Exceptional Points of Degeneracy and Branch Points for Coupled Transmission Lines—Linear-Algebra and Bifurcation Theory Perspectives

George W Hanson,Alexander B Yakovlev,Mohamed A K Othman,Filippo Capolino

doi:10.1109/tap.2018.2879761

Abstract

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.

Highlights

When propagation in a coupled-waveguide system is described in terms of a system matrix, exceptional points of degeneracy are points in the parameter space of such a system at which simultaneous eigenvalue and eigenvector degeneracies occur [1]
exceptional points of degeneracy (EPD) are usually viewed from a linear algebra standpoint, and are associated with systems described by matrices with Jordan blocks [1, 13], it has been observed that they represent points in configuration space where multiple branches of spectra connect, and are linked to branch points in the space of control variables [17, 18]
We have examined several aspects of EPDs on two coupled transmission lines, demonstrating that in the framework of the eigenvalue problem the eigenvalue degeneracies are always coincident with eigenvector degeneracies, such that all eigenvalue degeneracies correspond to EPDs

Summary

INTRODUCTION

When propagation in a coupled-waveguide system is described in terms of a system matrix, exceptional points of degeneracy are points in the parameter space of such a system at which simultaneous eigenvalue and eigenvector degeneracies occur [1]. It is important to point out that EPDs are manifest in the parameter space of a system’s eigenstates’ temporal evolution (e.g., such as certain coupled resonators with loss and gain), or of a system’s eigenstates’ spatial evolution. This latter case represents the evolution of eigenwaves in a given spatial direction, such as in a multimode waveguide with prescribed loss and gain, which is investigated in this paper, where the multimode waveguide is a pair of uniform coupled transmission lines. We show and discuss in detail the connection of EPDs with previous work on fold-point and branch-point singularities in waveguiding systems [20,21,22,23,24,25,26,27,28] associated with mode degeneracies and mode interactions, which provides a complementary viewpoint for understanding EPDs

COUPLED TRANSMISSION-LINE FORMULATION

EPD from a Linear Algebra Perspective

EDPs leading to branch points in the complex-frequency plane

CONCLUSIONS