Paraxial wave propagation: Operator techniques

The similarity between the Schr\"odinger equation and the paraxial wave equation permits numerous analogies linking these fields, which is pivotal in advancing both quantum mechanics and wave optics. In this study, we demonstrate the application of operator techniques to an electromagnetic field characterized by the function $f(x + ay)$, leveraging the structural analogies between these equations. Specifically, we employ initial conditions defined by Airy and Bessel functions to illustrate the practical implementation of these techniques.