介质光栅中传播常数的界
Bounds on Propagation Constants in Dielectric Gratings
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中文总结 AI 辅助
该研究针对无损耗介质衍射光栅，从完全耦合矢量本征问题的傅里叶级数表述出发，分别推导了正则模式和幽灵模式归一化纵向传播常数的解析界，为相关模式分析提供了理论依据。
中文摘要 AI 辅助
我们建立了无损耗介质衍射光栅中模式的归一化纵向传播常数β=k_z/k_0的解析界。若ε_max表示最大相对介电常数，我们证明尽管不存在标量色散关系，正则模式（β²∈ℝ）满足β²≤ε_max；进一步证明，由相关本征问题的非厄米性质所允许的所谓幽灵模式（β²∉ℝ）满足|Re(β)|≤√ε_max/2。这两个界均直接从完全耦合矢量本征问题的傅里叶级数表述得到。
英文摘要
We establish analytical bounds on the normalized longitudinal propagation constants $β=k_z/k_0$ of modes in lossless dielectric diffraction gratings. If $ε_{\max}$ denotes the maximum relative permittivity, we show that regular modes ($β^2\in\mathbb{R}$) satisfy $β^2\leε_{\max}$ despite the lack of a scalar dispersion relation. We further show that the so-called ghost modes ($β^2\notin\mathbb{R}$), admitted by the non-Hermitian nature of the associated eigenproblem, satisfy $|\mathrm{Re}(β)|\le\sqrt{ε_{\max}}/2$. Both bounds are obtained directly from the Fourier-series formulation of the fully coupled vectorial eigenproblem.
