26th Australian Institute of Physics Congress and the Australian and New Zealand Conference on Optics and Photonics

A Scalable Algebraic Framework for Bifurcation Analysis in Kerr Resonator Systems

Not scheduled

15m

Belinda Hutchinson Building (The University of Sydney )

Contributed Oral ANZOS | Nonlinear Photonics (ANZCOP NLPH)

Speaker

Lewis Hill (Max Planck Institute for the Science of Light)

Description

Optical bistability and spontaneous symmetry breaking (SSB) are fundamental nonlinear phenomena in Kerr resonators, underpinning applications including optical switching, logic operations, random number generation, and photonic information processing [1,2]. While these effects are well understood in simple configurations, analytical access to their stationary states and bifurcations becomes increasingly challenging in systems with multiple interacting nonlinear degrees of freedom, including counterpropagating, polarization, multimode, and coupled-resonator configurations.

Here, we present a scalable algebraic framework for Kerr resonator systems [3]. Using techniques from nonlinear algebra, including resultants and Gröbner bases, we derive compact polynomial representations that encode all homogeneous stationary solutions of coupled nonlinear cavity equations. Optical bistability and SSB are identified directly from the roots of these polynomials, while bifurcation points are obtained analytically from their discriminants.

We apply the framework to Kerr ring and Fabry–Pérot resonators, obtaining analytical descriptions of both intensity and amplitude stationary states under balanced and imbalanced pumping conditions. The resulting polynomial structure provides direct access to symmetry-broken, symmetry-restored, and multistable regimes, recovering known numerical results while substantially extending the analytical understanding of these systems. The scalability of the framework has recently been demonstrated through its application to Kerr degenerate optical parametric oscillators, providing analytical access to the stationary states underlying Hopf bifurcations, period-doubling, and chaotic dynamics [4].

Furthermore, the polynomial representation can be interpreted as a characteristic polynomial of an auxiliary linear non-Hermitian system, establishing a direct correspondence between nonlinear bifurcations and exceptional points [5]. The approach offers a powerful general tool for understanding and engineering certain nonlinear phenomena.

References:
[1] A. Ghosh et al., Laser Photon. Rev. 20, e01500 (2026).
[2] L. Quinn et al., Opt. Lett. 48, 3741–3744 (2023).
[3] J. D. Mazo-Vásquez et al., arXiv:2512.14168 (2025).
[4] L. O. Trinchão et al., arXiv:2605.18690 (2026).
[5] L. Hill et al., Commun. Phys. 9, 58 (2026).