General complex envelope solutions of coupled-mode optics with quadratic or cubic nonlinearity

TL;DR

This paper derives exact analytic solutions for complex field envelopes in coupled-mode optical systems with quadratic or cubic nonlinearities, using Weierstrass elliptic functions, applicable to various nonlinear optical phenomena.

Contribution

It provides the first compact form of general solutions involving Weierstrass functions for coupled-mode systems with quadratic and cubic nonlinearities.

Findings

Solutions encompass sum and difference frequency generation

Includes polarization and parity-time dynamics

Applicable to optical processing applications

Abstract

The analytic general solutions for the complex field envelopes are derived using Weierstrass elliptic functions for two and three mode systems of differential equations coupled via quadratic $\chi_2$ type nonlinearity as well as two mode systems coupled via cubic $\chi_3$ type nonlinearity. For the first time, a compact form of the solutions is given involving simple ratios of Weierstrass sigma functions (or equivalently Jacobi theta functions). A Fourier series is also given. All possible launch states are considered. The models describe sum and difference frequency generation, polarization dynamics, parity-time dynamics and optical processing applications.