Dynamical tunneling-assisted coupling of high-$Q$ deformed microcavities using a free-space beam

We investigate the efficient free-space excitation of high-$Q$ resonance modes in deformed microcavities via dynamical tunneling-assisted coupling. A quantum scattering theory is employed to study the free-space transmission properties, and it is found that the transmission includes the contribution from (1) the off-resonance background and (2) the on-resonance modulation, corresponding to the absence and presence of high-Q modes, respectively. The theory predicts asymmetric Fano-like resonances around high-$Q$ modes in background transmission spectra, which are in good agreement with our recent experimental results. Dynamical tunneling across Kolmogorov-Arnold-Moser tori, which plays an essential role in the Fano-like resonance, is further studied. This efficient free-space coupling holds potential advantages to simplify experimental conditions and excite high-$Q$ modes in higher-index-material microcavities.

Figure 1

Scanning electron microscope (SEM) view of a deformed silica microtoroid cavity. Here $R$ and $\varphi$ are the polar coordinates in the cavity plane. The red arrow denotes the laser beam. Inset: False color illustration of a resonant mode distribution obtained with wave simulation. The two black arrows denote the dominantly directional emission toward the 180${}^{\circ }$ far-field direction. The strength outside the cavity is magnified for a clear view.

Figure 2

(a) A typical PSOS of the deformed microcavity. The red solid line denotes the critical line defined as $\sin \chi =1/n$. The orange dotted line stands for a KAM torus. (b), (c) Normalized Husimi projection of the resonant mode and the excitation state, respectively.

Figure 3

(a) The schema of the scattering process described by Eq. (17). (b) Stimulated transmission spectrum of the free-space excitation process obtained by the boundary element method. (c) Experimental spectrum (black) and the fitted oscillations (red).

Figure 4

Calculated transmission spectra with $\kappa /2\pi ,\gamma /2\pi =0.003$ GHz and $t/r=0.2$. The phase shift between the two amplitudes varies from 0 to $7\pi /4$.

Figure 5

Experimental transmission spectra. The on-resonance line shapes are (a) EIT-like and (b) asymmetric Fano resonance. Insets: Enlarged views of the on-resonance transmission.

Figure 6

Calculated transmission spectra with ${\gamma }_t/2\pi =0.006$ GHz and $t/r=0.2$, and with $K$ varying from 0 to 60, for (a) EIT-like line shape and (b) Lorentz-like line shape. The background is set to the same level for each graph.

Figure 7

(a) Effective potential $V_{\mathrm{eff}}$ against $\sin \chi$. The solid and dashed curves correspond to the cases of silica ($n=1.45$) and GaAs ($n=3.3$) microcavities, respectively. The red triangles mark the potential at the critical angle, where the potentials are both $\hslash \omega$. (b), (c) Husimi projections of the excitation state inside the cavity at nonresonant frequency with the deformation parameter $\eta$ of the cavity setting as $0.5$ and $1.5$, respectively. These two figures are plotted in the same scale. Orange dotted curves and red solid lines denote KAM tori and critical-refraction lines.