As an analog model of general relativity, optics on some two-dimensional (2D) curved surfaces has received increasing attention in the past decade. Here, in light of the Huygens–Fresnel principle, we propose a theoretical frame to study light propagation along arbitrary geodesics on any 2D curved surfaces. This theory not only enables us to solve the enigma of “infinite intensity” that existed previously at artificial singularities on surfaces of revolution but also makes it possible to study light propagation on arbitrary 2D curved surfaces. Based on this theory, we investigate the effects of light propagation on a typical surface of revolution, Flamm’s paraboloid, as an example, from which one can understand the behavior of light in the curved geometry of Schwarzschild black holes. Our theory provides a convenient and powerful tool for investigations of radiation in curved space.

中文翻译：

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任意二维弯曲空间中的光传播理论

作为广义相对论的模拟模型，一些二维（2D）曲面上的光学在过去十年中受到越来越多的关注。在这里，根据惠更斯-菲涅耳原理，我们提出了一个理论框架来研究光在任何二维曲面上沿任意测地线的传播。该理论不仅使我们能够解开先前存在于旋转表面人工奇点处的“无限强度”之谜，而且使研究任意二维曲面上的光传播成为可能。基于这一理论，我们研究了光传播对典型旋转表面的影响，以弗拉姆抛物面为例，从中可以理解光在史瓦西黑洞弯曲几何形状中的行为。