Folded chaotic whispering-gallery modes in nonconvex, waveguide-coupled planar optical microresonators

Chaotic whispering-gallery modes have significance both for optical applications and for our understanding of the interplay between wave phenomena and the classical ray limit in the presence of chaotic dynamics and openness. In strongly nonconvex geometries, a theorem by Mather rules out the existence of invariant curves in phase space corresponding to rays circulating in whispering-gallery patterns, so that no corresponding modes of this type are expected. Here, we discuss numerical computations of the electromagnetic fields in planar dielectric cavities that are strongly nonconvex because they are coupled to waveguides. We find a family of special states which retains many features of the chaotic whispering-gallery modes known from convex shapes: An intensity pattern corresponding to near-grazing incidence along extended parts of the boundary, and comparatively high-cavity $Q$ factors. The modes are folded into a figure-eight pattern, so overlap with the boundary is reduced in the region of self-intersection. The modes combine the phenomenology of chaotic whispering-gallery modes with an important technological advantage: The ability to directly attach waveguides without spoiling the $Q$ factor of the folded mode. Using both a boundary-integral method and the finite-difference time-domain technique, we explore the dependence of the phenomenon on wavelength in relation to cavity size, refractive-index contrast to the surrounding medium, and the degree of shape deformation. A feature that distinguishes folded from regular whispering-gallery modes is that a given shape will support high-$Q$ folded chaotic whispering-gallery modes only in certain wavelength windows.

Figure 1

Spatial structure of a folded chaotic whispering-gallery mode (insets), and $Q$ factor of the mode versus deformation parameter $\epsilon$ at refractive index $n=2.4$. The resonator deformation is defined in Eq. (2); attached waveguides of finite lengths are shown horizontal; their lengths are slightly unequal to remove reflection symmetry. False color represents field intensity. The real part of $k$ (not shown) decreases approximately linearly from 14 to 12.5 as $\epsilon$ increases.

Figure 2

(a)–(c) Three self-intersecting unstable periodic orbits with angles of incidence $sin\chi \ge 0.6$ in the billiard shape given by Eq. (2) with $\epsilon =0.43$. The lengths $L_1$, $L_2$ of the orbits are slightly different. The same topology of orbits persists for a wide range of deformations. Panel (c) is obtained from (b) by adjusting the initial conditions to generate eight instead of three reflections in the top half. For the rectangular unstable periodic orbit in (d), the increasing proximity to the corners of the waveguide aperture is illustrated for increasing $\epsilon$. (e) A ray trajectory launched in the cavity (green dot) and escaping through the waveguide opening (red dot). It retains the folded character of the orbits in (a) and (b) for a short time.

Figure 3

Poincaré surface of section with the curve parameter $\varphi$ as position and $sin\chi$ as momentum variable. Here, $\chi$ is the angle between incident rays and the surface normal, so that $sin\chi =1$ corresponds to grazing incidence. Ray trajectories show up as point clouds in the chaotic sea, or as one-dimensional lines for regular motion. Shown as insets are a stable bouncing-ball orbit (left) and an unstable periodic orbit (right). Arrows point to the corresponding phase-space locations; in particular, the six black dots indicate the location of the six bounce points for the periodic orbit. Shaded boxes indicate the intervals of $\varphi$ which describe the attached waveguides. Rays entering these regions will escape the resonator. The thin horizontal lines mark the critical angle for total internal reflection $|sin{\chi }_c|=\frac{1}{n}$ ($n=2$ here). Here and in all subsequent results, $n$ can be viewed as the interior refractive index while the exterior refractive index is unity.

Figure 4

Small-cavity limit of the folded chaotic WGM, showing only low-$Q$ factors as the transverse width of the mode is comparable to the cavity size. Below $k\approx 7$, the wavelength is too long to observe the WGM-like concentration of intensity near the top and bottom of the cavity. The horizontal waveguide stubs are attached in a way that smoothly matches the shape given by Eq. (2) for the vertical lobes. The refractive index is $n=2.4$ inside and $n=1$ outside, and the vertical lobes are described by Eq. (2) with $\epsilon =0.444$. In this and the following plots, one can discern even and odd parity with respect to reflections at the horizontal axis. Although we intentionally break reflection symmetry across the vertical axis by making the horizontal waveguide lengths unequal, the modes still show approximate antinodes (left) or nodal lines (right) along the vertical axis. This is because the cavity supporting most of the intensity is still left-right symmetric.

Figure 5

Wave-number sweeps of the $Q$ factor for different refractive indices $n$, discarding low-$Q$ modes with $\kappa \ge 0.05$ (for $n=1.6$, only modes with $\kappa \ge 0.2$ are shown). The deformation is $\epsilon =0.444$. The appearance of distinct $Q$-factor peaks as a function of quasibound-state wave number is most pronounced at the largest refractive index, $n=2.4$, and becomes nearly unobservable at $n=1.6$ (therefore, data for $n=1.6$ were not collected beyond $nk\approx 106.5$). The peaks that do remain observable are approximately at the same values of $nk$ for all $n$.

Figure 6

$Q$-factor scan versus interior wave number at deformation $\epsilon =0.52$ and refractive index $n=2.4$, showing modes with decay rate $\kappa <0.05$.

Figure 7

Field intensities for the modes corresponding to the first four $Q$-factor peaks in Fig. 5 for $n=2.4$, $\epsilon =0.444$. The $Q$ factors are (a) 2500, (b) 2100, (c) 2400, and (d) 4500.

Figure 8

Field intensities for folded WGMs at different deformations: (a) $\epsilon =0.52$, (b) $\epsilon =0.6$, (c) $\epsilon =0.7$. The refractive index is $n=2.4$. In (c), a smaller wave number is chosen, whereas (a) and (b) have comparable wave numbers. The narrowing of the horizontal stubs is a result of the requirement that its tangents must match the curve described by Eq. (2) at the corners.

Figure 9

(a) Folded chaotic whispering-gallery mode as obtained in a finite-difference time-domain computation for the same deformation $\epsilon =0.444$ and refractive index $n=2.4$ shown in Fig. 5. The wave number corresponds to the top of the first $Q$-factor peak, $k\approx 13.6$ ($Q=2612$). Waveguide and resonator structure are underlaid as green shading, and the false-color scale represents the electric field. (b) Same deformation and refractive index, but approximately 10 times shorter wavelength, $k\approx 116.0$ and $Q=9898$. (b) Discussed in Sec. 4.

Figure 10

Two different modes with thinner waveguides, obtained by introducing a vertical offset between the two lobes of Eq. (2) and adjusting $\epsilon$ to match them smoothly to the horizontal waveguides. The refractive index is $n=2.4$. In (a) $k\approx 13.3$ and $Q\approx 1580$, in (b) $k\approx 22.4$ and $Q\approx 2550$.

Figure 11

(a) Stable and unstable manifolds around an unstable periodic orbit. The zoomed-in region shown in (a) corresponds to the top left quadrant of the Husimi projection in (b). The real-space wave intensity of the mode in (b) is shown in the inset to (a). It is the same mode as shown in Fig. 7. Shaded regions in (b) mark the waveguides, which were treated as escape windows in (a). The thin horizontal lines mark $|sin{\chi }_c|=\frac{1}{2.4}$. Refractive escape is not considered in the ray simulation for (a), to get a more complete picture of the manifolds. Note the correspondence in shape between the phase-space structure and the areas of large Husimi weight.

Figure 12

Comparison of Husimi plots for the same states shown in Fig. 8. Only the positive-$sin\chi$ half of the phase space is displayed (shaded regions and horizontal lines as in Fig. 11). The low wave number ($k\approx 17.2$) in (c) leads to lower phase-space resolution than in (a) and (b) where $k\approx 31.4$ and $k\approx 31.2$.