In many important practical situations, beam propagation through a nonlinear media can be accurately modeled by the following nonlinear Schrödinger equation (NLS)
i Q_{η} + Δ_{⊥}Q + Q^{2} Q = 0  (NLS) 
with a function Q_{0}(ρ) as an incident radially symmetric pulse Q(ρ, η=0)= Q_{0}(ρ) (here Q_{η} = ∂Q / ∂_{η} and the Laplacian Δ_{⊥} is given by Δ_{⊥}= ∂^{2}/∂ρ^{2} + (1/ρ)(∂/∂ρ), η=z / L_{df} is a normalized propagation distance, and ρ= (x^{2}+y^{2})^{1/2} / R_{0} is the normalized distance from the center of the pulse in the transverse twodimensional plane, L_{df} is the diffraction length and R_{0} is the beam’s FW1/eM radius).
NLS equation is a paraxial approximation of the following nonlinear Helmholtz (NHL) equation [1,2]
E_{zz}(x,y,z,t) + Δ_{⊥}E + (k_{0})^{2}(1+(2n_{2}/n_{0})E^{2}) E = 0  (NHL) 
derived by assuming the following:

Electric field is written in the following SVEA form, with Q(x,y,z,t) slowly varying along propagation direction z, (λ_{0} / 2πR_{0})^{2}Q_{ηη} ≈ 0, which is satisfied when λ_{0} << R_{0}:
E(x,y,z,t) = R_{0} k_{0} (2n_{2}/n_{0})^{1/2} Q(x,y,z,t) exp[ i (ω_{0}t– k_{0}z) ]  where backpropagation term Q*(x,y,z,t) is excluded
See also: Nonparaxial approximation, Slowly varying envelope approximation (SVEA).
References  
[1]  G. Fibich A and S. Tsynkov B, Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions, J. Comp. Phys. 210, (2005), 183224. 
[2]  G. Baruch, G. Fibich, and Semyon Tsynkov, Simulations of the nonlinear Helmholtz equation: arrest of beam collapse, nonparaxial solitons and counterpropagating beams, Opt. Express 16, 1332313329 (2008). 
SimphoSOFT® propagation mathematical model is based on paraxaial approximation. 
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