Semianalytic Representations of a Fourth-Order Mutual Coherence Function for a Laser Beam in a Turbulent Medium

New (linear and nonlinear) integral-functional equations, the solutions of which are a truncated spectral characteristic of a fourth order coherence function of a laser beam propagating in a turbulent medium are derived. The linear integral-functional includes an auxiliary function W on which the solution of the given equation does not depend (it is invariant with respect to the choice of W). For the first time formally strict and fairly simple representations are obtained for the fourth-order coherence function and its truncated spectral characteristics. In searching for representations of these functions, in particular, we have used the invariance of the solutions of the second order differential equation in partial derivatives for the four-point coherence function and a new linear integral-functional equation with respect to the choice of the function W. It is shown that by a special choice of this function and the parameters determining the choice of the position of the “observation” points and the transverse cross sections of the laser beam, it is possible to obtain different exact, asymptotic, and semianalytic representations for the truncated spectral characteristics, integral characteristics of the fourth-order coherence function and the function itself. Here semianalytic representations are taken to be ones that contain information on the unknown quantities partially and in inexplicit form, but which simultaneously make it possible to find them in analytic (or numerical) form by direct use of such (e.g. iterative) constructive procedures. In particular, a semianalytic representation is the recurrence formula obtained with which an analytic expression and numerical values for the above functions can be found effectively for a laser beam propagating in a turbulent medium.