ABSTRACT

The extrinsic and intrinsic curvature of a two-dimensional waveguide influences wave propagation therein. While this can already be apprehended from a geometric point of view in terms of geodesics generalizing straight lines as the shortest distance between any two points, in wave optics interference phenomena strongly govern the field evolution, too. Radii of curvature in the order of the wavelength of light modify the local effective refractive index by altering the mode profile. Macroscopic radii only influence light propagation for non-vanishing intrinsic (or Gaussian) curvature. A positive Gaussian curvature leads to refocusing and thus an imaging behavior, whereas negative Gaussian curvature forces the field profile to diverge exponentially. These effects can be explained by an effective transverse potential acting on the electromagnetic field distribution’s envelope. This can also be extended to nonlinear beam propagation. In this review paper we give a thorough introduction to differential geometry in two-dimensional manifolds and its incorporation with Maxwell’s equations. We report on first fundamental experiments in this newly emerging field, which may lead to applications in integrated optical circuits. The close conceptual analogy to phenomena in four-dimensional spacetime with constant curvature as well as toy-models of the Schwarzschild metric and a wormhole topology are also discussed.

摘要

二维波导的外部曲率和固有曲率会影响其中的波传播。尽管从几何学角度已经可以将测地线概括为任意两点之间的最短距离，这在几何学上已经可以理解，但在波光学中，干涉现象也强烈地控制着场的发展。通过改变模式轮廓，光波长的曲率半径改变了局部有效折射率。宏观半径仅影响光的传播，以消除固有的（或高斯）曲率。正的高斯曲率会导致重新聚焦，从而导致成像行为，而负的高斯曲率会导致场轮廓呈指数发散。这些作用可以通过作用在电磁场分布的包络线上的有效横向电位来解释。这也可以扩展到非线性光束传播。在这篇综述文章中，我们对二维流形中的微分几何及其与麦克斯韦方程组的结合进行了详尽的介绍。我们报告了这个新兴领域的第一个基础实验，这可能会导致在集成光学电路中的应用。还讨论了与具有恒定曲率的二维时空现象的紧密概念类比，以及Schwarzschild度量的玩具模型和虫洞拓扑。在这篇综述文章中，我们对二维流形中的微分几何及其与麦克斯韦方程组的结合进行了详尽的介绍。我们报告了这个新兴领域的第一个基础实验，这可能会导致在集成光学电路中的应用。还讨论了与具有恒定曲率的二维时空现象的紧密概念类比，以及Schwarzschild度量的玩具模型和虫洞拓扑。在这篇综述文章中，我们对二维流形中的微分几何及其与麦克斯韦方程组的结合进行了详尽的介绍。我们报告了这个新兴领域的第一个基础实验，这可能会导致在集成光学电路中的应用。还讨论了与具有恒定曲率的二维时空现象的紧密概念类比，以及Schwarzschild度量的玩具模型和虫洞拓扑。