Counting statistics of chaotic resonances at optical frequencies: Theory and experiments

A deformed dielectric microcavity is used as an experimental platform for the analysis of the statistics of chaotic resonances, in the perspective of testing fractal Weyl laws at optical frequencies. In order to surmount the difficulties that arise from reading strongly overlapping spectra, we exploit the mixed nature of the phase space at hand, and only count the high-$Q$ whispering-gallery modes (WGMs) directly. That enables us to draw statistical information on the more lossy chaotic resonances, coupled to the high-$Q$ regular modes via dynamical tunneling. Three different models [classical, Random-Matrix-Theory (RMT) based, semiclassical] to interpret the experimental data are discussed. On the basis of least-squares analysis, theoretical estimates of Ehrenfest time, and independent measurements, we find that a semiclassically modified RMT-based expression best describes the experiment in all its realizations, particularly when the resonator is coupled to visible light, while RMT alone still works quite well in the infrared. In this work we reexamine and substantially extend the results of a short paper published earlier [L. Wang et al., Phys. Rev. E 93, 040201(R) (2016)].

Figure 1

(a) Schematic representation of the free-space coupled cavity system: the cavity field is excited by visible or infrared laser, while the transmitted signal is detected by an oscilloscope. Key: PLC = polarization controller; FC = fiber coupler; OL = optical lens; PR = photon receiver. (b) A typical transmission spectrum with the high-$Q$ regular modes highlighted.

Figure 2

(a) (inset) Sketch of the deformed microcavity with an inner absorber, characterized by the angle ${\theta }_a$, and (main) corresponding Poincaré surface of section ($\widehat{\varphi }\equiv \varphi /2\pi$) with deformation factor $\eta =11.7\%$. The solid red line indicates the angle of total internal reflection, while the dashed curve is given by an absorption angle ${\theta }_a$ such that $r\simeq 0.85$. Different shades of color indicate loss to the absorber (lighter) and by refraction into air (darker). (b) Poincaré section of the microcavity with $\eta =4.2\%$. The dashed curve is given by an absorption angle ${\theta }_a$ such that $r\simeq 0.77$.

Figure 3

(a), (c), (e) Number of chaotic states $n_{\gamma }$ vs the rescaled absorber-to-cavity ratio $\xi$, obtained by classical, RMT and Weyl law respectively. (b), (d), (f) The corresponding expectations for $|a_{\omega }{|}^2$. The red vertical line corresponds to the critical angle, $sin{\theta }_c\simeq 0.69$. Here $sin{\theta }_{\mathrm{th}}=0.6$. (c)–(f) adapted from Ref. [38].

Figure 4

Survival probability in the chaotic region (log scale). Points: average survival probability $P\left(t\right)$ of a ray in the microcavity vs $t$ (in units of Poincaré time) at: (a) $\eta =11.7\%,\xi =0.13$, from ${10}^6$ randomly started trajectories. Line: $P\left(t\right)\propto \exp (-t/{\tau }_d), {\tau }_d=6$ (from Ref. [38]); (b) $\eta =4.2\%,\xi =0.1, {\tau }_d=14$. Insets: the long-time simulation showing algebraic decay.

Figure 5

(a) Image of the microcavity obtained by scanning electron microscopy (SEM). (b) SEM cross-section image of the toroidal part, taken at an angle of ${56}^{\circ }$ with the horizontal direction. (c) Finite-element method simulation of a fundamental TE mode (color scale in arbitrary units). The white solid curve is the boundary of the cavity. (d) Finite-element method simulation of the light propagating inside the 2-$\mu \mathrm{m}$-thick silica waveguide bonding with a thick silicon layer (color scale in arbitrary units). (e) Fraction of remaining energy in silica vs the propagating distance.

Figure 6

Normalized transmission and top-view optical images of the cavity with $r\simeq 0.81$ [(a) and (e)], 0.77 [(b) and (f)], 0.70 [(c) and (g)], 0.64 [(d) and (h)]. Inset of (a) shows background noise. Insets of (b)–(d) show the high-$Q$ modes. Reflection of the silica-to-silicon interface results in a brighter color for the silicon pillar in the optical image (boundary shown by red dashed curves). Scale bar is 50 $\mu \mathrm{m}$. Adapted from Ref. [38].

Figure 7

Dots: number of high-$Q$ regular modes ($n_{\omega }$) observed in the transmission spectra of the microcavity coupled to infrared light, as a function of rescaled absorber-to-cavity ratio $\xi$. Blue dashed curve: best fit of the classical prediction (25) [together with Eq. (29)]. Left: $\eta =11.7\%$; right: $\eta =6\%$. Here $sin{\theta }_c\simeq 0.69$ and $sin{\theta }_{\mathrm{th}}=0.6$.

Figure 8

Number of high-$Q$ regular modes ($n_{\mathrm{reg}}$) observed in the transmission spectra of the microcavity (dots), as a function of rescaled absorber-to-cavity ratio $\xi$. Blue dashed and red solid curves are respectively RMT- and semiclassical prediction best fits. (a), (b), (c) Infrared light; (d), (e) visible light. Here $sin{\theta }_c\simeq 0.69$ and $sin{\theta }_{\mathrm{th}}=0.6$. Insets: the area where the two curves differ most.

Figure 9

Statistics of the quality factor for the chaotic rays in the deformed microcavity, from a ray-dynamics simulation.

Figure 10

Dots: average linewidth of the excited regular modes vs. their number from a free-space coupling experiment (cf. Fig. 6). Each data point represents one experiment with a different size of the silicon pillar. Solid line: best-fit curve of Eq. (41). (a) $\eta =4.2\%, \lambda =635$ nm, with fitting parameters ${\gamma }_{\mathrm{reg}}=419\mathrm{MHz}$, and $\kappa =78$. (b) $\eta =6\%, \lambda =635$ nm, ${\gamma }_{\mathrm{reg}}=378\mathrm{MHz}$, and $\kappa =120.5$.

Figure 11

Normalized transmission and top-view images of the cavity coupled by fiber taper [(a) and (b); notice the fiber beside the cavity] and free-space laser beam [(c) and (d)]. The resonances in (a) have $Q$ factors typically of the order of ${10}^6–{10}^7$. Here the absorber-to-cavity ratio is $r\simeq 0.83$.