Analog gravity by an optical vortex: Resonance enhancement of Hawking radiation

Propagation of coherent light in a Kerr nonlinear medium can be mapped onto a flow of an equivalent fluid. Here we use this mapping to model the conditions in the vicinity of a rotating black hole as a Laguerre–Gauss vortex beam. We describe weak fluctuations of the phase and amplitude of the electric field by wave equations in curved space, with a metric that is similar to the Kerr metric. We find the positions of event horizons and ergoregion boundaries, and the conditions for the onset of superradiance, which are simultaneously the conditions for a resonance in the analog Hawking radiation. The resonance strongly enhances the otherwise exponentially weak Hawking radiation at certain frequencies and makes its experimental observation feasible.

Figure 1

Graphical solution of equation (17). The bell-shaped curve (in black) represents the Laguerre–Gauss profile of the laser beam, while the three colored curves (dotted green, solid blue, and dashed red) represent the squared radial velocity $v_r^2\left(r\right)$ for three different values of the focal length $f$, corresponding to $f<f_c$, $f=f_c$, and $f>f_c$, respectively. For $f=-500\mathrm{mm}$ (upper, dotted, green curve) there are no solutions. For $f_c=-555$ mm (middle, solid, blue curve) there is one solution. For $f=-600$ mm (lower, dashed, red curve) there are two solutions. In the latter case, two event horizons appear, thus introducing a subsonic (region II, shaded in blue in the figure) and a supersonic (regions I and III) region for the flow. The radial intensity profile of the vortex, with vorticity $n=8$, corresponds to $I=2$ W, $g=5.5\times {10}^{-4}$ m/W, $w_0=1$ mm, and ${\beta }_0=(2\pi /7.80)\times {10}^7{\mathrm{m}}^{-1}$. These parameters allow a broad range of frequencies to satisfy the requirement $L^{-1}<\nu <l_n^{-1}$, where $L$ is the length of propagation in the nonlinear medium, and $l_n$ is the nonlinearity length defined in Eq. (32).

Figure 2

Graphical solution of Eq. (20). The bell-shaped curve (in black) represents the Laguerre–Gauss profile of the laser beam. The dashed red (upper) curve represents the squared total velocity $v_0^2\left(r\right)$ and defines the position of the inner ($r_{e-}$) and outer ($r_{e+}$) ergoregions. The solid blue (lower) curve corresponds to the squared radial velocity $v_r^2\left(r\right)$ and defines the position of the inner ($r_{h-}$) and outer ($r_{h+}$) horizons. There are two ergoregions, one lying between $r_{h-}$ and $r_{e-}$, and the other lying between $r_{e+}$ and $r_{h+}$. The parameters used here are the same as in Fig. 1, except for ${\beta }_0=(\pi /7.80)\times {10}^7{\mathrm{m}}^{-1}$ and $f=-450$ mm. The flow in regions I and III is supersonic, whereas in region II it is subsonic.

Figure 3

A schematic depiction of the flow structure of a vortex. Two solid blue circles represent the outer ($h+$) and inner ($h-$) event horizons, separating the supersonic regions I and III from the subsonic region II (shaded blue area). Two dashed red circles show the borders of the outer ($e+$) and inner ($e-$) ergoregions. The arrows show the radial component of the outgoing flow.

Figure 4

${\gamma }_a$ as a function of the focal length $f$ for the outer horizon. The red dots correspond to the actual numerical value of ${\gamma }_a$, while the blue solid line is a spline interpolation. As can be seen, in the vicinity of the critical focal length $f_c\simeq 520\text{mm}$, ${\gamma }_a$ shows a typical resonant behavior. Note that, while Eq. (42) contains an actual divergence for $f=f_c$, in the interpolation shown in this picture, this does not appear because it is instead replaced by a resonance. The divergence, in fact, is an artifact of the approximated analysis carried out to obtain Eq. (42) and it is not present if the full solution is taken into account. The radial intensity profile of the vortex, with vorticity $n=6$, corresponds to $I=2\mathrm{W}$, $g=5.5\times {10}^{-4}\mathrm{m}/\mathrm{W}$, $w_0=1$ mm, and ${\beta }_0=(2\pi /7.80)\times {10}^7{\mathrm{m}}^{-1}$.

Figure 5

The squared wave vector $k^2\left(r\right)$ in the subsonic region II for the parameters $I=2$ W, $g=5.5\times {10}^{-4}$ m/W, $w_0=1$ mm, ${\beta }_0=(2\pi /7.80)\times {10}^7{\mathrm{m}}^{-1}$, and $f=6$ m. The vorticity of the beam is chosen $n=20$ then $r_{h_{-}}=0.925$ mm and $r_{h_{+}}=3.03$ mm. We assume that the registered frequency of the fluctuation is $\nu =3{\text{m}}^{-1}$ and we consider also three possible vorticities of the emitted radiation: for $m=2$ (black, dashed line) the graph is everywhere positive, corresponding to propagating wave with $k\left(r\right)>600{\mathrm{m}}^{-1}$. For $m=3$ (solid, blue line) there is a barrier in the middle of the region II where the wave becomes evanescent. For $m=6$ (dotted, green line) the barrier becomes high and broad and covers the most part of the region II, thus leaving a narrow area near the black horizon free. The shaded red region, encapsulated by the two red dots corresponding to $r=2.73$ mm and $r=3.35$ mm, corresponds to the region (for the case $m=3$), where the wave vector is evanescent, i.e., $k^2\left(r\right)<0$. The green dot at $r=3.78$ mm corresponds to the starting point of the region in which the wave vector for $m=6$ corresponds to a propagating wave, i.e., $k^2\left(r\right)>0$.

Figure 6

The squared wave vector $k^2\left(r\right)$ in the subsonic region II for the parameters: $I=2$ W, $g=5.5\times {10}^{-4}$ m/W, $w_0=1$ mm, ${\beta }_0=(2\pi /7.80)\times {10}^7{\mathrm{m}}^{-1}$, and $f=6$ m. The vorticity of the beam is chosen $n=9$, so that $r_{h_{-}}=0.925$ mm and $r_{h_{+}}=3.03$ mm. Under these conditions, the resonance frequencies are ${\nu }_{-}=5.803516989{\mathrm{m}}^{-1}$ and ${\nu }_{+}=0.5408657347{\text{m}}^{-1}$, respectively. We assume that the registered value of the fluctuation is $\nu =2{\mathrm{m}}^{-1}$. Here, $m=2$ and the potential forms a barrier, reaching the value $V\left(r_m\right)=8\times {10}^5{\mathrm{m}}^{-2}$ in its maximum not far from the midpoint $r_m=(r_{h_{+}}-r_{h_{-}})/2$ of region II. The light red box in the figure shows the region where the wave vector is evanescent; namely, $k^2\left(r\right)<0$.