# Rayleigh length determination for Laguerre-Gaussian Modes

Recently I have measured the Rayleigh length of a Gaussian electron beam probe in a scanning electron microscope, using the function:

$$w(z) = w_0 \times \sqrt{1 + (z/z_r)^2}$$

Where $w$ is beam radius, $w_0$ is beam waist, $z$ the displacement from the position of the beam waist (set as the origin) and $z_r$ the Rayleigh length.

What I intend to do next is convert the Gaussian beam into a higher Laguerre-Gaussian mode (a vortex beam) and do the same measurement for Rayleigh length, but what I am unsure of is whether the beam width function above will apply to other beam modes like the one I am about to attempt.

The beam still has a Gaussian envelope, except that there is a phase singularity in the center (giving rise to a "ring" intensity cross-section, as opposed to a spot from a Gaussian beam). I have no reason to believe that it should disperse differently to a Gaussian beam however I can't be sure. Can anyone verify this for me?

I did see another question related to this one mentioning a Gouy phase shift but again not sure here.