[evlatests] Some Thoughts on Polarizer-Induced 'closure' errors

[Barry Clark]

If you assume 3C 286 has zero circular polarization (perhaps safe,
perhaps not), you can derive the true polarizations of our realized
'R' and 'L' IFs by watching the (D term corrected) RR and LL amplitudes
as the polarization rotates around (if 'R' is really elliptical, the
RR will go up when the source polarization lies on the long axis of
the ellipse.

Rick Perley wrote:
>     Our recent tests have clearly shown that (at least for the data 
> taken recently), the limitations to imaging capabilities at C and L 
> bands are very nicely corrected by applying a 'baseline-based' 
> calibration, in addition to the usual antenna-based calibration.   
> George has long advocated that the origin of this lies in a 2nd order 
> term, ignored in standard gain packages.  These scribblings are an 
> attempt to quantify (and support) his hypothesis. 
> 
>     In general, the four complex outputs of a correlator (RR, RL, LR, 
> LL) depend on both the polarization state of the incoming radiation and 
> the polarization states of the two antennas.  For observations of a 
> totally unpolarized point source with two antennas whose parallactic 
> angles are the same, the output of the 'RR' correlator product is quite 
> simple:
> 
>     RR = G1.G2*.C1.C2*[1+D1.D2*].I
> 
>     In this expression, G1 and G2 are the complex gains of the RCP 
> electronics for the two antennas, while C1, C2, D1, and D2 describe the 
> antenna polarizations.   The C's and D's are not independent -- they 
> depend on the same quantities (the magnitude and orientation of the 
> antenna polarization ellipse), but I write it in this form to 
> demonstrate how part of the antenna polarization effects can be factored 
> into antenna-based terms and absorbed into the standard calibration. 
> 
>     Formally, D = tan(beta).exp(2i phi), and C = cos(beta).exp(i phi), 
> where beta is the deviation of the antenna polarization ellipse angle 
> from perfect circularity.  For a perfect polarizer, beta = 0, so that 
> |C| = 1, and |D| = 0.   The angle phi represents the orientation of the 
> antenna polarization ellipse, in the antenna frame. 
> 
>     We have excellent evidence that the antenna polarizations are very 
> stable.  The usual least-squares solution for the (true) gain terms 
> (which are in general highly variable) should 'soak up' the (nearly 
> unity) C terms in the preceding equation, so that after standard 
> calibration, we are left with a residual:
> 
>     RR = [1+D1.D2*].I  (where I is the flux density of the point source). 
> 
>     It is clear that the term in brackets can't be factored into a 
> product of two antenna-based terms.  Providing the polarizers are 'good' 
> (small |D|), the influence of this term is usually very small:  A good 
> polarizer provides |D| ~ .01, so the influence of this neglected term is 
> 1 part in 10^4 -- generally not perceptible.  But for a 'not-so-good' 
> polarizer, |D| ~ 0.1, so the effects on the correlation are now at the 
> 1% level, which is about  the (worst-case) level we see in the data for 
> the L and C band observations.   Indeed, the L and C band polarizers 
> have D term amplitudes generally between 5 and 10%, so the generally 
> observed offsets are of the correct magnitude.  I also emphasize here 
> that the X-band data have never required baseline-based calibration to 
> reach super-high dynamic range, and the (old, narrow-band) polarizers on 
> these systems have very low cross-polarization. 
> 
>     The AIPS program BLCAL makes baseline-based solutions -- it solves 
> for the term in square brackets for each baseline.  The observations 
> show that this operation, when done with a single solution averaging 
> over the entire length of the observation,  is highly (nearly 
> perfectly!) effective in removing the image artifacts, indicating that 
> the origin is highly stable indeed. 
> 
>     Vivek reports that the distribution of these closure offsets shows 
> much greater variation in the real part than in the imaginary.   The 
> term in square brackets can be expanded into its real and imaginary 
> parts and, making the assumption that the polarization is not too large, 
> we get, to first order:
> 
>     Real = 1 + beta1.beta2
>     Imag = 2.beta1.beta2.(phi1-phi2). 
> 
>     (the numbers following the angles represent subscripts for the two 
> antennas involved). 
> 
>     Hence the smaller variations in the imaginary part are an expected 
> consequence, providing the two antennas polarization ellipses have 
> similar orientations -- which is an intended feature. 
> 
>     Although it might be thought that we can 'fix' this problem by 
> simply incorporating the results of a standard 'PCAL' analysis, it's not 
> so.  The outputs from the standard polarization analyses have to be 
> referenced to some assumed standard, as the first-order responses in the 
> (RL) and (LR) correlations are proportional to *differences* in the D 
> terms.  So the 'D' terms we usually see are not the 'real' ones -- they 
> are all referenced to some (unknown) standard value.  It doesn't matter 
> in polarization imaging, since the assumed offsets cancel out, but it 
> *does* matter in the corrections needed to the parallel-hand 
> responses.   To do this in a more complete way (than simply running the 
> BLCAL route), we need to measure the absolute (non-referenced) values of 
> the antenna polarizations.    There are various ways to do this -- but I 
> think the easiest is to use the 'rotate the receiver trick'.  The 
> trouble here is that this is labor intensive -- o.k. for a one-off 
> experiment.  But if these terms do vary significantly in time, and we 
> need to track them to do high-fidelity imaging, a more automated 
> approach will be needed. 
> 
> 
> 
>    
> _______________________________________________
> evlatests mailing list
> evlatests at listmgr.cv.nrao.edu
> http://listmgr.cv.nrao.edu/mailman/listinfo/evlatests