Relativistic Hermite polynomials and Lorentz beams

The link between relativistic Hermite polynomials and Lorentz beams is shown. That suggests introducing new optical fields. The paraxial propagation properties of such fields are studied in detail. They are finally put in relation to the so-called Weber-Hermite beams, which emerged within a certain class of general solutions of the 1D paraxial wave equation in Cartesian coordinates as a result of a recent re-analysis of such an equation.