Wave optics on helicoidal graphene nanoribbons

We present an exact theoretical investigation of the optical properties of helicoidal graphene nanoribbons by deriving the spatial and frequency-dependent refractive index n(ξ, ω˜) within the continuum medium approximation. The Helmholtz equation is formulated on a helicoidal surface parameterized by intrinsic coordinates and is transformed into a Schrodinger-like equation, where the surface curvature gives rise to an effective geometric potential. This framework enables the precise computation of the refractive index n(ξ, ω˜), allowing for a detailed and general analysis of its behavior in all physically relevant regimes. The results demonstrate that the curvature-induced optical response is pronounced in the visible frequency range, indicating a significant geometric influence on wave propagation. In the high-frequency limit ω˜ → ∞, the refractive index asymptotically approaches unity (n → 1), and the gamma rays propagate as if in a vacuum, showing complete insensitivity to the background curvature. These findings underscore the crucial role of geometry in modulating the optical behavior of nanoscale materials.