Operator-based propagation of Whittaker and Helmholtz-Gauss beams

We introduce a compact operator-based technique that solves the paraxial wave equation (PWE) for a broad class of structured light fields. Using the spatial evolution operator to propagate two families of physically apodized inputs, Gaussian-apodized Whittaker (WG) integrals and Gaussian-apodized Helmholtz (HG) fields, we derive closed-form expressions that retain the Gaussian width and therefore describe finite-energy beams. The method unifies and extends the HG families and readily generalizes to non-separable non-diffracting architectures. To physically validate our analytical expressions, we experiment on superposed Bessel-Gauss beams. The precise agreement between physical dynamics and theoretical predictions confirms that this operator approach yields exact, artifact-free phase and amplitude functions, demonstrating that it is a practical alternative to standard diffraction-integral treatments.