Semi-Analytical Representations of Fourth-Order Mutual Coherence Function for Laser Beam in Turbulent Medium

New (linear and non-linear) integro-functional equations are derived. The solutions of these equations are truncated spectral characteristics of fourth-order mutual coherence function of a laser beam propagating in a turbulent medium. The linear integro-functional equation includes some function W on which the solution of this equation does not depend (that is, it is invariant under any choice of function W). Firstly, formally rigorous and fairly simple representations for the fourth-order mutual coherence function and its truncated spectral characteristics are obtained. In finding these and other representations for these functions, we used, in particular, the property of invariance of solutions of a second-order partial differential equation for fourth-order mutual coherence function and new linear integro-functional equation with respect to any choice of function W. It is shown that by means of a special choice of this function and the parameters that determine the positions of the observation points and the cross sections of the laser beam, it is possible to obtain various exact, asymptotic, and semi-analytical representations for the truncated spectral characteristics, integral characteristics of the fourth-order mutual coherence function, and this function itself. In addition, semi-analytical representations are understood as representations that contain information about unknown values in a partially implicit form, but at the same time allow them to be found in an analytical (or numerical) form through the use of any (for example, iterative) constructive procedures. In particular, a semi-analytical representation is the recursive formula obtained in the article. It can be used to effectively find analytical expressions or numerical values for the above-mentioned functions on various cross sections of a laser beam propagating in a turbulent medium.

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