Exceptional Points of Degeneracy in Electromagnetic Periodic Waveguides and the Role of Symmetries

We investigate the role of reflection and glide symmetry in periodic lossless waveguides on the dispersion diagram and on the existence of various orders of exceptional points of degeneracy (EPDs). We use an equivalent circuit network to model each unit-cell of the guiding structure. Assuming that a coupled-mode waveguide supports $N$ modes in each direction, we derive the following conclusions. When $N$ is even, we show that a periodic guiding structure with reflection symmetry may exhibit EPDs of maximum order $N$ . To obtain a degenerate band edge (DBE) with only two coupled guiding structures, reflection symmetry must be broken. For odd $N, N+1$ is the maximum EPD order that may be obtained, and an EPD of order $N$ is not allowed. We present an example of three coupled microstrip transmission lines and show that breaking the reflection symmetry by introducing glide symmetry enables the occurrence of a stationary inflection point (SIP), also called frozen mode, which is an EPD of order three.