Rigorous Bound on the Violation of Dynamic Reciprocity Induced by Four-Wave Mixing

Dynamic reciprocity imposes stringent performance constraints on nonlinear optical devices such as isolators and circulators. The seminal result by Shi et al. establishes that nonlinear optical devices relying on the intensity-dependent refractive index obey dynamic reciprocity for small signals with spectrally distinct fields. However, it has also been recognized that it is possible to violate dynamic reciprocity by exploiting frequency mixing processes. In this paper, we establish a rigorous upper bound on this violation that is independent of device geometry. We demonstrate that this bound captures the parameter scalings of realizable physical systems, and that under some conditions dynamic reciprocity violation can grow unbounded to achieve arbitrary nonlinear isolation. These results provide an analytically robust version of dynamic reciprocity, as well as theoretical guidance for the development of power efficient nonlinear optical isolators and circulators.