On the Twistability of Partially Coherent, Schell-model Sources

In this paper, the problem of assessing the twistability of a given bona fide cross-spectral density is tackled for the class of Schell-model sources, whose shift-invariant degree of coherence is represented by a real and symmetric function, {denoted as} $\mu (-\text{bfr})=\mu (\text{bfr})$. By employing an abstract operatorial language, the problem of determining the highly degenerate spectrum of a twisted operator ${\widehat{W}}_u$ is addressed through a modal analysis based on {the} complete knowledge of the spectrum of the {\em sole} twist operator ${\widehat{T}}_u$, as found by R. Simon and N. Mukunda. [J. Opt. Soc. Am. A $\mathbf{\text{15,}}$ 1361 (1998)]. To this end, the evaluation of the complete tensor of the matrix elements \bra n',\ell'|\hat W_u|n,\ell\ket is carried out within the framework of the so-called {\em extended Wigner distribution function}, a concept recently introduced by M. {VanValkenburgh} [J. Mod. Opt. $\mathbf{\text{55,}}$ 3537 - 3549 (2008)]. As a nontrivial application of the algorithm developed here, the analytical determination of the spectrum of saturated twisted astigmatic Gaussian Schell-model sources is also presented.