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Lasers are not (usually) cryogenic, are are only sometimes low noise.
The gross noise sources include:
Mode hopping where the system suddenly changes which cavity mode has the highest gain, this can be made worse where any of the output beam gets back into the laser output as very small thermal and vibration effects can cause massive changes in the phase of the reflected light....
Thermal lensing, where the very high power density in the medium causes local changes in refractive index, this is particularly fun where an intracavity harmonic generator is in use as the conversion efficiency is very sensitive to power density.
Note that there are a few numbers that broadly define laser beam quality:
Coherence length - Over how many meters is the bean coherent, this sets the depth of field for holography and implies things about level of phase noise.
M squared - Tells you about the divergence and numerical aperture your downstream optics will require.
The mode number tells you what the far field beam profile is.
The fact that a laser is (or is not) single longitudinal mode tells you if the thing is lasing on one frequency or a few closely spaced ones.
There are things you can do intracavity to narrow the line width (And thus lower the noise), an intracavity etalon for example will act the same as adding a quartz crystal to an oscillator by narrowing the range over which the cavity has gain.
You can also play various mode locking tricks but those are an advanced subject.
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