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This thesis analyzes metasurfaces and waveguides in geometric optics. In the first and second chapters, we give a mathematical approach to study metasurfaces. A metasurface is a surface together with a function called phase discontinuity. The phase discontinuity is chosen so that the metasurface produces a desired reflection or refraction job. We give analytical conditions between the curvature of the surface and the set of refracted directions to guarantee the existence of phasedoi:10.34944/dspace/3362 fatcat:qecxv3q3hrfqdeuyelnvw4mmcm