You will notice that a lot of the questions on Maker Forums is on the beam pattern of a CO2 lasers.
I found a nice source that explains what the TEM pattern is and why it happens. It is also a nice explanation on lasers in general.
“The laser can be treated as a resonant cavity = 1.6 X 106 wavelengths long and 1.2 X 104
wavelengths wide. Such a resonant cavity has many allowed modes of oscillation ; they have
negligible axial electric and magnetic fields and are designated as transverse electromagnetic modes
(T EM ). They can be characterized as T EMmnq modes, where m, n, and q are mode numbers that
take on integral values 0,1 ,2, … The longitudinal mode number is q, and m and n are the transverse mode numbers.”