Lattice Resonances in Periodic Arrays of Time-Modulated Scatterers

TL;DR

The paper develops a dispersion-aware framework that merges the dipolar approximation with time-Floquet theory to study lattice resonances in periodic arrays of time-modulated scatterers. By modeling each scatterer as a time-varying harmonic oscillator via $f(t)=1+\Delta\cos(\Omega t)$, it identifies modulation conditions that drive amplification in isolated scatterers and shows that collective lattice resonances in an array enable amplification at substantially lower modulation strengths due to enhanced light-matter coupling and increased lifetimes. The analysis highlights the crucial role of Rayleigh anomalies and their replicas in shaping the amplification spectrum and demonstrates that the lattice resonances provide a practical route to dynamic, gain-free amplification across microwave to visible plasmonic platforms. Overall, the work offers a simple, broadly applicable method for dynamic control and nonreciprocal functionalities in time-modulated photonic lattices.

Abstract

Lattice resonances are collective optical modes supported by periodic arrays of scatterers, arising from their coherent interaction enabled by the underlying periodicity. Owing to their collective nature, these resonances produce optical responses that are both stronger and spectrally narrower than those of individual scatterers. While such phenomena have been extensively studied in conventional time-invariant systems, recent advances in time-varying photonics present new opportunities to exploit and enhance the extraordinary characteristics of these collective modes. Here, we investigate lattice resonances in periodic arrays of time-modulated scatterers using a simple framework based on the dipolar approximation and time-Floquet theory, where each scatterer is modeled as a harmonic oscillator with periodically varying optical properties. We begin by analyzing the response of an individual scatterer, leveraging our model to identify the complex eigenfrequencies that define its dynamics. We show that, for the appropriate modulation amplitude and frequency, the imaginary part of one of these eigenfrequencies vanishes, leading to amplification. Building on this, we extend our analysis to a periodic array to investigate the effect of the interplay between temporal modulation and lattice resonances. In contrast to isolated scatterers, the collective nature of lattice resonances introduces a markedly more intricate spectral dependence of the amplification regime. Notably, this amplification emerges at substantially lower modulation strengths, facilitated by the enhanced light-matter interaction and increased lifetime provided by these collective resonances. Our work establishes a simple theoretical framework for understanding collective lattice resonances in time-modulated arrays, enabling dynamic control and amplification of these modes.

Figures (7)

• Figure 1: (a) Schematic representation of the system under consideration, consisting of a square array of identical scatterers. The array lies in the $xy$ plane, is surrounded by vacuum, and is excited by a plane-wave electric field that propagates parallel to the $z$ axis and is polarized along the $x$ axis. (b) Absorbance spectra for time-invariant arrays with various combinations of period $a$ and nonradiative damping rate $\gamma$, as indicated in the legend. For reference, the orange dashed curve shows the corresponding results for an individual scatterer. The inset provides a zoomed view around the resonance of the individual scatterer. In all cases, we assume $\kappa = 0.15\omega_{\rm r}$.
• Figure 2: Real (a) and imaginary (b) parts of the complex eigenfrequencies $\tilde{\omega}$ of a time-modulated individual scatterer, plotted as a function of the modulation amplitude $\Delta$. Each colored curve corresponds to a different modulation frequency $\Omega$, as indicated in the legend. (c) Values of $\Delta$ and $\Omega$ for which one eigenfrequency crosses the real axis, with ${\rm Re}\{\tilde{\omega}\}/\Omega = 0$ (solid curve) or ${\rm Re}\{\tilde{\omega}\}/\Omega = \pm 1/2$ (dashed curve). In all cases, we assume $\gamma = \omega_{\rm r}/40$.
• Figure 3: Absorption cross section of a time-modulated individual scatterer as a function of the excitation and modulation frequencies. We consider two modulation amplitudes: $\Delta = 0.363$ (a) and $\Delta = 0.925$ (b), corresponding to the two minima in Figure \ref{['fig2']}(c). The black dashed lines indicate the conditions $\Omega = \omega / n$ and $\Omega = 2\omega / (2n\pm1)$, while the dotted gray lines correspond to $\Omega = (\omega\pm\omega_{\rm r})/n$. In both panels, $\sigma_{\rm abs}$ is normalized to $a^2$ (with $a = 1.1\lambda_{\rm r}$), and we set $\gamma = \omega_{\rm r}/40$. (c) Schematic illustration of the amplification process for the singularities discussed in the text. The upper row shows cases arising from the condition $\Omega=\omega/n$, while the lower row corresponds to those associated with $\Omega = 2\omega/(2n \pm 1)$. Each schematic illustrates the relationship between $\omega$ and $\Omega$, as well as their approximate connection to $\omega_{\rm r}$ at the onset of the corresponding singularity.
• Figure 4: (a,c) Absorbance of an array of time-modulated scatterers as a function of the excitation frequency and modulation amplitude. (b,d) Spectra of the absolute value of the absorbance for different values of $\Delta$, as indicated in the legend. Panels (a) and (b) show results for $\Omega = \omega$, whereas in panels (c) and (d), $\Omega = 2\omega$. In all cases, $a = 1.1\lambda_{\rm r}$ and $\gamma = \omega_{\rm r}/40$. The black dashed lines indicate the position of the first Rayleigh anomaly of the array, while the gray dotted lines correspond to replicas of higher-order Rayleigh anomalies.
• Figure 5: Relative contribution of individual harmonics to the absorbance of the time-modulated array. Panels (a) and (b) correspond to $\Omega = \omega$ and $\Omega = 2\omega$, respectively. In both panels each color represents a different excitation frequency, as indicated in the legend, corresponding to the singularity peaks of the results for $\Delta = 0.6$ (green curve) shown in Figures \ref{['fig4']}(b) and (d). In all cases, $a = 1.1\lambda_{\rm r}$ and $\gamma = \omega_{\rm r}/40$.