Optical components built from thin-film layered structures are technologically very important. Applications include but are not limited to energy conversion and conservation, data transmission and conversion, space technology, imaging and so on. In practice these structures are defined by various parameters such as the refractive-index profile, the layer thickness and the period. The problem to find the combination of parameters which yield the spectral response closest to a given target function is referred to as optimal design. This dissertation considers several topics in the mathematical modeling and optimal design of these structures through numerical optimization. A key step in numerical optimization is to define an objective function the measures the discrepancy between the target performance and that of the current solution. Our first topic is the impact of the objective function, its metric in particular, on the optimal solution and its performance. This is done by numerical experiments with different types of antireflection coatings using two-material multilayers. The results confirm existing statements and provide a few new findings, e.g. some specific metrics can yield particularly better solutions than others. Rugates are optical coatings presenting continuous refractive-index profiles. They have received much attention recently due to technological advances and their potential better optical performance and environmental properties. The Fourier transform method is a widely used technique for the design of rugates. However, it is based on approximate expressions with strict assumptions and has many practical limitations. Our second topic is the optimal design of rugates through numerical optimization of objective functions with penalty terms. We found solutions with similar performance and novel solutions by using different metrics in the penalty term. Existing methods used only local basis functions such as piece-wise constant or linear functions for the discretization of the refractive-index profile. Our third topic is the use global basis functions such as sinusoidal functions in the discretization. A simple transformation is used to overcome the difficulty of bound constraints and the result is very promising. Both multilayer and rugate coatings can be obtained using this method. Diffraction gratings are thin-film structures whose optical properties vary periodically along one or two directions. Our final topic is the optimal design of such structures in the broadband case. The objective functions and their gradient are obtained by solving variational problems and their adjoints with finite element method. Interesting phenomena are observed in the numerical experiments. Limitations and future work in this direction are pointed out.
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