Polarization Calibration

TIP Last Update: JeffMangum - 13 Dec 2007

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Contents


Polarization Calibration Memos (ALMA, EVLA, etc.)


Instrumental Polarization Status (December 2007)

Richard Hills has produced a summary of the current status of instrumental polarization measurements. Several important issues raised:
  • The polarization purity specification (i.e. the amount of allowed cross-polarized signal) for the frontends is < -20dB. Since it is the voltage response of each element of the array that matters, so the requirement on the frontend should have been set at -40dB to achieve this polarization purity requirement. Furthermore, the specification should really have been even lower than this in order to allow for some contributions from other parts of the system.
  • The antenna does not contribute significantly to the instrumental cross polarized signal.
  • Several of the frontend systems do not meet the -20dB specification and are asking for change requests to relax the specification to -16dB. Richard recommends that this request not be granted as all realistic avenues for improving the cross polarization performance of the frontends has not been investigated.

-- JeffMangum - 11 Dec 2007

On 2007/12/12 the CCB accepted the change request proposed by ESO to lower the cross-polarized signal leakage spect to -16dB. Richard Hills produced a minority opinion objecting to this change.

-- JeffMangum - 13 Dec 2007

Calibration Plan Section Revision

Text for revised Polarization Calibration section from Steve Myers...

5 Polarization Calibration

5.1 General Polarization Calibration Issues
 
ALMA will use linearly polarized feeds because they have a wider usable
bandwidth than circularly polarized feeds, and can provide complete coverage
of all millimeter wavelength atmospheric windows with a reasonable number of
receivers.  Cotton (1998) treated the problem of polarization calibration for
the MMA in detail.  A more recent treatment for ALMA can be found in Appendix
C to the Report of the ALMA Science Advisory Committee March 2000 Meeting.
The general problem of calibrating polarization in interferometry has been
described fully by Hamaker, Bregman and Sault [RD7], [RD8], who develop a
Jones matrix approach (for an earlier similar approach, see Schwab [RD4]).
The main detail that we must be concerned with here is that the measurement of
linear polarization is corrupted by contamination from Stokes I.  For linearly
polarized feeds, this corruption is in the form of a gain stability term (as
opposed to circularly polarized feeds, where the corruption arises from a
leakage term).  Another point of note is that it is not easy to distinguish
circular polarization from the instrumental polarization terms when using
linearly polarized feeds. We first consider the dominant on-axis instrumental
polarization (e.g. that exhibited in observations of a point source in the
center of the field), then generalize to dealing with the instrumental
polarization over the entire primary beam (e.g. needed for observations of
extended sources and for mosaicing).

For linearly polarized feeds, two (presumably) orthogonal polarizations X and
Y are measured, with X and Y oriented at some angle in the focal plane of the
telescope (e.g. with X aligned with the elevation axis).  After correlation,
four products are available: XX, YY, XY, YX.  In the absence of instrumental
effects, the products are related to the Stokes parameters by:

(1)   XX = 0.5*[ I + Q*cos(2*chi) + U*sin(2*chi) ]
      YY = 0.5*[ I - Q*cos(2*chi) - U*sin(2*chi) ]
      XY = 0.5*[ - Q*sin(2*chi) + U*cos(2*chi) + i*V ]
      YX = 0.5*[ - Q*sin(2*chi) + U*cos(2*chi) - i*V ]

where chi is the parallactic angle (this assumes all feeds have the same focal
plane orientation).  

For each polarization p, antenna i has an associated gain g_pi and (on-axis) 
polarization leakage (from the opposite polarization) d_pi.  We now expand
the observed polarization products to linear order in the instrumental and
polarization terms to illustrate the leading order gain and leakage effects:

(2)   XX = 0.5*g_X1*g_X2*[ I + Q*cos(2*chi) + U*sin(2*chi) ]
      YY = 0.5*g_Y1*g_Y2*[ I - Q*cos(2*chi) - U*sin(2*chi) ]
      XY = 0.5*g_X1*g_Y2*[ (d_1X-d_2Y*)*I - Q*sin(2*chi) + U*cos(2*chi) + i*V ]
      YX = 0.5*g_Y1*g_X2*[ (d_2X*-d_1Y)*I - Q*sin(2*chi) + U*cos(2*chi) - i*V ]

Note that if the gains are known accurately, then the
intensity I can be recovered by observing the two parallel hands,

(3)   XX + YY = I

with the error proportional to the linear polarization and the gain errors.
In the absence of any source linear polarization Q,U it is even simpler as
either of the parallel hands is a proxy for I.  Note that unless the
observations are made over a range of parallactic angles chi, one cannot
separate Q and U with only parallel hands (chi=0 only gives Q), with

(4)   XX - YY = Q*cos(2*chi) + U*sin(2*chi) 

for a gain-stable system.  Circular polarization is obtained by differencing
the cross-hands,

(5)   XY - YX = d*I + i*V      d = 0.5*( d_1X - d_2Y* - d_2X* + d_1Y )

again assuming stable gains.  Note that a fraction of I given by the
polarization leakage shows up as an error in V, so to meet the 0.1% spec one
needs d-term calibration stability upon calibration transfer of 0.1% (this
does not imply the magnitude of the d-terms need to be 0.1%).  In both cases,
differential gain instability shows up directly as a limitation on the
polarization determination (e.g. the spec of 0.01% on gain fluctuations and
0.05% on the difference between polarizations on a given antenna).  In the
absence of gain errors,

(6)   XY + YX = e*I - Q*sin(2*chi) + U*cos(2*chi)  
                              e = 0.5*( d_1X - d_2Y* + d_2X* - d_1Y )

and thus if the leakage is calibrated (and I known), then for a single
parallactic angle chi the system of equations given by XX-YY and XY+YX can be
solved for Q and U.  Obviously, for a calibrated system, with g's and d's
known, then the system XX,YY,XY,YX can be solved for I,Q,U,V.

For calibration, the matrix form of the full (not linearized) equations are
used (see the Hamaker et al. papers) which are solved iteratively using the
difference between observed and model visibilities. This is the implementation
in the ALMA Offline AIPS++ software.  The quantities that must be solved for
are 2x2 matices: the antenna gains G (two complex matrix elements per antenna
g_pi, g_qi on the diagonals) and the leakages D (two complex matrix elements
per antenna, d_ip and d_iq on the off-diagonals).

Thus, because the linear polarization is entangled with the total intensity,
and in general many sources exhibit linear polarization at the percent level,
there are many times when all four cross correlations per baseline will need
to be performed, which will reduce the available bandwidth by a factor of 2
and the sensitivity by sqrt(2).  We consider several ALMA use cases which
demonstrate when we may need to consider all four cross correlations and when
we may use approximations to make use of just the two parallel hand cross
correlations.

Case 1: Amplitude calibration is performed by knowing precisely the gains and
system temperatures of the antennas (not by looking at an astronomical
source), and phase (delay) calibration is performed on a quasar (or a
combination of radiometric [WVR] plus a quasar).  The quasars will generally
be a few percent linearly polarized, but may be as much as 10-20% polarized,
and hence Stokes Q and U will influence the parallel hand visibilities.  These
sources have almost no circular polarization.  For a point source, the linear
polarization of the calibrator will not affect the phase, only the amplitude.
Note here that we are only concerned with imaging, as the calibration is
known.

We further consider the subcases:

Case 1.1: Total intensity imaging with no polarization in the target source.
Many millimeter spectral line sources will have little or no linear
polarization.  Nothing special needs to take place, as the parallel hands will
basically contain Stokes I (see above).
  
Case 1.2: Total intensity imaging with appreciable linear polarization in
the target source.  The linear polarization in the target source will corrupt
the parallel hand visibilities in a systematic way.  However, when the XX and
YY visibilities are added together, the linear polarization corruptions cancel
out.  This is acceptable for low to moderate dynamic range total intensity
observations, but may not be sufficient for high dynamic range total intensity
observations, as residual gain errors will limit the cancellation of the
linear polarization and adding the XX and YY correlations results in a
condition in which gain errors no longer close, limiting the use of
self-calibration.  High dynamic range total intensity imaging of a source with
appreciable linear polarization may require full polarization calibration and
imaging.

Case 1.3: Polarization imaging.  A bright calibration source must be observed
to determine the instrumental polarization leakage or "D" terms.  If the
calibrator has known (or zero) linear polarization and no circular
polarization, the D terms can be determined in a single snapshot.  If the
calibrator has unknown linear polarization, the calibrator must be observed
through sufficient parallactic angle coverage to permit separation of the
calibrator and the D terms.  Application of the D terms will permit the
polarization imaging.

Case 2: Amplitude calibration is performed by assuming the flux density of
some astronomical sources is known precisely, and using measurements of that
source to set the voltage to flux density conversion scaling.  If the source
of flux density is not polarized, there is no problem.  If it is linearly
polarized, then the parallel hand visibilities will vary systematically with
parallactic angle, the XX and YY visibilities varying in opposite senses.  In
this case, we are dealing with imaging of the calibration sources in order to
arrive at a calibration solution.

There are several options:

Case 2.1: For total intensity observations of a target source at low to
moderate SNR, the array-wide XX and YY gain ratios can be determined and
corrected for.

Case 2.2: High SNR total intensity observations will require accounting for
the different parallactic angles of each antenna, which will result in
imperfect cancellation when using the array-wide gain ratios.  In this case,
the full polarization calibration will need to be performed on the quasar,
even if there is no interest in polarization.  Full polarization observations
of the source are only needed if desired.  In all cases in which the cross
hand visibilities are explicitly used, the X-Y phase offset must be monitored
for each antenna.  As there is no simple way to determine the X-Y phase offset
astronomically, ALMA could inject a tone into the feeds, as the AT does.
Cotton (1998) points out that it is difficult to generate a millimeter RF
tone, and that injecting an IF tone further downstream in the electronics is
simpler, though not as good instrumentally (it does not calibrate the portion
of the offset which occurs before the IF).  On the other hand, we could derive
an RF signal from the LO and inject it into the feeds for the X-Y phase
calibration.  Note that tone injection is not currently planned for ALMA, but
may have to be investigated if it is found that astronomical calibration is
insufficient.

The choice of a flux density calibrator may also interact with the
polarization calibration.  Unresolved asteroids which are not azimuthally
symmetric will have some time dependent linear polarization, which will
complicate the flux density calibration.  If stars are used for a flux
standard, they may display some circular polarization, which would require
that another source be used for the D term calibration.

As stated above, the general full polarization calibration requires good
coverage in parallactic angle to separate the constant instrumental
polarization (D term) signal from the sinusoidally varying astronomical
polarization signal.  This causes some concern since ALMA is envisioned to be
predominantly a near-transit instrument with real time imaging capability.  If
instrumental polarization calibration is required for many observations, it
may be prudent to keep a database of the instrumental polarization solutions
at the various frequencies and bandwidths and rely upon that whenever
possible.  Unlike the VLA, the ATNF compact array shows essentially no time
variability in the instrumental polarization (less than 1:10000 over 12 hours,
with variations of 0.1% over months).  Given the constraints of ALMA, time
constant instrumental polarization is certainly an important design goal.  An
analysis of the cause of the VLA unstability would also be useful.

One way around the complication of good parallactic angle coverage is to use
sources of known polarization (one special case of which is totally
unpolarized sources).  Holdaway, Carilli, and Owen (1992) have demonstrated
that it is possible to solve for the instrumental polarization for a
single snapshot, (i.e., a single parallactic angle) if the source polarization
is known in advance.  So, it would be beneficial to ALMA observing to identify
bright, compact sources with known polarization or no polarization for use as
polarization calibrators.  Unfortunately, such sources are currently
completely unknown at millimeter wavelengths - all quasars have variable
polarization angle.  Perhaps sources with emission dominated by dust (which is
polarized) or planetary observations will suffice, and should be investigated
as part of the commissioning or verification phase (or earlier on existing
millimeter arrays).  Some study of what level of polarization is acceptable as
"unpolarized" is also warranted, as the symptom of source polarization in
d-terms is an offset (which may be easily calibratable later in the
processing).

5.2    Polarization Beam Calibration

5.2.1  Description of the Problem

Because the polarization response of the telescope varies across the primary
beam, there is an "off-axis" leakage pattern that must be dealt with in
addition to the "on-axis" leakage described previously. As in the on-axis
case, this manifests as a corrupting leakage of total intensity flux into the
cross hand visibilities. For "classic" synthesis observations of small sources
near the beam center, the polarization primary beam is not an issue. However,
as many as 50% of ALMA's observations may require mosaicing ie., the source
fills the primary beam), so there must be a strategy for dealing with the
effects of the polarization beam for ALMA's polarization observations.

As total intensity leaks into the polarization visibilities through the
polarization beam, there are two principal effects that must be dealt with: 1)
spurious polarization appearing in sources located off-axis, and 2) polarized
sidelobes which will extend across the image.  These effects are intrinsically
asymmetric (generated by the feeds being offset from the optical axis, in
addition to possible variations between antennas), and in general will
be time-variable (e.g., as the asymmetric polarization beam rotates with
parallactic angle over the course of the observation).  We attempt to quantify
the severity of these effects for ALMA.

In the first case, we consider the measurement of an inherently unpolarized
source at the half-power point of the primary beam of a single pointing.
Let E be the amplitude of the polarization beam at the half-power point.
In a linear mosaic constructed by summing dirty images apodized by the primary
beams, an unpolarized source at the half power point of one pointing would be
near the center of an adjacent pointing (where the polarization beam is zero
if the on-axis polarization calibration has been done correctly).  The first
pointing's image of the target source, after multiplying by the primary beam,
would show an apparent polarization of E/2, while the second pointing would
still show zero polarization.  In the linear mosaic, the off-axis image is
summed in at weight 0.5, so after adding the images together and normalizing,
the apparent polarization of the source would be E/4.  If the observations
are not snapshots, then rotation of the beam over the course of a track will
reduce the spurious polarization by an additional factor of 2, to E/8.  

As a example, consider the VLA.  The VLA antennas have nearly identical
polarization beam patterns which are constant in time, with apparent
fractional polarization of about 5% at the primary beam half power point.  For
a long-track mosaic observation of our source, the apparent rms polarization
will be approximately 0.6%.  For many applications, this level of spurious
polarization will be acceptable, and the VLA can achieve this level without
doing anything beyond on-axis polarization calibration (e.g. Dubner et
al. 1996, AJ, 111, 304).  As the ALMA feeds will also be off-axis in the focal
plane, we can expect similar levels of cross-polarization.

In the second effect, the polarization beam observed for bright sources will
not be localized, but will spread like the point spread function sidelobes.
Again, let E be the (fractional) amplitude of the polarization beam. Consider
a source of flux density S observed with rms fractional side lobe level of
sigma_PSF --- the polarized side lobe level in a snapshot will be S * E *
sigma_PSF.  As described above, this will be reduced by a further factor of 8
in a long-track mosaic.  For a 100 mJy source, 0.03 rms sidelobes, and
polarization beam of about 0.05 fractional polarization at the half power
point, one will be left with confusing polarization error emission of order
0.15 mJy rms scattered across a snapshot image, or 0.02 mJy in a mosaic image.
Note that with N sources, the confusing polarization error will increase as
N^1/2.

The scattering of spurious polarization signals across the image by point
spread function sidelobes has been addressed in principle by Carilli &
Holdaway (VLA Test Memo 163).  The spurious polarization signal at the
location of the bright problem source was imaged in a snapshot-by-snapshot
basis and removed from the visibility data.  The residual visibilities could
then be imaged without contamination by the sidelobes of the time-variable
spurious polrization signal.  The drawback is that this method does not give
the correct polarization at the location of the bright problem source.

ALMA has a extremely demanding polarization spec of 0.1% --- one
should be able to detect polarization flux which is 0.1% of the total
intensity emission, providing it is above thermal noise; note that this
must be qualified, as it may be possible to achieve 0.1% fractional
polarization imaging at the location of a bright feature, but impossible to
measure the polarization at 0.1% of a nearby feature which is 1% as bright as
the bright feature. It is clear that something more must be done above the
procedure outlined above to meet this level of performance.

There are two other possibilities for correcting the spurious polarization: if
we had excellent knowledge of the polarization beam, if it were stable with
time (as it seems to be for the VLA), and if we had a model for the total
intensity of the source, we could calculate the spurious polarization signal
and subtract it directly from the visibilities.  For example, Bill Cotton has
measured the polarization beam at 1.4 GHz and used that to correct snapshot
observations in the NRAO VLA Sky Survey, rotating the beam to the parallactic
angle of the snapshot observation [Condon et al. 1998, AJ 115, 1693].  It is
not difficult to extend this to long track observations.  Or, in the general
case, if the polarization beam were not so well understood or were time
variable, a direction-dependent gain solver could be implemented to solve for
the polarization beam at those positions sampled by the bright problem sources
during the course of imaging and deconvolution. This is the approach taken by
AIPS++ and will be available in the ALMA Offline Data Processing package.

5.2.2 Impact of Polarization Beam Calibration

The polarization beam calibration requires no extra hardware.  

The full beam correction will require a battery of test observations
during ALMA commissioning, and then fewer test observations
intermittently to monitor the polarization beams. A set of
``standard'' polarization beams will be measured and made available to
the users and to the science pipeline. These will have to be monitored
(in case of time variability) and updated when necessary (e.g. when an
optics change is made).  See also the Primary Beam chpater.

Occasionally, a demanding polarization observation may not be
supported by the standard polarization beams determined by the ALMA
staff, and the observer will need to determine polarization beams
which are unique to their observing setup (or one will have to be
measured and provided by the ALMA staff). At this time it is unknown
how often this will occur. It should be the goal that the set of
standard beams cover at least 75% of projects.  

Determination of the polarization primary beam is equivalent to a
standard holography run, but would not require a raster out into the
far sidelobes as in a holography observation. The time required to
perform this observation will be frequency dependent: at low
frequencies where good sources are available and the noise is low, the
observation could be carried out in 1-10 minutes. At higher
frequencies, it is estimated that an hour will be required.

5.2.3 Further Work for Polarization Beam Calibration

There are remaining open questions regarding the polarization beam:

1. What is the level of off-axis polarization in the ALMA beam?

2. Can a single beam model be used for all antennas, or
does each antenna need its own beam?

3. Will the polarization beam change with time, with elevation angle, across
frequency bands, or within frequency bands?  

These issues, which probably cannot be addressed until we perform sensitive
measurements on the antennas, will impact how often beam measurement must be
done, and how many instances of the beam must be archived. This will impact
the required use of ALMA test time, and use of the archive.

The most important work for polarization beam calibration is the measurement
of the polarization beam. This could be performed on the ALMA prototype
antennas to give us an idea of what it will look like for the final ALMA
antennas, but as the feeds are evaluation feeds and not production feeds, we
expect the final ALMA polarization beams to be different. In any event, such
measurements will establish the magnitude of the problem.  In the meantime,
optics modeling could be used to estimate the polarization effects, and to
provide a baseline to compare with the data.

Furthermore, implementation of advanced imaging algorithms
incorporating the full polarization primary beam will be necessary, as
argued above, to meeting the ALMA specification of 0.1%. There are
ALMA software requirements to this effect, and these are part of the
scope of the software being developed in AIPS++ for the ALMA project.
The ALMA Imaging and Calibration Group should review the progress of
of the development of this software, and to participate in
the testing, and to monitor that the algorithm and software will reach the
required level of accuracy.

--
JeffMangum - 05 Oct 2004

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Topic revision: 2010-08-17, JeffMangum
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