Intensity correlation functions of lightwaves in a turbulent medium: an exact version of the two-scale method

An exact functional integral representation for the two-point intensity correlation function is first obtained by solving the moment equation and is regarded as a multiscale representation. The variable functions of integration therein involved can be effectively limited to a set of functions so that the entire phase term of the integrand becomes stationary against their arbitrary variation, exactly according to the Lagrange variational principle in dynamics. This means an approximation of the multiscale by a two-scale; thereby an exact version of the integral representations by the conventional two-scale method is obtained specifically for a collimated beam wave, including both spherical and plane waves as special cases. The result is valid independent of the medium scale that has been a basic parameter in the two-scale method and also of the medium fluctuation intensity. Nevertheless, it leads to the same zeroth-order expression as that obtained with the latter method and also to a similar first-order expression when making a formal expansion. The result is free from any expansion and is presented with a set of unperturbative equations of closed form, which can be solved iteratively. With exactly the same procedure, the three-point intensity correlation and the twofrequency intensity correlation are also obtained for the same incidence beam wave. These provide an accurate solution of the moment equation to investigate tough problems, including very large values of the scintillation index observed up to 6.