The ABCD matrices for reflection and refraction for any incident angle and surface

Abstract A method to derive the ABCD matrices for reflection and refraction for any diopter under arbitrary incident angle is developed. The result is benchmarked on a commercial software for an optical system compound of two cylindrical lenses. It is used to simulate the general astigmatism and calculate the resulting Orbital Angular Momentum of a laser beam that propagate through a pair of cylindrical lenses. It more generally useful to ease and simplify such kind of simulations in any optical system by avoiding use of complicated commercial softwares.