  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.
