Frozen Surface Modes on a Moving Interface

Abstract

Spatio-temporal modulation enables synthetic motion at effective velocities approaching the speed of light, providing new regimes for light-matter interaction. Traditional Cherenkov-type effects arise when the velocity of an emitter matches or exceeds the phase velocity of electromagnetic modes supported by a medium. Here, we study dispersive systems in which phase and group velocities differ markedly. Specifically, we explore the case of group-velocity matching for surface waves, where the emitter moves at the same velocity as the flow of energy. This gives rise to frozen surface modes which are stationary in the emitter frame, accompanied by resonant energy accumulation. The result is a dramatic increase of the local density of optical states, the power extracted from the emitter, and the optomechanical forces and torques it experiences. Since surface modes naturally exhibit slow group velocities, this is accessible at lower relative speeds than phase-velocity effects. This phenomenon provides a route to enhanced light-matter interaction via real or synthetic motion.

Figures (4)

• Figure 1: Dispersion relation of surface plasmon excitation (solid) and emitter spectrum (dashed) in (a) the static case where the emitter and interface are stationary; (b) in the emitter reference frame $S$ with stationary emitter and moving interface; (c) in the surface reference frame $S'$ with stationary interface and moving emitter. The red dots indicate the group-velocity matching condition (frozen mode) and the green crosses indicate the phase-velocity matching condition (Cherenkov radiation).
• Figure 2: Regimes of mode excitation for an $x$-polarized line dipole placed at $h=0.05\lambda_0$ above a moving surface, where $\lambda_0=2\pi c/\omega_0$. (a) Tilted dispersion relation of a lossless Drude surface with $v=0.25c$, corresponding to Fig. \ref{['fig:moving_dipole']}(b). The dashed lines correspond to (i) $\omega_0<\omega_{\text{fsm}}$, (ii) $\omega_0=\omega_{\text{fsm}}$ and (iii) $\omega_0>\omega_{\text{fsm}}$, where $\omega_{\text{fsm}}$ is the frequency of the frozen surface mode in Eq. \ref{['eq:frozen mode']}. (b) Magnetic field distribution for the three cases in panel (a), where the arrows represent the phase velocity of each mode. Mode 2 is a backwards propagating mode, emitted to the left with its phase fronts moving to the right. (c) Spatial spectrum of the magnetic field shown in panel (b) at $z=0$. Modes 1 and 2 coalesce at $\omega_0=\omega_\text{fsm}$ corresponding to the frozen mode with a broadened spectrum.
• Figure 3: Color map of (a) LDOS, (b) power extracted, (c) lateral optical force, and (d) torque experienced by an x-polarized line dipole, as a function of $\omega_0$ and $v$, for $h = 0.05\lambda_0$. The dipole moment is normalised to emit unit power in free space. The dashed lines correspond to the frozen mode condition, where enhancements in the response are observed. The effect of small losses are included in the Drude model for $\varepsilon_r$, which regularizes the response. The results are plotted using a signed logarithmic color scale to show the sign reversal in panels (c) and (d).
• Figure 4: (a) Intersection of the Doppler shifted emitter spectrum (lilac plane) with the three-dimensional SPP dispersion (yellow) in $(k_x',k_y',\omega')$ space, corresponding to the surface frame $S'$. Their intersections (solid lines) result in the excited SPPs (i) below, (ii) at, and (iii) above the frozen-mode condition. The dashed line corresponds to $k_y'=0$, representing the 2D results. The modes coalesce at the frozen mode condition, where group-velocity matching is achieved (red dot). (b) Surface charge density $\sigma(t',x',y')$ at $z'=0$ and $t'=0$ at the frozen-mode condition. The frozen wave-packet is accompanied by conical and cylindrical waves, reflecting the presence of two segments of the dispersion relation in panel (a)(ii). An interactive visualization corresponding to panel (a) is available at https://frarodfo.github.io/moving-interface-frozen-modes/ and archived in Ref. rodriguez_fortuno_2026_mifm.