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

TL;DR

This work develops an exact algebraic framework to characterize stationary states and their bifurcations in Kerr ring and Fabry–Pérot resonators, unifying optical bistability and spontaneous symmetry breaking under nonlinear-algebraic methods. By employing polynomial resultants and Gröbner bases, the authors obtain compact multi-parameter polynomial conditions for circulating intensities and amplitudes, and extract bifurcation points via discriminants. They further interpret these bifurcations as exceptional points of an auxiliary non-Hermitian system, providing a deep link between nonlinear dynamics and non-Hermitian physics. The framework delivers analytic control over Kerr-type nonlinearities, enabling precise predictions of bifurcation lines and symmetry-breaking behavior with potential applications in photonic-circuit design and all-optical information processing.

Abstract

High-quality Kerr resonators are a key platform for studying nonlinear optical phenomena, where bifurcations such as optical bistability and spontaneous symmetry breaking are both of theoretical and practical significance. In this work, we present an analytical framework, which allows finding the stationary states and their bifurcations for the propagating fields in Kerr ring and Fabry-Perot resonators. Using tools from nonlinear algebra, namely, polynomial resultants and Groebner bases, we derive compact polynomial expressions describing the system full solution in both intensity and amplitude representations. The bifurcations are derived from these expressions, and are additionally characterized as exceptional points of an auxiliary linear non-Hermitian system. This work unifies key phenomena in Kerr resonators under the broader framework of nonlinear algebra and offers better control of nonlinear optical systems and the design of photonic devices, enabled by full analytic control.

Figures (6)

• Figure 1: Kerr ring and Fabry-Pérot resonator configurations. Kerr ring configurations: (a) two laser beams counter-propagate within the resonator after entering through an optical coupler, and (b) a single laser beam pumps elliptically polarized light into the resonator, with right- and left-circularly polarized components co-propagating within it. (c) Fabry-Pérot resonator configuration, where elliptically polarized light circulates back and forth due to reflections at the boundaries of the cavity. In all three scenarios, the output intensity is measured after leaving the resonator.
• Figure 2: Real and imaginary parts of the circulating intensities. Real (a,c) and imaginary (b,d) parts of the roots $P$ of $p(P)$ as functions of the cavity detuning in (a,b) with $P^\mathrm{in} = 6$, and the input intensity in (c,d) with $\theta = 7$. In both cases $A=1$ and $B=2$. Solid lines in (a,c) represent real, non-negative $P$ corresponding to physical solutions, while dashed lines indicate roots with nonzero imaginary part, corresponding to unphysical solutions. Symmetric and asymmetric solutions are shown in black and gray, respectively. Green triangles mark the bifurcations at the SSB points, while red diamonds and yellow squares denote bifurcations at the optical bistability limits for asymmetric and symmetric solutions, respectively. The bifurcations occur at the transition points between physical and unphysical solutions, marked by the dotted vertical lines.
• Figure 3: Bifurcation lines in parameter space. Bifurcation lines for SSB and optical bistability in the parameter space spanned by the cavity detuning $\theta$, and the input intensity $P^\mathrm{in}$, with $A=1$ and $B=2$. The dot-dashed red and dashed yellow lines correspond to the limits for optical bistability in the asymmetric and symmetric cases, respectively; the points at which these lines merge are marked with an empty diamond and an empty square, respectively. The solid green line corresponds to the SSB line. The black dashed lines correspond to the parameter scans in Fig. \ref{['fig:fig2']}. Green, yellow, and red shaded areas correspond to the symmetry-broken, optical bistability of the symmetric, and optical bistability of the asymmetric solution, respectively.
• Figure 4: Real and imaginary parts of the field amplitudes. Same as Fig. \ref{['fig:fig2']}, but for the filed amplitudes $E$, setting $E^\mathrm{in} =\sqrt{P^\mathrm{in}} \cdot \text{e}^{\text{i} \pi/4}$. The dashed lines correspond to solutions not fulfilling the physicality constraint.
• Figure 5: Parameter space spanning imbalanced parameters, and different parameter scans. (a) Parameter space spanned by the ellipticity angle $\chi$ and the difference of cavity detuning $\Delta$, at which bifurcations occur (solid black lines). The dotted orange line corresponds to the points at which the symmetry between the intensities is recovered, i.e., where $P_1 =P_2$. Dashed lines correspond to the parameter scans in (b-d). Circulating intensities as functions of (b) the ellipticity for $\Delta = -2$, (c,d) as functions of the difference of cavity detuning with $\chi = 0.5$, and $\chi=1.45$, respectively. Light- and dark-red lines correspond to the circulating intensities. The dashed black lines and black dots in panels (b-d) correspond to the position of the bifurcations in (a). All panels assume $A=1$, $B=2$, $\theta=5$ and $P^\mathrm{in}=10$.