Could a spatial filter improve a heterodyne signal? Consider two beams of light at slightly difference frequencies that are interfered at a detector. The signal of interest is contained in the phase of the observed signal. As the beams travel around they pass through a variety of optics, which slightly distort the beams' wavefronts. As I understand it, a spatial filter is supposed to clean up the wavefronts. What would be the effect of the spatial filter on the observed signal? Would it give a less noisy measurement of the signal phase?
 A: It depends somewhat on the specifics of your measurement, but there are some cases in which it will improve your SNR to clean the beam up.  This could be accomplished with the use of a spatial filter or a mode cleaner; the mode cleaner will do a better job but will add complexity.  
To qualify the first statement a bit; generally your signal will be contained in the beat note between the fundamental modes ($TEM_{00}$) of your two beams. It is possible that the higher order modes of the two beams can be picked up on common, in which case they can interfere and give you signal.  If the two beams travel around different paths, they will pick up different distortions which will add higher order mode components which will not interfere with each other at your detector.  This means that these components of the light will add noise, in the form of shot noise, but will not add signal, decreasing your SNR.  
You should be careful though, particularly with the spatial filter.  If the beams are mostly $TEM_{00}$ to begin with you may end up making the SNR worse by using a tricky to align spatial filter.  In addition, the fast lenses used in spatial filters will introduce spherical aberrations which could also cause headaches.
