# Spherical wave solution for paraxial equation in far field region

I am reading this paper. It consider the far-field region for the paraxial equation, $$(\partial^2_x+\partial^2_y-2ik\partial_z)u(x,y,z)=0$$. Within the Rayleigh range, the solution will be described by Gaussian beams.

However, in the far-field region, the beams will diverge and will become spherical-like waves. It is assumed that the center of curvature of such a spherical wave is located at the origin (x,y,z)=(0,0,0). Then it reads the solutions may be factored as $$u(x,y,z)=\left \{\frac 1 z \exp \left [ -i \frac k {2z} (x^2+y^2)\right ] \right \} w(x,y,z) ,$$where the factor in braces corresponds to the transverse distribution of a paraxial spherical wave whose amplitude is given by the function $$w(x,y,z)$$. See Eq.(2) in the above paper.

I know the spherical wave diminishes as $$1/r$$, but I do not understand the exponential term $$\exp \left [ -i \frac k {2z} (x^2+y^2)\right ]$$. What does it represent?

It's a binomial expansion approximation to the $$r$$ in $$e^{ikr}$$. $$r= \sqrt{x^2+y^2+z^2} = z\sqrt{ 1+ (x^2+y^2)/z^2}\\ \approx z\left(1+ \frac 12 (x^2+y^2)/z^2+\dots\right) = z+ \frac 1{2z}(x^2+y^2)+ \ldots$$ The approximation is acceptable as long as one is much closer to the $$z$$ axis than the origin. This is the essenece of the paraxial appoximation after all.