Interface States in Space-Time Photonic Crystals: Topological Origin, Propagation and Amplification

TL;DR

This work addresses the challenge of topologically characterizing space-time photonic crystals with travelling-wave modulations by leveraging a Lorentz transformation to a comoving frame, where a conserved joint $P'T'$ symmetry yields a quantized spatiotemporal Zak phase, providing a $\mathbb{Z}_2$ topological classification. The authors show that two STPhC phases (trivial and obstructed) arise, with interface states predicted at boundaries between opposite phases and characterized via bulk properties such as band crossings and energy density. They derive and analyze two boundary geometries—spatiotemporal and purely spatial—finding distinctive phenomena: a propagating interface along moving boundaries for ST boundaries, and frequency-converted replicas with broadband amplification for spatial boundaries, both tied to the ST Zak phase and robust against perturbations. Overall, the work provides a general framework for topology in time-varying media and suggests broad applicability across photonics and other wave systems, including potential extensions to higher dimensions and different platforms.

Abstract

Studying the topology of spatiotemporal media poses a fundamental challenge: their remarkable properties stem from breaking spatial and temporal symmetries, yet this same breaking obscures their topological characterization. Here, we show that space-time symmetries persist in crystals with travelling-wave modulation, enabling the study of their topological properties and the prediction of spatiotemporal interface states. Using a Lorentz transformation to the frame comoving with the modulation, we identify a conserved joint parity-time-reversal symmetry in the new variables that enforces the quantization of the Zak phase, elevating it to a $\mathbb{Z}_2$ topological invariant. We then calculate the associated interface states and uncover unique features arising from time-varying effects, including selective directional excitation, propagation along moving boundaries, frequency-converted replicas, and broadband amplification even in the absence of momentum gaps.

Figures (4)

• Figure 1: Spatiotemporal photonic crystal in the laboratory and comoving frames. (a) Travelling-wave modulated permittivity in the lab-frame. A blue arrow signals the direction of the continuous space-time translation symmetry that defines the unit cell highlighted in the orange shaded region. The white arrow corresponds to the lattice vector. (b) Band structure of a STPhC for a modulation speed $c_g=\Omega/g=0.2c_0$, modulation strength $\alpha=0.3$ and $\epsilon_m=\mu_m=1.2$. Grey and black dashed lines correspond to the dispersion relation in free space and in the unmodulated material, respectively. Black arrow represents the reciprocal lattice vector $\textbf{p}$. (c) Modulated permittivity in the comoving frame. The previous magnitudes are now spatial-like, but a new magneto-electric coupling $\xi'(x')$ appears. (d) Band structure of the same STPhC in the comoving frame, with the black arrow now representing the transformed reciprocal vector $\textbf{p}'$.
• Figure 2: Distinct topological phases in STPhCs. (a)-(b) Energy density distribution over the unit cell of the first band $m = 1$, for $\alpha > 0$ and $\alpha < 0$. (c)-(d) Absolute value of the electric field eigenfunction $|E'_{k'_{\text{gap}},m = 1}(x')|$ located at the first band gap shown in Fig. \ref{['fig:band_eps_dist']}(d) for $k' < 0$, considering $\alpha>0$ and $\alpha<0$. (e)-(f) $|E'_{k'_{\text{gap}},m = 2}(x')|$ at the same gap for the second band $m=2$, considering $\alpha > 0$ and $\alpha < 0$. All the parameters are the same as in Fig. \ref{['fig:band_eps_dist']}.
• Figure 3: Interface states between a trivial ($\alpha>0$) and a topological ($\alpha<0$) spatiotemporal slab, considering (a)-(e) a spatiotemporal boundary and (f)-(j) a spatial boundary. (a) Permittivity profile $\epsilon(x,t)$ of the two-slab configuration for $c_g = 0.3 c_0$. (b) Transmittance spectrum of two spatiotemporal slabs whose boundary moves along with the modulation, as a function of the modulation speed $c_g$. An interface state with resonance frequency following the bulk-predicted $\omega^{\text{ST}}_r$ is observed. (c) Horizontal cross section of the transmittance at $c_g = 0.3 c_0$. A clear peak inside the gap corresponding to the interface state is visible. (d) Same magnitude for $c_g = 0.57 c_0$. (e) Intensity distribution of the interface state observed when exciting the left slab with a plane wave of frequency $\omega^{\text{ST}}_r$. A propagating interface state is observed. (f) Permittivity profile $\epsilon(x,t)$ of the two spatial slabs for $c_g = 0.3 c_0$. (g) Transmittance spectrum as a function of $c_g$ for two slabs of the STPhC with purely spatial boundaries. An interface state is also present for the whole modulation interval. Furthermore, the frequency conversion induced by the temporal modulation enables replicas of the interface states at $\omega_r^{\text{S}} \pm \Omega$. (h) Horizontal cross section of the transmittance for $c_g = 0.3 c_0$, revealing the interface state inside the gap as well as the replicas at $\omega^{\text{S}}_r - \Omega$. (i) Same magnitude for $c_g = 0.52 c_0$, where broadband amplification ($T(\omega)>1$) is observed. (j) Intensity distribution of the interface state for a static boundary, showing no propagation.
• Figure 4: Robustness of interface states studied in the two-slab configuration of Fig. \ref{['fig:int_states']}(a) by perturbing the system with an increased modulation strength in the second slab by $(\alpha+\Delta)$, with $\Delta$ the perturbation parameter. (a) Deviation of the numerically obtained resonance frequency $\omega_i$ from the bulk-predicted $\omega^{\mathrm{ST}}_r$ as we increase $\Delta$ and $c_g$. (b) Zoom in at the peaks of transmittance observed inside the gap for $c_g = 0.1 c_0$, with a vertical line showing $\omega_r^{\text{ST}}$ as a reference. (c) Same magnitude for $c_g = 0.5 c_0$. (d) Analytical resonance frequency $\omega^{\mathrm{ST}}_r$ for different modulation strengths ($\alpha + \Delta$) as a function of the modulation speed $c_g$.