Lifetime statistics in chaotic dielectric microresonators

We discuss the statistical properties of lifetimes of electromagnetic quasibound states in dielectric microresonators with fully chaotic ray dynamics. Using the example of a resonator of stadium geometry, we find that a recently proposed random-matrix model very well describes the lifetime statistics of long-lived resonances, provided that two effective parameters are appropriately renormalized. This renormalization is linked to the formation of short-lived resonances, a mechanism also known from the fractal Weyl law and the resonance-trapping phenomenon.

Figure 1

Sketch of a stadium-shaped dielectric microresonator with $L=R$.

Figure 2

(Color online) (a) Positions of dimensionless complex resonance frequencies with TM polarization for the stadium microresonator with refractive index $n=5$. The filled squares indicate the quasibound states shown in Fig. 3. (b) Same as (a) but zoomed in onto the long-living resonance frequencies.

Figure 3

Near-field patterns of (a) a long-lived and (b) a short-lived quasibound state with $\Omega =15.71886-i0.00337$ and $\Omega =15.74567-i0.0378$. As in Fig. 2, these states are computed for TM polarization and $n=5$.

Figure 4

(Color online) Distribution of decay rates (in dimensionless units) for resonances in the stadium resonator with refractive index $n=3.3$ (histogram), compared to the prediction of RMT (dashed curve) based on dynamical model (3) with $\tau =1.44{\tau }_0$ and $M=38$. Short-living resonances in the stadium which form bands in the complex-$\Omega$ plane (see Fig. 2) are neglected via a cutoff. Top panel: TM polarization, $\text{Im}\Omega >-0.04$. Bottom panel: TE polarization, $\text{Im}\Omega >-0.1$. The large ticks indicate the value of the average dimensionless decay rate (see Table ).

Figure 5

(Color online) Same as Fig. 4, but for higher refractive index, $n=5$. The cutoffs are $\text{Im}\Omega >-0.015$ (TM) and $\text{Im}\Omega >-0.025$ (TE).

Figure 6

(Color online) Comparison of lifetime distributions in the RMT model for various numbers of channels $M$, with $n=5.0$ (top panel: TM; bottom panel: TE). The insets show the $M$ dependence of the RMT average ${⟨\text{Im}\theta ⟩}_{\text{RMT}}$.