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We design a three-way silicon optical waveguide with the Bloch dispersion relation supporting a stationary inflection point (SIP). The SIP is a third order exceptional point of degeneracy (EPD) where three Bloch modes coalesce forming the frozen mode with greatly enhanced amplitude. The proposed design consists of a coupled resonators optical waveguide (CROW) coupled to a parallel straight waveguide. At any given frequency, this structure supports three pairs of reciprocal Bloch eigenmodes, propagating and/or evanescent. In addition to full-wave simulations, we also employ a so-called “hybrid model” that uses transfer matrices obtained from full-wave simulations of sub-blocks of the unit cell. This allows us to account for radiation losses and enables a design procedure based on minimizing the eigenmodes’ coalescence parameter. The proposed finite-length CROW displays almost unitary transfer function at the SIP resonance, implying a nearly perfect conversion of the input light into the frozen mode. The group delay and the effective quality factor at the SIP resonance show an $N^3$ scaling, where N is the number of unit cells in the cavity. The frozen mode in the CROW can be utilized in various applications like sensors, lasers and optical delay lines. 

We will discuss two kinds of exceptional points of degeneracy in waveguides and their respective application in lasers. Such exceptional points occur in waveguides with balanced loss and gain (e.g., PT symmetry), and in waveguides without loss and gain (e.g., periodic Si waveguides). Waveguides with such exceptional points have a strong degeneracy of their wavenumbers and polarization states that enables specific wave physics, only found in these degenerate systems. We will discuss advantages and disadvantages of both concepts to conceive laser regimes, related to high power, high spectral purity, high efficiency, etc, and show some realistic designs involving Si ridge waveguides. 

A microstrip-technology three-way waveguide has been conceived to display exceptional modal degeneracy in the presence of periodic gain and radiation losses satisfying parity-time ( PT ) glide-symmetry. A third order exceptional point of degeneracy (EPD) is obtained in the modal dispersion relation, where three Floquet-Bloch eigenmodes coalesce at a single frequency, in their eigenvalues and polarization states. In the proposed structure, PT -glide-symmetry is achieved using periodically spaced lumped (radiation) loss and gain elements, each shifted by half a period. This particular degeneracy can be utilized in the design of devices such as radiating arrays with distributed amplifiers, arrays of oscillators, and sensors. 

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.