  Finite width of the analog event horizon is determined by the nonlinearity
length in the Kerr nonlinear optical system, which is discussed here, or by the
healing length in Bose-Einstein condensates. The various eigen modes 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 within the
leading order correction in the nonlinearity length. The Hawking radiation is
found to be that of a grey body with the emissivity larger 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.
