An extended parabolic equation and its application to the calculation of transverse and longitudinal mutual coherence functions in atmospheric turbulence

An extended, wide forward angle scattering version of the parabolic equation is considered and an operator expression for the solution of the generalized nmth moment of the electromagnetic wave field is obtained. Here, 'generalized' connotes the consideration of both the transverse as well as the longitudinal spatial moments of the wave field. A unified solution for the generalized second-order moment, i.e. the mutual coherence function (MCF), is found. The solution is applied to the case of Kolmogorov turbulent fluctuations within the atmosphere. In addition to demonstrating an interesting decaying oscillatory behaviour of the longitudinal MCF in atmospheric turbulence, it is found that the use of the extended parabolic equation yields negligible corrections to the transverse MCF, as calculated from the parabolic equation in the paraxial approximation.