Finite-difference approximations of the equation for laser beam propagation

Finite-difference approximations are sought for the Schroedinger equation appearing in the paraxial Fresnel approximation for propagation of a nonpolarized laser beam which satisfy the condition that the transverse power be constant. This condition at the same time guarantees stability of the numerical method. The first method is a renormalization method. The second method uses a variational principle and is more accurate than the first method, but requires more calculations. A combination of the two methods is also possible.