Exceptional Points and Lasing Thresholds: When Lower-Q Modes Win

TL;DR

The paper addresses which cavity mode reaches the lasing threshold first under uniform pumping and shows that exceptional points (EPs) can invert the conventional hierarchy dictated by quality factor and initial modal-gain slope. A non-Hermitian two-dimer toy model yields eigenfrequencies $\Omega_n^{\pm}=\Delta \omega_n/2+i(g/2-\gamma_n)\pm\sqrt{\kappa_n^2+(\Delta \omega_n-ig)^2/4}$, illustrating how mode-gain rates swap beyond an EP so a lower-$Q$ mode can preempt a higher-$Q$ rival at threshold. This effect is then demonstrated in a uniform-pumping polygonal microcavity, where two lower-$Q$ quasinormal modes coalesce at an EP and the resulting hybrid mode experiences a larger gain increase, crossing the threshold earlier. Robustness analyses and practical implementations—such as a composite four-cavity system and microring platforms—show the phenomenon survives realistic perturbations (e.g., corner rounding up to $r_{ ext{rounding}}\approx 0.01R$) and highlight the important role of non-Hermitian physics in lasing dynamics and mode control.

Abstract

One of the most fundamental questions in laser physics is the following: Which mode of an optical cavity will reach the lasing threshold first when gain is applied? Intuitively, the answer appears straightforward: When a particular mode is both temporally well confined (i.e., exhibits the highest quality factor) and experiences initially the largest increase of the modal gain, it is naturally expected to lase first. However, in this work, we demonstrate that this intuition can fail in surprising ways. Specifically, we show that in the presence of non-Hermitian degeneracies, known as exceptional points, the expected mode hierarchy can be dramatically altered. These spectral singularities can give rise to counterintuitive mode switching, where a mode with a lower quality factor and initially smaller increase of modal gain reaches the lasing threshold ahead of a more favorable competitor. Remarkably, this effect can occur even under spatially uniform pumping, underscoring the subtle and profound influence of non-Hermitian physics on lasing dynamics.

Figures (13)

• Figure 1: A schematic illustration of the main concept of this work shows the distribution of complex eigenfrequencies $\Omega$ of a laser cavity below the lasing threshold. The solid square and circles indicate the initial positions of three different eigenfrequencies in the absence of gain. The curved trajectories depict how these eigenfrequencies evolve as gain is introduced. Initially, the red mode (shown by the solid red square) has the lowest losses (i.e., the highest quality factor) and moves most rapidly (see dashed red line) toward the real axis as gain increases. However, at a certain gain level, two lower-quality modes (shown in solid blue lines) coalesce at an exceptional point (EP). Beyond this point, one of the resulting hybrid modes advances more quickly toward the real axis, eventually becoming the first to reach the lasing threshold—despite originating from lower-$Q$ modes. Ideally, the modes involved in this process should lie under the maximum value of a gain curve that excludes other modes but as we will see, this condition is not a necessary one.
• Figure 2: A simplified model illustrating the key effect discussed in this work can be constructed using two distinct laser systems, each consisting of two coupled cavities, as shown in (a). In both systems, the left cavity serves as the active laser element, while the right cavity acts as a passive reservoir. (b) The lasing mode of the first system (dashed red lines) has initially, i.e., in the absence of gain, a higher quality factor (lower loss $|\text{Im}\Omega|$) than any of the modes in the second system. However, as the gain $g$ increases, the modes of the second system (solid blue lines) coalesce at an EP. Beyond this point, one of the resulting hybrid modes accelerates toward the lasing threshold and surpasses the initially higher-$Q$ mode of the first system. (c) The rate of modal gain increases as a function of applied gain shows a clear phase transition experienced by the blue mode, ultimately allowing it to lase before the blue mode. Although this model involves two separate laser systems, it provides an intuitive demonstration of the underlying mechanism. In the main Letter and Supplemental Material SM, we show how this effect can be realized within a single laser system.
• Figure 3: A schematic illustration of the concept discussed in this work is shown in the context of a single laser cavity under uniform pumping. In the passive regime, mode ① is initially more confined within the cavity volume, resulting in a higher quality factor compared to mode ②. As gain $g$ is applied, mode ① initially experiences a stronger increase of modal gain and approaches the lasing threshold more quickly. However, if mode ② coalesces with another quasinormal mode (not shown here) to form an EP at a certain gain threshold $\tilde{g}$, the resulting hybrid mode can become more spatially confined than the mode ①. As a result, mode ② may experience a stronger increase of modal gain and ultimately reach the lasing threshold first.
• Figure 4: An illustration of the proposed microcavity. The polygonal shape is given by the vertices $V_{1-11}$ with relative coordinates $(x/R,y/R)$ being $(1, 0.59637)$, $(0.88924, 0.59637)$, $(-1, 0.02815)$, $(-1, -0.59637)$, $(-0.33022, -0.59637)$, $(-0.30598, -0.65828)$, $(-0.28174, -0.59637)$, $(-0.21017, -0.59637)$, $(-0.15, -5.6009)$, $(-0.08983, -0.59637)$, and $(1, -0.59637)$, respectively.
• Figure 5: Complex frequency plane. The frequency trajectories of modes in a polygonal cavity are shown for a change of the imaginary part of the refractive index from $n_{\text{imag}}=0$ (filled circles) to $n_{\text{imag}}=-0.04977$ (empty circles). The inset is a magnification with results of the individual simulations shown by thin dots.