Statistical properties of electromagnetic Gaussian Schell-model beam propagating in uniaxial crystals along the optical axis

Within the angular-spectrum representation, we study the partially coherent beam propagating in uniaxial crystals along the optical axis. By a method of vortex expansion, we derive the analytical solution for the cross-spectral density (CSD) function of an electromagnetic Gaussian Schell-model (EGSM) beam. We demonstrate that the analytical expression of CSD function can be written into a quasi-coherent-mode representation, whose basis vectors are constructed by the elegant Laguerre-Gaussian (eLG) functions. Several limits of the analytical solution are examined and good agreements with previous theories are obtained. Moreover, we calculate the energy density and degree of polarization (DOP) of the EGSM beam, from which the effects of coherent degree on the propagating properties are revealed. It is found that the energy conversion between circularly polarized components becomes rapid when the degree of coherence is decreasing. For all degree of coherence, the energy density and DOP exhibit a similar saturated behavior in the far field.