Čerenkov radiation in vacuum from a superluminal grating

Nothing can physically travel faster than light in vacuum. There are several ways proposed to bypass the light barrier and produce Čerenkov radiation (ČR) in vacuum. In this paper, we theoretically predict ČR in vacuum from a spatiotemporally modulated boundary. We consider the modulation of traveling wave type and apply a uniform electrostatic field on the boundary to generate electric dipoles. Since the induced dipoles stick to the interface, they travel at the modulation speed. When the grating travels faster than light, it emits ČR. In order to quantitatively examine this argument, we need to calculate the field scattered at the boundary. We utilize a dynamical differential method, which we developed in a previous paper, to quantitatively evaluate the field distribution in such a situation. We can confirm that all scattered fields are evanescent if the modulation speed is slower than light while some become propagating if the modulation is faster than light.

Figure 1

Single interface system composed of a dielectric and vacuum. The interface is under the spatiotemporal modulation of traveling wave type (1). The orthonormal tangential vectors ${\vec{t}}_{1,2}$ can be calculated from the interface profile. The permittivities above and below the interface are denoted as ${\epsilon }^{>}$ and ${\epsilon }^{<}$, respectively. We apply a uniform electrostatic field ${\vec{\mathcal{E}}}_{\mathbf{x},z}^{\mathrm{in}}$ on the modulated interface so that there are induced dipoles which travel on the interface due to its profile of the traveling wave type. The velocity of the grating $v_{\mathrm{ph}}=\mathrm{\Omega }/g$ can be tuned by two independent parameters so that it can exceed the speed of light. When the speed is faster than light, the induced dipoles emit Čerenkov radiation.

Figure 2

Integration paths leading to the boundary conditions at the modulated interface. The modulation-induced source, ${\vec{j}}_{\mathbf{x}}^{\mathrm{sou}}$, comes into contribution.

Figure 3

Snapshots (middle row) and their cross sections (upper and lower rows) of the field distributions in the subluminal and superluminal regimes. The intensity of the total electric field $|{\vec{\mathcal{E}}}_{\mathbf{x},z}{|}^2$ is plotted. (i) Subluminal regime in the dielectric and vacuum sides, $v_{\mathrm{ph}}=\mathrm{\Omega }/g=0.2c<c/\sqrt{\epsilon }$. (ii) Superluminal in the dielectric side and subluminal in the vacuum side, $c/\sqrt{\epsilon }<v_{\mathrm{ph}}=0.8c<c$. (iii) Superluminal regime in both sides, $c<v_{\mathrm{ph}}=1.2c$. The modulation parameters are given as follows: the spatial frequency $g=2\pi \left[\mu {\mathrm{m}}^{-1}\right]$, the modulation depth $2A=100\left[\mathrm{nm}\right]$. The input amplitude is $E_{\mathrm{in}}=\sqrt{Z_0\times 100\mathrm{mW}{\mathrm{cm}}^{-2}}\approx 0.614\left[\mathrm{mV}\mu {\mathrm{m}}^{-1}\right]$. The permittivities are ${\epsilon }^{>}=1$ and ${\epsilon }^{<}=\epsilon =2.25$. The cutoff is $m_c=10$ so that we take $2m_c+1=21$ diffracted waves into account. The colorbars for (ii) and (iii) are common and shown on the right of (iii). Note that the scale of the colorbar of (i) is different from those of (ii) and (iii), and the horizontal axes and the vertical axes of the color plots are normalized by the spatial period $g$ of the modulation.