Fundamental limits on $\chi^{(2)}$ second harmonic generation

TL;DR

This paper extends theoretical performance limits for electromagnetic wave phenomena to include nonlinear processes like second harmonic generation, providing bounds that align with computationally optimized structures and applicable to various frequency conversion methods.

Contribution

It generalizes the quadratic constrained quadratic program framework to nonlinear optics under the undepleted pump approximation, enabling performance bounds for nonlinear frequency conversion.

Findings

Derived bounds match features of optimized structures

Framework applicable to various nonlinear frequency conversion processes

Provides a convex relaxation approach for nonlinear photonics

Abstract

Recent advances in fundamental performance limits for power quantities based on Lagrange duality are proving to be a powerful theoretical tool for understanding electromagnetic wave phenomena. To date, however, in any approach seeking to enforce a high degree of physical reality, 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. The formulation can be straightforwardly modified to treat other frequency-conversion processes, including Raman scattering and four-wave mixing.