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Metalenses and Refraction Problems in Optics
Altiner Sahin, Irem
Altiner Sahin, Irem
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Thesis/Dissertation
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2025-05
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Mathematics
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https://doi.org/10.34944/jerg-s720
Abstract
This thesis studies several refraction problems involving metalenses using optimal transport techniques. Metalenses are ultra-thin surfaces that are composed of nanostructures to focus light. Mathematically they can be represented by a pair (Γ, Φ) where Γ is a surface in R^3, and Φ is a C^1 function defined in a neighborhood of Γ, called phase discontinuity. The main goal of this research is to study the existence of phase discontinuity functions in various refraction problems. After introducing mathematical tools to study metalenses in the first chapter, in the second chapter, we study the existence of phase functions that refracts a set of general directions into another for a general metasurface z = u(x). We provide necessary and sufficient conditions in the existence theorem for the existence of phase discontinuity functions for any C^2 metasurface and any given directions; and provide examples of special cases. The examples also contain the construction of phase functions of a special refraction job involving two planar metalenses. The existence theorem is also proved in higher dimensions when the metasurface is a subset of R^{n+1}. To conclude, we derive an optical path equation for single and multiple metalenses. In the following chapters, we deliver results on the existence of phases that refract a region to another while conserving energy between the regions. In the third chapter, we first consider the point source and collimated beam cases into near and far field, for a general metasurface z = u(x). We provide conditions for existence and uniqueness of these phases and discuss the relations of these problems with optimal transport. Finally, in the fourth chapter, we use optimal transport more directly to find conditions of existence of energy conserving phases for any given general directions of rays. We also discuss a refraction problem involving two metalenses, z = f(x) and z = g(x), stacked on top of each other, and conclude the thesis with the design of refracting-reflecting metalenses.
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