Quality factors and dynamical tunneling in annular microcavities

The key characteristic of an optical mode in a microcavity is its quality factor describing the optical losses. The numerical computation of this quantity can be very demanding for present-day devices. Here we show for a certain class of whispering-gallery cavities that the quality factor is related to dynamical tunneling, a phenomenon studied in the field of quantum chaos. We extend a recently developed approach for determining dynamical tunneling rates to open cavities. This allows us to derive an analytical formula for the quality factor which is in very good agreement with full solutions of Maxwell’s equations.

Figure 1

(Color online) Inverse quality factors $1/Q$ for the annular microcavity with refractive index $n_{\text{eff}}=2.0$. Shown is the theoretical prediction (solid line), which is the sum of the direct tunneling contribution (dotted line) and the dynamical tunneling contribution [dashed line, Eq. (15)] and numerical data (dots) for angular quantum number $m=19$ and radial quantum number $n=1$ vs the hole position $d$. The insets show the resonant state at $d=0.2475$ and $d=0.33$.

Figure 2

(Color online) (a) Annular cavity with $R=1$, $R_2=0.22$, and $d=0.33$. (b) Corresponding Poincaré section $\left(s,p=\text{sin}\chi \right)$ for $n_{\text{eff}}=2.0$, where $s$ is the arclength along the boundary and $\chi$ is the angle of reflection. Between the critical lines (dashed) with $\left|p_c\right|=1/n_{\text{eff}}$ is the leaky region, where the condition for total internal reflection is not fulfilled. (c) Circular cavity and (d) its completely regular phase space.

Figure 3

(Color online) Inverse quality factors $1/Q$ for the annular microcavity with $n_{\text{eff}}=2.0$. Shown is the theoretical prediction (solid line) which is the sum of the direct tunneling contribution (dotted line) and the dynamical tunneling contribution [dashed line, Eq. (15)], and numerical data (dots) for $m=7,\dots ,21$ and $n=1$ at $d=0.33$. The insets exemplarily show the resonant states of angular quantum number $m=7$ (left) and $m=21$ (right).

Figure 4

(Color online) Inverse quality factor $1/Q$ for the annular microcavity. Shown is the theoretical prediction (solid line) which is the sum of the direct tunneling contribution (dotted line) and the dynamical tunneling contribution (dashed line), and numerical data (dots) for $m=7,\dots ,21$, $n=1$ at $d=0.33$ and $n_{\text{eff}}=2.3$. The inset shows a Poincaré section of the classical phase space, where the critical line $p=p_c$ is marked (dashed line).

Figure 5

(Color online) Husimi functions (gray scale and contour lines) of the modes with angular quantum number (a) $m=14$ and (b) $m=18$ superimposed onto part of the Poincaré section of the classical phase space, where the critical line $p=p_c$ is marked (dashed line). The radial quantum number is $n=1$ and the refractive index is $n_{\text{eff}}=2.3$ as in Fig. 4.