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

[George Moellenbrock]

Thanks Rick, for the nice little summary.  I would make just
a few additional points:

> 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.

Formally, Ci is the fraction (in voltage) of the intended polarization 
that is transmitted by the system, CiDi is the fraction of the other 
polarization that leaks in.  Writing a parallel hand correlation in this 
manner is merely an expression of the fact that conventional net gain 
calibration nominally describe the gain of imperfect polarization states, 
and not of the intended (pure) states.

>    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,

This is true, but actually it is not only this that the gain calibration 
soaks up.... (read on)

> 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.

It is true that the [1+D1.D2*] cannot in general be identically factored 
as simple G1.G2*-ish multiplicative gains, i.e., it doesn't close. 
HOWEVER, some of this term DOES leak into the gain term along with the Cs. 
Indeed any non-closing contribution will do this at some level, which is 
among the reasons we don't do BLCAL early in the processing. To understand 
this tendency for the [1+D1.D2*] term, consider the case of all Di = D. 
Then all visibilities are affected by the same [1+D.D*] term, and it very 
neatly factors out as a single additional (and unknowable) global scale 
factor in the gain solution.  So, we can conclude that gain calibration 
alone compensates more efficiently for the subtle D*D effects the more 
nearly parallel the like polarizations are among antennas, even if they 
are not particularly pure.  If the two polarizations on each antenna are 
also very nearly orthogonal, then we can tolerate any degree of impurity 
and do well in I dynamic range with gain calibration alone.  (This 
strictly applies to entirely unpolarized sources, as Q,U, and V 
contributions will tend to upset the otherwise delicate balance.  I think
Barry's suggestion amounts to detecting the imbalance induced by
a polarized source.)

So, Rick's scaling argument about |D| actually applies not to the absolute 
Ds, but to some "rereferenced" version enforced implicitly by the gain 
calibration.  Total intensity dynamic range is limited by the relative 
degree of polarization impurity, to the tune of sqrt(N)/<D*D>, where these 
are the rereferenced Ds and N is the number of baslines.  For post-gain 
cal |"D"| ~ 2%, and 66 baselines, that is ~20000:1.  For 351 baselines,
you get closer to 50000:1.

Alas, the D information available in the parallel-hands and cross-hands 
are different from each other ("referenced" differently), and neither is 
an absolute measure of the true impurity. Both are relative:  the 
cross-hands measure the relative orthogonality of opposite polarizations 
among antennas, and the (subtle D*D signature within the) parallel-hands 
measure the relative parallel-ness of like polarizations among antennas. 
One must pay attention to the terms coupling the D to an external 
polarization reference (e.g., a source of known [and strong] 
polarization), but detecting these terms is limited by SNR in the 
cross-hands (D*D*P is a very small fraction of I), and by gain stability 
in the parallel hands (time-dependent gain calibration will confuse the 
D*P terms that rotate with parallactic angle---just ask the ATCA which has 
linears).

The best hope for an absolute measure of the Ds comes from introducing 
differential feed rotation between antennas as can be obtained by 
including VLBI antennas that will have differential parallactic angle 
rotation, or by physically rotating feeds on one or more (but not all) 
antennas.  Then the arbitrary referencing of the cross-hands is
avoided.

Sanjay's GMRT approach is also useful, at least for unpolarized sources. 
An ~equivalent strategy is to take the BLCAL solutions, subtract 1 and 
solve them for ordinary gains that will be interpretted as the D*D 
contributino on application.  So long as the original gain calibration 
hasn't obliterated things, this extracts an antenna based solution from 
the baseline-based one, and restores our honesty a little....


> 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.

What is the highest achieved so far at X-band?  Whereas L,C have been 
limited to a few 10000s:1, I thought X-band was maybe doing 70000 or 
80000:1, but not more.  This must be a result of nice parallelness that 
remains obscured by our usual measure of this---the cross-hands, which are 
detecting R-L orthogonality.  That a few % (residual) D-terms should keep 
us to <50000:1 is otherwise hard to overcome without BLCAL.

As for the observation that the BLCAL solutions are mostly in the real 
part...

>    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.

I wonder if this argument applies as neatly for the re-referenced Ds 
described above....

Note that all that has really been established here is that the magnitude 
of the non-closing effects is consistent with what we probably expect for 
the polarization contribution.  I.e, with channelized continuum and 
sampling in the antenna (i.e., good bandpass calibration, no averaged-over 
ripple, etc.), the polarization effects have stepped to the front of the 
line.  We should not be surprised by the BLCAL-less dynamic range floor 
we've reached (are we surprised by X-band?).  There are no doubt other 
subtle dynamic range limiting contributors---closing and non-closing--- 
lurking deeper....

-George

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