[evlatests] D*P contributions to total intensity

[Barry G. Clark]

All very complicated, and really needs to be written down carefully.
However, I can perhaps convince you of one simple truth - I is I.

We are used to making the claim that I = RR+LL.  But it is true
in any orthogonal polarization system.  Suppose we have a system
P = (a*R + b*L)
Q = (A*L - b*R)
where a^2 + b^2 = 1.
a=1, b=0 is pure circular, and a=b is orthogonal linears, and we are
somewhere in the middle, around b~0.1 or a bit less.
If in that system, we define the usual I
I' = PP + QQ
   = (a^2) RR + (b^2) LL + ab(RL + LR)
      + (a^2) LL + (b^2) RR - ab(RL + LR)
   = I
Therefore, the 'absolute' D terms, correcting our mean polarization
to real RR and LL, do not inter into making images of I.  They do
enter, at the few percent of source polarization level, in imaging
Q, U, V.  (Watch out for the last of those - it can hurt you.)

What 'D terms' do is so account for antenna polarizations, which differ
from antenna to antenna and which are not even orthogonal on one
antenna, correcting the measurements to those that would have been
made in a system in which all antennas had identical, orthogonalally
polarized feeds.

I don't see how you can call the fact that q and u can be bigger
than I remarkable - this sort of thing shows up all the time -
for instance, you often see stronger fringes in the bottom of
an absorption line than in the continuum.  If the Q and U maps
get bigger than the I map, then you can start to worry.

>     Plots of the cross-power visibility spectrum of Cygnus A, in all
> Stokes parameters have shown the remarkable fact that the Q and U
> visibilities are often a substantial fraction of -- and can even
> exceed
> -- the I visibility.  This situation has long been known for
> observations of distributed galactic emission.  What I want to
> emphasize
> here is that it will be a common situation for observations of highly
> polarized emission in general.
>
>     There's no surprise in this.  But what I want to emphasize here is
> that this provides another explanation (and a good one!) for our
> troubles in deriving high-fidelity images of objects like Cygnus A.
> The
> reason is the leakage between Q and U into I.  It works like this:
>
>     The observed correlation in (say) RR is written (ignoring issues
> of
> parallel hand calibration, and assuming that V = 0):
>
>        Vrr = (1 + Dr1Dr2*)I + Dr1(Q-iU) + Dr2*(Q+iU).
>
>     where I, Q and U are the visibilities for Stokes' I, Q, and U, and
> Dr1 is (for example) the complex coupling from LCP into RCP for
> antenna
> 1.  We normally argue that since the D's are a few percent, and both Q
> and U are a few percent of I, that the cross products between Ds, and
> between D and Q (or U) are of order 0.1% or less, and hence
> negligible.
>
>     But for highly polarized extended objects, the argument that the Q
> or U visibilities are negligible is incorrect -- they are often
> compariable to, and can on occasion exceed the I visibility.  Take the
> case where the I visibility hits a null (I = 0), while the Q and U
> visibilities do not.  (This is a common situation).   The measured
> Vrr,
> rather than being zero, becomes a scrambled version of the  polarized
> flux visibility.   Unless a correction is made, the derived 'I'
> visibilities will be in error, sometimes by significant amounts.
> This
> is a non-self-calibrateable error, which will lead to image
> degradation
> in the regime where dynamic ranges of thousands - to - one are
> desired.
>
>     So far as I know, the inversion from the 'RR' and 'LL'
> visibilities
> to derive the 'I' visibility takes no account of this leakage.
> Clearly, for precise imaging of objects like Cygnus A, a fuller
> inversion will be needed.
>
>     It is still unclear to me whether the 'relative' Ds that are
> determined as a matter of course via standard techniques are
> sufficient
> for this application, or whether the true Ds are needed.   I think
> 'true' Ds are needed, but others are invited to argue otherwise!
>
>
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