Scattering at Interluminal Interfaces

TL;DR

This work addresses scattering at interluminal space-time interfaces, where the interface velocity $v_m$ lies between the wave speeds in the two media. It develops a general solution by a symmetric decomposition into subluminal- and superluminal-limit interfaces and a space-time impulse-response framework, unifying the treatment of Case I and Case II and bridging the subluminal and superluminal regimes. The authors derive analytical expressions for the scattered waves (reflected, later-backward, and later-forward) and validate them via benchmark cases and full-wave simulations, revealing velocity-independent coefficients in Case I and the emergence of shock waves in Case II, as well as time-reversal properties. The results provide fundamental insight into interluminal physics and enable new space-time wave-control functions with potential optical and microwave implementations.

Abstract

Scattering at interluminal modulation interfaces, where a sharp space-time perturbation moves at a velocity lying between the wave velocities of the two surrounding media, has remained an open problem for decades. This regime is somewhat reminiscent of the Cherenkov regime, in which the velocity of a charged particle exceeds the phase velocity of light in a medium. However, because it involves two media and a moving interface, it gives rise to richer and more complex scattering dynamics, with a single scattered wave when the incident wave propagates in the same direction as the interface and three scattered waves when they propagate in opposite directions. Existing studies address only limited non-magnetic configurations, and a general formulation has yet to be established. In this paper, we present a complete and general solution to scattering in the interluminal regime using a symmetric decomposition approach based on subluminal and superluminal limit interfaces, together with a space-time impulse response. This approach provides clear physical insight into the scattering features of the interluminal regime. Our results bridge the long-standing gap between the subluminal and superluminal regimes and elucidate the fundamental mechanisms underlying interluminal scattering.

Figures (11)

• Figure 1: Wave scattering at different modulation interfaces. (a) Subluminal regime, where the interface moves slower than the wave. (b) Superluminal regime, where the interface moves faster than the wave. (c) Interluminal regime, where the interface moves at a velocity between the wave velocities. The right panel illustrates three incident cases, each generating a single scattered wave: reflected for $\psi_{\mathrm{i}a}$, later-backward for $\psi_{\mathrm{i}b}$ and later-forward for $\psi_{\mathrm{i}c}$. The subscripts 'r', 't', '$\zeta$' and '$\xi$' denote the reflected, transmitted, later-backward and later-forward waves, respectively; 'I' and 'II' indicate the scattered waves in Cases I and II.
• Figure 2: Dispersion diagrams showing the frequency transitions at interluminal interfaces [Fig. \ref{['fig:regimes']}(c)] for (a) Case I ($v_{\mathrm{m}}<0$) and (b) Case II ($v_{\mathrm{m}}>0$).
• Figure 3: Different polyline decompositions of a contramoving ($v_{\mathrm{m}}<0$) interluminal alternating and corresponding wave scattering. (a) Staircase decomposition combining pure-space and pure-time interfaces. (b) Asymmetric subluminal-limit and pure-time decomposition Shui_2014_one. (c) Symmetric decomposition combining subluminal-limit and superluminal-limit interfaces. The red solid line indicates the target interface trajectory and the red dashed lines represent its polyline approximations.
• Figure 4: General solution for a contramoving interluminal interface ($v_{\mathrm{m}}<0$) in Fig. \ref{['fig:polylines']}(c). (a) Symmetric decomposition within a single step, with $\tau_{\mathrm{i}}\to 0$. The light-blue region indicates the portion of the wave interacting with the subluminal-limit interface, while the dark-blue region indicates the portion interacting with the superluminal-limit interface. (b) Extension to arbitrary incidence using the space-time impulse response method, with $z_{0}$ denoting the initial position of the interface at $t=0$.
• Figure 5: Application of the $-v_2$-subluminal and $-v_1$-superluminal decomposition of Fig. \ref{['fig:impulse']}(a) to different modulation velocities, with $v_1=c$ and $v_2=0.27c$. (a) $v_{\mathrm{m}}=-0.38c$. (b) $v_{\mathrm{m}}=-0.58c$. (c) $v_{\mathrm{m}}=-0.84c$.