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

[Rick Perley]

    I think it's a bit more complicated -- but this idea should work.  
As I see it ...

    The I-vector is a constant in the complex plane.  To this is added 
two contributions which are proportional to the product of the (complex) 
polarization (Q+iU) with the desired (true) D-term and a time-variable 
unit length rotator vector which reflects the parallactic angle.  The 
two added (polarization-dependent) contributions rotate in opposite 
directions.  The amplitude of each of these contributions (one from each 
of the two antennas comprising the baseline) is a fraction |P|*|D| of 
the total intensity:  For P = 12% (for 3C286), and D = 8%, we should see 
about a 1% modulation of the total intensity.  As noted abover, there 
are two of these -- but it should be relatively easy to separate them. 

    This is certainly worth trying.  We'll have to be careful about 
trying to prevent the parallel-hand calibration from 'soaking up' some 
of the desired effect.

    Unhappily, 3C286 is notably circularly polarized, but I don't think 
this will cause much trouble, as the effect of V doesn't rotate with 
parallactic angle...



Barry Clark wrote:
> 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. 
>>
>>
>>
>>    
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