Four-wave mixing and enhanced analog Hawking effect in a nonlinear optical waveguide

We study in detail the scattering of light on a soliton propagating in a waveguide which has been proposed as an experimental system in which one could observe the analog Hawking effect. When not applying the rotating-wave approximation, we show that the linearized wave equation governing perturbations has the same structure as that governing phonon propagation in an atomic Bose condensate. By taking into account the full dispersion relation, we then show that the scattering coefficients encoding the production of photon pairs are amplified by a resonance effect related to the modulation instability occurring in the presence of a continuous wave. When using a realistic example of a silicon nitride waveguide on a silica substrate, we find that a soliton of duration 10 fs would spontaneously emit about one photon pair for every cm it travels, which makes the effect readily observable. This result is confirmed by numerically solving the equation encoding the Kerr nonlinearity and governing the evolution of the full field (soliton plus perturbations). We discuss the link with previous works devoted to the analog Hawking effect where the pair creation rates were about six orders of magnitude smaller.