An Algebraic Approach to Bifurcations in Kerr Ring and Fabry-Perot Resonators

TL;DR

This paper develops an algebraic framework using nonlinear algebra tools to analytically determine stationary states and bifurcations in Kerr resonators, unifying optical bistability and symmetry breaking analysis.

Contribution

It introduces a novel algebraic method employing polynomial resultants and Groebner bases to analyze bifurcations in Kerr resonators, applicable to broader nonlinear systems.

Findings

Derived compact polynomial expressions for system solutions.

Identified bifurcations as exceptional points in a non-Hermitian system.

Unified analysis of optical bistability and symmetry breaking.

Abstract

Nonlinear phenomena such as optical bistability and spontaneous symmetry breaking play a central role in Kerr resonators, and are increasingly exploited in photonic integrated circuits for all-optical information processing. In this work, we present an analytical framework allowing to find the stationary states and their bifurcations for the propagating fields in Kerr ring and Fabry-Perot resonators, which can be generalized to other nonlinear systems. Using tools from nonlinear algebra, namely, polynomial resultants and Groebner bases, we derive compact polynomial expressions describing the system's full solution in both intensity and amplitude representations. The bifurcations follow directly from these expressions, and are additionally characterized as exceptional points of an auxiliary linear non-Hermitian system. Together, these results unify optical bistability and spontaneous symmetry breaking within a single analytical framework, and offer a route toward improved control of nonlinear optical systems, and the design of photonic devices.