General recipe for flatbands in photonic crystal waveguides.

We present a general recipe for tailoring flat dispersion curves in photonic crystal waveguides. Our approach is based on the critical coupling criterion that equates the coupling strength of guided modes with their frequency spacing and results in a significant number of the modes lying collectively in the slow-light regime. We first describe the critical coupling scheme in photonic crystal waveguides using a simple coupled mode theory model. We also determine that canonical photonic crystal waveguides natively correspond to strongly coupled modes. Based on these analyses, our design recipe is as follows: Tune the profile of the first Fourier component of the waveguide periodic dielectric boundary to lower the coupling strength of the guided modes down to its critical value. We check that this generalized tuning may be accomplished by adjusting any desired optogeometric parameter such as hole size, position, index etc. We explore the validity of this general approach down to the narrow two-missing rows waveguides. The interest of this method is to circumvent most of the common trial-and-error procedures for flatband engineering.