Finite width of the optical event horizon and enhancement of analog Hawking radiation

Coherent light propagating in a bulk Kerr nonlinear defocusing medium obeys nonlinear Schrödinger (NLS) equation, which is similar to the Gross-Pitaevskii equation for Bose-Einstein condensates (BECs). An equivalent hydrodynamic approach allows one to consider propagation of light as a flow of an equivalent "luminous fluid." An analog optical event horizon can be formed when the flow velocity of this fluid equals the local sound velocity, determined by the nonlinear term in the NLS equation. The analog event horizon is characterized by a finite width, also determined by the nonlinearity length, or by the healing length in Bose-Einstein condensates. The various eigenmodes of fluctuations are found in the immediate vicinity of the event horizon and the scattering matrix due to the finite width horizon is calculated to be within the leading order corrections in the nonlinearity length. The Hawking radiation is found to be enhanced with respect to that of a Planck's black body spectrum and is characterized by the emissivity greater than one. A procedure of paraxial quantization of the fluctuation field is discussed and its connection to the conventional quantization of the electromagnetic field is demonstrated. Quantum fluctuations of the electric field energy and those of its flow are calculated.