Backscattering in nonlinear microring resonators via a Gaussian treatment of coupled cavity modes

Systems of coupled cavity modes have the potential to provide bright quantum optical states of light in a highly versatile manner. Microring resonators, for instance, are highly scalable candidates for photon sources. Thanks to CMOS fabrication techniques for their small footprint and the relative ease of coupling many such microrings together. However, surface roughness of the waveguides and defects in the coupler geometry routinely induce splitting of the cavity modes due to backscattering and backcoupling. The parasitic back-propagating mode in a microring leads to hybridization of the modes, altering the linear and nonlinear properties of a system of coupled cavity modes and ultimately constraining the fidelity of quantum light sources that can be produced. In this paper, we derive a comprehensive general model for Gaussian nonlinear processes in systems of coupled cavity modes based on an effective field Hamiltonian and a dispersive input-output model. The resulting dynamics of the equations of motion are evaluated in a Gaussian process formalism via symplectic transformations on the optical modes. We then use this framework to numerically model and explore the problem of backscattering in microring resonators in physically relevant parameter regimes, involving the splitting of various resonances, and we calculate the consequent impurity and heralding efficiency of various heralded photon schemes. We provide a perturbative explanation of the observations and assess the correspondence between spontaneous and stimulated processes in these systems.