Erf beam dynamics: effect of diffraction and apodization

Recently, a new class of self-similar beams was shown to solve the paraxial Helmholtz equation while preserving both transverse shape (up to scaling) and on-axis intensity during propagation, combining features of eigenmodes (Hermite-Gaussian and Laguerre-Gaussian) and nondiffracting beams (cosine and Bessel). The resulting field, described by the error function (erf), is called the erf beam. We present a simple generation scheme using a π-phase step diffractive optical element illuminated by a plane wave. Using Fresnel diffraction theory, we derive the propagated field in closed form and demonstrate equivalence with the reported self-similar solution. To explain intensity preservation, we compute the Poynting vector and show that energy follows elliptical trajectories in the transverse plane, continuously redirecting power toward the optical axis to compensate diffraction. Since a plane wave is idealized, we introduce two practical realizations: an erf-Gaussian beam (via Gaussian apodization) and a truncated erf beam (finite aperture restriction). Closed-form propagation expressions are derived for both. These beams retain the essential features of the ideal solution over finite propagation distances, offering a feasible route toward experimental implementation.