Approximate analytical propagation equations of laser beams through a paraxial optical ABCD system with aperture

By using the Fourier series expansion, approximate analytical propagation equations of laser beams through a paraxial optical ABCD system with different apertures are derived. The hardedged aperture can be regarded as a special case treated here, while the other apertures can also be dealt with as a hardedged apterture. Numerical calculations show the advantage of our method and the consistency with the results by means of the Collins formula.