Rotation-induced mode coupling in open wavelength-scale microcavities

We study the interplay between rotation and openness for mode coupling in wavelength-scale microcavities. In cavities deformed from a circular disk, the decay rates of a quasidegenerate pair of resonances may cross or anticross with increasing rotation speed. The standing-wave resonances evolve to traveling-wave resonances at high rotation speed, however both the clockwise (CW) and counterclockwise (CCW) traveling-wave resonances can have a lower cavity decay rate, contrary to the intuitive expectation from the rotation-dependent effective index. With increasing rotation speed, a phase locking between the CW and CCW wave components in a resonance takes place. These phenomena result from the rotation-induced mode coupling, which is strongly influenced by the openness of the microcavity. The possibility of a nonmonotonic Sagnac effect is also discussed.

Figure 1

Splitting of the real (a) and imaginary (b) parts of the complex frequencies for a pair of $\left|m\right|=8$ resonances at $k_{0}R\simeq 5.3923-0.0114i$ in a circular microdisk of refractive index $n=2$, plotted as a function of the normalized rotation speed. Symbols show numerical solutions of Eq. (3) and solid lines give the perturbation result (4) with ${\eta }_8\simeq 1.2478+0.0665i$. (c) Enhancement of the Sagnac effect in open circular microdisk cavities given by $\mathrm{Re}\left[{\eta }_m\right]$. Connected dots and diamonds are for $n=2,3$, respectively.

Figure 2

Normalized expansion coefficients ${\alpha }_m$ defined in Eq. (8) for a pair of CW-dominated (a) and CCW-dominated (b) resonances at $\overline{\Omega }={10}^{-3}$, which have a dominant angular momentum $m=-8,8$, respectively. The cavity is a limaçon with $\epsilon =0.41$ and $n=2$, shifted along $\theta ={180}^{\circ }$ by $R\epsilon$ such that $\rho \left(0\right)=\rho \left({180}^{\circ }\right)$. The insets show the intracavity intensity patterns of these resonances, which are almost chiral symmetric. They cannot be distinguished by eye from those of the CW and CCW waves in the corresponding standing-wave resonances at rest (not shown). (c) The angular dependence of their external field intensities at $r=3R$. Note that they are not mirror images of each other about the symmetry axis of the cavity (dash-dotted line), especially near $\theta =0\left({360}^{\circ }\right),{180}^{\circ }$. (d) Same as (c) but for the CW waves (red) and CCW waves (black) in the corresponding standing-wave resonances at rest. They are mirror images of each other about the symmetry axis of the cavity.

Figure 3

Mode coupling of Pair 1 shown in Fig. 2. (a) and (b) Real and imaginary parts of the complex frequencies of the resonances that evolve into the CW- (red solid) and CCW-dominated (black dashed) ones. The angular dependence of their external field intensities is shown in (d). The symbols in (c) show their splitting $(k_{\mathrm{CCW}}-k_{\mathrm{CW}})R$ (real part: diamonds; imaginary part: circles) in the logarithmic scale (${log}_{10}$), where the solid lines are given by the coupled-mode theory (16) with $g=4.99+0.30i$ and the dashed lines show the complex frequency splitting of the $\left|m\right|=8$ resonances in a circular cavity of the same radius.

Figure 4

The same as Fig. 3 but for Pair 2 with a dominant $\left|m\right|=10$ [(a) and (b)] and Pair 3 with a dominant $\left|m\right|=11$ [(c) and (d)]. Solid lines are given by the coupled-mode theory (16) with $g=5.92-0.15i,5.90-0.071i$, respectively. Dashed lines in (a) and (c) show the splitting of the real part of the frequencies for the corresponding resonances in a circular cavity of the same radius. In (b) the cavity decay rates anticross and $|\mathrm{Im}\left[k_{\mathrm{CCW}}\right]|>|\mathrm{Im}\left[k_{\mathrm{CW}}\right]|$. In (d) they cross and $|\mathrm{Im}\left[k_{\mathrm{CCW}}\right]|>|\mathrm{Im}\left[k_{\mathrm{CW}}\right]|$ for $\overline{\Omega }\gtrsim {10}^{-6.2}$.

Figure 5

(a) and (b) Crossing of the real parts of a pair of quasidegenerate resonances, constructed using Eq. (15) and with $g=6-1i$, $\Delta k_{0}R=(0.1+6i)\times {10}^{-5}$, $k_0^{-}R=5-0.01i$. (c) and (d) A pair of quasidegenerate resonances reach an exceptional point at $\overline{\Omega }={\overline{\Omega }}_c={10}^{-5}$, constructed using Eq. (15). The parameters are the same as in (a) and (b) except for $g=6-0.1i$.

Figure 6

Relative phase (a) and amplitude (b) of the dominant angular components $m=\pm 8$ in Pair 1 as a function of $\Omega$. Red solid and black dashed lines represent the resonances that evolve into CW- or CCW-dominated ones, respectively. Parts (c) and (d) show the same for the dominant angular components $m=\pm 10$ in Pair 2. Dash-dotted lines in (a) and (c) are given by $\Delta \phi =1.11,-0.471$, respectively, which are given by Eq. (22) from the coupled-mode theory.