Propagation and transformation of a light beam on a curved surface

Abstract

Starting from the wave equation with a non-zero space curvature, a generalized coordinate-independent expression for the evolution of a light beam on a curved space is derived. By defining the propagation axes, the expression reduces to integrable Green functions without an inevitable singular point. With a Gaussian incident field, the stationary status and refocusing effect of the light field on different shapes of curved surfaces are discussed. Different from a constant diffusion behavior in a flat space, the field experiences a periodical diffraction and refocusing spontaneously with no additional optical elements. To be more specific, we noticed that the laser field on a curved surface experiences a fractional Fourier transform, with a propagation angle to be the transform order. We hope our theoretical results can provide some references for the practical application in a curved surface space.