We report on the derivation of a spectral element method whose originality comes from the use of a hierarchical basis built with modified Legendre polynomials. We restrict our work to TM polarization, which is the most challenging. The validation and convergence are carefully checked for metallic dielectric gratings. The method is shown to be highly efficient and remains stable for huge truncation numbers. All the necessary information is given so that non-specialists can implement the method.

中文翻译：

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修正勒让德多项式的谱元法用于层状光栅模态分析

我们报告了一种频谱元素方法的推导，该方法的独创性来自使用修改后的Legendre多项式构建的层次基础。我们的工作仅限于TM极化，这是最具挑战性的。仔细检查金属介电光栅的有效性和收敛性。该方法被证明是高效的，并且对于大量的截断数仍然保持稳定。给出了所有必要的信息，以便非专业人员可以实施该方法。