Abstract
The asymptotic behaviour of the light power at large distance in a random waveguide system with a short correlation length and a mathematical mechanism of the asymptotic behaviour are clarified. The discussion is based on the coupled mode theory. First, for the light propagation in an ordered waveguide system a new description in terms of the light power is presented. A solution of the integro-differential equation describing the light power is expressed as a contour integral in the Laplace transform domain. Singularities of the integrand are branch points and the branch cut integral determines the asymptotic behaviour of the solution. The light power decreases in inverse proportion to the distance. Secondly the description is extended to the case of a random waveguide system. The differential equation of the recurrence type describing the incoherent power is reduced to the integro-differential equation and it is shown that the kernel is the product of the kernel for an ordered system and the damping term. The equation is solved by using the same procedure as that for an ordered system and a contour integral representation of the solution is obtained. Singularities of the integrand are poles and branch points. The poles arise from the damping term of the kernel and the residues of the poles determine the asymptotic behaviour of the solution. The incoherent power decreases in inverse proportion to the square root of the distance.
