Wigner distribution function of Gaussian beam through an apertured and misaligned ABCD optical system

By expanding the hard aperture function into a finite sum of complex Gaussian functions, approximate analytical expressions of Wigner distribution function for fundamental mode Gaussian beam through an apertured and misaligned ABCD optical system are derived. Compared with the aligned case, results show that the misaligned optical system will introduce in the output Wigner distribution function a coordinate shift in space-frequency plane. To examine the analytical results, some numerical simulations are carried out and the absolute errors between analytical results and numerical integral ones are presented. It is shown that the approximate analytical results are proper and ascendant.