Slow and frozen light in optical waveguides with multiple gratings: Degenerate band edges and stationary inflection points

We show that a waveguide with multiple gratings can have a modal dispersion relation which supports frozen light. This means that light can be coupled efficiently to low group velocity modes of an optical waveguide or can even have finite coupling to zero group velocity modes. These effects are associated with stationary points in the dispersion of the form ω-ω _o ⊃(k⊃-k _o⊃)m, for integer order m>1, around a center frequency ω _o and wave number k _o. Stationary points of any order can be created, not only regular band edges (m=2), but also degenerate band edges (m>2 and even) and stationary inflection points (m odd). Using the perturbation theory of matrices in Jordan normal form, the modes and their properties are calculated analytically. Efficient coupling is shown to stem from evanescent modes which must accompany the presence of high-order stationary points with m>2.