Propagation of an Airy-Gaussian beam passing through the ABCD optical system with a rectangular aperture

By expanding the hard rectangular aperture function into a finite sum of complex Gaussian functions, we obtain the analytical formula for the propagation of an AiG beam passing through a paraxial ABCD optical system. According to the derived formulas, the propagation properties of an AiG beam through a paraxial ABCD optical system with a hard rectangular aperture are illustrated numerically. As a special case, the fractional Fourier transformation (FrFT) system is taken into consideration. The derived analytical formulas provide a convenient and effective way for studying the propagation and transformation of an AiG beam through an apertured general optical system.