Propagation in waveguides containing random irregularities: the second moment equation

The parabolic moment equations can be used to describe wave propagation in a waveguide, with an arbitrary refractive index profile, that contains weakly scattering irregularities. These equations remain valid even in the case of multiple scatter. An exact solution can be given if the guide has a parabolic profile, when the intensity on the axis of the guide exhibits periodic foci which are rapidly blurred as scattering increases. Exact solutions cannot be obtained when the profile is non-parabolic. The paper presents an approximate method of solving the second moment equation for an arbitrary profile. A simplified form of the solution useful in practice is given, and the method is illustrated for a waveguide with a cubic profile. The physical significance of the solution is discussed.