Pseudohyperboloidal Laser Resonator: A New Architecture for Laser Radiation Sources with Geometry-Defined Annular Outcoupling

Can mirror geometry determine not only optical feedback in a laser cavity, but also where its radiation emerges? We explore this possibility using pseudohyperboloidal (PHB) reflectors: radially translated hyperbolic surfaces defined by exact equations. The study exemplifies Geometric Wave Engineering (GWE), a proposed approach to designing wave responses through prescribed boundaries within established electrodynamics. The PHB construction has external focal circles and a truncation forming an annular opening. For a specially matched family of meridional rays, we derive an exact map between reflections and predict interception by the opening. The theorem applies to selected trajectories, not to every ray or electromagnetic mode. Electromagnetic simulations reveal passive resonances. With a continuously pumped, saturable two-level medium, the cavity sustains bounded oscillations after the initial coherent source is switched off. We compare PHB with a specified unstable telescope and nearby deformed profiles under matched numerical gain, aperture and measurement conditions. In long-duration comparisons, exact PHB produces greater local annular output than the tested controls and greater broadband flux than the telescope through common external planes on three finite grids. One deformation can yield higher absolute far-plane flux, whereas exact PHB achieves more flux relative to its recorded internal field-energy diagnostic. This connection between an exact geometrical transport rule and an active electromagnetic response offers a testable route to jointly engineering laser feedback and outcoupling. Numerical convergence, three-dimensional modes, pump-energy accounting and collected power require further examination before assessing applications to powerful molecular lasers.