Fundamental Limits on Second Harmonic Generation

Recent advances in fundamental performance limits for power quantities are proving to provide a powerful theoretical tool for understanding the optimization of wave phenomena. To date, however, in any approach allowing power conservation to be enforced locally, the linearity of the wave equation plays a critical role. In this manuscript, we generalize the current quadratically constrained quadratic program framework for evaluating linear photonics limits to incorporate nonlinear processes under the undepleted pump approximation. Via the exemplary objective of enhancing second harmonic generation in a (free-form) wavelength-scale structure, we illustrate a model constraint scheme that can be used in conjunction with standard convex relaxations to bound performance in the presence of nonlinear dynamics. Representative bounds are found to anticipate features observed in optimized structures discovered via computational inverse design. Extensions of our results to other frequency-conversion processes, including Raman scattering and four-wave mixing, are straightforward.