# Enhanced nonlinear frequency conversion and Purcell enhancement at exceptional points

## TL;DR

This paper derives formulas for enhanced nonlinear frequency conversion and Purcell effect at exceptional points in nonlinear cavities, demonstrating significant emission rate enhancements for embedded emitters.

## Contribution

It introduces analytical formulas for radiative emission at exceptional points in nonlinear resonant cavities and demonstrates a photonic crystal design achieving this enhancement.

## Key findings

- Up to 100-fold increase in emission rate at EPs.
- Analytical formulas for nonlinear emission at exceptional points.
- Design of a photonic crystal slab supporting an EP.

## Abstract

We derive analytical formulas quantifying radiative emission from subwavelength emitters embedded in triply resonant nonlinear $\chi^{(2)}$ cavities supporting exceptional points (EP) made of dark and leaky modes. We show that the up-converted radiation rate in such a system can be greatly enhanced---by up to two orders of magnitude---compared to typical Purcell factors achievable in non-degenerate cavities, for both monochromatic and broadband emitters. We provide a proof-of-concept demonstration by studying an inverse-designed 2D photonic-crystal slab that supports an EP formed out of a Dirac cone at the emission frequency and a phase-matched, leaky-mode resonance at the second harmonic frequency.