Scattering at Space-Time Interfaces between Dispersive Media

Abstract

Dynamic modulation of material properties in space and time enables powerful control over wave propagation, yet existing theories largely rely on idealized, nondispersive models. In realistic media, frequency dispersion can strongly reshape wave dynamics, especially near resonances in highly dispersive platforms such as epsilon-near-zero materials. Here, we develop a general frequency transition theory for electromagnetic scattering at moving interfaces between dispersive media. From phase continuity, we derive nonlinear frequency transition relations and show that dispersion fundamentally reshapes the space-time scattering landscape, enabling additional propagating solutions with no counterpart in nondispersive systems. Applied to Drude, Lorentz and double-Drude media, the theory reveals how resonant dispersion, material loss and negative-index branches reorganize the scattering channels. For the two-wave scattering class, we further introduce a mixed-domain formulation that combines time-domain interface kinematics with frequency-domain constitutive relations, yielding closed-form scattering coefficients. These results establish a unified framework for dispersive space-time scattering and open opportunities for dispersion-based transition engineering in realistic materials.

Figures (5)

• Figure 1: Dispersion-mediated space-time modes in a lossless Drude medium [Eq. \ref{['eq:Refractive_Index_Drude_Media']}], with $n_{\infty,1} = 1$, $\omega_{\text{p},1} = 5$, $n_{\infty,2} = 1.22$ and $\omega_{\text{p},2} = 10$. (a) Scattering regimes as a function of the modulation velocity, $v_{\text{m}}$, and the incident frequency, $\omega_{\text{i}}$. The dashed lines indicate the corresponding nondispersive velocity limits. (b) Space-index representation of the different space-time states with associated spectral transition diagrams, where some of the solutions are crossed out because they do not satisfy the selection conditions in Eqs. \ref{['eq:Selection_Conditions']}. The dotted curves represent dispersion relations evaluated at fixed values of the imaginary part of the frequency solutions.
• Figure 2: Dispersion-mediated space-time modes for the Lorentz dispersion model [Eq. \ref{['eq:Refractive_Index_Lorentz_Media']}] with parameters $n_{\infty,1} = 1$, $\omega_{\text{p},1} = 5$, $\omega_{0,1} = 3$, $\gamma_{1} = 0.001$, $n_{\infty,2} = 1.22$, $\omega_{\text{p},2} = 10$, $\omega_{0,2} = 5$ and $\gamma_{2} = 0.001$. The dashed lines indicate the nondispersive velocity limits.
• Figure 3: Dispersion-mediated space-time modes in a double-Drude medium [Eq. \ref{['eq:Refractive_Index_Double_Drude_Media']}], with $n_{\infty\text{e},1} = 1$, $\omega_{\text{pe},1} = 0$, $n_{\infty\text{m},1} = 1$, $\omega_{\text{pm},1}=0$ (vacuum) and $n_{\infty\text{e},2} = 1$, $\omega_{\text{pe},2} = 10$, $n_{\infty\text{m},2} = 1$ and $\omega_{\text{pm},2}=10$. (a) Scattering regimes as a function of the modulation velocity, $v_{\text{m}}$, and the incident frequency, $\omega_{\text{i}}$. The dashed lines indicate the corresponding nondispersive velocity limits. (b) Space-index representation of the different space-time states with associated spectral transition diagrams, where some of the solutions are crossed out because they do not satisfy the selection conditions in Eqs. \ref{['eq:Selection_Conditions']}. The dotted curves represent dispersion relations evaluated at fixed values of the imaginary part of the frequency solutions.
• Figure 4: Application to the Drude dispersion model, with the same parameters as in Fig. \ref{['fig:Number_of_Waves_Drude']}, and fixed modulation velocity, $v_{\text{m}}/c = -0.4$. (a) Frequency transitions. (b) Scattering amplitudes. The dashed line at $\omega = 10$ is the incident frequency used in the simulation of Fig. \ref{['fig:FDTD_Validation_Drude']}.
• Figure 5: Comparison of scattering at a moving interface in dispersive and nondispersive media. (a) Space-time diagram for the dispersive case at $\omega_{\text{i}} = 10$ and $v_{\text{m}}/c=-0.4$, with the same parameters as in Fig. \ref{['fig:Number_of_Waves_Drude']}. (b) Corresponding space-time diagram for the nondispersive case, with refractive indices equal to the high-frequency limits of (a). (c) Fourier spectra of the scattered fields: analytical predictions for both dispersive and nondispersive cases, along with numerical results obtained from a dispersive FDTD simulation.