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

[Rick Perley]

    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.