Mode Switching Through Exceptional Points Induced by Lasing-Inversion Coupling

TL;DR

This work addresses how gain–loss coupled cavities near an exceptional point can exhibit a second threshold that activates a second mode and a frequency comb; they introduce a Bloch temporal coupled-mode theory and a Bogoliubov–de Gennes (BdG) description of lasing–inversion coupling, revealing lasing-inversion hybrid (LIH) modes and an LIH-induced exceptional point (LIH-EP). They show that a Hamiltonian Krein–Hopf bifurcation and the LIH-EP topology govern the emergence and switching between single-mode lasing and frequency combs, with a quartic effective model enabling analytic identification of the LIH-EP and its parameter dependence. Importantly, increasing the passive-mode loss lowers the second threshold and broadens the comb, suggesting practical routes to reconfigurable dual-comb sources for applications such as dual-comb spectroscopy and adaptive optical communication.

Abstract

The gain-loss coupling in optical cavities induces exceptional points (EPs), where two optical modes coalesce. The large modal overlap near an EP intensifies gain competition, favoring single-mode lasing. Recent studies further revealed self-modulation closer to the EP that transforms the lasing mode into a frequency comb. Such EP-enabled comb formation suggests a previously unaccounted-for mechanism that overcomes the strong gain competition and drives a second mode to threshold. Here, using a Bloch coupled-mode theory derived from first principles, we show that the second threshold arises from dynamical couplings among the population inversion, the lasing field, and a dark cavity mode. The lasing-inversion coupling produces extra EPs, whose spectral structure governs switching among single-mode lasing and frequency combs with different repetition rates. This above-threshold mode-switching mechanism enables new opportunities for tunable photonic systems, including adaptive optical communication links and dual-comb spectroscopy.

Figures (3)

• Figure 1: Laser behavior of gain--loss coupled cavities near an exceptional point.a Eigenfrequency trajectories of the two optical modes $\mathrm{O_I}$ (red) and $\mathrm{O_{II}}$ (green), obtained from $\mathbf{H}_0(\nu_1)$, for two values of the passive cavity loss. Symbols denote B-CMT results, while solid lines show Maxwell--Bloch simulations. Yellow arrows indicate increasing pump strength $\nu_1$, and $\nu_{\mathrm{th}1}$ marks the first lasing threshold. Inset: schematic of the gain--loss coupled-cavity system. b Lasing spectra above the first threshold for different passive losses and pumping strengths, showing the transition from a fixed point (single-mode lasing, left panel) to limit cycles (frequency combs, middle and right panels). The field intensity is normalized by the factor $f=\hbar\sqrt{\gamma_\perp \gamma_\parallel}/(2R)$. The values of all system parameters are included in Supplementary section II.
• Figure 2: Lasing--inversion hybrid (LIH) modes above the first threshold.a The eigenvalue spectrum of $\mathbf{H}_{\mathrm{BdG}}$ for small relative pumping $\Delta\nu$. The lasing mode pinned at $\omega=0$ (red), the non-self-conjugate inversion fluctuation pinned at $-i\gamma_\parallel$ (gray), the $\mathrm{LIH}_{\mathrm{I}}$ modes (blue), and the Hamiltonian Krein-Hopf Bifurcation (black circle) are shown. b Eigenvalue trajectories of the $\mathrm{LIH}_{\mathrm{I}}$ (blue) and $\mathrm{LIH}_{\mathrm{II}}$ (green) modes as functions of pumping $\Delta\nu$ beyond the bifurcation. The left column corresponds to $\gamma_2=0.38~\mathrm{ps}^{-1}$, while the right column corresponds to $\gamma_2=0.42~\mathrm{ps}^{-1}$. Circles denote eigenvalues of $\mathbf{H}_{\mathrm{BdG}}$ given by Eq. \ref{['HBdG']}, while crosses indicate the roots of the quartic equation $Q(\omega,\Delta\nu)=0$ defined in Eq. \ref{['Q']}.
• Figure 3: Mode switching through LIH-induced exceptional points.a Trajectories of the LIH eigenvalues $\tilde{\omega}_{\mathrm{I}}$ (blue) and $\tilde{\omega}_{\mathrm{II}}$ (green) obtained from the quartic equation $Q(\omega,\Delta\nu)=0$ for different values of the passive cavity loss $\gamma_2$. Triangles indicate crossings of the eigenvalues with the real axis, marking the second lasing threshold. The red dot in the middle panel denotes the LIH-EP frequency $\omega_{\mathrm{EP}}$ satisfying Eq. \ref{['LIH-EP']}. The inset in the left panel shows a zoomed-in view of $\tilde{\omega}_{\mathrm{I}}$ near threshold, highlighting the PHS-EP bifurcation shown in Fig. \ref{['fig_LIH']}a. Yellow arrows indicate increasing pump strength $\Delta\nu$. b Real and c imaginary parts of the LIH eigenvalue surfaces as functions of $\gamma_2$ and $\Delta\nu$. Blue and green curves correspond to the $\gamma_2$ slices shown in a. The red sphere marks the LIH-EP, and the black dashed line denotes the second lasing threshold defined by $\mathrm{Im}(\omega)=0$.