EFFECTIVE-INDEX METHOD FOR THE ANALYSIS OF OPTICAL WAVE-GUIDE 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 errors of the methods for the calculation of the propagation constant and the coupling coefficient and are 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 one provides a lower bound. A set of complete bounds can thus be obtained by applying the effective-index method twice. A much more accurate result can also be obtained from a linear combination of these two results. The application of the effective-index method to arrays of nonrectangular waveguides is also discussed and illustrated with numerical examples for arrays of circular waveguides.