Marginally Unstable Periodic Orbits in Semiclassical Mushroom Billiards

Optical mushroom-shaped billiards offer a unique opportunity to isolate and study semiclassical modes concentrated on nondispersive, marginally unstable periodic orbits. Here we show that the openness of the cavity to external electromagnetic fields leads to unanticipated consequences for the far-field radiation pattern, including directional emission. This is mediated by interactions of marginally unstable periodic orbits with chaotic modes. We also show that the semiclassical modes are robust against perturbations to the shape of the cavity, despite the lack of structural stability of the corresponding classical orbits.

Figure 1

Bottom corner of the hat of the mushroom cavity. The ray represents the trajectory of an orbit with reflection angle $\theta =90°-\alpha$ that hits the undercarriage with angle ${\alpha }^{\prime }\le \alpha$. When close to the corner, the curvature can be ignored and ${\alpha }^{\prime }=\alpha$. The inset shows a period-4 MUPO and the entire mushroom cavity with the parameters $R$, $r$ and $h$ measured from the center of the diameter of the hat. This point is also the origin of the cylindrical and Cartesian coordinates used throughout this Letter, in which $z$ represents the axis perpendicular to the plane of the billiard.

Figure 2

Spatial intensity $|H_z{|}^2$ for the period-4 MUPO mode with $\lambda =41.0\mathrm{nm}$ and $Q=13000$ in a polymer microcavity. The intensity was averaged over one oscillation period.

Figure 3

(a) Far-field radiation pattern $|H_z^f({\varphi }_j){|}^2$ for the high $Q$ period-4 MUPO mode shown in Fig. 2. (b) Magnification of $|H_z^f({\varphi }_j){|}^2$ for $0°\le \varphi \le 30°$. The far-field pattern is strongly bidirectional. The angular resolution is $\Delta \varphi =0.008°$.

Figure 4

(a) Calculated ray diagram indicating the point of incidence on the undercarriage $d$, reentry angle $\beta$, and emission angle $\varphi$ in a mushroom cavity. (b) Values of ${\alpha }^{\prime }$ and $d$ that result in emission angles $8.5°\le \varphi \le 11°$ (shaded area). The corresponding values of $\beta$ (black curve) agree with the results shown in Fig. 2.

Figure 5

(a) Fourier transformation for the period-4 MUPO modes excited by $\lambda =550.92\pm 0.5\mathrm{nm}$ in a semiconductor microcavity. There are two eigenmodes within this excitation range, at wavelengths $\lambda =550.61\mathrm{nm}$ and $\lambda =550.91\mathrm{nm}$. The $Q$ values for these modes are 39 000 and 140 000, respectively. (b),(c) Far-field pattern $|H_z^f({\varphi }_j){|}^2$ corresponding to (b) $\lambda =550.61\mathrm{nm}$ and (c) $\lambda =550.91\mathrm{nm}$.

Figure 6

(a),(b) Spatial intensity $|H_z{|}^2$ for the period-4 MUPO modes at (a) $\lambda =550.61\mathrm{nm}$, $Q=39000$ and (b) $\lambda =550.91\mathrm{nm}$, $Q=140000$ in a semiconductor microcavity. The intensity was averaged over one oscillation period. Different times were selected to yield the same maximum intensity for both plots.