Propagation of hollow Gaussian beam through a misaligned first-order optical system and its propagation properties

Propagation properties of hollow Gaussian beam through a misaligned first-order ABCD system is studied using the generalized Huygens-Fresnel diffraction integral, augmented matrix. It is shown that, as a hollow Gaussian beam passes through the misaligned first-order ABCD system, the beam shape is not preserved, the out-put beams have differences when passing different misaligned optical systems. We can adjust the size of dark region through adjusting the misaligned transverse vector E.