Effective-index method for the analysis of optical waveguide couplers and arrays: an asymptotic theory

An asymptotic analysis of the effective-index method applied to linear and plane arrays of identical rectangular waveguides is detailed. Two different ways of applying the method, depending on how the effective index is calculated, are considered. Expressions which are accurate at a large normalized frequency are derived for the error of the methods in calculating the propagation constant and the coupling coefficient. The expressions are also used to highlight the effects of the dimensions of the array on the accuracy of the methods. It is shown that when the element waveguides of a linear array are weakly coupled, one method provides an upper bound for the coupling coefficient while the other provides a lower bound. A set of complete bounds can thus be obtained by applying the effective-index method twice