Wave localization in pseudo-hyperboloids

Wave Localization in Pseudohyperboloid Structures

Second-order pseudohyperboloid geometries demonstrate ray trajectory confinement in the equatorial region, as confirmed by Monte Carlo simulations. Previous calculations validated statistical accumulation of trajectories in the central area with maximum cross-sectional width. Now, the key challenge is to verify whether this localization persists at the wave level, where interference, mode structure, and diffraction effects are accounted for.

The geometry creates an energy landscape: the equatorial zone with expanded cross-section acts as a potential minimum for waves, while narrowing at the edges forms barriers. This aligns with the ray picture, where trajectories repeatedly return to the center.

Key Results from Ray Tracing Model

Monte Carlo ray tracing revealed consistent confinement across various configurations. Here’s the retention data:

| Rating | Equatorial Radius, R | Slit Half-Width, a | Focal Curvature, b | Ray Confinement (%) | Focal Zone Capture (LDOS, %) |

|--------|----------------------|-------------------|--------------------|---------------------|-------------------------------|

| 1 | 40.0 | 0.10 | 1.00 | 93.1 | 14.09 |

| 2 | 30.0 | 0.05 | 0.50 | 89.7 | 13.77 |

| 3 | 20.0 | 0.05 | 0.50 | 88.9 | 15.22 |

These parameters illustrate the dependence of the effect on geometry: larger radius enhances confinement, while narrow slits improve localization.

Wave Interpretation of the Equatorial Zone

In the approximate wave description, the transverse size determines local density of states (LDOS). The maximum cross-section at the equator makes this region favorable for modal configurations. The symmetry of the two funnels reinforces energy recirculation, forming a quasi-trap.

Energy injected through the neck of one funnel redirects the field toward the center. Breaking symmetry (offset by L/2) weakens confinement, opening an axial leakage channel—predominant in the ray limit.

Modes Depending on Wavelength

System behavior varies significantly with wavelength:

• Long wavelength (diffraction regime): modes and interference dominate. Localization persists as enhanced state confinement.
• Comparable wavelength: competition between geometry and diffraction. Requires full-wave simulation for refinement.
• Short wavelength (geometric optics limit): closely matches Monte Carlo results, with multiple reflections concentrated at the equator.

Symmetric structures enhance resonance; asymmetry promotes directional leakage.

Key Takeaways

• The pseudohyperboloid confines up to 93% of rays in the equatorial zone via geometric design.
• Wave hypothesis: the center is a minimum in the energy potential, supported by agreement with ray dynamics.
• Symmetry strengthens localization; asymmetry opens an axial escape path.
• Effect is stable under Monte Carlo analysis but requires full-wave validation.
• Outlook: a geometric principle for wave devices, pending rigorous proof.

Limitations and Next Steps

The hypothesis is strong due to convergence across independent methods, but remains non-full-wave. Required next steps include:

• 3D FDTD or FEM simulations for mode profiles and LDOS.
• Directional pattern analysis under asymmetric conditions.
• Assessment of energy input/output and losses.
• Sensitivity to defects and material properties.

The term "wave attractor" is used conditionally: it refers to a geometrically induced localization region, not a strict mathematical attractor.

— Editorial Team