Spherical wave solution for paraxial equation in far field region

Question

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?

Answer 1

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.