These elements were investigated for their potential use in then upcoming integrated optical circuits. The first idea to exploit the inherent beam converging properties of two-dimensional curved waveguiding surfaces has sparked knowledgeable effort in the 1970s under the term geodesic lenses. ![]() We then continue to summarize the endeavors and advances of recent research activities in the field. It is our aim here to give a straightforward introduction to the theory of differential geometry in two-dimensional curved space and beam evolution therein. While borrowing some of the same mathematical formalism, the complexity of optics in curved space is strongly alleviated by restricting light propagation to two dimensions. Above that, the consequences seem only relevant in the context of astronomy and over the vastness of space. ![]() The level of abstraction and mathematical complications of this theory might not make for a good first impression. This cannot solely be blamed on improved fabrication techniques, but might in part be due to an unease that one befalls when leaving the intuitive realm of Euclidean geometry and diving into the muddle that can be curved space – a topic which most physicists initially encounter during their courses on Einstein’s General Relativity. Still, fully embracing the underlying potential of the concept and its practical implications has only recently become of interest in the optics community, with lots of groundwork being made over the last few years. The strong formal analogy between Maxwell’s equations in a dielectric medium and their covariant formulation in curved space has been well known for a long time. 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. We report on first fundamental experiments in this newly emerging field, which may lead to applications in integrated optical circuits. In this review paper we give a thorough introduction to differential geometry in two-dimensional manifolds and its incorporation with Maxwell’s equations. This can also be extended to nonlinear beam propagation. These effects can be explained by an effective transverse potential acting on the electromagnetic field distribution’s envelope. A positive Gaussian curvature leads to refocusing and thus an imaging behavior, whereas negative Gaussian curvature forces the field profile to diverge exponentially. Macroscopic radii only influence light propagation for non-vanishing intrinsic (or Gaussian) curvature. Radii of curvature in the order of the wavelength of light modify the local effective refractive index by altering the mode profile. 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. ![]() The extrinsic and intrinsic curvature of a two-dimensional waveguide influences wave propagation therein.
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