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.

中文翻译：

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使用完美的导电光栅实现高衍射效率的优化方法。

这项工作介绍了应用于一维周期光栅的一阶和二阶优化方法的实现，数值示例和实验收敛性研究。通过边界积分方程和形状导数，可以优化光栅的轮廓，以使横向极化的给定衍射模式的衍射效率最大化。我们对三种不同的优化方法进行了全面比较：一阶方法（梯度下降）；一种基于牛顿迭代的二阶方法，其中通常的牛顿步长被Hessian矩阵的频谱分解所给出的特征值的绝对值所取代，以处理非凸性；以及Broyden-Fletcher-Goldfarb-Shanno（BFGS）算法，一种准牛顿法。提供了数字示例来验证我们的主张。此外，为在Littrow配置中实现高效率而设计了两个光栅轮廓，然后将它们与高效商用光栅进行了比较。提供了从数值实验得出的结论和建议，以及未来的研究途径。