Optimization methods for achieving high diffraction efficiency with perfect electric conducting gratings

Abstract

This work presents the implementation, numerical examples, and experimental 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 is optimized such that it maximizes the diffraction efficiency for given diffraction modes for transverse electric polarization. We provide a thorough comparison of three different optimization methods: a first-order method (gradient descent); a second-order approach based on a Newton iteration, where the usual Newton step is replaced by taking the absolute value of the eigenvalues given by the spectral decomposition of the Hessian matrix to deal with non-convexity; and the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm, a quasi-Newton method. Numerical examples are provided to validate our claims. Moreover, two grating profiles are designed for high efficiency in the Littrow configuration and then compared to a high efficiency commercial grating. Conclusions and recommendations, derived from the numerical experiments, are provided as well as future research avenues.

Fast solver for quasi-periodic 2D-Helmholtz scattering in layered media

“…Future work considers: (i) including cut-off frequencies in our analysis, (ii) extending our results to three dimensional Helmholtz equations and Maxwell's equations on periodic domains and (iii) applications in uncertainty quantification [40] and shape optimization [6]. Each of these subfigures present error convergence curves for the two scenarios of refraction indices considered and specified in Table 1.…”

Shape optimization for the strong routing of light in periodic diffraction gratings

“…A similar approach from Aylwin et al [36] has recently appeared for the computation and optimization of periodic diffraction gratings consisting of metallic surfaces by using boundary integral formulation.…”

“…Diffraction gratings can be modeled by using integral equation based formulations [34]; namely, by using volume [10] or boundary integral formulations [7,9,11,[35][36][37]. In these formulations, quasi-periodicity and outgoing conditions are imposed naturally by the use of the outgoing quasi-periodic Green's function.…”

Supplementary document for Diffraction efficiency optimization for multilayered parametric holographic gratings - 5309531.pdf

“…where N is the number of variables subject to optimization. Far-field derivatives with respect to the design parameters are computed as in [1] (also see [2][3][4][5][6][7]), then the derivatives of the efficiency are computed through the relation…”