PATH-INTEGRAL ANALYSIS OF PROPAGATION IN A WAVE-GUIDE WITH RANDOM INHOMOGENEITIES

The method of path integration is applied to the analysis of wave prop agation in both a graded-index optical waveguide and in an otherwise h omogeneous infinite medium whose refractive indices have random statis tical inhomogeneities superposed upon a regular variation of refractiv e index with suitable averaged properties. We use techniques originall y employed in the context of electron propagation in a set of random s catterers to calculate the averaged Green function describing paraxial wave propagation in a medium whose refractive index has statistical i nhomogeneities. The concept of an averaged density of modes is introdu ced, and the paper presents detailed calculations of this quantity for two limiting cases. In the first, the correlation length associated w ith the distribution of inhomogeneities is zero along the direction of propagation. In the second, the Feynman variational technique is used to describe the propagator in a medium whose statistical inhomogeneit ies have an infinite correlation length along the direction of propaga tion. Comments are made about the intermediate case which is of greate r relevance to real waveguides.