An efficient and unified method for band structure calculations of 2D anisotropic photonic-crystal fibers

IF 1.4 2区数学 Q1 MATHEMATICS Calcolo Pub Date : 2024-03-23 DOI:10.1007/s10092-024-00572-6

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Abstract

In this article, band structure calculations of two dimensional (2D) anisotropic photonic-crystal fibers (PhCFs) are considered. In 2D PhCFs, Maxwell’s equations for the transversal electric and magnetic mode become decoupled, but the difficulty, arising from the anisotropic permittivity \({{\varvec{\varepsilon }}}\) and/or permeability \({{\varvec{\mu }}},\) plaguing the frequency-domain finite difference method, especially the original Yee’s scheme, is our top concern. To resolve this difficulty, we re-establish the connection between the lowest order finite element method with the quasi-periodic condition and Yee’s scheme using 2D non-orthogonal mesh, whereby the decoupled Maxwell’s equations in 2D anisotropic PhCFs are readily discretized into a generalized eigenvalue problem (GEP). Moreover, we spell out the nullspace of the resulting GEP, if it exists, and explicitly construct the Moore–Penrose pseudoinverse of the singular coefficient matrix, whose smallest positive eigenvalues can be solved by the inverse Lanczos method. Extensive band structures of 2D PhCFs are calculated and benchmarked against reliable results to demonstrate the accuracy and efficiency of our method.

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二维各向异性光子晶体光纤带状结构计算的高效统一方法

摘要本文考虑了二维（2D）各向异性光子晶体光纤（PhCFs）的带状结构计算。在二维光子晶体光纤中，横向电模和磁模的麦克斯韦方程是解耦的，但由于各向异性的介电常数\({{\varvec{\varepsilon }}} 和/或磁导率\({{\varvec{\mu }}}, \)困扰着频域有限差分方法，尤其是最初的 Yee 方案，这是我们最关心的问题。为解决这一难题，我们重新建立了准周期条件下的最低阶有限元方法与使用二维非正交网格的 Yee 方案之间的联系，从而将二维各向异性 PhCF 中的解耦麦克斯韦方程轻松离散为广义特征值问题 (GEP)。此外，我们还阐明了所得到的广义特征值问题的空域（如果存在的话），并明确构建了奇异系数矩阵的摩尔-彭罗斯伪逆，其最小正特征值可通过逆 Lanczos 方法求解。我们计算了二维 PhCF 的广泛带状结构，并以可靠的结果为基准，证明了我们方法的准确性和效率。

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来源期刊

Calcolo 数学-数学

CiteScore

2.40

自引率

11.80%

发文量

审稿时长

>12 weeks

期刊介绍： Calcolo is a quarterly of the Italian National Research Council, under the direction of the Institute for Informatics and Telematics in Pisa. Calcolo publishes original contributions in English on Numerical Analysis and its Applications, and on the Theory of Computation. The main focus of the journal is on Numerical Linear Algebra, Approximation Theory and its Applications, Numerical Solution of Differential and Integral Equations, Computational Complexity, Algorithmics, Mathematical Aspects of Computer Science, Optimization Theory. Expository papers will also appear from time to time as an introduction to emerging topics in one of the above mentioned fields. There will be a "Report" section, with abstracts of PhD Theses, news and reports from conferences and book reviews. All submissions will be carefully refereed.