Optimization methods for achieving high diffraction efficiency with perfect electric conducting gratings
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This work presents the implementation, analysis, and convergence study of first- and second-order optimization methods applied to one-dimensional periodic gratings. Through boundary integral equations and shape derivatives, the profile of a grating (taken to be a perfect electric conductor) is optimized such that it maximizes the diffraction efficiency for a given diffraction mode. We provide a thorough comparison of two optimization methods: a first-order one based on gradient descent and a second-order approach based on Newton iteration. For the latter, two variations have been explored; in one option, the first Newton method replaces the usual Newton step for the absolute values of the spectral decomposition of the Hessian matrix to deal with non-convexity, while in the second, a modified version of this Newton method is considered to reduce computational time required to compute the Hessian. Numerical examples are provided to validate our claims.
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