Wigner distribution function of Lorentz and Lorentz–Gauss beams through a paraxial ABCD optical system

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 Lorentz and Lorentz–Gauss beams through a paraxial ABCD optical system is derived. The properties of the WDF of Lorentz and Lorentz–Gauss beams propagating in free space are demonstrated. The normalized WDFs of Lorentz and Lorentz–Gauss beams at the different spatial points are depicted in the several observation planes. The influences of the beam parameters on the WDF of Lorentz and Lorentz–Gauss beams in free space are also analyzed at different propagation distances. The special WDF of a Lorentz beam results in its higher angular spreading than the Gaussian beam.