Exceptional Points of Degeneracy and Branch Points for Coupled Transmission Lines—Linear-Algebra and Bifurcation Theory Perspectives
Exceptional Points of Degeneracy and Branch Points for Coupled Transmission Lines—Linear-Algebra and Bifurcation Theory Perspectives
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耦合传输线的简并点和分支点——线性代数和分岔理论的视角
DOI：
10.1109/tap.2018.2879761
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发表时间：
2019
影响因子：
5.7
通讯作者：
F. Capolino
中科院分区：
文献类型：
--
作者：
G. Hanson;A. Yakovlev;M. Othman;F. Capolino
我们提供了一个新的角度来研究例外退化点(EPD)，将当前的线性代数观点与分叉理论联系起来。我们将这些概念应用于与在支持两个模式(在每个方向上)的波导中的传播相关的EPD，被描述为耦合传输线。我们证明了EPD是与连接多个色散谱分支的折叠分叉相关的色散函数的奇点。这提供了导波结构中已知的各种模式相互作用现象与最近在量子力学、光子学和超材料系统中观察到的有趣效应之间的重要联系，这些效应是以代数EPD形式描述的。由于分歧理论只涉及特征值，我们还通过将系统的特征向量投射到特征值来建立与线性代数观点的联系，分析表明，两个特征值的合并自动地导致两个各自的特征向量的合并。因此，对于所研究的双耦合传输线问题，特征值的简并性显式地隐含了EPD。此外，我们还比较详细地讨论了EPD在复频率平面上定义分支点的事实，给出了这些分支点的简单公式，并证明了奇偶时间(PT)对称性导致实值EPD出现在实频率轴上。
We provide a new angle to investigate exceptional points of degeneracy (EPD) relating the current linear-algebra point of view to bifurcation theory. We apply these concepts to EPDs related to propagation in waveguides supporting two modes (in each direction), described as a coupled transmission line. We show that EPDs are singular points of the dispersion function associated with the fold bifurcation connecting multiple branches of dispersion spectra. This provides an important connection between various modal interaction phenomena known in guided-wave structures with recent interesting effects observed in quantum mechanics, photonics, and metamaterials systems described in terms of the algebraic EPD formalism. Since bifurcation theory involves only eigenvalues, we also establish the connection to the linear-algebra point of view by casting the system eigenvectors in terms of eigenvalues, analytically showing that the coalescence of two eigenvalues results automatically in the coalescence of the two respective eigenvectors. Therefore, for the studied two-coupled transmission-line problem, the eigenvalue degeneracy explicitly implies an EPD. Furthermore, we discuss in some detail the fact that EPDs define branch points in the complex frequency plane, we provide simple formulas for these points, and we show that parity-time (PT) symmetry leads to real-valued EPDs occurring on the real-frequency axis.