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\newcommand{\memnum}{125}
\newcommand{\memtit}{Post-Correlation Basis Conversion in \AIPS}
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\title{\AIPS\ Memo 125 \\
  Post-Correlation Basis Conversion in \AIPS}

\author{Rick Perley and Eric Greisen\\National Radio Astronomy
  Observatory, Socorro, NM}		
\date{\today}

\maketitle

\begin{abstract}
Calibration and imaging of interferometric visibility data are much
simpler when the receiver systems are designed to provide output
voltages proportional to the circular components of the electric
field.  On the other hand, the antenna performance (sensitivity, and
polarization purity over wide bandwidths) is much better for data
taken with linear-basis receivers.  In this memo, we describe how
post-correlation visibility data can be converted from its original
linear basis to a circular basis. Doing so enables better calibration
and imaging while maximizing sensitivity and polarization purity.
Examples are given using VLA and MeerKAT data.
\footnote{This memo has also appeared as EVLA Memo 229.}
\end{abstract}

\section{Motivation}
Polarimetry requires antennas which are themselves polarized -- each
producing two output signals representing two different polarization
states.  In principle, it doesn't matter what these two polarization
states are, so long as they are different.  In practice, the processes
of calibration and imaging are simplified if the two states are
orthogonal\footnote{In this context, orthogonal means the antenna's
  two polarization states are oppositely directed in the Poincar\'e
  sphere.}  -- usually either opposite circular (RCP and LCP) or
orthogonal linear (for example, vertical and horizontal).

Thus, engineers design the receiver systems (antennas, receivers,
amplifiers, and signal transmission) such that the two output voltages
provided to the correlator represent, as closely as possible, either
the opposite circular electric field components of the incoming
radiation, or the orthogonal linear components.  If done well,
subsequent calibration is straightforward, and the imaging results
will be the same -- all else being equal.

But all else is {\em not} equal.  Linear basis data are usually more
difficult to accurately calibrate, since virtually all calibrator
sources used in the meter-to-millimeter wavelength range are
significantly ($\sim$2 -- 10\%) linearly polarized.  This linearly
polarized flux couples directly with the linear feeds, causing the
output correlations to vary sinusoidally with parallactic angle -- a
situation which can be easily handled if the polarization state is
known in advance.  Unfortunately, the great majority of these
calibrator sources are variable, both in total intensity and in
fractional polarization, on timescales varying from days to months,
with amplitudes of a few to tens of percent.  Monitoring the flux
density and polarization characteristics of the more than 1000 VLA
calibrators on such timescales with the VLA's eight receiver bands is
simply not feasible. Thus, the polarization state of the calibrators
used in any observation will not, in general, be known in
advance\footnote{For observations taken over a considerable length of
  time, the polarization states of the calibrators can be determined
  from the data.  However, this cannot be done for short-duration
  observations, an increasingly common mode of observation.}.
Furthermore, at low frequencies ($<\sim$3 GHz), the ionosphere
dynamically rotates the observed plane of polarization on timescales
of seconds to hours, thus requiring estimation of the ionospheric
Faraday rotation measure (IFRM) -- a significant complicating factor.

These calibration complications are avoided if the data are taken in a
circular basis, for which the knowledge of the linear polarization
state of the calibrators is not needed for gain calibration, as the
linear polarized state of the radiation is entirely encoded in the
cross-hand correlations.  And while the circular component of the
source radiation is directly coupled to circular feeds, the fractional
circularly polarized flux, V/I, is always less than 1\%\ for calibrator
sources, and can be safely ignored in standard gain
calibration.

Unfortunately, producing wide-band dual-polarization signals accurately
proportional to the electric field's circular components is technically
difficult.  To our knowledge, all modern wide-band receiver systems
utilized in astronomy natively produce output voltages proportional to
the linear components of the electric field.  To obtain the desired
circular-basis voltages, an additional device which provides the
appropriate 90 degree phase shifts and signal combinations must be added
to the receiver system.  These devices are not perfect -- they add
noise, and are difficult (or impossible) to design for good performance
over very wide bandwidths. Thus, while there are strong reasons to
prefer a circular basis for calibration and imaging, there are also
strong engineering arguments for remaining in the linear basis.

It is important to emphasize that these statements also hold true for
self-calibration -- a critical calibration step needed for
high-fidelity imaging.  With linear systems, the self-calibration
operation will in general require source models in three Stokes
parameters (I, Q, and U), while for circular systems, only a Stokes
I  model is required.  We reemphasize also that at low frequencies,
ionospheric effects rotate the incoming plane of polarization on
timescales of seconds to hours.  For linear systems, these rotations
must be accounted for in both standard and self-calibration
operations, while for circular systems, these rotations can be
ignored\footnote{Of course, for polarimetric imaging, the ionospheric
  rotations must be accounted for in both systems.}.

We illustrate these points in the next section.

\section{Interferometric Response to Polarized Radiation}

Astronomers use the Stokes parameters\footnote{Originally defined by
  George Stokes in 1852 for use in optics, and extended to astronomy
  by Chandrasekhar in 1946.} (I, Q, U, and V) to describe the
polarization state of electromagnetic radiation. They are real
numbers, which greatly facilitate their use in visual presentations
for resolved sources.

The correlations provided by an interferometer are functions of the
two antennas' polarization states and the source polarization state.
The most general expression relating these correlations to the
antennas and source polarizations was published in 1963 by Morris,
Radakrishnan, and Seielstad\footnote{Ap.J. {\bf 139}, 551, 1963}. The
relationship can be written as:
\begin{eqnarray}
R_{mn}&=& G_{mn}\{[\cosdc - \sinsc]e^{-i\Delta\Psi}(\mathcal{I+V})/4\nonumber\\
&&\hspace{5mm} +[\cossc - \sindc]e^{-i\Sigma\Psi}(\mathcal{Q}+i\mathcal{U})/4\nonumber\\
&&\hspace{5mm} +[\cossc + \sindc]e^{i\Sigma\Psi}(\mathcal{Q}-i\mathcal{U})/4\nonumber\\
&&\hspace{5mm} +[\cosdc + \sinsc]e^{i\Delta\Psi}(\mathcal{I-V})/4\}
\end{eqnarray}
where $\mathcal{I}, \mathcal{Q}, \mathcal{U}$, and $\mathcal{V}$ are
the Stokes visibilities of the source structure in Stokes I, Q, U, and
V, $\chi_m$ is the antenna ellipticity for signal channel m,
$\Sigma\Psi = \Psi_m + \Psi_n$ and $\Delta\Psi = \Psi_m - \Psi_n$ are
the sum and difference of the two antennas' ellipticity position
angles for the correlation pair under consideration.  Note that the
antenna ellipticity position angle, $\Psi$, is in sky coordinates, so
will in general include both the antenna ellipticity angle $\psi$,
measured in the antenna coordinate frame, and the antenna parallactic
angle, $\Psi_p$.  $G_{mn}$ is a correlator-based gain for the pair
under consideration, and is normally factored into a product of the
two component antennas' gains: $G_{mn} = G_mG_n^*$.

This rather fearsome expression produces very simple results when pure
linear and pure circular basis antennas are considered.

\subsection{Pure Circular}
In this case, the antenna ellipticities for the right and left
polarizations are $\chi_r = -\pi/4$, and $\chi_l = \pi/4$, so the
visibilities become:
\begin{eqnarray}
  R_{rr} &=& G_{r1}G^*_{r2}(\mathcal{I} + \mathcal{V})/2\\
  R_{rl} &=& G_{r1}G^*_{l2}e^{-i2\Psi_p}(\mathcal{Q} + i\mathcal{U})/2\\
  R_{lr} &=& G_{l1}G^*_{r2}e^{i2\Psi_p}(\mathcal{Q} - i\mathcal{U})/2\\
  R_{ll} &=& G_{l1}G^*_{l2}(\mathcal{I} - \mathcal{V})/2
\end{eqnarray}
where $\Psi_p$ is the antenna parallactic angle, a known quantity, and
which for simplicity we have assumed is the same for the two antennas
(a condition which is very close to true for the VLA and
MeerKAT)\footnote{The antenna ellipticity angle dependence is absorbed
  into the gain phase factors.}.

Note that the parallel-hand responses are solely functions of Stokes
$\mathcal{I}$ and $\mathcal{V}$, while the cross-hand responses are
solely functions of Stokes $\mathcal{Q}$ and $\mathcal{U}$.  Since the
fractional circular polarization V/I is less than 1\%\ for all
calibrator sources, we can usually ignore the $\mathcal{V}$
visibility for the purposes of calibration, thus simplifying the
process of gain calibration, which then needs to know only the value
of $\mathcal{I}$, which for a point source at the phase center is a
real number, equal to the flux density.

From Equations 2 -- 5, the Stokes visibilities are found to be:
\begin{eqnarray}
\mathcal{I} &=& R'_{rr} + R'_{ll}\\
\mathcal{V} &=& R'_{rr} - R'_{ll}\\
\mathcal{Q} &=& e^{-i2\Psi_p}R'_{rl} + e^{i2\Psi_p}R'_{lr}\\
i\mathcal{U}&=& e^{i2\Psi_p}R'_{rl} - e^{-i2\Psi_p}R'_{lr}
\end{eqnarray}
where the prime indicates calibrated visibilities.  The $\mathcal{I}$
and $\mathcal{V}$ visibilities utilize only the parallel-hand
correlations, while the $\mathcal{Q}$ and $\mathcal{U}$ visibilities
utilize only the cross-hand correlations.

Contrast this simple situation to that for pure linear systems, shown
next.


\subsection{Pure Linear}

In this case, the antenna ellipticities are zero: $\chi = 0$, and the
antenna polarization ellipse position angles are aligned with the
dipole feeds, and are orthogonal.  Thus: $\Psi_v=\Psi_p, \Psi_h = \Psi_p +
\pi/2$, where we are assuming the antennas have az-el mounts.
Equation 1 then gives:
\begin{eqnarray}
  R_{vv} &=& G_{v1}G^*_{v2}[\mathcal{I} + \mathcal{Q}\cos(2\Psi_p) + \mathcal{U}\sin(2\Psi_p)]/2\\
  R_{vh} &=& G_{v1}G^*_{h2}[-\mathcal{Q}\sin(2\Psi_p) + \mathcal{U}\cos(2\Psi_p) + i\mathcal{V}]/2\\
  R_{hv} &=& G_{h1}G^*_{v2}[-\mathcal{Q}\sin(2\Psi_p) + \mathcal{U}\cos(2\Psi_p) - i\mathcal{V}]/2\\
  R_{hh} &=& G_{h1}G^*_{h2}[\mathcal{I} - \mathcal{Q}\cos(2\Psi_p) - \mathcal{U}\sin(2\Psi_p)]/2
  \end{eqnarray}
again assuming the antennas' parallactic angles are the same and the
antenna feeds are correctly aligned\footnote{Inclusion of these
  offsets is straightforward, but needlessly complicates the equations,
  and obfuscates the central points.}.

All four correlations are now functions of three Stokes' visibilities.
In particular, both linear visibilities $\mathcal{Q}$ and
$\mathcal{U}$ are involved in the two parallel-hand correlations.
Gain calibration using these high-signal correlations thus requires
knowledge of the source Q and U flux values -- which are not usually
known in advance.

These equations give solutions:
\begin{eqnarray}
\mathcal{I} &=&   R'_{vv} + R'_{hh}\\
i\mathcal{V} &=&  R'_{vh} - R'_{hv}\\
\mathcal{Q} &=&(R'_{vv} - R'_{hh})\cos(2\Psi_p) - (R'_{vh} + R'_{hv})\sin(2\Psi_p) \\
\mathcal{U} &=&(R'_{vv} - R'_{hh})\sin(2\Psi_p) + (R'_{vh} + R'_{hv})\cos(2\Psi_p)
\end{eqnarray}

In contrast to the pure circular case, all four correlations are in
general required to derive the $\mathcal{Q}$ and $\mathcal{U}$ Stokes
visibilities.

\section{Post-Correlation Basis Conversion}

We noted in the introduction that there are strong engineering reasons
to utilize linear-basis antennas, due to superior sensitivity,
bandwidth, and polarization purity.  However, for the processes of
calibration and self-calibration, circular-basis signals are much
preferred.  Can we have both?  Is it possible to convert the
linear-based visibilities provided by an interferometer to
circular-based visibilites?  The answer -- happily -- is yes, and is
useful to do so, provided certain conditions are met.

It is useful to first consider how modern receiver systems generate
outputs proportional to circular polarization.  To do this, the
voltage from the horizontal dipole is advanced (phase shifted) by 90
degrees, then added to the vertical dipole signal, creating a voltage
proportional to the right-circular voltage component.  Similarly, to
generate the left-circular component, the horizontal voltage must be
delayed (phase retarded) by 90 degrees, then added to the vertical
dipole signal.  The necessary 90-degree phase shifts must be the same
across the full bandwidth of the receiver, and the input signal
amplitudes must be balanced.  For the JVLA, at L, S, and C bands, a
microwave coaxial hybrid coupler, following the OMT, does the
combining. These devices are remarkably wideband -- a 2:1 BWR -- but
are also rather poorly balanced, producing output voltages with
significant frequency-dependent ellipticity.  At X, Ku, K and Ka
bands, a corrugated phase-shifter combined with a 45-degree rotation
transition before the OMT is employed.  These provide better performance
than the coaxial couplers, but have a narrower BWR -- 1.5:1.  At Q-band,
a waveguide septum polarizer providing native circularly polarized
outputs is used (and are utilized at all bands on the VLBA).  This is a
much simpler and highly compact device, but of much narrower BWR --
1.2:1.

In terms of the analytic signals $V_v$ and $V_h$, corresponding to the
linear-basis components of the astronomical signal as measured by an
alt-az antenna, the operations described above are given by a simple
matrix equation $V_{cir} = J V_{lin}$:
\begin{equation}
  \left(\begin{array}{*{1}{c}}
    V_r\\
    V_l
  \end{array}\right)
  =
  \left(\begin{array}{*{2}{c}}
    1&i\\
    1&-i\\
  \end{array}\right)
  \left(\begin{array}{*{1}{c}}
    V_v\\
    V_h
  \end{array}\right)
\end{equation}
where the 2 x 2 coupling matrix J is known as the Jones matrix, and
the column voltage vectors $V_{cir}$ and $V_{lin}$ are the antenna
responses for the circular and linear bases.

These simple equations hide an important caveat: For the conversion to
be useful, the amplitudes of $V_v$ and $V_h$ must be accurately
balanced, so that their ratio for an unpolarized source is near unity,
and the phase difference between them must be close to zero.  The
desired accuracy and stability of these limits will be discussed
later.

We represent the four correlations derived from a single baseline by a
column vector $R$, each element being defined by a complex
multiplication -- for example, $R_{rr} = V_{r1}V_{r2}^*$.  The
relation between the output circular-basis vector $R_c$ and the input
linear-basis vector $R_l$ is found from the outer product of the two
antennas' Jones matrices: $R = J_1 \otimes J^*_2$, giving:
\begin{equation}
  \left(\begin{array}{*{1}{c}}
    R_{rr}\\
    R_{rl}\\
    R_{lr}\\
    R_{ll}
  \end{array}\right)
  =
  \left(\begin{array}{*{4}{c}}
    1&-i&i&1\\
    1&i&i&-1\\
    1&-i&-i&-1\\
    1&i&-i&1
  \end{array}\right)
  \left(\begin{array}{*{1}{c}}
    R_{vv}\\
    R_{vh}\\
    R_{hv}\\
    R_{hh}
  \end{array}\right)
\end{equation}

It is of course possible to execute the inverse operation --
converting a circular-basis dataset into the linear-basis.  For
completeness, we give the inverse transform (ignoring scaling
factors):

\begin{equation}
  \left(\begin{array}{*{1}{c}}
    R_{vv}\\
    R_{vh}\\
    R_{hv}\\
    R_{hh}
  \end{array}\right)
  =
  \left(\begin{array}{*{4}{c}}
    1&1&1&1\\
    i&-i&i&-i\\
    -i&-i&i&i\\
    1&-1&-1&1
  \end{array}\right)
  \left(\begin{array}{*{1}{c}}
    R_{rr}\\
    R_{rl}\\
    R_{lr}\\
    R_{ll}
  \end{array}\right)
\end{equation}


\subsection{Implementation into \AIPS}

The basis conversions defined above have been implemented in the \AIPS\
task {\tt VH2RL}, and the results (from linear to circular) tested using
both VLA P band data, and MeerKAT UHF band data.

As noted above, it is important that good amplitude and phase matching
of the linear-basis data be obtained before combination.  In practical
terms, this means that a pure linear input signal gives the same
output amplitude and phase when observed oriented with the vertical
and horizontal dipoles, or, equivalently, observation of an
unpolarized source provides the same output amplitudes in both signal
channels.  Similarly, the signal phase paths, from sky to correlator,
must be the same for both polarization channels.  Combination of the
signals with deviations from these conditions will result in an
elliptically polarized signal, which will show up as D terms in the
calibration.  These conditions are the same as those imposed on analog
conversion of the voltage signals done at the antenna.  These points
are further developed in Section 5.

\section{Examples Using VLA and MeerKAT Data}

Here we show some examples, using data taken with these two arrays,
focusing on the calibration steps required to ensure proper amplitude
and phase matching prior to basis conversion.

\subsection{A VLA P-band Observation}
We show results from C-configuration data taken 13Dec2022 with the
VLA's linearly-polarized P-band system, as part of a lunar polarimetry
experiment.  At these low frequencies, there are very few sources with
high linearly-polarized emission, and only four for which we know the
intrinsic polarization properties\footnote{The details of our
  determination of these properties will be given in a following EVLA
  Memo.}.  These are the Moon, the compact and bright hotspots of
DA240 and 3C303, and the quasar 3C345\footnote{The planets Mars and
  Venus have known polarization properties, but are too small and weak
  to be used by the VLA at these low frequencies.}.

The VLA's P-band system nominally covers a 256 MHz-wide span of
frequencies, from 224 to 480 MHz.  Due to RFI, only half of this can
be usefully employed for science observations when the array is in the
D and C configurations. For the lunar polarization program data
utilized here, we retained two 64-MHz-wide bands which are relatively
free of RFI: the lower one spanning 288 -- 352 MHz, the upper spanning
384 -- 448 MHz.  For each of these blocks, the correlator was
configured to provide the data in four spectral windows, each 16 MHz
wide, with 1-second averaging and 125 kHz spectral resolution.  The
results shown here are from the lower band.

The goal of this program was to provide data for improving our
calibration procedures for low-frequency polarimetric observations.
The target object observed was the Moon, whose polarization properties
are exactly known, but which is also heavily resolved and of low
surface brightness, so cannot serve as any sort of calibrator.  The
primary gain calibration sources observed were 3C147 and 3C295, both
of which are very strong (over 50 Jy) and unpolarized.  Additionally,
the polarized sources DA240 (whose northern hotspot is nearly 20\%\
polarized) and the unresolved quasar 3C345, (whose nucleus is
polarized at about the 2 -- 4 \%\ level), and a nearby amplitude and
phase calibrator, J1021+2159, whose polarization state was unknown,
were included.

To enable successful conversion of the linear-basis visibilities to a
circular basis, the parallel-hand gains must be equalized, and any
phase offset between the two channels removed.  For the VLA, no
attempt is made to ensure these requirements are met via gain and
phase adjustments to the on-line systems, so they must be done
post-correlation to the linear-basis visibility data.

In what follows, we describe the minimal pre-conversion calibration
process.  The two essential operations are to equalize the antenna
gains of the two polarization channels, and to remove any phase
difference between them.  Both operations must be done as a function
of frequency, as spectral variations in amplitude and phase are
ubiquitous in these systems.

Following flagging to remove non-functional antennas and RFI
contamination, the data were decimated by a factor of 80 (10 in time,
and 8 in frequency), to 10-seconds and 1 MHz to facilitate calibration
and imaging.  We then performed the following operations:
\begin{enumerate}
    \item{{\tt BPASS}}. This program solves for the spectral variations
      in antenna amplitude and phase.  For the purpose of gain
      balancing, the flux density of the calibrator source is not needed
      -- we chose the default 1 Jy -- but it is critical that the
      calibrator source chosen be unpolarized.  For simplicity, we
      utilized a single scan of 3C147.  Because our goal is to balance
      the V and H gains, and we did not utilize the gain calibration
      program {\tt CALIB}, the {\tt BPASS} program must be used in its
      absolute mode -- no normalization of the resulting spectral
      amplitude solutions.  The phase solutions are with respect to the
      reference antenna, and will include the parallel-hand delays.

      Example solutions for three antennas are shown in the left hand
      panel of Fig.~\ref{fig:label1}.  The amplitude solutions show the
      characteristic drop-outs due to the attenuation at the edges of
      the four spectral windows.  The slopes in the phase solutions
      are due to differential delays between the the target and
      reference antenna, and are different for the two polarization
      channels. Application of the bandpass solutions removes spectral
      amplitude and phase variations for each polarization channel,
      leaving only the spectral phase differences between the two
      polarization channels of the reference antenna.  These are
      removed by the following two programs.
      \begin{figure}[ht!]
  \centerline{\hbox{
      \includegraphics[width=3.5in]{Lin-Bpass.eps}
      \includegraphics[width=3.5in]{Lin-XPhase.eps}}}
  \centerline{\parbox{6.5in}{
      \caption{\small {\bf Left Panel:} Showing the absolute bandpass
        solutions from 3C147, assuming unit flux density, for antennas
        ea16, ea17, and ea18.  The left column shows the vertical
        channel, the right column the horizontal channel. The upper
        section of each panel shows the phase, the lower section the
        amplitude.  The slope in the phase results from the delay of
        each antenna w.r.t. the reference, ea06. The notches in the
        amplitude gains are due to the anti-aliasing function of the
        WIDAR correlator. {\bf Right Panel:} Showing the cross-hand
        phase solutions, using DA240 as the calibrator source.  A 85
        degree slope, due to the cross-hand delay, has been removed
        via application of {\tt RLDLY}.}
      \label{fig:label1}}}
\end{figure}
    \item{{\tt RLDLY}}. The delay removal included in the {\tt BPASS}
      operation is not sensitive to the delays between the two
      polarization channels, which will appear as a phase slope in the
      cross-hand phase spectra.  Due to this, it proves generally
      necessary to utilize {\tt RLDLY} to find and remove this delay.  In
      principle, a polarized source should be used, but in practice, due
      to the small but significant misorientations of the dipoles, a
      strong unpolarized source serves as well.  We used the unpolarized
      source 3C147 for this determination.  The resulting delay was
      3.7 nsec, corresponding to a phase slope of 85 degrees over the
      64 MHz total bandwidth.
    \item{{\tt VHDIF}}. This program removes the residual cross-hand
      (V-H) phase of the reference antenna.  This phase is calculated
      from the expression $\tan\phi = -V/U$, so requires the observation
      of a strongly polarized compact source\footnote{The curious reader
      might wonder why {\tt VHDIF} cannot also remove the cross-hand
      delay, thus bypassing the need for {\tt RLDLY}\@.  The answer is
      that {\tt VHDIF} can, but gets into trouble when the absolute
      value of the cross-hand phase is close to 90 degrees, since the
      value of $\mathcal{U}$ is then close to zero.  A large delay value
      greatly increases the chance that some channels will be close to
      this condition.  If, by bad luck, the cross-hand phase for all
      channels is close to 90 degrees following {\tt RLDLY}, one must
      insert with {\tt CLCOR} an offset phase to the H channel, to
      enable {\tt VHDIF} to find a good solution.}. We used the eastern
      hotspot of DA240, which provides nearly 500 mJy of polarized flux.
      The position angle of the polarized signal does not need to be
      known for this operation.  However, two solutions, differing by
      180 degrees, are possible, and imaging of the highly polarized
      calibration source is needed to determine which is the correct
      value\footnote{Application of the wrong solution will generally
      result in no visible polarized emission.}. {\tt VHDIF} includes a
      {\tt DOINVERS} adverb, which adds 180 degrees to the solutions. We
      refer the reader to Section 5.3 of EVLA Memo \#207 for
      details. The resulting (correct) cross-hand phase is shown in the
      right-hand panel of Fig.~\ref{fig:label1}.
\end{enumerate}
We emphasize that the use of a single observation to perform the
amplitude and phase balancing requires the signal transmission system
having good gain stability over the length of an observation.  In
fact, the VLA's amplitude and phase gains are usually very stable over
time.  Significant variations in the V-to-H gain ratios over time can
be corrected by utilizing multiple observations of a strong, unpolarized
source\footnote{Note that poor atmospheric phase stability is not a
  problem at this stage, since these fluctuations are the same in both
  polarization channels.}.

At this point, the linear-basis data are sufficiently balanced in
amplitude and phase to enable accurate conversion to a circular basis.
This was done by utilizing the new \AIPS\ program {\tt VH2RL}\@.
Following this, the standard calibration routine for a circular-basis
dataset was followed:
\begin{enumerate}
\item{{\tt FRING}}.  The delay solutions are all very small -- of order
  0.1 nsec -- since the {\tt BPASS}, {\tt RLDLY}, and {\tt VHDIF}
  operations, used to balance the linear gains, have already removed all
  delays.  The phase solutions, for four antennas, are shown in the
  left-hand panel of Fig.~\ref{fig:label2}. These phase solutions show
  significant temporal variations, since the phase solutions in the
  linear-basis data were made from a single bandpass observation.  The
  phase fluctuations increase with increasing baseline, as expected from
  P-band observations in C-configuration due to ionospheric
  irregularities.
    \begin{figure}[ht]
  \centerline{\hbox{
      \includegraphics[width=3.5in]{Circ-Phase.eps}
      \includegraphics[width=3.5in]{Circ-Amp.eps}}}
  \centerline{\parbox{6.5in}{
      \caption{\small {\bf Left Panel:} Showing the R phase gain
        solutions for four antennas, using all four calibration
        sources.  Note the phase stability significantly worsens with
        baseline length.  The sources are color-coded: Red shows
        3C147, blue for 3C295, green for the lunar calibrator
        J1021+2159.  The second observation of 3C147 shows zero phase,
        as it was this scan used to calibrate the bandpasses in the
        linear-based data.  {\bf Right Panel:} Showing the amplitude
        gain solutions from the four calibration sources. Most of the
        scatter is due to the low flux density of the gain calibrator
        J1021+2159.}
      \label{fig:label2}}}
  \end{figure}
\item{{\tt BPASS}}.  The solutions are near unity amplitude (deviations
  less than 0.5\%) and zero phase (deviations less than 0.5 degrees), as
  the spectral variations were removed in the linear-basis amplitude
  and phase balancing.
\item{{\tt CALIB}}. All four calibrator sources can now be included for
  detailed gain solutions, as their linear polarizations will not
  affect the parallel-hand gains. The resulting gains -- following
  determination of their spectral flux densities-- are shown in the
  right-hand panel of Fig.~\ref{fig:label2} for these four antennas.
  The values are not near unity, as a 1-Jy flux was assumed in the
  linear {\tt BPASS} calibration\footnote{Since these solutions are in
    voltage units, the source flux density is equal to the square of
    the values -- $\sim$53 Jy.}.
\item{{\tt RLDLY}}. This step removes the cross-hand delay between the R
  and L channels. The observed delay is less than 0.5 nsec, as the
  cross-hand delays were removed in the linear calibrations.
\item{{\tt PCAL}}. This solves for the cross-hand leakages. For low
  frequency applications, it is necessary to use unpolarized sources, as
  the program cannot separate a time-varying (due to ionospheric Faraday
  rotation) astronomical polarized signal from a time-invariant
  receiver leakage\footnote{In the absence of ionospheric rotations,
    the program can solve for the astronomical source polarizations
    and the instrumental polarizations, provided the observations span
    a sufficient range in parallactic angle.}.  We used 3C147 and
  3C295 for this example. Because the linear feeds provide quite high
  polarization purity, we expect the resulting circularly converted
  data to also show high purity.  Examples for three antennas are
  shown in the left-hand panel of Fig.~\ref{fig:label3}, which are
  indeed quite low -- typically 2\%.
\begin{figure}[t]
  \centerline{\hbox{
      \includegraphics[width=3.5in]{Circ-Pol.eps}
      \includegraphics[width=3.5in]{Circ-XPhase.eps}}}
  \centerline{\parbox{6.5in}{
      \caption{\small {\bf Left Panel:} Showing the cross-polarization
        solutions for three typical antennas, with RCP on the left,
        LCP on the right. In each of the six panels, the phase
        solutions are on top, amplitude solutions below.  All
        amplitude values are small -- typically 2\%.  {\bf Right
          Panel:} Showing the cross-hand phases required to align the
        source polarization position angles.}
      \label{fig:label3}}}
\end{figure}
\item{{\tt TECOR}}.  The observed position angles for polarized emission
  at these low frequencies can be greatly rotated by Faraday rotation
  in the ionosphere.  For circular-basis data, these rotations show up
  as phase offsets in the cross-hand correlations.  It is important
  that the cross-hand correlations take account of these rotations
  prior to the use of any cross-hand (RL and LR correlators) data.
  This program calculates and applies these phases.
\item{{\tt RLDIF}}. This program adjusts the R-L phase to obtain the
  correct alignment of the EVPA.  This operation requires knowledge of
  the EVPA, as a function of frequency, of one strongly polarized
  source.  We used DA240, for which the source RM is known to be 3.22
  $\mathrm{rad/m}^2$, and whose intrinsic (zero-wavelength) EVPA is
  $-114$ degrees\footnote{Refer to the following EVLA Memo for the
    derivation of these values.}.  The resulting phase solutions are
  shown in the right-hand panel of Fig.~\ref{fig:label3}.
\end{enumerate}
In the preceding description, it will be noted that the {\tt FRING},
{\tt BPASS}, and {\tt RLDLY} functions all returned unit amplitude and
zero phase values -- a result of the preceding gain balancing of the
linear-basis data. Hence executing the programs was not actually needed
here.  Indeed, had we utilized the known flux of 3C147 in the
linear-basis calibration, and solved for gain as a function of time
using that source, the {\tt CALIB} operation in the circular-basis
calibration would also have been superfluous.  Presuming good initial
gain calibration of the linear-basis data, only the {\tt PCAL}, {\tt
  TECOR}, and {\tt RLDIF} operations are needed in the circular-basis
data\footnote{Note, however, that if it is desired to utilize a known
  linearly-polarized calibrator in the gain calibration process of the
  linear-basis data, {\tt CALIB} will need to be run.}.

Imaging results from these procedures will be shown in a following memo.

\subsection{MeerKAT Data Example}

The MeerKAT array's UHF-band utilizes a low-frequency,
linearly-polarized receiver system covering $\sim$540 -- 1100 MHz.  We
have utilized a lunar observation taken with this array on 22Jun2021
to test our conversion routines.  Unlike the VLA's P-band system, the
MeerKAT UHF receivers have an on-board calibration system which
removes nearly all the cross-hand delay.  This simplifies the
processing needed for conversion to a circular basis.

These MeerKAT data are much less affected by RFI than the VLA's P-band
system, so we show here the entire bandwidth, spanning 544 through
1088 MHz.  The MeerKAT correlator provides the data in a single
`spectral window'. As \AIPS\ expects data broken into multiple spectral
windows, we converted the data to a 16-SPW dataset, with each SPW
spanning 34 MHz.  Following essential flagging, the data were
decimated to 24 second averaging, and 2.125 MHz channel width for
faster processing.

The {\tt BPASS} operation was done on a single scan of J1939-6342, an
unpolarized and unresolved standard flux calibrator.  The flux density
was known in advance, and was utilized for the solutions.  The
resulting bandpasses, for three selected antennas, are shown in the
left-hand panel of Fig.~\ref{fig:MeerBP}.
\begin{figure}[ht]
  \centerline{\hbox{
      \includegraphics[width=3.5in]{Meer-Lin-Bpass.eps}
      \includegraphics[width=3.5in]{Meer-Lin-VHPhase.eps}}}
  \centerline{\parbox{6.5in}{
      \caption{\small {\bf Left Panel:} Showing the absolute bandpass
        solutions from J1939-6342, for antennas m000, m001, and m003.
        The left column shows the vertical channel, the right column
        the horizontal channel. The upper section of each panel shows
        the phase, the lower section the amplitude.  The slope in the
        phase results from the delay of each antenna w.r.t. the
        reference, m002. {\bf Right Panel:} Showing the cross-hand
        phase solutions, using 3C286 as the calibrator source.  There
        is only a small residual cross-hand delay in these data.}
      \label{fig:MeerBP}}}
\end{figure}
As the cross-hand delays in these data had already been removed by the
on-line system, the {\tt RLDLY} program, needed for the VLA data, was
not required.  We thus went straight to the {\tt VHDIF} operation,
utilizing 3C286, to remove any residual cross-hand phase.  The results
are shown in the right-hand panel of Fig.~\ref{fig:MeerBP}.

The data were then transformed to a circular basis via {\tt VH2RL}, and
the standard circular calibration procedures performed.  As with the VLA
data, the {\tt FRING} and {\tt BPASS} operations provided close to zero
phase and unit amplitude corrections.  Because the gain normalization
operation ({\tt BPASS}) in the linear dataset utilized the known flux
density of the calibrator, the {\tt CALIB} operation on the circulars
provided near unit amplitude solutions as well.  Examples of the phase
and amplitude solutions are shown in Fig.~\ref{fig:MeerGains}.
\begin{figure}[ht!]
  \centerline{\hbox{
      \includegraphics[width=3.25in]{Meer-Cir-Phase.eps}
      \includegraphics[width=3.25in]{Meer-Cir-RAmp.eps}}}
  \centerline{\parbox{6.5in}{
      \caption{\small {\bf Left Panel:} Showing the phase gain solutions
        for four antennas, using all three calibration sources.  The
        sources are color-coded: Red shows 3C286, blue for the nearby
        lunar calibrator J1733-1304, and green for the flux calibrator
        J1939-6342.  The first observation of J1939-6342 shows zero
        phase, as it was this scan used to calibrate the bandpasses in
        the linear-based data.  {\bf Right Panel:} Showing the
        amplitude gain solutions from the three calibration sources.
        The values are near unity, as the known flux density of
        J1939-6342 was used in the gain normalization process.}
      \label{fig:MeerGains}}}
\end{figure}
The notable phase offsets for 3C286 and J1939-6342 are likely due to
incorrect baseline coordinates -- these sources are both at very
different declinations than the moon and its nearby calibrator.  The
{\tt RLDLY} operation was performed next -- a small delay of 0.31 nsec
was removed from the data.  Following this, {\tt PCAL} was run, and the
results are shown in the left-hand panel of Fig.~\ref{fig:MeerLinPol}.
\begin{figure}[ht!]
  \centerline{\hbox{
      \includegraphics[width=3.5in]{Meer-Cir-XPol.eps}
      \includegraphics[width=3.5in]{Meer-Cir-RLPhase.eps}}}
  \centerline{\parbox{6.5in}{
      \caption{\small {\bf Left Panel:} Showing the cross-polarization
        solutions for three typical antennas, with RCP on the left,
        LCP on the right. In each of the six panels, the phase
        solutions are above, amplitude solutions below.  All amplitude
        values are very small -- typically 0.5\%.  {\bf Right Panel:}
        Showing the cross-hand phases required to align the source
        polarization position angles, using the polarized calibrator
        3C286. Because these data were taken at low frequency, the
        Faraday rotation correction program TECOR was run before
        {\tt RLDIF}\@.}
      \label{fig:MeerLinPol}}}
\end{figure}
It will be noted that the cross-polarizations in these MeerKAT data
are considerably smaller than those on the VLA data.  This is true for
nearly all the MeerKAT antennas, which appear to have been designed to
prevent significant coupling between the polarized channels.

The final operation was running {\tt RLDIF}, using the known polarized
calibrator 3C286\@.  The required R-L phase solutions are shown in the
right-hand panel of Fig.~\ref{fig:MeerLinPol}.  Examples of the
resulting images will be shown in a following memo.

\section{Inclusion of Antenna Cross-Polarization}

We have noted earlier the requirement that the linearly-polarized data
be accurately amplitude and phase balanced prior to conversion to a
circular basis.  Here we discuss the preferred accuracy.

Sadly, antennas are not purely polarized. Inevitably, some of the
signal in one polarization channel couples into the other channel.  In
terms of the antenna's ellipticity, there is a small deviation from
the pure state.  Thus, for a linearly-polarized antenna, the antenna
ellipticity has a small value $\chi \ll 1$, while for a
circularly-polarized antenna, the ellipticity is slightly less (in
absolute value) than $\pi/4$.

In the standard treatments (e.g. Thompson, Moran, and Swenson), these
leakage signals are quantified by D terms\footnote{Analysis shows
  that these terms are related to the antenna ellipticity and position
  angles by: $D_r = \tan\beta_re^{i2\phi_r}$, and $D_l =
  \tan\beta_le^{-i2\phi_l}$, where $\beta_r = \pi/4 + \chi_r$, and
  $\beta_l = \pi/4 - \chi_l$, and $\psi_r$ and $\psi_l$ are the
  antenna ellipse major axis orientations, referenced to the antenna
  frame.}.  So, for example, they write, for the relation between the
observed visibility, $V^{obs}$, and the true visibility, $V$:
\begin{equation}
  \left(\begin{array}{*{1}{c}}
    V_v^{obs}\\
    V_h^{obs}
  \end{array}\right)
  =
  \left(\begin{array}{*{2}{c}}
    1&D_{hv}\\
    D_{vh}&1\\
  \end{array}\right)
  \left(\begin{array}{*{1}{c}}
    V_v\\
    V_h
  \end{array}\right)
\end{equation}
where the 2 x 2 Jones matrix describes the leakage of signals from one
polarization channel to the other.  Again forming the outer product
operation, we obtain the relations between the observed correlated
products, and the true products:
\begin{equation}
  \left(\begin{array}{*{1}{c}}
    R_{vv}^{obs}\\
    R_{vh}^{obs}\\
    R_{hv}^{obs}\\
    R_{vv}^{obs}
  \end{array}\right)
  =
  \left(\begin{array}{*{4}{c}}
    1&D^*_{hv2}&D_{hv1}&D_{hv1}D_{hv2}^*\\
    D^*_{vh2}&1&D_{vh1}D_{vh2}^*&D_{hv1}\\
    D_{vh1}&D_{vh1}D_{hv2}^*&1&D_{hv2}^*\\
    D_{vh1}D^*_{vh2}&D_{vh1}&D_{vh2}^*&1
  \end{array}\right)
  \left(\begin{array}{*{1}{c}}
    R_{vv}\\
    R_{vh}\\
    R_{hv}\\
    R_{hh}
  \end{array}\right)
\end{equation}

The leakages in the linear-basis correlations $R_l$ will show up as D
term leakages in the circular converted correlations $R_c$. In
general, analysis shows that the amplitudes of the resulting
circular D terms will be similar to those in the originating
linear D terms.  Thus, the $\sim 2$\%\ amplitudes of these terms
shown in the preceding section in VLA data are thus believed to
reflect the purity of the linear systems.

There is one special exception to the preceding statements -- the
leakages in the linear basis data due to differential antenna
rotation.  This may be caused by mis-orientation of the linear feed
dipoles, or by differential parallactic angles.  This situation is
worthy of a more detailed analysis.  Presume an antenna's orthonal
feeds or receiver are misoriented by an angle $\theta$ with respect to
the reference direction.  For the linear-basis Jones matrix, we then
write for the observed output voltages
\begin{equation}
  \left(\begin{array}{*{1}{c}}
    V^{obs}_v\\
    V^{obs}_h
  \end{array}\right)
  =
  \left(\begin{array}{*{2}{c}}
    \cos\theta&\sin\theta\\
    -\sin\theta&\cos\theta\\
  \end{array}\right)
  \left(\begin{array}{*{1}{c}}
    V_v\\
    V_h
  \end{array}\right)
\end{equation}
Comparison to Equation 21 shows that the D terms are real, with leakage
amplitude equal to $\sin\theta$ which, for the usual situation where
the misorientations are small, is approximately equal to $\theta$.  A
CCW misalighnment gives a positive D term, a CW misalignment a
negative term\footnote{A more careful analysis takes into account the
  $\cos\theta$ loss term in the matrix, which after gain calibration
  results in the D term amplitudes being given by $\tan\theta$,
  which in the small-angle limit gives the same result cited above.}.

For the circular case, Equation 21 after insertion into the defining
conversion matrix (Equation 18), and forming the circular products,
produces a very different relation:
\begin{equation}
  \left(\begin{array}{*{1}{c}}
    V^{obs}_r\\
    V^{obs}_l
  \end{array}\right)
  =
  \left(\begin{array}{*{2}{c}}
    e^{-i2\theta}&0\\
    0&e^{i2\theta}\\
  \end{array}\right)
  \left(\begin{array}{*{1}{c}}
    V_r\\
    V_l
  \end{array}\right)
\end{equation}
-- the off-diagnonal D terms are all zero!  This special case of
misaligned dipoles transforms the rotation into phase offsets in the
parallel hand voltages in the circular basis\footnote{This is easy to
  understand: Rotating a circularly polarized antenna changes the
  phase of a circularly polarized signal, but not the amplitude.}.
This result illustrates a great advantage of the circular-basis system
-- all rotations, including receiver misalighments, parallactic angle
orientations, and ionospheric rotations, transform to phase rotations
in the data, leaving the correlation amplitudes unchanged.

We illustrate these points in Figure~\ref{fig:D-terms}.  Here we show
the calculated D terms for two VLA antennas with the original
linear-basis data (left panel) and the corresponding terms when the
data are converted to a circular basis (right panel).  The two
antennas are chosen to illustrate the effect of a large positive
rotation (ea03, in the upper panels), and a large negative rotation
(ea07, in the lower panels), with respect to the reference antenna,
ea06.  According to the preceding analysis, antenna rotation --
relative to the orientation of the reference antenna -- is manifested
in the linear system by a real D-term whose amplitude is given by
$\sin\theta \sim \theta$, while in the circular system, it shows up as
a phase term in the parallel-hand gains, which is accounted for the
the gain calibration process.
\begin{figure}[ht]
  \centerline{\hbox{
      \includegraphics[width=3.7in]{Lin-VDTerm.eps}
      \includegraphics[width=3.7in]{Cir-DTerms.eps}}}
  \centerline{\parbox{6.5in}{
      \caption{\small The calculated cross-polarizations (D terms)
        for two VLA antennas with significant dipole misorientations
        w.r.t. the reference, for the originating linear-basis data,
        and the converted circular-basis data. {\bf Left Panel:} Ea03
        (top) and ea07 (bottom) for the linear-basis.  For each, the
        imaginary part is plotted on top, the real part below. Note
        the imaginary parts for both antennas are very small,
        typically 2\%, while the real parts are large, typically 10\%,
        and constant in frequency, as expected for a rotation. {\bf
          Right Panel:} The corresponding D-terms when the data are
        transformed to the circular basis, for the R polarization.
        All values are now small, less than $\sim2$\%.}
      \label{fig:D-terms}}}
\end{figure}

In Fig.~\ref{fig:D-terms}, each of the four panels shows the imaginary
(top) and real (bottom) values of the complex D leakage terms.  The
two left panels show the leakages for the V polarization for antenna
ea03 (above) and ea07 (below).  The two right panels show the
corresponding leakages for these antennas after conversion to
circular.  The linear-based terms show small imaginary, but large real
components, reflecting their origin as misorientations of the antenna
dipoles.  The values of +0.1 and -0.1 suggest rotations of about +6
and -6 degrees, respectively.  This interpretation is not unique --
the coupling could be due to processes further down the signal chain.
However, the fact that the real parts are nearly constant over the
full frequency range, and that the values for the H polarization
(not shown) are of the same amplitude and of opposite sign, provide
very strong evidence in favor of the cause being due to dipole
misorientation\footnote{Plus the fact that visual observation of these
  feeds show misorientations of a few degrees are common!}.

The right-hand panels show, on the same scale, the corresponding D
terms in the circular system for the converted R polarization.  All
values are very small -- typically $\sim$1\%, showing that the
circularly-converted data are very nearly pure .  The effects of the
antenna mis-orientations are moved to the phases, and have been
removed from the parallel-hand correlations as part of the regular
gain calibration.  Indeed, the cross-polarization terms are now so
small that it may suffice to not calculate them at all for reasonable
polarimetry.

We noted in the last section that the MeerKAT UHF-band linear and
circular-converted data show much smaller cross-polarizations than the
VLA P-band data.  Both the distribution in rotations among the MeerKAT
antennas is smaller, and the residual circular D-terms are smaller
than those in the VLA's P-band data.  Examples of this are shown in
Fig.~\ref{fig:MeerXPol} for the MeerKAT data.  Note that the amplitude
scales are a factor of six smaller than the corresponding values for
the VLA.
\begin{figure}[ht]
  \centerline{\hbox{
      \includegraphics[width=3.7in]{Meer-Lin-XPol.eps}
      \includegraphics[width=3.7in]{Meer-Cir-CrossPol.eps}}}
  \centerline{\parbox{6.5in}{
      \caption{\small The calculated cross-polarizations (D terms)
        for two MeerKAT antennas (m008, top, and m032, bottom) with
        significant dipole misorientations w.r.t. the reference, in
        the linear and circular bases.  For all panels, the imaginary
        part is plotted on top, the real part below. {\bf Left Panel:}
        The V cross-polarization for the linear basis. Note the
        imaginary parts for both antennas are very small, typically
        0.05\%, while the real parts are larger, typically 2\%, and
        constant in frequency, as expected for a rotation.{\bf Right
          Panel:} The same antennas as seen in a circular basis, for
        the R polarization.  All values are now very small, less
        than $\sim0.05$\%.}
      \label{fig:MeerXPol}}}
\end{figure}

The $\sim$0.02 value for the imaginary component in the figure for
m007 informs us that a feed rotation of $\sim 1$ degree is present --
this is the largest value we could find in the solutions.  The spread
in the MeerKAT antenna feed orientations is much smaller than that for
the VLA!  Examination of the right panel shows that the typical values
for both real and imaginary parts of the circular D-terms is nearly
negligible -- less than 0.005 for nearly all channels.  This is so
good that it seems likely that antenna polarization calibration is
hardly needed at all!

\section{Summary}

If, as we believe, wide-band receivers providing linear-basis signals
are more sensitive, and of higher polarization purity than those after
conversion to a circular basis by an antenna-based imperfect analog
device, it is clearly preferable to utilize linear-basis data for
astronomical observations.  However, there are significant
complications in gain calibration of these linear-basis signals, both
in regular gain calibration, and in self-calibration of polarized
target sources.  The basic complications in utilizing linear-basis
data directly for calibration arise from the need to know the
polarization state of the calibrators.  Not only is this not known in
advance for the more than 1000 calibrators, but this state is variable
on timescales of days to years for nearly all the commonly-used
calibrators.  Further complicating this at low frequencies are the
highly variable (seconds to hours) rotations induced by Faraday
rotation in the Earth's ionosphere.

All these complications can be bypassed by utilizing data in a
circular basis, as the linear-polarization information is solely
located in the cross-hand correlations.  All calibration operations
utilizing the high-signal parallel-hand data can then proceed without
concern for source polarization characteristics.   These advantages
also apply to the vital process of self-calibration of target sources.

We have shown in this memo how the nearly-pure linear-based data can be
easily transformed to equally pure circular-based data, following
straightforward amplitude and phase balancing.  The resulting
circular-based data then enable superior calibration, utilizing all
potential calibration sources, without regard to their unknown
polarization states, and imaging.  A new \AIPS\ task {\tt VH2RL}
generates the converted data, which we have tested with VLA P-band, and
MeerKAT UHF-band data.  Results from these, and other, datasets will be
reported in an upcoming memo.

\section{Acknowledgements}
We thank Oleg Smirnov and Wes Grammar for useful comments and corrections.

\end{document}
