Discrete paraxial approximation

Using a new numerical paraxial approximation to the scalar Helmholtz equation in optics, one obtains, with solutions of the type u(r)e -iK0S(r) , first a discrete eikonal equation and second a discrete ray equation both valid for any arbitrary stepsize unlike the discrete equations deduced directly from the usual eikonal and ray equations which requires a very small step-size (about wavelength). The properties of these equations are discussed and some applications given to light propagation in waveguides.