  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, their small footprint and the relative ease of coupling
many such microrings together, however, surface roughness of the wave-guides,
and defects in the coupler geometry routinely induce splitting of the cavity
modes due to backscattering and backcoupling. The parasitic back-propagating
mode in the microring leads to hybridisation of the modes, altering the linear
and nonlinear properties of this 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 the
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, we calculate the consequent impurity and heralding
efficiency of various heralded photon schemes, we explore a perturbative
explanation of the observations and assess the correspondence between
spontaneous and stimulated processes in these systems.
