actually....
Ideally, the exiting beam is slightly convergent... The picture is from the first LGS run, last December- I wasnt at Keck yet so I really dont know the details of the laser at the time - I suspect it may have been a pointing drift or adjustment... You do not see divergence of the beam...
The "spot" is 30-50 cm in diameter at the sodium layer 90 km up... its not so much a spot as it is a cylinder; the sodium layer is ~10 km thick....