattering cross section of a plane wave, incident along the $x$ direction, calculated by the static EBCM (solid line), and its $P\ell = H2$ component (dashed line). We can see two resonances, $\text{QM}_{1}$ and $\text{QM}_{2}$, corresponding to $P\ell = H2$, in the displayed spectral region. In the margin, we show the imaginary part of the $y$ component of the electric field on the $y$-$z$ section passing through the center of the sphere, for scattering at these resonances. (b): First inelastic ($n = \pm 1$) scattering cross section components for incoming frequency on resonance $\text{QM}_{1}$, as we vary the modulation frequency. The right (left) parts correspond to $n = -1$ ($n = 1$). The full (dashed) lines show the dynamic EBCM (Born approximation) results. A sharp dip for $n = -1$ is clearly visible. The dotted line depicts the absolute value of the radial overlap integral $|D_{H2}|$ from Eq. \ref{['eq:overlap_sphere_H']}. (c): Same as (b), but with higher modulation amplitude.
• Figure 2: (a): Cylindrical scatterer with $ε = 80$ and $\mu = 1$, of radius $r$ and height $h$, illuminated by an s-polarized plane wave perpendicular to the cylinder axis. (b) and (c): Static scattering cross sections ($m = 0$ component) for two different cylinder aspect ratios, plotted as a function of the dimensionless frequency $ωr/c$. The results are obtained using the static EBCM with angular-momentum cutoffs $\ell_{\rm{max}} = 10$ and $\ell_{\rm{cut}} = 12$stefanouLightScatteringPeriodically2023a. The four resonances indicated by arrows occur at: $ωr/c = 0.6269$ ($1$), $ωr/c = 0.6579$ ($2$), $ωr/c = 0.6525$ ($1'$), and $ωr/c = 0.67875$ ($2'$). Resonance ($2'$) corresponds to the supercavity mode. The normalized electric-field distribution inside the cylinder in the $y$-$z$ plane, perpendicular to the incident direction $x$, is shown in the margin for each resonance.
• Figure 3: Elastic ($n = 0$) and first inelastic ($n = \pm 1$) $m = 0$ components of the scattering cross section for the two different aspect ratios shown in Fig. \ref{['fig:static_m=0_s_pol']}. We consider an incoming plane wave with frequency corresponding to resonances ($2$) and ($2'$), respectively, and vary the dimensionless modulation frequency $Ωr/c$. (a): $r/h = 0.64$, $n = 0$, $η = 0.001$. (b): $r/h = 0.64$, $n = \pm1$, $η = 0.001$. (c): $r/h = 0.703$, $n = 0$, $η = 0.001$. (d): $r/h = 0.703$, $n = \pm1$, $η = 0.001$. (e): $r/h = 0.703$, $n = 0$, $η = 0.01$. (f): $r/h = 0.703$, $n = \pm1$, $η = 0.01$. The full (dashed) lines show the dynamic EBCM (Born approximation) results. For the figures on the right, the red and green curves show $n = -1$, while the orange and blue curves show $n = +1$. It can be seen that, for $r/h = 0.703$, a peak appears when we have resonance-to-resonance transition; that is, when the modulation frequency is such that the $n = 1$ component corresponds to resonance ($1'$). On the contrary, for $r/h = 0.64$ a sharp dip appears when the $n = 1$ component corresponds to resonance ($1$).
• Figure 4: Elastic ($n = 0$, left-hand diagrams) and inelastic ($n =1$, right-hand diagrams) components of the scattering cross section as a function of the modulation amplitude $η$ for the transitions ($2$) $\rightarrow$ ($1$) and ($2'$) $\rightarrow$ ($1'$), manifested in the right-hand diagrams of Fig. \ref{['fig:scan_mod_freq_born']}. The full (dashed) lines show the dynamic EBCM (Born approximation) results. For both aspect ratios, the Born approximation breaks down at sufficiently large $η$.