Geometric wave engineering of ring-localized states in open pseudo-hyperbolic cavities

Open macroscopic cavities typically exhibit transient chaos and chaotic escape 1, 2, limiting local intensity and precipitating thermal breakdown in high-power optics and plasma confinement architectures. Here we investigate a geometry-driven route to suppress this escape in a class of non-axially generated pseudo-hyperbolic resonators 3. By rotating a canonical hyperbola around an offset axis, we obtain an open three-dimensional cavity with a spatially structured radius function and a pair of ring-shaped focal zones above the equatorial gap. Throughout the manuscript, all lengths are expressed in dimensionless units normalized to a reference scale ξ; physical dimensionalization is recovered by fixing the product k₀ξ at the operating wavelength. For the optimal topology identified in our parameter scan (R = 20. 0, a = 0. 05, b = 0. 50), non-sequential stochastic ray dynamics yield a global energy retention of 88. 9% and a local energy concentration of 15. 22 ± 0. 25% in the gap region, where the reported uncertainty is dominated by systematic effects rather than Monte Carlo statistics. To interpret this localization beyond the geometric-optics limit, we derive an effective one-dimensional Helmholtz formalism 4, 5 in the adiabatic domains of the cavity, under Dirichlet boundary conditions corresponding to TM-polarized modes in a perfectly conducting cavity. The leading-order geometry-induced potential scales as Vₑff ∝ 1/r (x) ² 4, providing a steeply rising barrier in the horn regions and a low-potential equatorial trapping zone. The reduced wave model predicts a one-dimensional confinement fraction of ~14. 5%, of the same order as the stochastic ray result; the two measures probe different observables and their numerical proximity is treated here as qualitative consistency rather than as a quantitative match. Within the limits of the reduced wave model and the macroscopic-ray approximation, these findings identify a geometry-controlled localization mechanism in an open empty cavity and motivate further investigation by full-wave electromagnetic simulation and experiment.