Model for chaotic dielectric microresonators

We develop a random-matrix model of two-dimensional dielectric resonators which combines internal wave chaos with the deterministic Fresnel laws for reflection and refraction at the interfaces. The model is used to investigate the statistics of the laser threshold and linewidth (lifetime and Petermann factor of the resonances) when the resonator is filled with an active medium. The laser threshold decreases for increasing refractive index $n$ and is smaller for TM polarization than for TE polarization, but is almost independent of the number of out-coupling modes $N$. The Petermann factor in the linewidth of the longest-living resonance also decreases for increasing $n$ and scales as $\sqrt{N}$, but is less sensitive to polarization. For resonances of intermediate lifetime, the Petermann factor scales linearly with $N$. These qualitative parametric dependencies are consistent with the random-matrix theory of resonators with small openings. However, for a small refractive index where the resonators are very open, the details of the statistics become nonuniversal. This is demonstrated by comparison with a particular dynamical model.

Figure 1

Probability distribution $P\left(\gamma \right)$ of the scaled decay rate ${\gamma }_m=-2\mathrm{Im}{\theta }_m$, obtained from the eigenvalue equation (20). In the upper panels the refractive index is $n=1.5$, while in the lower panels it is $n=3.6$. The left panels are computed in random-matrix theory (RMT), while the right panels are computed in the quantum kicked-rotator model. The matrix dimension is $M=200$ (solid curves) and $M=100$ (dashed curves). The labels TM and TE discriminate the different polarizations. Each curve is based on ${10}^5$ realizations of $U$.

Figure 2

Probability distribution $P\left({\gamma }_0\right)$ of the extremal (smallest) scaled decay rate ${\gamma }_0$. The symbols and parameters are the same as in Fig. 1.

Figure 3

Ensemble average $⟨{\gamma }_0⟩$ of the scaled extremal decay rate ${\gamma }_0$ as a function of refractive index $n$, for TM and TE polarization (squares and circles, respectively). Solid symbols are obtained in random-matrix theory, while open symbols are obtained in the quantum kicked-rotator model. Each data point is based on ${10}^4$ realizations of $U$ with matrix dimension $M=200$.

Figure 4

Probability distribution $P\left(\kappa \right)$ of the scaled Petermann factor $\kappa =K∕M$. The symbols and parameters are the same as in Fig. 1.

Figure 5

Conditional probability distribution $P(\kappa \mid \gamma )$ of scaled Petermann factors $\kappa =K∕M$ at a fixed value $\gamma \approx 2.0$ of the scaled decay rate. Results are shown for both polarizations and two matrix dimensions. The refractive index is $n=1.5$.

Figure 6

Probability distribution $P\left({\tilde{\kappa }}_0\right)$ of the scaled extremal Petermann factor ${\tilde{\kappa }}_0=K_0∕\sqrt{M}$ of the longest-living resonance. The symbols and parameters are the same as in Fig. 1.

Figure 7

Ensemble average $⟨{\tilde{\kappa }}_0⟩$ of the scaled extremal Petermann factor ${\tilde{\kappa }}_0=K_0∕\sqrt{M}$ as a function of refractive index $n$. The symbols and parameters are the same as in Fig. 3.