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Wave dynamics on curved surfaces has attracted growing attention due to its close resemblance to the warped space time governed by general relativity. It also opens up opportunities for designing functional optical devices such as geodesic lenses. In this work we study the wave dynamics on the surface of a torus, a shape of considerable interest due to its nontrivial topology. Governed by the conservation of angular momentum, light propagates on the torus in two different types of modes: one is able to twist around and sweep through the whole surface of the torus; the other is confined within a certain angular range along the torus latitude direction. The confined mode exhibits an interesting self focusing or imaging behavior, which, similar to a geometric lens, shows no dependence of wavelength and thus suffers no chromatic aberration. By changing the geometric parameters of the torus, both the focusing point and the focusing distance can be controlled. Our work provides a new approach to manipulation of light propagation on a curved surface under the conservation of angular momentum.
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