Ray geodesics and wave propagation on the Beltrami surface: optics of an optical wormhole

This study investigates ray geodesics and wave propagation on the Beltrami surface, with particular emphasis on the effective potentials governing photon dynamics. We derive the geodesic equations and analyze the Helmholtz equation within this curved geometry, revealing that the resulting potentials are purely repulsive. For ray trajectories, the potential is determined by wormhole parameters such as the throat radius radial optical distance $\textit{u}$, scale parameter $\textit{R}$, and the angular momentum of the test field. Near the wormhole throat, the potential remains constant, preventing inward motion below a critical energy threshold, whereas at larger radial distances, it decays exponentially, allowing free propagation. In the context of wave propagation, the potential exhibits a centrifugal barrier along with a constant repulsive term at large $\textit{u}$. The Beltrami surface, characterized by constant negative Gaussian curvature, serves as a model for graphene sheets and optical wormholes in condensed matter systems. These results allow us to determine the space- and frequency-dependent refractive index of the optical medium, and we report quantitative values of the refractive index for different radial optical distances considering both the visible light and X-ray frequency domains. Our findings provide a coherent framework for understanding photon behavior in such systems, with promising implications for material applications.