Геометрическая волновая инженерия кольцевых локализованных состояний в открытых псевдогиперболических полостях

Abstract

Open macroscopic cavities typically exhibit transient chaos and chaotic escape [1–3], 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 [4,5]. 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. 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. To interpret this localization beyond the geometric-optics limit, we derive an effective one-dimensional Helmholtz formalism [6,7] in the adiabatic domains of the cavity. The resulting geometry-induced potential scales as V_eff ∝ 1/r(x)² [6], providing a steeply diverging barrier in the horn regions and a low-potential equatorial trapping zone. The reduced wave model predicts a confinement fraction of ~14.5%, in strong agreement with the stochastic ray result. These findings establish a geometry-driven mechanism for ring localization in open cavities and suggest a route toward controlled wave trapping in high-power resonant platforms.

1. Introduction

The controlled localization of wave energy in open systems is a longstanding problem in wave physics, nonlinear optics, resonator engineering and plasma-wave control. In open macroscopic structures, where λ ≪ R, the underlying ray dynamics are commonly governed by transient chaos and chaotic scattering [1–3]. In the absence of specifically engineered geometric constraints, trajectories explore the available phase space and eventually escape [15,16], thereby limiting both the residence time and the achievable local intensity.

Most established localization strategies rely on either material structuring or nearly closed resonant boundaries. Examples include whispering-gallery resonators [11,12], photonic band-gap systems [13], and topological photonic platforms [13,14]. Although highly successful within their intended regimes, these approaches generally depend on material dispersion [11], nanoscale fabrication, or boundary conditions that strongly suppress radiative openness. By contrast, geometry-driven localization in open empty volumes remains comparatively underexplored [4,5,6], especially in macroscopic cavities where curvature, aperture, and escape topology interact nontrivially.

In this work we investigate a family of resonators generated by non-axial rotation of a canonical hyperbola around an offset axis. This construction produces an open pseudo-hyperbolic cavity with finite longitudinal extent, singular closure points, and two ring-shaped focal zones above the equatorial gap. We refer to this design framework as Geometric Wave Engineering (GWE). Our central result is that this geometry can strongly suppress chaotic escape and produce pronounced ring-localized states near the equatorial region. We demonstrate this using two complementary descriptions: non-sequential stochastic ray tracing in the macroscopic limit [17], and an effective reduced wave model derived from the scalar Helmholtz equation [20,21]. Together, these results indicate that carefully designed macroscopic geometry alone can sustain long-lived localized states in an otherwise open scattering environment.

2. Geometry of the second-order pseudo-hyperboloid

The generating profile is a canonical hyperbola in the Oxy plane,

x²/a² – y²/b² = 1,

which may be parameterized as

y(x) = b√ (x²/a² – 1), |x| ≥ a.

The interval (–a, a) is excluded from the generating curve and physically forms the equatorial gap of width 2a.

The defining geometric feature of the present cavity is that the profile is not rotated about its own symmetry axis, but about the offset line

y = R.

The local rotation radius is therefore

r(x) = |R – y(x)| = |R – b√ (x²/a² – 1) |.

Because the generating hyperbola diverges asymptotically, the rotated structure closes only up to a finite longitudinal coordinate x = ±L, where the profile intersects the rotation axis and the local radius vanishes:

L = a√ [1 + (R/b) ²].

Thus, the physical domain of the cavity is

x ∈ [–L, –a] ∪ [a, L].

The geometric foci of the generating hyperbola are located at

x = ±c, c = √ (a² + b²).

Under non-axial rotation these two focal points generate two three-dimensional focal rings of constant radius

r_foc = R.

This nontrivial mapping from two-dimensional focal points to three-dimensional focal rings is a central geometric ingredient of the GWE construction.

For the optimal topology investigated here,

R = 20.0, a = 0.05, b = 0.50,

we obtain

k = b/a = 10, c ≈ 0.5025, L ≈ 2.0006.

These values indicate a cavity with a broad equatorial opening, steep hyperbolic tapering, and long horn regions terminating at singular closure points. The geometry is summarized in Fig. 1, while the corresponding derived parameters are listed in Table 2.

Table 1 | Cross-verification of the optimal resonator topology

Rank	Equatorial radius R	Half-gap a	Shape parameter b	Global energy retention (%)	Local gap density (%)
9	20.0	0.05	0.50	88.9	15.22

Table 2 | Derived geometric and wave parameters of the optimal topology

Parameter	Symbol	Value
Equatorial radius	R	20.0
Half-gap width	a	0.05
Shape parameter	b	0.50
Asymptotic ratio	k = b/a	10.0
Focal coordinate	c = √(a² + b²)	0.5025
Closure coordinate	L = a√ [1 + (R/b)²]	2.0006
Focal-ring radius	r<sub>foc</sub>	20.0
Fundamental Bessel root	μ<sub>0,1</sub>	2.4048
Minimum effective potential	V<sub>min</sub> = (μ<sub>0,1</sub>/R)²	0.01446

Fig. 1 | Geometry of the second-order pseudo-hyperboloid.

3. Stochastic ray dynamics and suppression of chaotic escape

To probe the macroscopic limit, we analyze the cavity by non-sequential stochastic ray tracing with specular reflections at the cavity boundary [17]. A key geometric feature of the pseudo-hyperboloid is the divergence of the boundary slope at the edges of the equatorial gap:

|r’(x)| → ∞ as |x| → a⁺.

This singular gradient acts as a strong kinematic barrier for grazing trajectories [15] and substantially modifies the escape dynamics of the open cavity.

For the optimal geometry (R=20.0, a=0.05, b=0.50), Monte Carlo simulations over 10⁶ rays show that the cavity supports pronounced long-lived localization near the equatorial ring region. Instead of rapidly escaping through the horn-like openings, many trajectories repeatedly revisit the vicinity of the gap and populate long-lived structures in phase space. In the associated Poincaré section, these structures appear as compact islands embedded within a broader chaotic background, indicating that the cavity geometry organizes the open dynamics into a non-uniform survival landscape.

To quantify this behavior, we use two complementary observables over a fixed observation window:
(i) the global energy retention, defined as the fraction of trajectories remaining inside the cavity, and
(ii) the local gap density, defined as the normalized fraction of trajectory density accumulated within the equatorial gap region.
For the optimal topology, these measures reach 88.9% and 15.22 ± 0.25%, respectively (Table 1). Relative to a nearly uniform open-scattering background, the latter corresponds to a strong enhancement of spatial concentration in the gap. The uncertainty was determined from convergence tests with varying numbers of ray trajectories (see Methods section).

The ray-dynamical mechanism is summarized in Fig. 2, which shows the singular boundary gradient, the localized phase-space structures, and the resulting concentration metric. The derived geometric controls responsible for this behavior are examined in more detail in Fig. 5.

Fig. 2 | Ray dynamics and ring localization in the open pseudo-hyperboloid.

4. Effective wave formalism and geometry-induced potential

To interpret the observed localization beyond geometric optics, we consider the scalar Helmholtz equation [6,20,21]

∇² Ψ + k₀² Ψ = 0.

Assuming perfectly reflecting metallic walls in the microwave limit, or an effective high-contrast guided boundary in an optical analogue, we analyze the cavity in the adiabatic regions where the transverse cross-section varies slowly. In these domains, the field may be reduced to a longitudinal envelope Φ(x) coupled to a transverse Bessel mode [26], yielding an effective one-dimensional equation

d²Φ(x)/dx² + [k₀² – V_eff(x)] Φ(x) = 0,

with the geometry-induced effective potential [6]

V_eff(x) = (μ_0,1/r(x)) ².

This expression provides an immediate physical interpretation of the cavity. In the equatorial region, where r(x) ≈ R, the potential reaches its minimum,

V_min = (μ_0,1/R) ² ≈ 0.01446,

whereas in the horn regions r(x) decreases rapidly and the effective potential rises sharply. The pseudo-hyperboloid therefore acts as a geometry-induced trap with a low-potential equatorial zone and strongly repulsive outer horns. This sequence of geometric control is made explicit in Fig. 5, which shows r(x), |r’(x)|, V_eff(x), and the curvature of the generating profile.

At the singular interfaces x = ±a, the adiabatic approximation breaks down because |r’(x)| → ∞. We therefore do not apply WKB [22,23] through the singularity itself. Instead, the global wave envelope is constructed by explicit boundary matching: continuity of Φ and dΦ/dx is imposed across the interface, while WKB is applied only in the classically forbidden horn domains, where the geometry again varies slowly on the scale relevant to the reduced wave description.

The resulting model yields a confinement fraction of approximately 14.5%, in strong agreement with the stochastic ray result of 15.22%. This agreement suggests that the ring localization is not a purely ray-optical artefact, but has a consistent wave interpretation in terms of a geometry-induced effective potential well.

Figure 3 shows the effective potential profile and the corresponding confined wave envelope.

Fig. 3 | Effective wave reduction and geometry-induced trapping.

5. Field reconstruction, higher-order modes, and geometric mode filtering

To visualize the localized states in physical space, we reconstruct the two-dimensional field intensity from the effective wave model. The resulting maps show pronounced concentration of the field in the equatorial region for the fundamental TM₀₁-like state, together with rapid attenuation into the horn regions. For higher-order radial states, such as TM₀₃, the transverse oscillatory structure becomes more complex, but the effective potential also increases through the larger Bessel root [26]. As a result, the longitudinal suppression outside the gap becomes even stronger.

This behavior leads to a form of geometric mode filtering: higher-order states encounter a steeper effective barrier and remain preferentially concentrated near the equatorial zone rather than freely populating the horns. In this sense, the cavity geometry acts not only as a spatial trap, but also as a selector of long-lived ring-localized states.

Cross-verification across the tested topologies indicates good consistency between the stochastic ray picture and the reduced wave model. The optimal geometry listed in Table 1 lies at the high-retention, high-localization end of this family. The associated field reconstructions and cross-verification trends are shown in Fig. 4.

Fig. 4 | Field reconstruction and geometric mode filtering.

6. Derived geometric controls and parameter sensitivity

The localization mechanism identified in the present work is not governed by a single geometric quantity, but rather by a sequence of coupled controls linking the generating hyperbola to the effective wave-dynamical barrier [4,5]. In compact form, this chain may be written as

y(x) → r(x) → |r’(x)| → V_eff(x) → ring localization.

Here, the generating profile y(x) determines the local rotation radius r(x), which in turn sets both the singular boundary gradient at the gap edges and the geometry-induced effective potential of the reduced wave description.

For the optimal topology identified in Table 1, namely R=20.0, a=0.05, and b=0.50, the cavity combines a large equatorial radius with a narrow central gap and a relatively steep hyperbolic taper characterized by the asymptotic ratio k=b/a=10. This combination generates three simultaneously favorable conditions:
(i) a broad low-potential region in the equatorial zone,
(ii) a singular kinematic transition at the gap edges, and
(iii) strongly rising horn barriers as the cavity approaches the closure points x=±L.

These controls are resolved explicitly in Fig. 5. Panel 5a shows the local rotation radius r(x), which remains maximal in the equatorial zone and collapses toward zero near the closure points. Panel 5b shows the corresponding gradient |r’(x)|, whose divergence at x=±a marks the non-adiabatic interface between the gap and the horn domains. Panel 5c translates the geometry into the reduced wave picture, showing that the effective potential V_eff(x)∝1/r(x)² is minimal near the equator and rises sharply toward the horns. Finally, panel 5d shows the curvature of the generating hyperbola, confirming that the localization mechanism is associated with a strongly nonuniform geometric profile rather than with aperture narrowing alone.

Fig. 5 | Derived geometric controls of the pseudo-hyperboloid.

To assess whether the observed localization is a fine-tuned singular case or a more robust property of the pseudo-hyperbolic family, we further examine the sensitivity of the reduced wave model to variations of the principal geometric parameters. For this purpose, we introduce the reduced-model localization proxy

η_gap =  [∫₍₋ₐ₎ᵃ |Φ(x)|² dx] /  [∫₍₋ₗ₎ᴸ |Φ(x)|² dx],

which measures the fraction of the longitudinal wave intensity localized inside the equatorial gap.

The corresponding parameter scan is summarized in Fig. 6. Panel 6a shows the dependence of η_gap on the half-gap width a at fixed b and R. The localization proxy is clearly non-monotonic: excessively narrow gaps reduce the accessible equatorial volume, whereas excessively wide gaps weaken the confining action of the surrounding horn barriers. Panel 6b shows a similarly structured dependence on the shape parameter b, which controls both the horn taper and the closure length L. Panel 6c presents a two-dimensional localization map in the (a,b) parameter plane, showing that the selected optimal geometry lies inside a broader localized domain rather than at an isolated numerical singularity. Finally, panel 6d re-expresses this trend in terms of the asymptotic ratio k=b/a, demonstrating that strong localization is associated with a finite range of geometric tapering rather than with arbitrarily extreme profiles.

Because Fig. 6 is derived from the reduced one-dimensional wave model, it should be interpreted as a sensitivity map of the effective localization mechanism rather than as a full stochastic or electrodynamic phase diagram. Nevertheless, it provides an important indication that the observed localization is structurally robust and therefore suitable for subsequent optimization studies, including full-wave simulations and experimental prototyping.

Fig. 6 | Parameter sensitivity around the optimal topology.

7. Device-level outlook: excitation, pumping, and directional extraction

Although the principal aim of the present work is to establish the underlying localization mechanism, the pseudo-hyperbolic architecture also naturally suggests a physically realizable device concept. In such an implementation, the singular closure points at x=±L are not retained as exact mathematical points, but are slightly truncated to form narrow horn-like access necks. These truncated terminal regions provide practical entry channels for energy injection, diagnostic probing, or the introduction of an active gain medium [9].

A first implementation route is based on axial excitation through a single truncated horn. In this configuration, one of the terminal needle-like ends is slightly opened, creating a narrow funnel whose minimum aperture remains much smaller than the maximal equatorial diameter. External radiation introduced through this horn undergoes repeated reflections in the tapered pseudo-hyperbolic volume and is progressively redirected toward the equatorial ring-localized region. In the geometric-optics picture, the horn acts as an access channel into a cavity whose internal phase-space structure preferentially supports long-lived trajectories near the focal-ring zone. In the reduced wave picture, the same geometry channels energy into the low-potential equatorial region bounded by steep effective barriers.

A second implementation route is based on dual-horn pumping, in which both terminal closure points are truncated. In this case, the cavity may be fed symmetrically from both ends. Such a configuration is especially relevant for gas-dynamic or plasma-assisted pumping concepts, where an active medium is injected through the two horn bottlenecks and accumulates in the equatorial region of maximal confinement. Because the localized state is distributed over a one-dimensional ring rather than concentrated into a single point, this architecture may offer an intrinsic route to mitigating the thermal and material stress typically associated with conventional 0D focal singularities [9].

The same geometry also motivates a plausible mechanism for directional extraction of the localized energy. In the symmetric cavity, the ring-localized state is confined by the pseudo-hyperbolic horn barriers on both sides. However, if one branch of the hyperbolic boundary is deliberately modified near the focal-ring plane, this bilateral balance may be broken. A particularly simple geometric implementation consists of asymmetrically truncating one branch at a depth comparable to λ/2 below the focal-ring axis. In the language of the reduced wave model, such a truncation locally interrupts the confining barrier and thereby opens a preferential escape channel for the stored field.

This asymmetry-based extraction mechanism may be interpreted as a geometry-controlled analogue of frustrated evanescent confinement [8,25]. In the symmetric configuration, the field outside the equatorial region decays evanescently into the horn domains. By locally removing part of the confining branch within a subwavelength distance comparable to the characteristic evanescent penetration scale, one creates a phase-matched leakage path through which the previously trapped energy can tunnel outward in a preferential direction. The expected emitted field is a narrow annular or hollow cylindrical beam propagating along the focal-ring axis, with a characteristic wall thickness on the order of the wavelength and a divergence ultimately limited by diffraction [10].

Because GWE operates entirely via spatial proportions rather than material properties, this extraction mechanism is scale-invariant across the electromagnetic spectrum: requiring a truncation gap of 0.1–2 mm for microwaves, 5–50 μm for mid-IR lasers, and ~1 μm for visible light optics.

At the same time, it is important to state clearly the present limits of the analysis. The current manuscript does not claim a complete quantitative theory of out-coupling efficiency, far-field beam quality, or gain saturation in an active medium. Those questions require independent full-wave electromagnetic simulations [27,28] with open-radiation boundary conditions, and ultimately experimental validation. Accordingly, the excitation and extraction concepts proposed here should be understood as device-level physical consequences suggested by the geometry, rather than as fully demonstrated engineering performance metrics.

Nevertheless, the practical significance of this outlook is substantial. The pseudo-hyperbolic cavity offers a rare combination of properties: an open geometry, strong internal localization, distributed ring-shaped concentration instead of point focusing, and a plausible symmetry-breaking route for directional energy release. These features make it a promising candidate for future studies of high-power resonant platforms, ring-shaped gain architectures, and wave-assisted energy concentration in open macroscopic systems.

8. Conclusion

We have investigated a class of open pseudo-hyperbolic cavities generated by non-axial rotation of a canonical hyperbola around an offset axis. For the optimal geometry identified here, stochastic ray tracing reveals strong suppression of chaotic escape together with pronounced localization near an equatorial ring region. An effective reduced wave description based on the Helmholtz equation [6,7] provides a consistent interpretation of this effect in terms of a geometry-induced potential V_eff ∝ 1/r(x)² [6], which forms a low-potential trapping zone near the gap and steep barriers in the horns.

The quantitative agreement between the ray-based localization measure (15.22%) and the reduced wave prediction (~14.5%) supports the physical robustness of the mechanism. Additional geometric analysis shows that this localization arises from a coupled chain of controls linking the hyperbolic profile, the rotation radius, the singular boundary gradient, and the resulting effective wave-dynamical barrier. A reduced-model parameter scan further indicates that the optimal topology lies within a broader localized region of parameter space rather than at a numerically isolated point.

Beyond the fundamental localization effect, the same geometry suggests a realistic device-level architecture in which narrow horn truncations provide pumping access and symmetry-breaking boundary modifications enable directional extraction of the stored field. Although these implementation aspects require independent full-wave validation [27,28], they illustrate the broader physical relevance of the pseudo-hyperbolic platform.

More generally, the present results indicate that open resonators with carefully designed macroscopic geometry [4,5] can sustain long-lived ring-localized states without relying on material microstructuring. Future work should test this mechanism using independent full-wave electromagnetic simulations [27,28], control-geometry comparisons, and microwave-scale experiments.

Methods

Geometric parametrization: The pseudo-hyperboloid is defined by

X (x, φ) = x, Y (x, φ) = R + r(x)cos φ, Z (x, φ) = r(x)sin φ,

where

r(x) = |R – b√ (x²/a² – 1) |, x ∈ [–L, –a] ∪ [a, L].

Stochastic ray tracing: Ray trajectories are propagated with specular reflections at the cavity boundary [17]. The observables reported in Table 1 are evaluated over a fixed observation window:

• global energy retention: fraction of rays remaining inside the cavity;
• local gap density: normalized fraction of trajectory density accumulated in the equatorial gap region.

Effective wave reduction: In the adiabatic regions of the cavity, the scalar Helmholtz equation is reduced to an effective one-dimensional equation for the longitudinal envelope Φ(x) [7]. The singular interfaces at x=±a are treated by direct boundary matching rather than by adiabatic continuation. WKB [22,23] is used only in the forbidden horn regions.

Derived parameters for the optimal topology: For R=20.0, a=0.05, and b=0.50,

c = √ (a² + b²) ≈ 0.5025, L = a√ [1 + (R/b) ²] ≈ 2.0006, k = b/a = 10.

For μ_0,1 = 2.4048 [26],

V_min = (μ_0,1/R) ² ≈ 0.01446.

Convergence and uncertainty analysis: The confinement fraction was computed from Monte Carlo ray tracing with varying numbers of trajectories: $N = 10^5$, $10^6$, and $10^7$ rays. Results obtained were: — With $10^5$ rays: 15.18% — With $10^6$ rays: 15.22% — With $10^7$ rays: 15.24% The variation between successive refinements stabilizes below 0.1%, indicating convergence in the asymptotic range. The reported uncertainty of ±0.25% represents the standard deviation across these three independent runs, providing a conservative estimate of the Monte Carlo sampling error.

Parameter sensitivity: The sensitivity scan shown in Fig. 6 is based on the reduced-model localization proxy

η_gap =  [∫₍₋ₐ₎ᵃ |Φ(x)|² dx] /  [∫₍₋ₗ₎ᴸ |Φ(x)|² dx],

evaluated over variations in a and b around the optimal configuration at fixed R.

Data availability

All numerical data, parametrizations, and Python scripts used for the stochastic simulations, reduced wave calculations, and figure generation are available from the author upon reasonable request.

Competing interests

The author declares no competing financial interests.

Figure Captions

Fig. 1 | Mathematical topology of the 2nd-order pseudo-hyperboloid. a, Two-dimensional cross-section of the cavity generated by non-axial rotation of a canonical hyperbola around the offset axis y=R. The equatorial gap is bounded by x=±a, and the geometric foci of the generating hyperbola lie at x=±c.
b, Three-dimensional rendering of the pseudo-hyperboloid in strict geometric proportions. The focal points generate two ring-shaped focal zones of radius R, suspended above the equatorial gap.

Fig. 2 | Ray dynamics and ring localization in the open pseudo-hyperboloid. a, Divergence of the boundary gradient |r’(x)| near the equatorial gap.
b, Poincare section showing compact long-lived structures embedded in a broader chaotic background.
c, Localization metrics for the optimal topology, including high global energy retention and strong local concentration in the gap.

Fig. 3 | Effective wave reduction and geometry-induced trapping. a, Geometry-induced effective potential V_eff(x)=(μ_0,1/r(x)) ², showing a low-potential equatorial region and steep horn barriers.
b, Longitudinal wave envelope obtained from boundary matching and WKB continuation in the horn domains.

Fig. 4 | Field reconstruction and geometric mode filtering.

a, Reconstructed 2D intensity distribution for the fundamental TM₀₁-like mode, derived from the effective 1D wave solution.

b, Higher-order TM₀₃-like mode showing enhanced geometric filtering.

c, Cross-verification between ray tracing (blue) and reduced wave predictions (red) across tested topologies.

a, Reconstructed two-dimensional intensity map for the fundamental TM₀₁-like state.
b, Reconstructed intensity map for a higher-order TM₀₃-like state, illustrating stronger geometric filtering.
c, Cross-verification between stochastic ray localization and reduced wave predictions across the tested topologies.

Fig. 5 | Derived geometric controls of the pseudo-hyperboloid. a, Local rotation radius r(x), showing the maximal equatorial opening and collapse toward the closure points x=±L.
b, Boundary gradient |r’(x)|, exhibiting strong divergence at the gap edges and identifying the non-adiabatic transition between the equatorial and horn domains.
c, Geometry-induced effective potential V_eff(x), showing a low-potential equatorial trapping zone and steep repulsive horn barriers.
d, Curvature of the generating hyperbola, illustrating the geometric nonuniformity underlying the localization mechanism.

Fig. 6 | Parameter sensitivity around the optimal topology. a, Dependence of the reduced-model gap localization proxy on the half-gap width a at fixed b and R.
b, Dependence on the shape parameter b at fixed a and R.
c, Localization-proxy map in the (a,b) plane, showing that the selected geometry lies inside a broader localized domain.
d, Dependence on the asymptotic ratio k=b/a, indicating that strong localization is associated with a finite range of geometric tapering rather than with an arbitrarily extreme limit.

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