METASURFACES AND WAVEGUIDES IN OPTICS
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 phase
... hase discontinuities. The approach contains both the near and far field cases. A starting point is the formulation of a vector Snell's law in the presence of abrupt discontinuities on the interfaces. Also, we derive the equations that the phase discontinuity function must satisfy in order for the metasurface to refract or reflect energy with a prescribed energy pattern, they are Monge-Amp\'ere partial differential equations, and we prove the existence of solutions. In the third chapter, we model energy losses in waveguides. In particular, we give quantitative estimates of the energy internally reflected in case of a straight guide and a circularly curved guide. We give a detailed ray tracing and internally reflected energy analysis for each striking point on the boundary of the guide.