# Symmetry-based analytical solutions to the \chi^{(2)} nonlinear directional coupler

## TL;DR

This paper derives exact analytical solutions for the complex  nonlinear directional coupler by exploiting symmetry, enabling better understanding and control of nonlinear optical processes like second harmonic generation and parametric amplification.

## Contribution

It introduces symmetry-based analytical solutions for the  nonlinear coupler, a nonintegrable system, applicable across phase matching conditions, advancing theoretical understanding.

## Key findings

- Exact solutions for symmetric  couplers derived
- Analysis of second harmonic generation and parametric amplification regimes
- Insights into the influence of field parity and power on device operation

## Abstract

In general the ubiquitous \chi^{(2)} nonlinear directional coupler, where nonlinearity and evanescent coupling are intertwined, is nonintegrable. We rigorously demonstrate that matching excitation to the even or odd fundamental supermodes yields dynamical analytical solutions for any phase matching in a symmetric coupler. We analyze second harmonic generation and optical parametric amplification regimes and study the influence of fundamental fields parity and power on the operation of the device. These fundamental solutions are useful to develop applications in classical and quantum fields such as all-optical modulation of light and quantum-states engineering.