Exceptional rings in nonlinear non-Hermitian planar optical microcavities: implementation, signal enhancement, and topology

TL;DR

This work studies exceptional rings in nonlinear non-Hermitian planar optical resonators driven by TE-TM splitting and circular dichroism. Starting from a linear ER in k-space, the authors show that Kerr-type nonlinearity splits the ring into an inner second-order ring and an outer third-order (n=3) ring, organized by an elliptic umbilic in nonlinear parameter space. They demonstrate enhanced and tunable perturbation responses at nonlinear ERs and unveil a rich topological landscape, including a quasi-toroidal structure in a higher-dimensional parameter space and preserved topological invariants under detuning. The findings offer a universal framework for engineering ERs and EPs in nonlinear non-Hermitian systems with potential applications in mode switching and sensing across photonic platforms.

Abstract

Non-Hermitian systems hosting exceptional points (EPs) exhibit signal enhancement and unconventional mode dynamics. Going beyond isolated EPs, here we report on the existence of exceptional rings (ERs) in planar optical resonators with specific form of circular dichroism and TE-TM splitting. Such exceptional rings possess intriguing topologies as discussed earlier for condensed matter systems, but they remain virtually unexplored in presence of nonlinearity, for which our photonic platform is ideal. We find that when Kerr-type nonlinearity (or saturable gain) is introduced, the linear ER splits into two concentric ERs, with the larger-radius ring being a ring of third-order EPs. Transitioning from linear to nonlinear regime, we present a rigorous analysis of (spectral and band) topologies and report enhanced and adjustable perturbation response in the nonlinear regime. Whereas certain features are specific to our system, the results on non-Hermitian topologies and nonlinearity-enhanced perturbation response are generic and equally relevant to a broad class of other nonlinear non-Hermitian systems, providing a universal framework for engineering ERs and EPs in nonlinear non-Hermitian systems.

Figures (10)

• Figure 1: Concept of exceptional ring in planar microcavities with circular dichroism and spin-orbit coupling. (a) Sketch of planar resonator with circular dichroism in the form $\Gamma_+\neq\Gamma_-$ and finite TE-TM splitting. The complex-valued in-plane resonator dispersions, $\mu(\bm{k})$, host an ER centered at $k=0$ in reciprocal space. Nonlinearity splits that ring into mulitple rings (marked as black donuts). (b,d) Isotropic in-plane dispersions of the circularly polarized modes split into TE and TM modes by their coupling. (c,e) In the ER regime, when circular dichroism is introduced, $\Gamma_+\neq\Gamma_-$, TE-TM splitting is substracted inside an exceptional ring (ER) forming in $k$-space at $|\bm{k}_{\mathrm{ER}}|=\sqrt{|\gamma|/|\Delta_\mathrm{LT}|}=1/\mathrm{\upmu m}$; the dispersions become circularly/elliptically polarized as shown by their $S_3$ Stokes vector element. Dispersions are rotationally invariant in $k$-space and $y$-axes in (b,c) are relative to the dispersion minimum.
• Figure 2: Exceptional ring in in-plane resonator dispersion in linear regime. Diagonal elements of response to left (a) and right (b) circularly polarized light, plotted along $k_x$. Comparing (a) and (b), the difference in linewidths as expected inside the exceptional ring are clearly visible [cf. Fig. 1(e)]. These full model calculation is in good agreement with the dispersions of the simplified 2x2 system in Eq. (1) (black circles). The displayed results are rotationally symmetric in $k$-space and energies are measured relative to the dispersion minimum.
• Figure 3: Exceptional rings in the complex-valued energy spectra in nonlinear regime at zero detuning ($\Delta = 0$). Central cuts of the (a) real and (b) imaginary parts of energy spectra of the linear eigenvalue problem (neglecting the $k^2$ dispersion). The respective surfaces are depicted in the insets with the exceptional ring (ER) marked in black. Central cuts of the (c) real and (d) imaginary parts of energy spectra of the nonlinear eigenvalue problem. The respective surfaces are depicted in (e-f). In contrast to the linear regime, the energy spectrum in the nonlinear regime hosts two exceptional rings, one with radius $|\bm{k}|^2=|\gamma|/|\Delta_\mathrm{LT}|=1~\mathrm{\upmu m}^{-1}$ (solid blue dot and labeled by EAR) and a second ring with radius $|\bm{k}|^2=\sqrt{\gamma^2+\alpha^2}/|\Delta_\mathrm{LT}|$ (solid violet dot and labeled by EXR). Between these two rings [marked in red in section B in panel (c-d)] four solutions are found. Inside the inner (outside the outer) ring two circularly (linearly) polarized solutions are found [section A and C in panel (c,d)]. The circles mark points where a solution crosses the boundary between two segments without tracing through an exceptional ring and the cross marks $k=0$. These markers are used in Fig \ref{['fig:4']}(b) to analyze the bifurcation behavior near the two rings. The results displayed are rotationally symmetric in $k$-space. Simulation parameters: $\{\alpha,g_\mathrm{c},g_\mathrm{x}\} = \{0.125,0.25,0\}~\mathrm{meV\upmu m}^2$.
• Figure 4: Elliptic umbilic structure of exceptional rings and exceptional points in nonlinear regime. (a) Sketch of the three sections A-C [as displayed in Fig. \ref{['fig:3']}(c)] with two, four, and two solutions, respectively. (b) The bifurcation behavior on the edges of ring B characterizes the inner (outer) ring as a ring of fold (cusp) points. (c) Radii of fold rings (turquoise lines) and cusp-rings (violet dots) for degnerate modes (segments marked as above) and for finite mode detuning $\Delta$. The umbilic structure shows that fold-rings are part of exceptional arcs and cusp-rings are located at exceptional nexuses. (d) Cross section in (c) for variable nonlinearity $\alpha$. The depicted elliptic umbilic predicts exceptional ring radii and type in the full parameter space. The central organization point represents a point on the linear exceptional ring discussed in Section \ref{['sec:linear']}.
• Figure 5: Enhanced perturbation response at nonlinear exceptional rings and points. In (a-b) a $3^\mathrm{rd}$-order exceptional point on the exceptional nexus ring [star in (d)] is traced for increasing $\alpha$. (a) The inverse signal-enhancement factor $(\mathrm{SEF})^{-1}$ decreases in the nonlinear regime. The coloring visualizes $\alpha$ in each curve (growing from black to yellow). (b) The maximum $\mathrm{SEF}_\mathrm{max}$ at the exceptional point as a function of $\alpha$. (c) Response $\Delta_\omega$ of the exceptional points on the exceptional arc ring (EAR) and exceptional nexus ring (EXR) on the elliptic umbilic along $\Delta=0$ [horizontal dashed line in (d) corresponds to $\epsilon=0$ in (c)] for a fixed nonlinearity $\alpha$. The response of the EAR significantly deviates from the square-root law of linear exceptional points. Here, small perturbations tune the system along the left exceptional arc of the umbilic [red line in (d)]. Larger perturbations tune it away from the arc due to its curvature [blue area in (d)] leading to growing response. (d) Colored cut through elliptic umbilic at finite nonlinearity $\alpha$. The coloring denotes the response function scaling factor $\eta$ when perturbing an exceptional point on the umbilic singularity structure. Detuning allows to realize different rings on the umbilic, which feature distinct perturbation responses. Perturbation from the lower to the upper arc (or vice versa) leads to characteristic square-root response $\eta_1$. Nexus perurbation leads to stronger response $\eta_2$. The perturbation $\epsilon$ is defined in relation to the respective point on the umbilic. Simulation parameters: $\{g_\mathrm{c},g_\mathrm{x}\} = \{2\alpha,0\}~\mathrm{meV\upmu m}^2$.