Path Integral Analysis of the Linearly Tapered Graded-Index Wave-Guide

In this paper we derive the propagator and coupling efficiencies of a linearly tapered graded-index waveguide section, using the method of path integration in the context of paraxial, scalar wave optics. The results of this analysis are then used to derive a design criterion for high-coupling-efficiency tapers. Finally, the propagator expression is used to examine the validity of both the local normal mode approximation and the paraxial approximation. The paraxial approximation is found to be useful for all linear tapers, whilst the local normal mode approximation is found to be of restricted validity.