The Hawking effect in dielectric media and the Hopfield model

We consider the so-called Hopfield model for the electromagnetic field in a dielectric dispersive medium in a framework in which one allows a space-time dependence of microscopic parameters, aimed to a phenomenological description of a space-time varying dielectric perturbation induced by means of the Kerr effect. We discuss the analogue Hawking effect, by first analyzing the geometrical optics for the Hopfield model, and then by introducing a simplified model which has the bonus to avoid many difficulties which are involved in the full Hopfield model, still keeping the same dispersion relation. Amplitude calculations are indicated, and generalized Manley-Rowe identities are derived in a quantum scattering theory framework. Our main result is an analytical calculation of the spontaneous thermal emission in the single-branch case, which is provided non perturbatively for the first time in the framework of dielectric black holes. An universal mechanism for thermality between optical black holes and acoustic black holes is also pointed out.