The pure Gaussian beam (M2~1) follows the diffraction limited divergence.The deviation from the Gaussian beam deteriorates beam profile quality and beam divergence. I believe that there is no perfect analytical formalism to demonstrate the relationship between M2 factor and TEMnp. Your comment is appreciated.
There is no perfect analytical formalism. However, one can use math in order to devise some relationship. In my opinion, the best is integral formula for the power P(R) of the beam included inside of a circle with radius R
$ P(R) = \int_0^{2 \pi} d \phi \int_0^R dr r I(r, \phi) $
By definition M = R/w , where Rw is radius for which P(Rw) = P(\infty)*( 1 - exp(-2))
and w is Rw of fundamental mode.
Then M can be easily calculated for any given analytical formula for TEMmn mode.
The best method is to use Matlab or Python (scipy, numpy and matplotlib).
BTW - do you know how to write math formulae in RG answers ???
m2 = (pi)(d)(Theta) / (4)(Lemda)
i am Calculated M2 Factor Using 4 Ne-Ne laser making Custer Method generally M2 for TEM00 is ~1.
for Using 4 He-Ne laser i got the is ~3 with STD 0.50.
there is no perfect relation between M2 and TEM.
Due to their circular symmetry, the LG modes are quite a nice way of looking at this -- see our recent paper
Divergence of an orbital-angular-momentum-carrying beam upon propagation
MJ Padgett, FM Miatto, M Lavery, A Zeilinger, RW Boyd
New Journal of Physics 17, 023011
I agree with all answers. To my mind, M2 is useful only for characterization of beam quality, when real beam profile is close to Gaussian TEM00 (M2
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