Theory of randomly inhomogeneous waveguides with slight material dispersion: I. Single-mode waveguides

This paper promotes the possibility of using the paraxial approximation for solving time-dependent wave equations describing the propagation of signals in slightly dispersive media. In particular, propagation of a Gaussian signal in a single-mode optical fibre is analysed under the assumption that the permittivity may be taken as a Gaussian random field. It is shown that the width sigma(t) of the signal grows in time according to the formula sigma(t) = {[sigma(0)](2) + C(1)t(2) (1+ C(2)t(2))}(1/2) where C-1 and C-2 are positive constants. (The value of C-1 is a manifestation of the very fact that the fibre represents an optically dispersive medium; the value of C-2 stems from the randomness of the permittivity characterized by some nonzero variance and by some finite correlation length.) Technically speaking, the paper utilizes the standard path-integral formalism. Within its framework, the author takes advantage of approximating the autocorrelation function of the permittivity by a quadratic function. With this idea, he shows that vital path-integral calculations can be carried through in a completely analytical way.