Taking the random electromagnetic cosh-Gaussian beam as a typical example of random electromagnetic beams, the analytical expressions for the cross-spectral density matrix element of random electromagnetic cosh-Gaussian beams propagating through non-Kolmogorov atmospheric turbulence are derived, and used to study the changes in the states of polarization (degree of polarization, orientation angle and degree of ellipticity) of random electromagnetic cosh-Gaussian beams in non-Kolmogorov atmospheric turbulence. It is shown that the states of polarization of random electromagnetic cosh-Gaussian beams in non-Kolmogorov atmospheric turbulence are different from those in free space. The degree of polarization decreases, and the orientation angle and degree of ellipticity increase with increasing structure constant. The on-axis degree of polarization and the degree of ellipticity appear to have an oscillatory behavior and the orientation angle has a rapid transition for the larger cosh-part parameter of random electromagnetic cosh-Gaussian beams in atmospheric turbulence.
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