Geometric Wave Engineering – Matched FDTD Evidence for Aperture-Side Near-Field Redistribution in a Pseudohyperboloidal Resonator Compared with a Straight-Generator Reference

We report finite-difference time-domain evidence that a pseudohyperboloidal resonator can redistribute the near-field electric-field-density proxy relative to a matched straight-generator reference resonator. The central result is not a demonstrated far-field collimated beam and not a unique topological Poynting-stagnation line. The supported result is narrower: in the investigated near-field diagnostics, the hyperbolic meridional generator shifts the slot-adjacent and focal-corridor working regions toward the output-side aperture more effectively than a matched straight-generator control with the same closure, aperture placement, source configuration, and diagnostic convention. A local radial Poynting-balance crossing, S_r ≈ 0, is observed in both the PHB and the straight-generator reference. Therefore, the existence of S_r ≈ 0 alone is not claimed as a unique property of the PHB wall. The PHB-specific signature is instead the spatial redistribution of the E²-density proxy and the aperture-side locking of the working layer relative to the selected radial sub-band. The article uses E² as an electric-field-density proxy, not as the full electromagnetic energy density. A complete E²-density proxy claim would require evaluation of both electric and magnetic energy contributions. The present claim is therefore formulated in terms of E²-field-map diagnostics and local Poynting-vector components. The comparison between m = 0 and m = 1 is reported as a mode/window-dependent indication, not as a completed one-parameter proof of fundamental mode dependence, because the available strict runs do not use identical random source configurations. A stricter future test requires identical source geometry, analytic mode excitation, or statistical source averaging. The strongest directional ordering is observed only at the near-field M1 diagnostic plane. Downstream M2 diagnostics show broadening of the extracted pattern, so the present results should not be interpreted as proof of far-field beam formation. These calculations establish a conservative computational priority record: a PHB geometry can be used as a passive near-field redistributor that changes how a right-side output aperture samples the local radial Poynting-vector field when compared with a matched straight-generator reference.