TEM00 is a dot, that's what we want, but sometimes they hop usually to a double dot and back, through the grin and focus lens you often don't notice. TEM10 and TEM20 are the ones I have seen sometimes with the collimating lens removed, but I still love 532nm, I use a wider longer focal length collimating lens in a slider to zoom to a tiny spot my 532 output, they are capable of very tight spots at distance.
The pump diode in a Diode Pumped Solid State ( DPSS ) 532nm laser is a 808nm that pumps a 2 part bonded crystal of ndyv04 and ktp, the yvo4 turns 808 into 1064nm and the kyp doubles the frequency from 1064nm to 532nm.
The number 532nm is the distance between waves ( 532 billionths of a meter ) so 532 is higher in frequency than 1064 as the waves are twice as close together.
There is a gradient index lens and ir filter and then a double concave double convex act as an expander to give a good looking beam over distance, it's actually needle thin coming out of the crystal but diverges more rapidly. So the expander fixes that.
During the crystal doubling mode hoping can occur, temperature and alignment can be a factor.
Now a 515nm or 520nm direct is like the 808nm pump, it's a semiconductor laser diode, a p/n junction emitting photons via of a substrate and is much more dependable, but the beam is not as clean or round.
However single mode direct greens are getting better all the time, I have been looking at the nichia 4216
HERE
The pics below is DTR's I think, and they overlap a bit, but the ancient 532 is round and clean.