Luminal mirror

When a refractive index modulation of dispersive medium moves at the speed of light in vacuum, an incident electromagnetic wave, depending on its frequency, either is totally transmitted with a phase shift, or forms a standing wave, or is totally reflected with the frequency upshift. The luminal mirror converts a short incident pulse into a wave packet with an infinitely growing in time local frequency near the interface and with an energy spectral density that asymptotically is the inverse square of frequency. If the modulation disappears, the high frequency radiation is released.

Figure 1

The electric field strength in the $(x,t)$ plane computed from (a) Eq. (13) and (b) Eq. (16), for $\xi ,\eta$ defined by Eqs. (6) and (7), ${\alpha }_{\delta }=1,{\alpha }_p=1,{\omega }_{pe}^2=10,\tau =1$. The color scale is saturated; black is for values beyond the range. The diagonal dashed line in (b) is the plasma-vacuum interface worldline. Vertical dashed lines correspond to zero wave group velocity of Eq. (11). Hollow arrows show the wave propagation direction.

Figure 2

The electric field strength profiles (a) and spectra (b) at $t=100$ computed from Eq. (13), red, and Eq. (16), black, for ${\alpha }_{\delta }=1,{\alpha }_p=1,{\omega }_{pe}^2=10,\tau =1$. The inset in (a) shows a zoomed region near $x=100$. The vertical dashed line shows where the wave group velocity of Eq. (11) is zero. Hollow arrows show the wave propagation direction. The dashed line in (b) is the asymptote for large $k$. $C_{p\delta }=\int |{\tilde{E}}_p|dk/\int |{\tilde{E}}_{\delta }|dk$.

Figure 3

Reflected wave behind the luminal plasma-vacuum interface seen in the numerical solution of Eq. (2). (a) The electric field strength $E_p$ and its magnitude product $k_{\mathrm{loc}}\left|E_p\right|$ with (b) its local wave number $k_{\mathrm{loc}}$ at $t=500$, shown for $\zeta =x-ct<0$. In (a), $E_p$ is linearly scaled but its ordinate axis is not shown, $E_{p,\max }\approx 0.67$; the solid and dashed curves are $\{\zeta ,E_p(\zeta <0)\}$ and $\{-\zeta ,E_p(\zeta >0)\}$, respectively. (c) The evolution of the maximum electric field strength (left axis) and wave number (right axis).

Figure 4

The electric field strength spectra at different time moments seen in the same numerical solution as in Fig. 3.

Figure 5

The numerical solution of Eq. (2) for the luminal plasma slab gradually disappearing from $t=400$ till $t=440$. (a) The beginning of the Gaussian pulse and plasma slab interaction, and the local wave number of the latest instance of the electric field (top). (b) The electric field in the disappearing plasma slab, and the almost coinciding profiles of the electric field local wave number (blue, red, black for $t=400,420,440$). (c) The electric field strength and the profile function isocurves $\rho =0.8,0.5,0.1$ in the $(x-ct,t)$ plane.