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.

中文翻译：

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光束在曲面上的传播和转换

从具有非零空间曲率的波动方程出发，推导了光束在弯曲空间上的演化的广义坐标独立表达式。通过定义传播轴，该表达式可简化为可积分的Green函数，而无需不可避免的奇异点。利用高斯入射场，讨论了光场在不同形状的曲面上的平稳状态和重聚焦效果。与在平坦空间中恒定的扩散行为不同，该场会经历周期性的衍射并自发地重新聚焦，而无需其他光学元件。更具体地说，我们注意到曲面上的激光场经历了分数阶傅立叶变换，且传播角为变换阶数。