This thesis has three parts. In the first part, we find an expression for a Green's function in a cylindrical dielectric optical fiber. We also find its asymptotic expansion, which proves that this Green's function is the physical solution, radiating away from the source.; In the second part of the thesis we are concerned with a problem of optimization. We consider a micro-structured optical fiber with a large air hole at its axis, and possessing a mode which travels mostly through this air hole. Starting with this fiber cross section, we optimize it to obtain a mode which is most focused at the fiber axis.; While working on this optimization problem, we realized a need for a level set method which preserves geometry and topology constraints. This is the subject of the third chapter of the thesis. We prove the effectiveness of our method, discuss the numerical issues in implementation, and provide two numerical examples.
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