Dielectric black holes induced by a refractive index perturbation and the Hawking effect

We consider a 4D model for photon production induced by a refractive index perturbation in a dielectric medium. We show that, in this model, we can infer the presence of a Hawking type effect. This prediction shows up both in the analogue Hawking framework, which is implemented in the pulse frame and relies on the peculiar properties of the effective geometry in which quantum fields propagate, as well as in the laboratory frame, through standard quantum field theory calculations. Effects of optical dispersion are also taken into account, and are shown to provide a limited energy bandwidth for the emission of Hawking radiation.

Figure 1

Scheme of the RIP geometry. $x_{+}$ and $x_{-}$ indicate the black hole and white hole horizon positions, respectively.

Figure 2

Branches relative to the dispersion relation (94) in the comoving frame. The dashed line corresponds to $\omega =-v{k}_x$. Comoving frequencies corresponding to the experiments in Ref. 26 are of the order of ${10}^{13}\mathrm{rads}/\mathrm{s}$.

Figure 3

Prediction of the Hawking emission spectral range accounting for the RIP velocity and the material refractive index $n$ from a complete Sellmeier equation. The two curves show, for the case of fused silica, $n_0$ (in blue) and $n_0+\delta n$ (in green) with $\delta n=1\times {10}^{-3}$. The shaded area indicates the spectral emission region predicted for a Bessel filament that has an effective refractive index $n=c/v_B$ where the Bessel peak velocity is $v_B=v_G/\cos \theta$ with cone angle $\theta =7$ deg and $v_G=d\omega /dk$ is the usual group velocity. The values and parameters used correspond to the experiments shown in Ref. 26.

Figure 4

The dispersion relation (96) in the comoving frame for a 2D model at a fixed point $x$ (only the branch relevant to horizon formation is shown). The shaded regions delimit the bands representing the values of $k_x$ for which a phase horizon exists. See Eqs. (110, 111), in which we chose values of the velocity $v$ and $\eta$ for illustrative purposes and that are not related to the experimental values used in Fig. 3.

Figure 5

Penrose diagram for the analogue metric (2).