Exceptional Points of Degeneracy and Branch Points for Transmission-Line Problems - Linear Algebra and Bifurcation Theory Perspectives

TL;DR

This paper explores exceptional points of degeneracy in coupled transmission lines, linking linear algebra and bifurcation theory, and reveals their significance as branch points affecting wave propagation and modal interactions.

Contribution

It introduces a dual approach to identify EPDs, connects them with bifurcation theory, and discusses their role as branch points, advancing understanding of wave phenomena in complex systems.

Findings

EPDs coincide with eigenvector degeneracies in coupled lines

EPDs are branch points in the complex-frequency plane

PT symmetry results in real-valued EPDs on the real axis

Abstract

We demonstrate several new aspects of exceptional points of degeneracy (EPD) pertaining to propagation in two uniform coupled transmission-line structures. We describe an EPD using two different approaches - by solving an eigenvalue problem based on the system matrix, and as a singular point from bifurcation theory, and the link between these two disparate viewpoints. Cast as an eigenvalue problem, we show that eigenvalue degeneracies are always coincident with eigenvector degeneracies, so that all eigenvalue degeneracies are implicitly EPDs in two uniform coupled transmission lines. Furthermore, we discuss in some detail the fact that EPDs define branch points (BPs) in the complex-frequency plane; we provide simple formulas for these points, and show that parity-time (PT) symmetry leads to real-valued EPDs occurring on the real-frequency axis. We discuss the connection of the linear algebra approach to previous waveguide analysis based on singular points from bifurcation theory, which provides a complementary viewpoint of EPD phenomena, showing that EPDs are singular points of the dispersion function associated with the fold bifurcation. This provides an important connection of 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 EPD formalism.