电磁学中的复本征模和本征频率
Complex Eigenmodes and Eigenfrequencies in Electromagnetics
摘要：
本文首次对一般有耗行波电磁结构的复本征模进行了综合处理，其中每单位长度传播相移（$β$）相关的复本征频率$Ω(β)$被映射到各种电磁结构的频率相关的复传播常数$γ(ω_0)$。严格的程序，首先计算复杂的本征模的均匀和周期性的电磁结构，使用全波模拟和已知的分析结果确认。两个映射程序进一步提出了任意均匀和周期性结构，其中已知的$\{Ω-β\}$关系表示使用多项式和傅立叶级数展开，分别。因此，在已知的$\{Ω-β\}$关系中用$\{ω_0, γ\}$代替$\{Ω,~jβ\}$，形成特征方程，然后对两个未知数数值求解该特征方程，表示结构的物理色散关系$ω_0(β)$和频率相关传播损耗$α(ω_0)$。映射过程中表现出各种情况下，包括无界均匀介质，矩形波导，Drude色散超材料和周期性的电介质叠层，其中精确的传播特性已成功地检索在所有情况下，在整个频率的通带和阻带。
英文摘要：
A first comprehensive treatment on complex eigenmodes is presented for general lossy traveling-wave electromagnetic structures where the per unit length propagation phase shift ($β$) dependent complex eigenfrequencies $Ω(β)$ are mapped to the frequency dependent complex propagation constant $γ(ω_0)$ for variety of electromagnetic structures. Rigorous procedures are presented to first compute the complex eigenmodes of both uniform and periodic electromagnetic structures which are confirmed using full-wave simulations and known analytical results. Two mapping procedures are further presented for arbitrary uniform and periodic structures, where the known $\{Ω-β\}$ relationship is expressed using polynomial and Fourier series expansions, respectively. Consequently replacing $\{Ω,~jβ\}$ with $\{ω_0, γ\}$ in the known $\{Ω-β\}$ relation, a characteristic equation is formed which is then numerically solved for the two unknowns, representing the physical dispersion relation $ω_0(β)$ and the frequency dependent propagation loss $α(ω_0)$ of the structure. The mapping procedure is demonstrated for variety of cases including unbounded uniform media, rectangular waveguide, Drude dispersive metamaterial and a periodic dielectric stack, where exact propagation characteristics have been successfully retrieved in all cases across both passbands and stopbands across frequency.
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