Hawking spectrum for a fiber-optical analog of the event horizon

Hawking radiation has been regarded as a more general phenomenon than in gravitational physics, in particular in laboratory analogs of the event horizon. Here we consider the fiber-optical analog of the event horizon, where intense light pulses in fibers establish horizons for probe light. Then, we calculate the Hawking spectrum in an experimentally realizable system. We found that the Hawking radiation is peaked around group-velocity horizons in which the speed of the pulse matches the group velocity of the probe light. The radiation nearly vanishes at the phase horizon where the speed of the pulse matches the phase velocity of light.

Figure 1

Space-time diagram of light-ray trajectories near the horizon given by Eq. (2). Straight lines (dashed orange) represent rays traveling with the fluid (copropagating waves), for which there is nothing special in the horizon. Curved lines (solid blue) represent rays traveling against the fluid (counterpropagating waves) and they are split into two at the horizon. The center black line is the horizon. The radius $r$ is written in terms of $r_S^{-1}$ units.

Figure 2

Space-time diagram of light rays near the horizon for the velocity profile in Eq. (7) for two cases: dispersionless (up) and dispersion $c\left(k\right)$ from Eq. (8) (down). In the first case the horizon is well defined, but in the second one is fuzzy as it depends on $k$. We show three pairs of rays that conserve $\omega -uk$, one with positive (blue) and the other with negative (green) frequency.

Figure 3

The Fourier transform of the sech $\left(t\right)$ pulse (solid blue), the real part of the half Fourier-transforms (dashed orange), the imaginary parts of left (dot-dashed green) and right (dotted red) Fourier transforms.

Figure 4

Dispersion relation $\beta \left(\omega \right)$ for counterpropagating waves using the model in Eq. (26) with two terms (solid blue). We also show the negative of this function (dashed red) that is useful to match the negative frequencies.

Figure 5

Dispersion relation in the comoving frame ${\omega }^{\prime }={\omega }^{\prime }\left(\omega \right)$ from Eq. (15) and $\beta \left(\omega \right)$ from Eq. (26). We show the function for the copropagating waves (solid blue), its negative (dot-dashed red) to match with negative-frequency waves, and the counterpropagating waves (dashed orange). We also show the two points of interest: the phase horizon $\left({\omega }_z,0\right)$ and the group horizon $\left({\omega }_h,{\omega }_h^{\prime }\right)$.

Figure 6

The dispersion relation for counterpropagating (solid blue) and copropagating waves (dashed orange); the negative of the counterpropagating (dot-dashed red) dispersion is also plotted as it indicates the matching with the negative frequency. The horizontal line (dotted green) represents the conservation of ${\omega }^{\prime }$ and the points shown satisfy the matching conditions for the unperturbed system. The labels 1, 2, 3, and $3^{\prime }$ correspond to the identification of the modes for the numerical solution.

Figure 7

Close up on ${\omega }_h^{\prime }$ from Fig. 6. The diagonal purple line shows the maximum slope reached by the pulse with $\delta n_{\text{max}}$ and defines the edge of frequencies ${\omega }_{\text{min}}^{\prime }$ from the three modes that are able to reach the horizon due to the Kerr effect when we consider the soliton. This range is shown in green shadow.

Figure 8

Full Hawking spectrum for the dispersion. The peaks are in ${\omega }_h$ and ${\omega }_{\text{max1}}$. The vertical gray lines mark the shown frequencies, the lines on the sides of ${\omega }_h$ are ${\omega }_{\text{min3}}$ and ${\omega }_{\text{min3"}}$. Also, to the right $\text{of}{\omega }_{\text{max2}}$ is ${\omega }_{\text{min2}}$ and to the left of ${\omega }_{\text{max1}}$ is ${\omega }_{\text{min1}}$, both indistinguishable from this scale. These lines limit the region for creation of Hawking radiation.

Figure 9

Close up of the Hawking spectrum around the horizon in ${\omega }_h$ and around the negative frequency horizon in ${\omega }_{\text{max1}}$. In both cases we observe that the particle production dips to zero exactly at the horizon. The dashed orange lines correspond to the same arbitrary value of ${\omega }^{\prime }$ and they will be correlated.

Figure 10

Norm conservation. The positive norm (solid blue) is the sum of the norm of modes 2, 3, and $3^{\prime }$, the negative norm (dashed green) is the norm of mode 1. We also show the sum of both of them and we check the norm conservation (dotted red). The norm is normalized according to the maximum (close to the horizon).

Figure 11

Relative difference between the positive- and negative-norm modes (relative error) in terms of ${\omega }^{\prime }$. As expected, the maximum value is at the horizon and it is less than 1%.