Approximate analytical solution for guided modes in one-dimensional arrays of identical dielectric rectangular waveguides for integrated photonics

<h2>Abstract</h2> Presented here is a new approximate Analytical solution for guided modes of one-dimensional arrays of identical rectangular dielectric waveguides and phased laser arrays (including, but not limited to one-dimensional photonic crystals). The well-known concept of an Analytical solution for a single rectangular step-index waveguide, based on separation of variables, has been modified and extended to the case of one-dimensional photonic arrays of identical waveguides. This approach proposes an analytical description of a full set of array modes, or supermodes (i.e., guided modes of a multiwaveguide system), in arrays of coupled strip-, embedded strip-, buried, strip-loaded-, rib-, inverted rib-, etc. waveguides of rectangular geometry, incorporating an arbitrary number of individual devices. It is applicable not only to step-index waveguides (as in the case of the existing Analytical solution for a single waveguide), but also to their graded-index counterparts. In practical computations, this Approach is numerically stable. It can be useful for fast, accurate, and comprehensive analysis and design of the integrated photonics components mentioned above. The accuracy of the new Analytical approach is examined against the solution of the two-dimensional Helmholtz equation obtained with the Finite Difference scheme and against the Coupled-Mode Theory.