Directional emission through dynamical tunneling in a deformed microcavity

We study directional emission through dynamical tunneling in a two-dimensional deformed microcavity. The tunneling emissions are characterized by free-space emergence with a finite distance $\delta$ from the cavity boundary. As the cavity deformation increases, it is shown numerically that one of two main outputs becomes dominant with a decreasing $\delta$ value. This asymmetrical enhancement of tunneling outputs can be understood by the influence of asymmetric short-time ray dynamics near the critical line on the dynamical tunneling process. In addition, the change of $\delta$ with deformation can be explained by the interference of tangential wave components brought by the dynamical tunneling.

Figure 1

(a) Schematic diagram of a refractive ray and a tunneling ray in a circular microcavity with a radius $R=1$. The tunneling ray appears with a positive $\delta$ value. (b) Tunneling modes (blue dots between the solid lines) and refractive modes (red dots). The red, upper line is $\text{Re}[\mathit{kR}]=m$, and the blue, lower line is $\text{Re}[\mathit{kR}]=m/n$. The modes can be also classified by the radial quantum number $l$, $l=1,2,\dots ,$ corresponding to each row from the bottom just above the blue, lower line.

Figure 2

(a) $M^{\mathrm{WG}}(1,60)$ ($\text{Re}[\mathit{kR}]\simeq 46.66$) and its emission Husimi function $H_E(x,y)$ probing waves propagating in the positive $x$ direction in the range of $x>R$. (b) $M^{\mathrm{WG}}(3,60)$ ($\text{Re}[\mathit{kR}]\simeq 54.30$) and its $H_E(x,y)$ in the range of $x>R$. (c) The red dots are the peak positions, $y_m=R+\delta$ in $H_E(x=2R,y)$ of WGMs. The open circles are the semiclassical expectations explained in the text. Here we take $R=1$. From the top, each row of dots belongs to $l=1,2,\dots$ mode groups, respectively.

Figure 3

(a) Rotated projection surface to get the emission Husimi function $H_E(\theta ,y)$ for a deformed microcavity as a function of $\theta$ and $y$. (b) The change of phase-space structure (PSOS) as a function of increasing deformation parameter $\varepsilon$. The orange line denotes the critical line for total internal reflection, $p_c=\sin {\chi }_c=1/n$.

Figure 4

Emission Husimi functions $H_E(\theta ,y)$ for quadrupole microcavities. The insets show directions of tunneling emissions with optical modes. (a) $\varepsilon =0.01$. (b) $\varepsilon =0.03$. (c) $\varepsilon =0.07$. (d) $\varepsilon =0.09$. For $\varepsilon =0.05$, see Figs. 5a and 5b. The white lines denote images of cavity boundary edges.

Figure 5

(a) A high-$Q$ mode when $\varepsilon =0.05$. (b) The emission Husimi function $H_E(\theta ,y)$ of the high-$Q$ mode shown in (a), showing an asymmetric tunneling emission, with tunneling output $A$ being stronger than $B$. (c) Boundary Husimi function $H(s,p)$ of the high-$Q$ mode, superimposed on the PSOS of ray dynamics. (d) Magnified view on a logarithmic scale of $H(s,p)$ around the critical line.

Figure 6

(a) Increasing asymmetry of tunneling emission with deformation (black dots). The red open circles show the asymmetry of two peaks in the Husimi tail profile $H(s,p_c)$. (b) Boundary positions of tunneling emission (solid squares for $A$ and open squares for $B$) compared with the maximum positions in the Husimi tail $H(s,p_c)$ (solid circles) and the peak positions of $H(s,p)$ (open circles). (c,d) The change after four bounces on the initial point sets [rectangular stripes in (c)] prepared on the PSOS at $\varepsilon =0.05$. The critical line is indicated by a dashed line.

Figure 7

(a) Decreasing $\delta$ of tunneling output $A$ with $\varepsilon$. The insets are normalized $H(s,p_c)$’s. (b) Tangentially emitting Gaussian-beam-like waves for $M^{\mathrm{WG}}(1,60)$. (c) Various window functions. (d) Calculated $H_E(x_0=1.5,y)$ using the window functions in (c), where $\delta$ is the distance between peak position and the cavity boundary, $y=1$.