How these laser show systems work
It's not a motor, it's a galvanometer: a system which produces a deflection proportional to current. It's very different from a stepper motor as there are no discrete 'steps'.
A webpage for an example professional galvo unit: http://www.thorlabs.de/newgrouppage9.cfm?objectgroup_id=3770
That does the courtesy of explaining both how it works
The angular orientation (position) of the mirror is optically encoded using an array of photocells and a light source, both of which are integrated into the interior of the galvanometer housing. Each mirror orientation corresponds to a unique ratio of signals from the photodiodes, which allows for the closed-loop operation of the galvo mirror system
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The Proportional Derivative (PD) servo driver circuit interprets the signals from the optical position detecting system inside the motor and then produces the drive voltage required to rotate the mirror to the desired position.
and also the angular precision of 0.0008° (15 µrad) if you use their special clean power supply.
The light show unit doesn't specify an angular resolution, only a scanner speed in "points per second".
According to this guy, a closed loop galvanometer gives you a feedback signal that you can use to detect the current position of the armature.
So, you send a signal to the galvanometer representing the position it should take, then amplify the difference between the feedback signal and the applied signal to correct the position until it matches your desired position.
The example I linked to uses a capacitive feedback, but there are also optical systems and potentiometer based systems.
So, it really is a galvanometer and not some kind of servo or other motor.
That site also goes into detail of building and using the position detector and the needed feedback loop. There are also plenty of technical terms and abbreviations that might help in finding more useful specifications on commercial units.
Accuracy and repeatability will depend on the mechanical parts (bearings) as well as the noise levels in the control circuitry. I couldn't find anything about that. I think you need to look for information on the ILDA test patterns to get started in that direction.
Angular resolution will depend (in part) on the resolution of whatever DAC you are using to drive the galvanometer and also on the driving electronics and feedback circuitry. It will also depend on the scan angle. The scan angle divided by the number of bits of your DAC will give you the theoretical resolution of the scanner.
To get your needed 50 microns at 3 cm distance, you would need an angular accuracy of about 0.095 degrees. Given a maximum deflection of 40 degrees (I find notes that that is the maximum for closed loop systems) then that is only about 420 steps - a 10 bit DAC ought to be enough, though I see units avalaible that take digital input at up to 24 bits.
Points per second is a measure of how fast the galvanometer can respond to changes in angle.