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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 eigenvalueproblem based on the system matrix, and as a singular point from bifurcationtheory, and the link between these two disparate viewpoints. Cast as aneigenvalue problem, we show that eigenvalue degeneracies are always coincidentwith eigenvector degeneracies, so that all eigenvalue degeneracies areimplicitly EPDs in two uniform coupled transmission lines. Furthermore, wediscuss in some detail the fact that EPDs define branch points (BPs) in thecomplex-frequency plane; we provide simple formulas for these points, and showthat parity-time (PT) symmetry leads to real-valued EPDs occurring on thereal-frequency axis. We discuss the connection of the linear algebra approachto previous waveguide analysis based on singular points from bifurcationtheory, which provides a complementary viewpoint of EPD phenomena, showing thatEPDs are singular points of the dispersion function associated with the foldbifurcation. This provides an important connection of various modal interactionphenomena known in guided-wave structures with recent interesting effectsobserved in quantum mechanics, photonics, and metamaterials systems describedin terms of the EPD formalism.

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