高对比度谐振器系统中的波导：理论与快速计算

中文摘要 AI 辅助

本工作研究非零内部 Neumann 频率附近、亚波长 regime 之外的高对比度谐振器系统中的导模。在正则外部 regime（参考波数下外部 Dirichlet 问题适定）中，我们通过将外部 Helmholtz Dirichlet-to-Neumann 映射压缩到内部谐振 Neumann 特征空间的迹上，引入了无限维频率相关电容算子。我们证明了当对比度 δ→0 时，连续问题的预解式范数收敛到该离散有效算子，并推导了紧致缺陷频率和线缺陷带函数的一阶渐近公式。随后，我们通过 Combes--Thomas 论证建立了电容系数的指数型非对角衰减，从而实现了离散算子的指数精度截断，并表明其保留的系数可通过局部 Helmholtz 问题计算。在物理频率下，对于增长的有限簇问题，该局部近似在一致稳定性假设下指数收敛。通过引入消失的复吸收项结合 Hermitian 对称化，可消除该稳定性假设。特别地，阶为√δ 的吸收参数，结合相互作用截断和阶为|logδ|的 patch 半径，足以保持缺陷本征频率一阶高对比度展开的 O(δ²) 精度，从而得到一种快速计算方法。针对偶极和四极共振的数值实验，验证了该离散模型对由材料或几何失谐产生的直波导和弯曲波导的精度、指数局域性和适用性。

英文摘要

In this work, we study guided modes in systems of high-contrast resonators near nonzero interior Neumann frequencies, beyond the subwavelength regime. In the regular exterior regime, where the exterior Dirichlet problem is well-posed at the reference wavenumber, we introduce an infinite-dimensional frequency-dependent capacitance operator obtained by compressing the exterior Helmholtz Dirichlet-to-Neumann map to the traces of the interior resonant Neumann eigenspaces. We prove the norm-resolvent convergence of the continuous problem to this discrete effective operator as the contrast $δ\to0$, and derive first-order asymptotic formulas for compact-defect frequencies and line-defect band functions. We then establish exponential off-diagonal decay of the capacitance coefficients by a Combes--Thomas argument, yielding an exponentially accurate truncation of the discrete operator, and show that its retained coefficients can be computed from local Helmholtz problems. At the physical frequency, this local approximation converges exponentially under a uniform stability assumption for the growing finite-cluster problems. The stability assumption can be removed by introducing a vanishing complex absorption together with a Hermitian symmetrization. In particular, an absorption parameter of order $\sqrtδ$, together with interaction truncation and patch radii of order $|\logδ|$, suffices to preserve the $O(δ^2)$ accuracy of the first-order high-contrast expansion of the defect eigenfrequencies, yielding a fast computational method. Numerical experiments for dipole and quadrupole resonances illustrate the accuracy, exponential locality, and applicability of the discrete model to straight and bent waveguides generated by material or geometric detuning.

发表机构

• ETH Zürich（苏黎世联邦理工学院）
• City University of Hong Kong（香港城市大学）
• Tsinghua University（清华大学）

机构由 AI 辅助整理，请以论文原文为准。

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39 pages