Thoughts on Improvements to Hertz-mode for April

Giles ... December 3, 2005


The older document "hertz_mode.html" outlines the problems with trying to
use Hertz mode to observe bright extended sources like GMCs with SHARP.

The following paragraph is what I said about the problem in that document:

___
To first order, ~2.5" pixel misalignment will merely degrade resolution
(perhaps we go from 9" -> 10") at edges.  But there is a bigger problem.  
Flux gradients, combined with pixel-pair misalignment, combined with
pointing drifts within a cycle, will cause a fake polarization.  It will
integrate down as it will be uncorrelated from cycle to cycle.  It will
probably be of order 10% per cycle.  For faint sources this will not be a
big problem, as the errors will be 10% per cycle anyway.  For these
sources we require 100 cycles (~ 10 hours) for a good measurement. For the
proposed survey of Orion, however, this is a killer.  It degrades our
signal-to-noise by a factor of 5.
___

However, I now realize that I overstated the problem a bit.  For the GMC
survey, we expect 2% errors after an hour of integration on any given
position.  For the polarization spectrum work, I think the requirements
will be similar.  This means we can tolerate 6% errors in one cycle.  
(Recall that the errors are random.)  Thus, errors of 10% per cycle will
not degrade our signal to noise by a factor of 5. In fact, if we can
reduce the problem by a factor of 4-6 we will be in pretty good shape.

I think we can do this by a combination of the second and third solutions
that I proposed "hertz_mode.html" :

The second solution was "Hardware fix: the biggest problem is relative
rotation between H and V subarrays.  This might be correctable by tipping
the combiner.  But we have not studied this carefully."

Looking at Hua-bai's notes posted on the web, he seems to think that he
might be able to get this reduced to one-quarter pixel (worst case).  
This should help a lot.  Maybe there are some tests that we can do in
January to see if this is feasible.  If we can measure the pixel pointing
offsets for 9 pixels in each sub-array this might give us enough
information to evaluate the feasibility.  Or possibly just scanning Mars
around in a Lissajous might give us this info.

The third solution was: "Correct for this effect: For some sources, the
TF_theta map can tell us the pointing for that particular value of theta,
and how it might be different from the pointing at other values of theta.  
Then we can correct for the fake polarization caused by this pointing
drift."

I think the combination of these two solutions can get us to the point
where the systematic error is not dominant.  I give a very crude
quantitative analysis below:

1. pointing shifts between hwp positions will give us a typical change in
flux of X % per pixel.  But, pixel gains are un-knowable at the 10% level,
so if X < 10, then the pointing shift will not be recognized as a pointing
shift.  In this case any fake polarization will not be correctable.

2. However, its not really that bad.  Since we have 144 pixels, each with
a gain that is uncertain to 10%, we will probably be able to detect a
pointing change that only causes flux changes averaging around 10%.  This
is because we use all 144 fluxes to determine just two numbers:  the RA
pointing shift and the dec pointing shift.  So I would argue that we *can*
detect pointing shifts that have X < 10, as long as X is not too much less
than 10.  Lets call this "averaging advantage factor" F1.  I'd guess its a
factor of 2-3.  By averaging in this way, we can detect pointing shifts
that are big enough to cause a typical flux change of X > (10/F1)%.  If we
can detect them, we can correct the polarization results for their
effects.

3. How much fake polarization will be created by a pointing change that 
changes the flux by (10/F1)% ?  If the flux gradient is uniform, the 
answer is zero, if the peak moves from being centered on the H pixel to 
being centered on the V pixel, then the answer could be of order 
2(10/F1)%.  On average, I would expect that the change in flux would be a 
bit bigger than the difference in the change of flux between the H and V 
pixels.  Here I am assuming we can make it so that the worst misalignment 
is 0.25 pixel, as in Hua-bai's memo.  Thus we gain another modest factor, 
call this F2.  Lets say F2 = 2.  So the spurious polarization is 
[10/(F1*F2)]%.

4. Using my above estimates for F1 and F2, I find that the spurious
polarization is ~2% per cycle, or ~0.7% after an hour.  This is about
three times lower than the statistical error, so it only increases our
noise by ~10% since they add in quadrature.