Inverse Vernier effect in coupled lasers

In this report we study the Vernier effect in coupled laser systems consisting of two cavities. We show that depending on the nature of their coupling, not only can the “supermodes” formed at overlapping resonances of these two cavities have the lowest thresholds as previously found, leading to lasing at these overlapping resonances and a manifestation of the typical Vernier effect, but also they can have increased thresholds and are hence suppressed, which can be viewed as an inverse Vernier effect. The inverse Vernier effect can also lead to an increased free spectrum range and possibly single-mode lasing, which may explain the experimental findings in several previous studies. We illustrate this effect using two coupled micro-ring cavities and a micro-ring cavity coupled to a slab cavity, and we discuss its relation to the existence of exceptional points in coupled lasers.

Figure 1

Schematics showing two types of coupled microcavities. Type I utilizes an explicit interferometer setup, while type II does not.

Figure 2

Inverse Vernier effect in two evanescently coupled micro-ring cavities. (a) Trajectories of the lasing thresholds versus the frequencies as the coupling $J$ increases from 0 to 1. The squares and dots mark the uncoupled lasing modes at $J=0$, respectively. The triangles show the coupled lasing modes at $J=1$. Arrows indicate the direction of motion as $J$ increases. $R$ and $0.9R$ are the radius of the larger and smaller cavities, respectively. (b) Lowest thresholds of the lasing modes at $J=0.5$, which are evolved from the uncoupled resonances in the larger ring cavity. Note the increased thresholds especially at the perfectly aligned resonances near $kR=3.3,6.6$. (c) shows the detuning of these uncoupled resonances with the nearest counterparts in the smaller ring cavity. The total loss in the two cavities are ${\kappa }_1={10}^{-4}$ and ${\kappa }_2=5\times {10}^{-4}$ respectively, corresponding to $Q$ factors of $1.5\times {10}^4$ and $3\times {10}^3$. The refractive index is $n=3$.

Figure 3

Inverse bifurcation of the lasing frequencies (a) and bifurcation of the corresponding thresholds (b) of the perfectly aligned modes 1 and 2 near $kR=3.3$ shown in Fig. 2, as the coupling increases from $J=0$. The bifurcations occur near $J=4\times {10}^{-3}$. The solid lines in (c) and (d) show the much weaker $J$ dependencies of the two low-threshold modes on the left of mode 1 in Fig. 2, with the thin one farther away from mode 1. The dots show the analytical approximations given by Eqs. (16) and (17).

Figure 4

Ratio $V$ of light amplitudes inside the two micro-ring cavities for modes 1 and 2 in Figs. 3 and 3. $V$ is defined by Eqs. (10) and (11). The insets in (a) illustrate their different intensity ratios at $J=3.5\times {10}^{-3}$ below the EP at $J_c=4\times {10}^{-3}$ and their identical intensity ratio $\left|V\right|=1$ at $J=7\times {10}^{-3}$ above the EP. (b) The identical phase of $V$ for modes 1 and 2, both below and beyond $J_c$.

Figure 5

Inverse Vernier effect in a slab cavity of length $L$ coupled with a micro-ring cavity of radius $R=L/1.8\pi$ [see the inset in (a)]. (a) Threshold changes for the lowest threshold modes near $kL=20$ at $J=0.5$. They originate from the uncoupled slab resonances, the loss of which is assumed to come from the radiation through two mirrors of reflectivity $r=0.99$ and lower than that in the micro-ring ($\kappa =5\times {10}^{-3}$). The threshold change is inversely correlated with the detuning from the nearest micro-ring resonance [see (b)].

Figure 6

Schematics showing the right-hand side of Eq. (B1).