Path-integral analysis of an arbitrarily tapered, multimode, graded-index waveguide; 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