Parametric processes in nonlinear structures with reflections: an asymptotic-field approach

The generation of engineered quantum states of light via nonlinear processes is fundamental for quantum technologies based on photons. Although embedding nonlinear materials within resonant structures allows for the enhancement and tailoring of photon properties, accurately modeling these quantum interactions remains a challenge. In this work, we apply the asymptotic-fields formalism, an approach based on scattering theory, to describe nonlinear optical processes within a Fabry-P\'erot cavity. Unlike previous applications of this formalism, we explicitly account for reflections in the system. We derive the interaction Hamiltonian and calculate photon-pair generation rates using perturbation theory. The versatility of this model is illustrated through three examples: (i) spontaneous parametric down-conversion in an idealized cavity with flat-response mirrors; (ii) the generation of counter-propagating photon pairs in a periodically-poled material; and (iii) spontaneous four-wave mixing in a cavity built with Bragg reflectors.