Shape-invariant anisotropic Gaussian Schell-model beams: a complete characterization

We present a complete analysis of shape-invariant anisotropic Gaussian Schell-model beams, generalizing the shape-invariant beams introduced earlier by Gori and Guattari (1983) and the recently discovered twisted Gaussian Schell-model beams. We show that the set of all shape-invariant Gaussian Schell-model beams forms a six-parameter family embedded within the ten-parameter family of all anisotropic Gaussian Schell-model beams. These shape-invariant beams are generically anisotropic and possess a saddlelike phase front in addition to a twist phase in such a way that the tendency of the latter to twist the beam in the course of propagation is exactly countered by the former. The propagation characteristics of these beams turn out to be surprisingly simple and are akin to those of coherent Gaussian beams. They are controlled by a single parameter that plays the role of the Rayleigh range; its value is determined by an interplay among the beam widths, transverse coherence lengths, and the strength of the twist parameter. The positivity requirement on the cross-spectral density is shown to be equivalent to an upper bound on the twist parameter. The entire analysis is carried out by use of the Wigner distribution, which reduces the problem to a purely algebraic one involving $4\times4$ matrices, thus rendering the complete solution immediately transparent