Fig. 1. Surface of section illustrating the different regions of phase space for a closed quadrupole billiard with boundary given by r(ϕ) = R(1 + ∊ cos 2ϕ) for ∊ = 0.072. Real-space ray trajectories corresponding to each region are indicated at right: a) A quasi-periodic, marginally stable orbit. b) A stable four-bounce “diamond” periodic orbit (surrounded by stability “islands” in the SOS) c) A chaotic ray trajectory. Orbits of type (b) have associated with them regular gaussian solutions as we will show below.

Fig. 2. Black background gives the surface of section for the quadrupole at ∊ = 0.17 for which the four small islands correspond to a stable bow-tie shaped orbit (inset). A numerical solution of the Helmholtz equation for this resonator can be projected onto this surface of section via the Husimi transform[28] and is found to have high intensity (in false color scale) precisely on these islands, indicating that this is a mode associated with the bow-tie orbit.

Fig. 3. (a) Vertical lines indicate wavevectors of bound states of the closed quadrupole resonator for ∊ = 0.17; no regular spacings are visible. (b) Spectrum weighted by overlap of the Husimi function of the solution with the bow-tie island as in Fig. 2. Note the emergence of regularly spaced levels with two main spacings Δk_long and Δk_trans . These spacings, indicated by the arrows, are calculated from the length of the bow-tie orbit and the associated Floquet phase (see Section 2.4 below). The color coding corresponds to the four possible symmetry types of the solutions (see Section 4 below). In the inset is a magnified view showing the splitting of quasi-degenerate doublets as discussed in Section 4.3. Note the pairing of the (+, +) and (+, -) symmetry types as predicted in section 4.2. The different symmetry pairs alternate every free spectral range (Δk_long ).

Fig. 4. Illustration of the reduction of the Maxwell equation for an infinite dielectric cylinder to the 2D Helmholtz equation for the TM case (E field parallel to axis) and k _∥ = 0.

Fig. 5. Coordinate system and variables used in the text displayed for the case of a quadrupolar boundary ∂D and the diamond four-bounce PO. A fixed coordinate system (X, Z) is attached to the origin. The “mobile” coordinate systems (x_m ,z_m ) are fixed on segments of the periodic orbit so that their respective z-axes are parallel to the segment, while their origins are set back a distance L_m (or nL_m for transmitted beam axes), so as to account for zeroth order phase accumulation between successive bounce-points. ξ ₁, ξ ₂ are the common local coordinates at each bounce (index m suppressed). Scaled coordinates are denoted by tildes, e.g. x̃_m = √kx_m . The coordinate transformations at each bounce m are given by z_i = L_m + ξ ₁ sin χ_i + ξ ₂ cos χ_i , z_r = L_m + ξ ₁ sin χ_i - ξ ₂ cos χ_i , z_t = nL_m + ξ ₁ sin χ_t + ξ ₂ cos χ_t and x_i = ξ ₁ cos χ_i - ξ ₂ sin χ_i , x_r = ξ ₁ cos χ_i + ξ ₂ sin χ_i , x_t = ξ ₁ cos χ_t - ξ ₂ sin χ_t , where i, r, t refer to the incident, transmitted and reflected solutions.

Fig. 6. Intensity of the TM solution for a bow-tie mode plotted in a false color scale, (a) calculated numerically and (b) from the gaussian optical theory with parameters m = 100, φ = 2.11391, N_μ = 1 and N = 4. Note the excellent agreement of the quantized values for kR (R is the average radius of the quadrupole).

Fig. 7. Schematic indicating a direct tunneling process (black arrow) and a chaos-assisted tunneling process (yellow arrow) which would contribute to splitting of bow-tie doublets.

Fig. 8. The numerically determined splittings of bow-tie doublets for a closed quadrupole resonator with ε = 0.14 (black dots) vs kR; the red line denotes the prediction of Eq. (60) for the average splitting, the blue line an estimate of the splitting based on the “direct” coupling. Note the large enhancement due to chaos-assisted tunneling and the large fluctuations around the mean splitting.