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Usually a laser beam is characterized with parameter $M^2$ which shows how much does the beam size diverge in comparison to the Gaussian beam.

$w=w_0\sqrt{1+M^2(\frac{z-z_0}{z_R})^2}$

But what in that case is the equation for the beam radius of curvature? Is it the same as for the classical Gaussian beam?

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The radius of curvature depends in detail on the beam profile and the aberration present in the beam. Indeed, in an aberrated beam, there is no simple definition of the radius of curvature. One of the problems with the $M$ parameter is that there is no simple relationship between it and the beam details; it is a very blunt characterization. If you want to know what is happenning with your laser, you need to measure its intensity and phase profile. From there, you can calculate the beam's evolution through space with e.g. the Fourier transform method I outline in this answer here. The most powerful and accurate instrument for this job is a point diffraction phase shifting interferometer.

What can be defined is the radius of the least squares best fit (or best fit by any other reasonable criterion) spherical surface to the wavefront. This will have the same value as given by the Gaussian formula with a very good approximation, with the approximation becoming better as the distance becomes much larger than the Rayleigh length. But remember, one needs to add the wavefront aberration to this wavefront to get accurate results from any serious calculation.

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