Fractional Fourier transforms of electromagnetic rectangular Gaussian Schell model beams

We extend the scalar rectangular Gaussian Schell model (RGSM) beams to the electromagnetic domain and obtain the analytical expression for the propagation of the electromagnetic RGSM (EM RGSM) beams through an ABCD optical system. As a practical application, we illustrate how the spectral density, the spectral degree of polarization and the spectral degree of coherence of the EM RGSM beams through the fractional Fourier transform (FRFT) optical system depend on the FRFT order p and the value of beam profile M. It is found that the periods of the on-axis spectral density, the on-axis degree of polarization and the transverse degree of coherence for the FRFT order are all 2. In addition, it is of interest that the profiles of the transverse spectral intensity and the degree of polarization in the standard Fourier transform plane (i.e., p=1) are shown to form flat rectangular region which is wider for larger values of M.