Ray and wave optics in twisted graphene nanoribbons

Abstract

We investigate ray trajectories and wave propagation on a continuous helicoidal surface describing twisted graphene nanoribbons (TGNRs), treating the surface as a distortion-free structure and neglecting lattice discreteness and variations in hopping parameters. This low-dimensional curved geometry is described by a radial coordinate u is an element of [-d/2, d/2] and an angular coordinate v is an element of [ - & ell;/2, & ell;/2], where d and & ell; are the ribbon's width and total length, respectively. We derive a closed-form expression for the angular velocity of rays, with an analytical solution for the full trajectory being achievable only for narrow helicoids (u4 approximate to 0). We then solve the Helmholtz equation on the twisted surface, transforming it into a one-dimensional Schr & ouml;dinger-like form, which yields a refractive index dependent on radial position and frequency, controlled by the number of twists along the ribbon. We identify critical thresholds that distinguish regimes where the refractive index is real or complex, determined by the radial distance, twist count, and frequency. These results demonstrate that the refractive index can be continuously tuned by varying the twist count, suggesting that TGNRs could serve as frequency-selective photonic filters or waveguides.