[mmaimcal] pointing-introduced phase errors scale with surface rms?

[Bryan Butler]

folks,

so, peter and i had a couple of conversations about this pointing
induced phase error today, and he's read through the JPL paper
carefully, and has set me straight on what's going on.

short story is that these types of phase errors should be small for
our antennas.

now, more discussion.  in the paper, they are considering phase errors 
which are induced by large scale errors (astigmatism and spherical 
aberration) across the reflector surface.  so, what you need to do is 
consider the wavelength divided by the equivalent surface rms (as also 
correctly pointed out by harvey).  peter tells me that the expected 
gravitational astigmatism is about 12um for our 12m antennas, so this 
ratio at 850 GHz is about 30.  this is about the same factor as for 
the 70m antenna at 3.5cm, where the equivalent surface rms in the paper 
is 1mm.  now, at 850 GHz, the pointing on our 12m antennas is about 
1/10th of a beam.  this is about 3 mdeg on a 70m aperture.  so, go to 
figure 17, and read off the group delay for the 70m antenna at 3 mdeg.
this is about: tau_g ~ 0.35 picosec.  convert to phase via:
tau_phi ~ tau_g / 2 => tau_phi ~ 0.17 ps.  convert to true phase via:
phi = 2 * pi * nu * tau_phi ~ 0.5 deg.  so, the effect, even at 850 
GHz, should be less than a degree of phase.

this answer is verified by considering figure 15 from the paper, which 
shows how phase error on a single antenna scales with frequency.  this 
figure is for the 34m antennas, with a 550 um equivalent rms.  to get 
this same factor of 30 in lambda/rms, we need to consider a frequency 
of about 18 GHz.  1/10th of a beam pointing for a 34m antenna at 1.65cm 
corresponds to about 12 arcsec, or 3.3 mdeg.  so, given the curve for 
antenna phase error in figure 15 for 5 mdeg pointing error at 18 GHz, 
this indicates that the phase error at 850 GHz on our 12m antennas 
should be of order tenths of a degree.  interferometric phase will be 
sqrt(2) worse, very similar to the number derived above.

some verification that this is the right answer also lies in figure 1.  
look at the lambda/30 rms case, for 0.1 pointing offset (1/10th of 
a beam).  you can see that the phase is _very_ small.

so, a red herring.

now, as darrel points out (and peter pointed out to me this morning),
even for a perfect antenna, there is a phase error induced from an 
error in pointing.  the equivalent path error is about:

  l ~ (D / 2) * [(theta**2) / 2]

for an antenna of diameter D and pointing error of theta.  for pointing 
error of 0.7 arcsec and diameter of 12m, this is only about 0.3 
picometers (pretty damn small!)...

for very large sources, i would guess that this gets much more 
complicated, as the phase errors in the aperture will map to structure 
errors in the recovered brightness distribution.  but this is what 
mark has been simulating all along, i think, just not modelling the 
full distribution of phase across the aperture.


	-bryan