Emission from dielectric cavities in terms of invariant sets of the chaotic ray dynamics

The chaotic ray dynamics inside dielectric cavities is described by the properties of an invariant chaotic saddle. The localization of the far-field emission in specific directions, recently observed in different experiments and wave simulations, is found to be a consequence of the filamentary pattern of the saddle’s unstable manifold. For cavities with mixed phase space, the chaotic saddle is divided in hyperbolic and nonhyperbolic components, related, respectively, to the intermediate exponential $(t<t_c)$ and the asymptotic power-law $(t>t_c)$ decay of the energy inside the cavity. The alignment of the manifolds of the two components of the saddle explains why even if the energy concentration inside the cavity dramatically changes from $t<t_c$ to $t>t_c$, the far-field emission changes only slightly. Simulations in the annular billiard confirm and illustrate the main results.

Figure 1

(Color online) (a) Annular billiard defined by two eccentric circles, with control parameters $q=0.65$ and $\delta =0.3$. Lines indicate one stable periodic orbit (at $\varphi =0$) and one MUPO (7,1) (polygonal form) [42]. Trajectories that do not cross the dotted circumference of radius $q+\delta$ belong to the whispering gallery. (b) Phase space representation of the closed billiard shown in (a), obtained using a Poincaré surface of section at the outer boundary (parametrized by $s=\varphi ∊[0,2\pi ]$). The stable periodic orbit in (a) is at the center of a chain of KAM islands. Black dots mark 8000 iterations of a single chaotic trajectory. The horizontal dashed lines denote $p_c=\pm 1∕n$ for $n=4$, as used in the following figures.

Figure 2

(Color online) Fraction of initial energy inside the billiard shown in Fig. 1 as a function of time. Different curves correspond to the refractive indices $n=\{10,4,2,1.5\}$ (from top to bottom). (a) Linear-log and (b) log-log scale. The dashed lines correspond to an exponential fitting for short times and $t_c$ indicates the transition time to a power-law. Rays were started uniformly distributed in the chaotic component.

Figure 3

(Color online) Energy along $p$ at time $t$ normalized by total energy at time $t$, for $t=0$, 1, 40, 80, 160, and 300 (top to bottom). Refractive index $n=4$ leads to $p_c=0.25$ and $t_c=80$ (see Fig. 2). Rays were started uniformly distributed in the chaotic component (outside the KAM island). Vertical arrows indicate minima of the distribution that systematically appear for all times. The dotted line indicates the reflection coefficient $R_{\mathrm{TM}}$ in Eq. (1), multiplied by 10 for clarity.

Figure 4

(Color online) (a) Closed system’s phase space [magnification of Fig. 1b]. (b) Hyperbolic component of the CS and (c) its unstable manifold for $n=4$. The method of Ref. 27 with $t^{†}=30<t_c=80$ was employed, with rays started away from the regions of regular motion. Notice that the unstable manifold enters $I$ in (c), a sign of rays leaving the cavity (only points with $i>0.01$ are plotted). Due to the time reversible symmetry and the spatial symmetry of the annular billiard, the stable manifold of the CS can be obtained from the unstable one by $(s,p)\mapsto (2\pi -s,p)$.

Figure 5

(Color online) Energy density in the phase space of the annular billiard for $n=4$ $(p_c=0.25)$ at two different times: (a) $t=40=t_c∕2$; (b) $t=160=2t_c$. (c) Magnification of (a) in the leak region; (d) magnification of (b) in the leak region. Rays were started uniformly distributed in the chaotic region.

Figure 6

(Color online) Far field intensity transmitted in the direction $\beta =\varphi -{\theta }_T$ measured from the center of the annular billiard at times $t=t_c∕2=40$ (dashed line) and $t=2t_c=160$ (solid line). Rays were started uniformly distributed in the chaotic region and $n=4$.