Path-Integral Analysis of An Arbitrarily Tapered, Multimode, Graded-Index Wave-Guide - the Inverse-Square-Law and Parabolic Tapers

The method of path integration is used to study tapered, graded-index waveguides in the context of paraxial, scalar wave optics. Closed form analytic results are obtained for the propagator, or Green's function, and coupling efficiencies of such structures. The results of this general theory are applied to tapers of parabolic and inverse-square-law shapes to derive closed form expressions for the lowest-order mode-coupling efficiency of these tapers. The results are compared with those obtained for the linear taper (discussed in a previous paper) in order to establish the suitability of each taper geometry for use in practical optical components. The inverse-square-law taper is found to be the least suitable one for single-mode devices.