[evlatests] D*P contributions to total intensity

[Barry Clark]

I think you just agreed with me, in the end, that if we have a system
of two orthogonal polarizations, and the observation is properly
calibrated, I is given by the sum of the two "parallel hand" products.
There is nothing special about circular polarization in this respect.
Your matrices can refer to leakage of any polarization basis; again
there is nothing special about circular.  Going to a different
polarization basis is multiplying by a rotation matrix on the Poincare
sphere.

The critical item, as you point out, is getting things calibrated,
which, I agree is problematic at the levels we are interested in.
The reason we love circular polarization is that the natural levels
of circular polarization in the sources we look at are a couple of
orders of magnitude lower than their linear polarization, so calibration
is a lot easier if your systems are negligibly different from circularly
polarized.  However, I think calibration can be done at the required
level if, as George suggested, you iterate between calibration and D
term determination for a cycle or two.

Rick Perley wrote:
>    Well, I agree it's complicated, and also that it needs to be written 
> down (which I am attempting to do).  But I don't agree that 'I = I'.
>    Starting with the famous general expression by Morris, Rad and 
> Seielstad (and reversing to sign of V to align with current 
> definitions), one can generate a compact matrix equation relating the 
> observed visibility vector (RR, LL, RL, LR) to the 'Stokes' Visibility 
> Vector (I+V, I-V, Q+iU, Q-iU).  (I've written these as row vectors, but 
> they are actually column vectors).  These are related by the so-called 
> Mueller matrix.  (A bad name, in my opinion, but never mind ...).  This 
> relation can easily be recovered by use of Jones matrices and the 'outer 
> product' rules.
>    The elements of the Mueller matrix are products of the terms 
> involving the antenna ellipticities, which I write as Cr, Cl, Sr, and 
> Sl.  Formally
> 
>    Cr = cos(beta_r)exp(-i phi_r)
>    Cl = cos(beta_l)exp(i phi_l)
>    Sr = sin(beta_r) exp(i phi_r)
>    Sl = sin(beta_l) exp(-i phi_l)
> 
>    The 'beta' terms are the deviations of the antenna ellipticities from 
> perfect circular, while 'phi' is the orientation of the antenna ellipses.
>    In general, the 16 terms in the Mueller matrix are non-zero (but some 
> can be made very small by good design), so that the inversion from the 
> visibility vector to the Stokes vector involves all observables.  By 
> taking shortcuts (like defining I' = RR + LL, and pretending this 
> represents true I), we incur errors, whose sizes depend on the size of 
> the various products of the Cs and Ss, and on the sizes of the terms in 
> the visibility vector.
>    A simple example should help.  Suppose we observe a source with is 
> 100% circularly polarized.  Then, I = V, and the two parallel hand 
> visibilities become very simple:
> 
>    RR = Cr1.Cr2* . I
>    LL = Sl1.Sl2* . I
> 
>    The physical interpretations of the Cs and Ss are simple:  C 
> represents the loss of RCP from R_in to R_out of the polarizer, (the 
> signal went into the L_out port), while S represents the mixing of LCP 
> from L_in to R_out.
>    If we now generate I' by adding RR and LL, we get:
> 
>    I' = (Cr1.Cr2* + Sl1.Sl2*).I
> 
>    Now -- in general, the term in brackets will not equal 1.  But, under 
> closer inspection (I'll leave the details out here), if the two antennas 
> involved have their polarization ellipses each equal (beta_r = beta_l), 
> and orthogonal (phi_r = phi_l + pi/2) then the magnitude of the term in 
> brackets is indeed equal to 1, and we would get the right answer for I 
> by taking that 'short-cut'.  Hence, taking the shortcut gives the right 
> answer only if each of the antenna feeds is truly orthogonal.  I suspect 
> this is true whatever the incoming state of the polarization is, but 
> have not attempted to show this.  As Barry noted, this is complicated 
> stuff...
> 
>    Note in the above that I've assumed the data are properly calibrated, 
> in the sense that the amplifier gain terms have been properly accounted 
> for without disturbing the polarimetric relations.  Just how we do this 
> -- since the raw RR and LL outputs are mixtures of the incoming 
> polarization states, and our calibrators in general have small but 
> non-negligible Q, U, and V contributions, is a bit of mystery to me ...
> 
> Barry G. Clark wrote:
>> 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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>>>
>>>     
>>
>>