The Wigner distribution function of a super Lorentz-Gauss SLG11 beam through a paraxial ABCD optical system

An orthonormal beam family of super Lorentz-Gauss (SLG) beam model is proposed to describe the higher-order mode beams with high divergence, which are generated by a high power diode laser. Here we consider the simplest case of the SLG beams, where there are four mutually orthogonal SLG beams, namely SLG(00), SLG(01), SLG(10), and SLG(11) beams. The SLG(00) beam is just the Lorentz-Gauss beam. Based on the Collins integral formula and the Hermite-Gaussian expansion of a Lorentz function, an analytical expression for the Wigner distribution function (WDF) of an SLG(11) beam through a paraxial ABCD optical system is derived. The properties of the WDF of an SLG(11) beam propagating in free space are demonstrated. The normalized WDFs of an SLG(11) beam at the different spatial points are depicted in several observation planes. The influence of the beam parameter on the WDF of an SLG(11) beam in free space is analyzed at different propagation distances. The second-order moments of the WDF of an SLG(11) beam in free space are also examined. This research reveals the propagation properties of an SLG(11) beam from another perspective. The WDFs of SLG(01) and SLG(10) beams can be easily obtained by using the WDFs of Lorentz-Gauss beam and the SLG(11) beam.