Space-time refraction of space-time wave packets

TL;DR

The paper develops a unified framework for space-time refraction of space-time wave packets at moving planar interfaces by identifying two invariants: conservation of the tangential wavenumber $k_x$ and a Doppler-type invariant $T = k_z - \omega/v_m$. It derives three explicit refraction laws for baseband, X--wave, and sideband STWPs, each connecting incident and transmitted group indices and spectral tilts and reducing smoothly to static or temporal limits. Two key phenomena emerge: a space-time optical push broom that compresses a broad range of incident group velocities into a narrow transmitted band for baseband STWPs, and velocity spectral optical fission where an incident X--wave splits into two propagation-invariant branches with distinct velocities. The framework suggests practical applications in time-gating, buffering, and velocity multiplexing in reconfigurable space-time photonic media, and offers a route to simulating dynamical and horizon-like effects in photonics.

Abstract

Space-time modulation of refractive index can produce synthetically moving interfaces with arbitrary apparent velocities, including superluminal motion, offering new ways to control light in dynamic media. On the other hand, space-time wave packets are structured waves whose spatio-temporal spectra lie on tilted space-time planes, so their group velocity can be programmed, including superluminal values, even in a uniform medium. Here we develop a general theory of space-time refraction for such structured waves at a planar moving interface and show how a single boundary reshapes their velocity content. By identifying the invariants of a translating boundary, we obtain refraction laws for baseband, X-wave, and sideband packets that apply for arbitrary interface velocities and connect smoothly to static and purely temporal limits. These laws reveal regimes of "space-time anomalous optical push broom," where a moving interface compresses a wide range of incident velocities into a narrow transmitted band, and "velocity spectral optical fission," where an incident X-wave splits into two propagation-invariant branches with distinct velocities. The combined freedom to prepare waves with superluminal group velocity and to prescribe equally unconstrained interface speeds points toward reconfigurable time gating, optical buffering, velocity multiplexing, and controlled emission in moving media, and provides a route to photonic settings capable of emulating dynamical effects traditionally associated with gravitational or quantum processes.

Figures (3)

• Figure 1: Interaction of STWPs with a moving refractive index interface.a, Three dimensional schematic of a STWP encountering a planar space-time (ST) interface that separates two homogeneous media with refractive indices $n_{1}$ and $n_{2}$. Labels 1 and 2 mark the packet before and after interacting with the moving interface, which travels along $z$ with velocity $v_{\mathrm m}$. The corresponding group velocities are $v_{\mathrm g1}$ and $v_{\mathrm g2}$. b, Two dimensional view of the same configuration. The STWP axis forms an angle $\phi_{j}$ with the interface normal, while its spectrum is characterised by the tilt angle $\theta_{j}$ in the $(k_{z},\omega)$ plane, with $v_{\mathrm g j}= c\tan\theta_{j}$. The spectral tilt $\theta_{j}$ determines the group velocity and is distinct from the incidence angle $\phi_{j}$. c, Space-time refraction in the $(k_{z},\omega)$ dispersion diagram at $k_{x}=0$ for normal incidence. The red and blue lines denote the light lines for indices $n_{1}$ and $n_{2}$, and the refraction plane imposed by the moving interface maps an incident spectral line with tilt $\theta_{1}$ to a transmitted line with tilt $\theta_{2}$. d, Spectral construction of the three STWP families considered in this work. Top row, intersections of the spectral planes with the $(k_{z},\omega)$ section, showing baseband STWPs, X--waves, and sideband STWPs; green lines indicate the corresponding light lines. Bottom row, full three dimensional $(k_{x},k_{z},\omega)$ representation, where green cones denote the light cones and coloured curves show the spectral loci that define each STWP family.
• Figure 2: Space-time anomalous optical push broom via refraction of baseband STWPs.a, Space-time refraction law for baseband STWPs at normal incidence for $n_{1}=1$, $n_{2}=2$, and $v_{\mathrm m}=0.6 c$. The mapping $\theta_{1}\!\to\!\theta_{2}$ compresses a wide span of incident group indices into a narrow transmitted range. Orange and blue arrows mark two representative inputs. Inset: geometry of a baseband STWP incident on a planar space-time interface. b, Spectral construction in $(k_{x},k_{z},\omega)$ space. Red and blue cones are the light cones of media $1$ and $2$; the yellow plane is the space-time refraction plane that maps the incident spectral locus to the transmitted one. Insets: $(k_{z},\omega)$ projections showing the change in spectral tilt. c, World lines of the interface and of the packet centres at $x=0$. Dashed black: interface trajectory. Solid orange and blue: group trajectories with the interface present. Dotted lines: reference propagation without the interface. The moving boundary sweeps both packets onto nearly the same group velocity. d, Field intensity snapshots $\lvert E\rvert^{2}$ in the $(x,z)$ plane. Top: input at $t=0$. Middle: separation of the two STWPs in a uniform medium. Bottom: co-propagation after crossing the space-time interface. The dashed vertical line marks the interface location, and arrows track the packet peaks (see Supplementary Video 1 for a full time animation).
• Figure 3: Velocity spectral optical fission of an X--wave at a moving space-time interface.a, Space-time refraction law for an X--wave at normal incidence for $n_{1}=1.5$, $n_{2}=1.0$, and $v_{\mathrm m}=0.5 c$. The mapping $\theta_{1}\!\to\!\theta_{2}$ becomes double valued over a narrow interval of incident tilts (green band), so one input X--wave produces two transmitted branches. Magenta and blue arrows mark two representative outputs. Inset: geometry of an X--wave incident on a moving space-time interface. b, Spectral construction in $(k_{x},k_{z},\omega)$ space. Green and blue surfaces are the light cones of media $1$ and $2$. The yellow plane is the space-time refraction plane, which intersects the $n_{2}$ cone along two distinct loci corresponding to the two transmitted X--waves. Insets: $(k_{z},\omega)$ projections showing the incident tilt $\theta_{1}$ and the two transmitted tilts $\theta_{2a}$ and $\theta_{2b}$. Only the physical octant $k_{x},k_{z},\omega>0$ is shown. c, World lines at $x=0$. Dashed black: interface trajectory. Orange: incident X--wave. Magenta and blue: centres of the two transmitted packets, which separate according to their distinct group velocities. d, Field intensity snapshots $\lvert E\rvert^{2}$ at $t=0$, $240\,(2\pi/\omega_{0})$, and $414\,(2\pi/\omega_{0})$. The vertical dashed line marks the interface. Magenta and blue arrows track STWP-1 and STWP-2, whose spatial separation reflects velocity spectral optical fission (see Supplementary Video 2 for a full time animation).