Lorentz-Violating Wormhole Optics

We study massless spin-1 field propagation in a static, circularly symmetric $(2+1)$-dimensional wormhole with spatial Lorentz-violating anisotropy characterized by the throat radius $a$ and deformation parameter $\eta$. The geometry is horizon-free, geodesically complete, and asymptotically flat, with negative Gaussian curvature localized near the throat. Using the fully covariant vector boson formalism and covariant Maxwell theory, we derive an exact Schr\"odinger-type radial equation with a curvature-induced effective potential. Recasting the dynamics in Helmholtz form yields an effective refractive-index profile, showing that the wormhole acts as an inhomogeneous optical medium with position-dependent refractive index and frequency-dependent confinement, where low-frequency modes are strongly trapped while high-frequency modes propagate almost freely. A differential-geometric correspondence with helicoidal surfaces is established via $1/[a^2(1-\eta)] \leftrightarrow w^2$, demonstrating that Lorentz-violation-induced curvature is mathematically equivalent to curvature generated by geometric twist and linking the model to twisted graphene nanoribbons as analog-gravity platforms. These results provide a geometric framework for curvature-driven localization, dispersion, and anisotropic wave propagation in topologically nontrivial $(2+1)$-dimensional backgrounds.