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.