Geometric and wave optics in a BTZ optical metric-based wormhole
We investigate the geometric and wave optical properties of a $(2+1)$-dimensional ultra-static spacetime conformally related to the static BTZ black hole, characterized by constant negative Gaussian curvature. The associated optical metric defines a hyperbolic wormhole geometry, wherein null geodesics experience a Pöschl--Teller-type repulsive effective potential that suppresses circular photon orbits and directs all trajectories toward the optical origin. In the wave regime, we reformulate the Helmholtz equation into a Schrödinger-like form, revealing a spatially localized effective potential that encodes curvature and angular momentum effects. The resulting refractive index $n(ρ,ω)$ is both spatially and spectrally dispersive, leading to a position-dependent critical frequency $ω_c(ρ)$ that delineates the boundary between propagating and evanescent modes. At high frequencies, the medium becomes asymptotically transparent, while for $ω< ω_c(ρ)$, waves undergo exponential attenuation. These results demonstrate intrinsic curvature-induced spectral filtering and provide a geometrically tunable framework for analog gravity systems and graphene-based photonic platforms.