National Centre for Radio Astrophysics
Tata Institute of Fundamental Research

Internal Technical Report: xxxx

A detailed study of the polarisation convention of the Giant Metrewave Radio
Telescope (GMRT)

Poonam Chandra, S. Suresh Kumar, Sanjay Kudale, Devojyoti Kansabanik, Barnali
Das, Preeti Kharb, Silpa Sasikumar & Biny Sebastian
Corresponding Author’s email id<EMAIL_ADDRESS>and
<EMAIL_ADDRESS>
2023

Objective: To check the polarisation convention of the GMRT and understand
its causes and implications on the scientific outcome

##  1 Defining the problem

This work aims to investigate the polarisation convention of the u/GMRT and
understand whether these follow the standard IAU/IEEE convention. These tests
have been done on several strong and highly polarised pulsars with known
polarisation properties at GMRT wavelengths. Before going into the tests and
experiments, will discuss some basic ideas related to polarisation,
polarisation convention, and polarisation calibration in Section 2.

##  2 Introduction to Polarimetry

###  2.1 Polarisation of electromagnetic radiation

Electromagnetic (EM) waves from astronomical objects have preferred
orientation or rotation of their electric field vectors; known as
polarisation. The phenomenon of polarisation arises as a consequence of the
fact that EM wave behaves as a two-dimensional transverse wave. Polarisation
is a fundamental property of EM radiation and provides a rich source of
information on the physical properties, such as magnetic fields, asymmetry in
the astronomical phenomenon, etc. In radio bands, many processes can produce
polarisation, which can be either circular, linear or elliptical in nature.
Generally, radio sources have low fractional polarisation, but some coherent
emission mechanisms may produce up to 100% polarisation.

The polarisation of an EM wave is defined by the motion of its electric field
vector as a function of time within a plane perpendicular to the direction of
propagation. This plane is known as the plane of polarisation. If the EM
radiation propagates in the $z$-direction, the plane of polarisation will be
the $xy$ plane, and the electric field $\boldsymbol{E}$ in trigonometric
notation can be written as

$\boldsymbol{E}(t,z)=\boldsymbol{E}(0,0)\cos(\omega t-kz-\phi)$ (1)

where $t$ is the time, $\omega$ is the angular frequency, $k=\omega/c$ is the
wave vector, and $\phi$ is an arbitrary phase. The $\boldsymbol{E}(t,z)$ can
be decomposed in two orthogonal components ($x$ and $y$).

$\begin{split}E_{x}(t)&=E_{x}(0)\cos(\omega t-\phi_{1})\\\
E_{y}(t)&=E_{y}(0)\cos(\omega t-\phi_{2})\end{split}$ (2)

The above notations are trigonometric notations of the electric field. They
can also be written in complex notations as,

$\begin{split}E_{x}(t)&=E_{x}(0)e^{i(\omega t-\phi_{1})}\\\
E_{y}(t)&=E_{y}(0)e^{i(\omega t-\phi_{2})}\end{split}$ (3)

The angle between the plane of oscillation of the electric vector and the
plane of polarisation is called the polarisation angle (PA). For elliptical
polarisation, PA is the position angle of the major axis of the polarisation
ellipse (see Fig. 1).

###  2.2 Linear and circular polarisation

Polarisation of a EM wave can be defined by $E_{x}(0)$, $E_{y}(0)$, $\phi_{1}$
and $\phi_{2}$. The tip of the electric field vector generally follows an
elliptical trajectory in the $xy$ plane; accordingly, the wave is denoted as
elliptically polarised. Hence elliptical polarisation is the most general
state of polarisation and linear and circular polarisation are special cases.

Linear polarisation occurs when the polarisation ellipse degenerates into a
line, that is the phase difference $\delta=\phi_{2}-\phi_{1}=0$. For
$\phi_{1}=\phi_{2}=0$, linear polarisation (without any loss of generality) is

$\begin{split}E_{x}(t)&=E_{x}(0)\cos(\omega t)\\\
E_{y}(t)&=E_{y}(0)\cos(\omega t)\end{split}$ (4)

The orientation of the ellipse in the plane of polarisation $xy$ plane depends
only on ratio of $E_{x}(0)$ and $E_{y}(0)$, which are constant in time. Thus
in other words the polarisation angle is constant with time for linear
polarisation.

In circular polarisation, the EM field vector has a constant magnitude and is
rotating at a constant rate in a plane perpendicular to the direction of the
wave. Thus circular polarisation occurs when the polarisation ellipse
degenerates into a circle, that is $\phi_{2}=\phi_{1}\pm\pi/2$. For
$\phi_{1}=0$, circularly polarised light is,

$\begin{split}E_{x}(t)&=E_{x}(0)\cos(\omega t)\\\ E_{y}(t)&=\pm
E_{y}(0)\sin(\omega t)\end{split}$ (5)

The tip of the electric field vector moves circularly in the $xy$ plane with a
frequency, $\omega$, and the sign of $E_{y}(t)$ determines the sense of
polarisation. A circularly polarised wave can rotate in one of two possible
senses: clockwise or counterclockwise.

Figure 1: Position angle of polarised light. X represents towards North and Y
represent towards East.

Polarisation angle of the plane of linear polarisation on the sky is measured
from North towards East (Fig 1). In this figure, the major axis of the
polarisation ellipse is oriented at an angle $\chi$ with respect to the $X$
axis ($X$ axis represents towards North). The polarisation angle $\chi$ is
zero at north; and $\chi$ positive while measured from North towards East.
Thus represented on an image of the sky, a line segment representing
polarisation rotates counterclockwise as $\chi$ increases, and $\chi=0^{o}$
corresponds to a vertical orientation 111Researchers studying the polarisation
of the Cosmic Microwave Background (CMB) have been defining polarisation angle
to increase clockwise on the sky. This effectively swaps the sign of Stokes U
and confuses astronomers studying Galactic polarisation using CMB satellite
data..

###  2.3 Stokes parameters

The polarisation state of an EM wave is be characterised by the amplitudes
$E_{x}(0)$ and $E_{y}(0)$ and the phase difference $\delta=\phi_{2}-\phi_{1}$.
In general, astronomical observations deal with light intensities rather than
with field amplitudes, thus it is convenient to quantify polarisation via
characteristic intensities, i.e. the Stokes parameters $I$, $Q$, $U$, and $V$.
Here $I$ is the total intensity, $Q$ and $U$ represent linear polarisation,
and $V$ quantifies the amount of circular polarisation.

Stokes parameters are related as,

$I^{2}=Q^{2}+U^{2}+V^{2}$ (6)

The linearly polarised intensity $L$ can be defined as

$L=\sqrt{(Q^{2}+L^{2})}$ (7)

The polarisation angle $\chi$, is defined in terms of Stokes Q and U as,

$\chi=\frac{1}{2}\tan^{-1}\left(\frac{U}{Q}\right);\,\,0^{o}\leq\chi\leq
180^{o}$ (8)

The Stokes parameters are defined in terms of auto and cross-correlations of
the $x$ and $y$ components of the EM field vectors, i.e. $E_{x}(t)$,
$E_{y}(t)$ in a Cartesian frame whose $z$ axis is along the direction of
propagation. In trigonometric co-ordinate system Stokes parameters are defined
as,

$\displaystyle I$ $\displaystyle=\langle|E_{x}|^{2}+|E_{y}|^{2}\rangle$ (9)
$\displaystyle Q$ $\displaystyle=\langle|E_{x}|^{2}-|E_{y}|^{2}\rangle$ (10)
$\displaystyle U$ $\displaystyle=2\langle|E_{x}||E_{y}|\cos\delta\rangle$ (11)
$\displaystyle V$ $\displaystyle=2\langle|E_{x}||E_{y}|\sin\delta\rangle$ (12)

In complex exponential notation defined in Equations 2 and 3, the Stokes
parameters are defined as,

$\displaystyle I$ $\displaystyle=\langle E_{x}E_{x}^{*}+E_{y}E_{y}^{*}\rangle$
(13) $\displaystyle Q$ $\displaystyle=\langle
E_{x}E_{x}^{*}-E_{y}E_{y}^{*}\rangle$ (14) $\displaystyle U$
$\displaystyle=\langle E_{x}E_{y}^{*}+E_{y}E_{x}^{*}\rangle$ (15)
$\displaystyle V$ $\displaystyle=-i[\langle
E_{x}E_{y}^{*}-E_{y}E_{x}^{*}\rangle]$ (16)

###  2.4 Jones and Muller matrices

Polarised light can be represented by a Jones vector, $\mathbf{e}$, and linear
optical elements can be represented by Jones matrices, $J$. More precisely,
the Jones vector represents the amplitude and phase of the electric field in
the $x$ and $y$ directions. One can then define the Jones vector as,

$\mathbf{e}\equiv\begin{bmatrix}E_{x}(0)e^{i\phi_{1}}\\\
E_{y}(0)e^{i\phi_{2}}\end{bmatrix}$ (17)

For example, the Jones vector for linear polarised light in the $x$ direction
is $\begin{pmatrix}1\\\ 0\end{pmatrix}$, Jones vector for linearly polarised
at 45° from the $x$-axis is $\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\
1\end{pmatrix}$, and Jones vector for right-hand circular polarised light is
$\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\ -i\end{pmatrix}$.

The Jones matrices are $2\times 2$ linear matrix operators that represent the
different instrumental and atmospheric effects. The measured Jones vectors
then can be written as $\mathbf{e_{out}J\mathbf{e}}$.

Jones vectors and Jones matrices can be written in a four-vector format. Every
Jones vector has its four-vector counterpart in terms of Stokes parameters,
${\mathbf{S}}=\begin{pmatrix}I\\\ Q\\\ U\\\ V\end{pmatrix}$. The instrumental
response then represented in terms of a $4\times 4$ matrix, known as Muller
matrix ($M$). The Mueller matrix describes the modification of the Stokes
vector, $\mathbf{S}$. The Mueller matrix relates the output Stokes vector,
$\mathbf{S_{\mathrm{out}}}$ to the input Stokes vector, $\mathbf{S}$, as
${\mathbf{S}_{\rm out}}={\mathbf{M}}.{\mathbf{S}_{\rm in}}$.

###  2.5 Parallactic angle

Figure 2: Variation of parallactic angle with time for the source B1929+10
from rising to set. (Figure taken from MSc thesis of Sanjay Kudale).

For the case of an alt-azimuth mounted telescope, the feed dipoles rotate in
the plane of the sky, and hence with respect to the plane of polarisation of
the incoming radiation. This rotation is characterized by an angle called the
parallactic angle, $\psi_{p}$ given as,

$\tan\psi_{p}=\frac{\cos\mathcal{L}\sin\mathcal{H}}{\sin\mathcal{L}\cos\delta-\cos\mathcal{L}\sin\delta\sin\mathcal{H}}$
(18)

where, $\mathcal{L}$, $\mathcal{H}$ are latitude and hour-angle of the
telescope, respectively and $\delta$ is the declination of the source. As the
telescope tracks a source across the sky the parallactic angle changes (Figure
2).

###  2.6 IAU/IEEE Convention of polarisation

Unfortunately, there are multiple conventions of polarisation based on
coordinates systems and Stokes parameters. To alleviate this ambiguity, the
Institute of Electrical & Electronics Engineers (IEEE) defined a universal
standard and published it in 1969. In 1974, the International Astronomical
Union (IAU) endorsed the IEEE standard and supplemental with the definitions
of Cartesian coordinates and the sign of Stokes parameter V.

In the polarisation convention defined by the IAU, the local $X$-axis points
towards North, the local $Y$-axis points towards East, and the local $Z$-axis
points inwards to the observer for a right-handed system (Left panel of Figure
3).

Figure 3: Left figure shows the coordinate sphere and $XYZ$ right-handed
coordinate system defined by the IAU. The perspectives can be different based
on whether seen from the position of the source or the position of the
observer. The right figure shows the right-handed coordinate system seen from
the position of the observer. Here the radiation from the source propagates in
the positive $Z$ direction.

The position angle is measured starting from the North direction and going
towards the East, i.e. increasing counter-clockwise, when looking at the
source (Figure 1). The orientation of the ellipse in the plane of polarisation
$XY$ plane is constant in time; the polarisation angle $\chi$ corresponds to
the angle between the positive $X$ axis and the semi-major axis of the ellipse
(counted in the counterclockwise direction) 222The convention astronomers
follow for the PA (Polarization Angle) goes back to the 19th century and it
has been in use for observations going from radio to gamma rays: the PA
increases counter-clockwise when looking at the source. This convention is
consistent with the one used for the Position Angle and it has been enforced
by the IAU with a Resolution by Commissions 25 and 40 at the IAU XVth General
Assembly in Sydney in 1973 (see Transactions of the IAU, Vol. XVB, pg. 166).

Figure 4: The orientation of the electric vectors for Stokes $Q$ and $U$ on
the $xy$ plane of the sky. The figure also shows the IAU/IEEE conventions for
RCP and LCP and Stokes V.

IAU defines a right-handed coordinate system as shown in Figure 4. Once this
convention is defined and $XY$ coordinate system is chosen, the Stokes
parameters can be defined unambiguously. In such a case from the Equations 9
to 16, $+Q$ is pointing towards $X$ (E-W) and $-Q$ towards $Y$ (N-S). The $+U$
is 45 degrees from $X$ towards $Y$ (i.e. NE to SW line) and $-U$ towards 45
degrees from $X$ towards $-Y$ (i.e. NW to SE line) (Figure 4).

The handedness of the elliptically polarised light describes the direction of
rotation of the electric field vector as seen by this observer. It is more
confusing to define Stokes $V$. As per IAU definition, for the right-handed
circular polarisation, the position angle of the electric vector at any point
increases with time. This implies that the $Y$ component of the field will lag
behind the $X$ component, i.e. $E_{y}=\mp iE_{x}$. In such a case Stokes $V$
is defined as positive for right-handed circular polarisation (IAU/IEEE),

${\rm Stokes}\,\,V=RR^{*}-LL^{*}$ (19)

Another way of putting this is that a right-hand circularly polarised
radiation is the one where for the incoming radiation, the electric field
vector rotates counter-clockwise. For left-hand circularly polarised
radiation, the electric field vector rotates clockwise for the incoming
radiation. Here one notes that the electric field vectors along the line of
sight, at any instant of time, form a left-handed screw.

However the alternate way of defining Stokes V, i.e. $V=LL^{*}-RR^{*}$ is also
prevalent, especially among the pulsar community. This convention, where
Stokes $V$ is positive for left-hand circularly polarised radiation, is known
as the “PSR/IEEE” convention. Both PSR/IEEE and IAU/IEEE conventions require
that $RR^{*}$ and $LL^{*}$ are defined according to the IEEE convention, it is
the definition of Stokes V that distinguishes these two conventions. This
needs to be kept in mind for the discussion in the rest of the article.

###  2.7 Polarisation for interferometers

The polarisation ellipse can be decomposed in any orthonormal basis in the
plane of polarisation. Generally in radio astronomy, the standard Cartesian
linear basis and a basis of circularly rotating unit vectors of opposite
handedness are used.

Radio receivers in radio telescopes are polarimeters by construction. The
receiver of a radio telescope is composed of two orthogonal dipoles. Each of
these two dipoles receives the corresponding polarisation component and
converts it into an electric voltage that can be recorded and processed
electronically. The signals received by the dipoles can be auto-correlated as
well as cross-correlated, and Stokes parameters can be measured.

The Stokes parameters $I$, $Q$, $U$, and $V$ are derived from the measured
voltages (coming from the celestial object) recorded by two receivers in the
telescope corresponding to two orthogonal polarisations: $X$ and $Y$
(representing $E_{x}$ and $E_{y}$, respectively) for telescopes with dipolar
feeds; or $R$ and $L$ (representing $E_{R}$ and $E_{L}$, respectively). For
IAU/IEE convention, the $R$ and $L$ are related to $X$ and $Y$ as,

$\begin{split}R=\frac{(X+iY)}{\sqrt{2}}\\\
L=\frac{(Y-iX)}{\sqrt{2}}\end{split}$ (20)

This implies:

$\displaystyle I$ $\displaystyle=XX^{*}+YY^{*}\equiv RR^{*}+LL^{*}$ (21)
$\displaystyle Q$ $\displaystyle=XX^{*}-YY^{*}\equiv RL^{*}+R^{*}L$ (22)
$\displaystyle U$ $\displaystyle=XY^{*}+X^{*}Y\equiv-i(RL^{*}-R^{*}L)$ (23)
$\displaystyle V$ $\displaystyle=-i(XY^{*}-X^{*}Y)\equiv RR^{*}-LL^{*}$ (24)

Here $X$ and $Y$ correspond to linear dipoles and $R$ and $L$ correspond to
circular dipoles.

##  3 The u/GMRT

The GMRT antennas are 45m alt-azimuth mounted parabolic prime-focus dishes,
with an elevation limit set to 17.5o (though the dishes can go down to 16o).
The different feeds for different observing bands are mounted on a rotating
turret placed at the primary focus. The low noise receiving system of GMRT
receives dual polarisations. For GMRT all feeds are linearly polarised. They
are then converted to circular feeds for bands 2, 3 and 4. The L-band feed is
kept as linearly polarised. The GMRT antennas have RF swap switch at the end
of common box, which when set in swap mode, interchange the signal path with
opposite polarisation electronics. Thus the polarisation channels can be
swapped whenever required.

Each antenna has two polarisers named as V and H, with a 90 degree phase
difference. For bands 2, 3 and 4, one dipole element is connected to the
center of the RF connector through a rod and the other element is connected to
the ground of the RF connector. The Balun has a shortening plate at
$\lambda/4$ to create 90 degrees phase difference. These are shown in Figure
5.

Figure 5: Top Left: The feed dipoles. One dipole element is connected to the
center connector of the RF connector through a rod and the other is connected
to the ground of the RF connector. The Balun built with the feed is has a
$\lambda/4$ short. Top Right: Back view of the cross-dipole. Bottom: View of
the co-axial feed and the cross dipole feed focusing towards the dish (dashed
blue circle). This is same for bands 2, 3 and 4.

The figures 5 and 6 show the cross dipole feed with the dipole output
connectors marked as V and H. The V is connected to the Vertical element and H
to the horizontal element. The V connector centre pin is connected to the
lower arm of the vertical dipole. The H connector centre pin is connected to
the horizontal dipole arm on the opposite side of the connector. The connector
side is placed near the python cable. The dipole arms for Horizontal
polarisation is mounted parallel to the Feed Position System (FPS) rotating
axis and the Vertical dipole arms is orthogonal to the FPS rotating axis at
prime focus of the parabolic dish. This convention is followed in all bands
and at all antennas of GMRT array both for legacy and upgraded systems of
GMRT.

In summary the H dipole element is parallel to FPS rotating axis and V dipole
element is perpendicular to FPS rotation axis. The feed to hood connecting
cable is of equal length and could be used to swap the cables if needed, which
can give polarisation swap (Figure 6).

Figure 6: Left: This figure shows the cross dipole. There are two connectors V
and H, corresponding to vertical and horizontal component, respectively. The
connector and element marked as H goes parallel to the rotating axis of FPS
system. The marking on the feed is aligned to the python side which is H
input. Right: The feed connection, if viewed from top of the dish.

GMRT antennas are prime focus antennas, i.e. the radiation falling on the feed
reverses its circular polarisation. We show this effect in Fig 7. If this
reflection is not taken into account, it will result in a reversal of circular
polarisation. We need to keep this in mind.

Figure 7: This figure shows the configuration of the GMRT feed. The radiation
from a celestial source first falls on the dish and then reaches the feed
after the reflection from the antenna dish. This will cause a reversal of the
circular polarisation

##  4 Initial Test Observations

###  4.1 Target properties

The aim of these test observations is to investigate the polarisation
convention of the u/GMRT and understand whether these follow the standard
IAU/IEEE convention mentioned above. The results of these test observations
has already been reported as NCRA technical report (REF. Barnali’s report) and
re-described here for completeness purpose. To test the convention we choose a
test source and carry out uGMRT observations in both the interferometric and
the phase-array mode. The chosen source is pulsar B1702–19 (RA:
$17^{h}05^{m}36^{s}.099$, Dec: $-19^{d}06^{m}38^{s}.60$, B1950), with a
relatively higher degree of circular polarisation ($\approx 40\%$) as per the
pulse profiles from EPN database 333
http://www.epta.eu.org/epndb/#gl98/J1705-1906/gl98_610.epn.

We also look at the published 1369 MHz Stokes V profile of the pulsar with the
Parkes radio telescope Johnston & Kerr (2018) and 1408 MHz profile with the
Lovell telescope Gould & Lyne (1998). Parkes Telescope explicitly mentions
that the telescope follows the PSR/IEEE convention (According to PSR/IEEE
convention Stokes V is the difference of the Left circular polarisation (LCP)
and right circular polarisation (RCP) and the sign is positive if LCP is
higher than RCP, i.e. V$=$LCP – RCP 444See Section 2 for more details of the
definitions.). Since the signs match in the published data with both the
Parkes and Lovell telescopes, it indicates that the convention adopted by
Lovell telescope is same as Parkes telescope, i.e. the PSR/IEEE convention. As
the EPN database uses Lovell telescope data for this pulsar, we infer that the
convention used for Stokes $V$ in the EPN database is also V$=$LCP – RCP.

If we convert the Stokes profiles from PSR/IEEE convention into IAU/IEEE
convention (V$=$RCP – LCP, see Section 2 for details), then the pulsar is
$\approx 40\%$ right circularly polarised. In Figure 8, we show the Stokes
$I$, $Q$, $U$, $V$ published profiles at 610 MHz (left panel) and 400 MHz
(right panel) for this pulsar, generated from the data available in the EPN
database (Gould & Lyne, 1998). To make it consistent with the rest of the
report, we follow the uniform IAU/IEEE convention, and Stokes V profile is
plotted as, V$=$RCP – LCP.

###  4.2 uGMRT observations

We observed PSR B1702–19 in band 4 (550–750 MHz) of the uGMRT on March 19,
2020 with 200 MHz bandwidth, divided into 2048 channels. 3C286 was used as the
flux density and bandpass calibrator, and J1822-096 (RA:
$18^{h}22^{m}28.71^{s}$, Dec: $-09^{d}38^{m}56^{s}.84$, J2000) was used as the
phase calibrator. The on-source time of the pulsar was around 40 minutes. The
data were recorded both in the interferometric and the pulsar phased array
mode. The same pulsar with the same set of calibrators were also observed in
uGMRT band 3 with 200 MHz bandwidth with default settings (first LO 500 MHz,
500–300 MHz) on May 20, 2020.

In the pulsar mode, the array was first phased using the phase calibrator
J1822-096. The data were then recorded on the target pulsar in the standard
format followed at uGMRT (co- and cross-polar voltage products, viz. $RR^{*}$,
$LL^{*}$, and real and imaginary parts of $RL^{*}$ 555For u/GMRT channel 1 is
labelled as $RR^{*}$ and channel 2 $LL^{*}$. For details see next section.)
with a time resolution of 327.68 microseconds. Since the rotation period of
the target pulsar is about 280 ms, time resolution of 0.33 ms is appropriate.
These data were analysed using the full polarisation data analysis pipeline of
the uGMRT described in Kudale (2008). $R$ and $L$ powers were separately
scaled by its average bandshape to take out the bandpass effect. The final
outputs were Stokes $I$, $Q$, $U$ and $V$ ($=RR^{*}-LL^{*}$). Note that no
polarisation calibration was applied for the phased array data.

The interferometric data were flagged and calibrated using standard tasks in
casa (e.g. Das et al., 2019). The $RR^{*}$ and $LL^{*}$ data of the target
were separately self-calibrated to make the final images. We present the
results obtained from these analysis in the following two subsections.

Figure 8: Left: The 610 MHz profiles for the pulsar B1702–19 for Stokes $I$
(magenta curve), $Q$ (green curve), $U$ (blue curve) and $V$ (orange curve)
obtained by Gould & Lyne (1998) using the Lovell telescope. Also shown are the
angle of polarisation (PA) on top panel and linear polarisation
$L=\sqrt{Q^{2}+U^{2}}$ in yellow in the bottom panel. Right: Same as the left,
but at 408 MHz for PSR B1702-19. Note that the convention for Stokes $V$,
here, is $V=RR^{*}-LL^{*}$ even though the convention used in the original
paper (Gould & Lyne, 1998) was $V=LL^{*}-RR^{*}$. This is done to avoid
confusion since GMRT convention for defining $V$ is $RR^{*}-LL^{*}$.

Figure 9: Top Panel: Left: Comparison of $RR^{*}$ and $LL^{*}$ flux densities
for the pulsar B1702-19, obtained by imaging the interferometric data, and
that for a few test sources in the field of view (FoV) in band 4 (550–750 MHz)
of the uGMRT. Right: The Stokes $I$ (magenta), $Q$ (green curve), $U$ (blue
curve) and $V$ (orange curve) profiles for the pulsar B1702–19 at 650 MHz as
measured by the uGMRT. Also shown is the linear polarisation
$L=\sqrt{Q^{2}+U^{2}}$ in yellow. Bottom Panel: Left: A comparison of $LL^{*}$
and $RR^{*}$ flux densities of the pulsar B1702-19 (red square) in band 3 of
the uGMRT, along with a few other test sources in the FoV, obtained from
imaging of interferometric data. Right: The I, Q, U, V profiles obtained from
uGMRT band 3 data. The colour scheme is same as that for band 4. Note that no
polarisation calibration was carried out in the phased array data, and Stokes
$V$ is defined as $V=RR^{*}-LL^{*}$.

####  4.2.1 Results from the interferometric mode observations

At uGMRT, channels 1 and 2 are designated as $R$ and $L$, respectively. In the
left panels of Figure 9, we show the results obtained from the interferometric
data in bands 3 (bottom panel) and band 4 (top panel). We plot the flux
densities in the $LL^{*}$ against the flux densities in the $RR^{*}$ for the
target (marked by red square) along with a few other test sources in the field
of view. The flux densities of the test sources were measured in $RR^{*}$ and
$LL^{*}$ to make sure that the inferred difference in the target flux density
is not due to any systematic. As seen in the Figure 9, these test sources have
equal flux densities at the $RR^{*}$ and $LL^{*}$, and hence the flux density
difference in the target in the $RR^{*}$ and $LL^{*}$ is real.

In band 4, $RR^{*}$ flux density of the pulsar is $15.8\pm 0.1$ mJy, whereas
the $LL^{*}$ flux density comes out to be $23.9\pm 0.2$ mJy. This gives
percentage circular polarisation $V/I=-20\%$, while using IAU/IEEE convention,
i.e. $V=(RR^{*}-LL^{*})/2$. This uGMRT observation suggests the pulsar is left
handed circularly polarised at band 4, whereas it is right handed circularly
polarised (Stokes V positive) under IAU/IEEE convention as per EPN database.
Therefore, it appears that the uGMRT channel 1, which is currently designated
as $R$) is LCP ($L$), and opposite for channel 2, if one follows IAU/IEEE
convention in band 4.

In band 3, the $RR^{*}$ flux density of the pulsar is $\sim 17$ mJy and
$LL^{*}$ flux density is $\sim 28$ mJy. This gives circular polarisation
fraction, $V/I=-24\%$. Thus even in this case, the observation using uGMRT
suggests that the pulsar is left circularly polarised. Therefore, on the face
of it, the uGMRT channel 1 is also LCP ($L$) and vice-versa for channel 2 in
band 3 as per IAU/IEEE convention.

####  4.2.2 Result from the pulsar mode observations

The phased array observations provide pulse profiles of Stokes parameters,
which can not be obtained from interferometric observations. These pulse
profiles can be compared with the EPN pulse profiles directly. In the right
panels of Figure 9, we show the pulse profiles for Stokes $I$, $Q$, $U$ and
$V\,(=RR^{*}-LL^{*})$. The summary of the results obtained for both bands are
presented below.

Band 4:
The maximum circular polarisation observed from uGMRT observation is $-40\%$.
By comparing these profiles (top right panel of Figure 9) with those in the
left panel of Figure 8 using the EPN database, we find that:

  1. 1.

The signs of the Stokes $Q$ profiles in the two datasets do not match, but the
Stokes $U$ profiles agree with each other.

  2. 2.

The sign of the uGMRT Stokes $V$ profile does not match with that of the EPN
Stokes $V$ profile.

  3. 3.

The sweeps of the polarisation angle (PA) are opposite in the uGMRT from the
EPN data.

Band 3:
The maximum circular polarisation observed from uGMRT observation is $-28\%$.
By comparing these profiles (bottom right panel of Figure 9) with those in the
right panel of Figure 8 using the EPN data base, we find that:

  1. 1.

The signs of Stokes $Q$ profiles match between the uGMRT and EPN data,
however, the signs of Stokes $U$ do not match. This is opposite to that of the
result in band 4.

  2. 2.

The sign of the uGMRT Stokes $V$ profile does not match with that of the EPN
Stokes $V$ profile.

  3. 3.

The sweeps of the polarisation angle (PA) are opposite in the two datasets.

###  4.3 Legacy GMRT archival data

####  4.3.1 Interferometric data

We also searched for the legacy GMRT interferometric data for pulsar B1702-19.
We found several datasets and downloaded observations taken on 15 May 2007
(P.I.: Bhaswati Bhattacharyya) at 610 MHz with 16 MHz bandwidth, and data
taken on 2 Sep 2007 (PI: Bhaswati Bhattacharyya) at 330 MHz. There were no
flux calibrators in both the observations, and hence the flux densities cannot
be believed in terms of their absolute values, though the relative flux
density in channels 1 and 2 can give significant information on Stokes V. We
found Stokes V ($RR^{*}-LL^{*}$) to be negative, and Stokes $V/I=-22$% at 610
MHz as shown in the left panel of Figure 10, which is similar to what we
reported for uGMRT band 4 data in Section 4.2.1. For 330 MHz data, we found
the $LL^{*}$ flux density to be greater than $RR^{*}$ flux density as well.
Here circular polarisation fraction, $V/I$ was found to be $-25$% as shown in
the right panel of Figure 10. This is close to the value found in uGMRT band 3
observation mentioned in Section 4.2.1.

Figure 10: Left panel: The percentage circular polarisation for the pulsar
B1702-19 (red square) and a few other sources in FoV at legacy GMRT 604 MHz
frequency for data taken in 2007. Right panel: The percentage circular
polarisation for the pulsar B1702-19 (red square) and a few other sources in
FoV at legacy GMRT 331 MHz frequency for data taken in 2007. Note that Stokes
V is defined as per IAU/IEEE convention, $RR^{*}-LL^{*}$.

These results have an important implication. It means that the sense of
polarisation remains unchanged between legacy GMRT and uGMRT interferometric
data. Hence, we conclude that the legacy GMRT data also followed convention
opposite to that of IAU/IEEE.

####  4.3.2 Pulsar mode data

In Sanjay Kudale’s Master’s thesis, a detailed comparison has been made for
many pulsars with the EPN database. The thesis concluded that the sense of $V$
polarisation matches the two data sets. The thesis uses the Stokes V
definition as $RR^{*}-LL^{*}$, whereas the EPN database uses $LL^{*}-RR^{*}$
as Stokes V. This difference was not considered in the thesis while comparing
the profiles with the EPN database. Hence, in principle, the thesis
establishes that the phased array data in legacy GMRT followed convention
opposite to that of IAU/IEEE. Since many of the datasets used in this thesis
were with the GMRT hardware backend (GHB), it means the sense of polarisation
has remained opposite to that of the IAU/IEEE from the beginning of GMRT, i.e.
GHB, GMRT Software Backend (GSB) as well as current uGMRT GWB correlator.

###  4.4 Implications of the initial test results

The above exercise gives the following results. The sense of circular
polarisation is the same for interferometric data as well as beam data. This
also remains unchanged in the GHB, GSB, and GWB data.

Thus, there seems to be a reversal in the sense of Stokes $V$ in the GMRT
convention with that of the IAU/IEEE convention. If taken on the face value,
the uGMRT LCP should be RCP and the RCP should be LCP, to follow the IAU/IEEE
convention. This is valid for both band 4 and band 3 as well.

###  4.5 Issue with Stokes Q and U

However, there is a bigger issue, which is the discrepancy of Stokes
parameters $Q$ and $U$. In band 4 the signs of the Stokes $Q$ profiles do not
match between uGMRT and EPN data, but the Stokes $U$ profiles agree with each
other. In addition, the sweeps of the polarisation angle (PA) are opposite in
the uGMRT vs EPN data. The Stokes V results suggest that $R$ and $L$ are
swapped, however, simply swapping $R$ and $L$ can not explain the behavior of
$Q$ and $U$. From Equations 22 and 23,

$\begin{split}Q=RL^{*}+R^{*}L,\quad U=-i(RL^{*}-R^{*}L),\quad
PA=1/2\arctan(U/Q)\end{split}$ (25)

It is evident from these equations that swapping $R$ and $L$ will give an
opposite sign of $U$ and not that of Stokes $Q$.

We next consider the possibility that $X$ and $Y$ are swapped (§2). This is
because the GMRT feeds are linear feeds that are converted into circular feeds
for bands 2, 3, and 4. Thus it is important to explore the possibility of a
swap between X and Y itself. For clarity, we denote uGMRT $R$ and $L$ as
$\tilde{R}$ and $\tilde{L}$ respectively. In addition polarisation angle (PA)
for GMRT can be written as $\tilde{PA}$. If $X$ and $Y$ are swapped, we will
get:

$\begin{split}\tilde{R}&=\frac{Y+iX}{\sqrt{2}}=i\frac{X-iY}{\sqrt{2}}=iL\\\
\tilde{L}&=\frac{Y-iX}{\sqrt{2}}=-i\frac{X+iY}{\sqrt{2}}=-iR\end{split}$ (26)

where, $R,\ L$ are IAU/IEEE convention.

In that case, we will get
$\tilde{V}=\tilde{R}\tilde{R^{*}}-\tilde{L}\tilde{L^{*}}=LL^{*}-RR^{*}$, which
is opposite to the EPN database values converted into IAU/IEEE convention. For
Stokes $Q$ and $U$, we will get:

$\begin{split}\tilde{Q}&=\tilde{R}\tilde{L}^{*}+\tilde{R}^{*}\tilde{L}=(iL)(iR^{*})+(-iL^{*})(-iR)=-(RL^{*}+R^{*}L)=-Q\\\
\tilde{U}&=-i(\tilde{R}\tilde{L}^{*}-\tilde{R}^{*}\tilde{L})=-i\\{(iL)(iR^{*})-(-iL^{*})(-iR)\\}=-i(RL^{*}-R^{*}L)=U\\\
\tilde{PA}&=1/2\arctan(\tilde{U}/\tilde{Q})=1/2\arctan(U/-Q)=-1/2\arctan(U/Q)=-PA\end{split}$
(27)

This is similar to what was observed in uGMRT band 4 observation. Thus, this
initial test at band 4 infers that the $X$ and $Y$ dipoles need to be swapped
to explain the test results of band 4 uGMRT data.

In band 3, the signs of Stokes $Q$ profiles match between the uGMRT and EPN
data, however, the signs of Stokes $U$ do not match. This is opposite to that
of the result in band 4. These results can be explained by a simple $R$ and
$L$ swap, but not using $X$ and $Y$ swapping as mentioned for band 4. Again,
if we denote uGMRT $R$ and $L$ as $\tilde{R}$ and $\tilde{L}$ respectively,
then $\tilde{R}=L$ and $\tilde{L}=R$. In this case

$\begin{split}\tilde{Q}&=\tilde{R}\tilde{L}^{*}+\tilde{R}^{*}\tilde{L}=LR^{*}+L^{*}R=Q\\\
\tilde{U}&=-i(\tilde{R}\tilde{L}^{*}-\tilde{R}^{*}\tilde{L})=-i(LR^{*}-L^{*}R)=-U\\\
\tilde{PA}&=1/2\arctan(\tilde{U}/\tilde{Q})=1/2\arctan(-U/Q)=-1/2\arctan(U/Q)=-PA\end{split}$
(28)

Thus, this initial test at band 3 infers that the $R$ and $L$ dipoles need to
be swapped to explain the test results of band 3 uGMRT data. Goal of current
tests: In the above section, requiring $X$ and $Y$ to swap in band 4 and $R$
and $L$ to swap for band 3 to explain the observed results are quite
incomprehensible in the first place. Since polarisation calibration is not
done in the previous experiment, it is not obvious whether the discrepancy
between band 3 and band 4 results is due to polarisation calibration issues or
intrinsic. In this report, we aim to find out the polarisation convention
adopted by the uGMRT (Gupta et al., 2017) and legacy GMRT in various bands
(Swarup et al., 1991).

##  5 Understanding the GMRT polarisation

With this background, we can start to investigate the results seen in bands 3
and 4 and understand the polarisation convention of the GMRT.

Multiple factors could affect true source polarisation of a radio wave. Once
radio waves emanate from the source, they travel through the interstellar
medium (ISM). The magnetic field of the ionized ISM can cause Faraday
rotation, causing the intrinsic polarisation of the source to change even
before it reaches the telescope. Radio waves then interact with the antenna
dish. Here they are reflected and brought to a focus at a feed. This
reflection will cause a reversal of polarisation and needs to be taken into
account. The signals received at the feeds are amplified and they encounter a
large number of electronic components before being recorded as final data
products. Thus various external and internal factors can affect the
polarisation (Figure 11). From the propagation of the radiation from the
astrophysical source of interest to the output of the correlator, there are
various stages where one needs to correct for the above effects. We check for
the following (in no particular order):

  * •

Faraday rotation

  * •

Parallactic rotation due to alt-az mount

  * •

Signal propagation from the front end (when it reaches the dish) to the
optical fibre

  * •

Optical fibre to back end (user-friendly data)

  * •

Calibration issues

  * •

Dependence on observational settings

Figure 11: Schematic of polarisation signal propagation in space, and through
electronics which affects the state of polarisation (MSc thesis: Sanjay Kudale
adapted from Hamaker & Bregman (1996)).

###  5.1 Faraday effects

If a linearly polarised light is propagating through a magnetized plasma, It
is found that the transmitted light is still linearly polarised, but that the
plane of polarisation is rotated by an angle proportional to the projection of
the magnetic field along the direction of the propagation of light. This is
called the Faraday rotation (or the Faraday effect). ISM generally has some
magnetic field. Magnetic fields in the ISM added to the presence of free
electrons, can change the polarisation properties of an electromagnetic wave
traveling through it.

In circularly polarised light the direction of the electric field rotates at
the frequency of the light, either clockwise or counter-clockwise. This
electric field causes a force on the ISM electrons. The motion thus affected
will be circular, and these circularly moving electrons create their magnetic
field in addition to the ISM magnetic field. The created field will be
parallel to the external field for one (circular) polarisation, and in the
opposing direction for the other polarisation direction. The result of this is
that the net magnetic field is enhanced in one direction and diminished in the
opposite direction. This results in left and right circularly polarised waves
propagating at slightly different speeds, known as circular birefringence.
Since a linear polarisation can be decomposed into the superposition of two
circularly polarised components of opposite handedness and equal amplitude,
the effect of a relative phase shift, induced by the Faraday effect, is to
rotate the orientation of a wave’s linear polarisation (Figure 12).

In contrast to the Faraday effect in solids or liquids, interstellar Faraday
rotation has a simple dependence on the wavelength of light $\lambda$. The
observed frequency-dependent polarisation angle, $\chi(\lambda^{2})$ is given
as $\chi=\chi_{0}+RM\lambda^{2}$, where $\chi_{o}$ is intrinsic source
polarisation angle, and the rotation measure (RM, in units of rad m-2) is
defined as,

$RM=0.81\int_{l}^{0}n_{e}B_{\parallel}.dl$ (29)

where $l$ is the distance travelled by the polarised emission (in parsecs),
$n_{e}$ is the number density of free electrons (in cm-3) and $B_{\parallel}$
is the magnetic field strength component along the line-of-sight (LOS) (in
$\mu$G). The accepted sign convention is that positive RM indicates a magnetic
field oriented toward the observer, and negative RM indicates a magnetic field
oriented away from the observer.

Figure 12: The Faraday rotation effect. One can see the rotation of the plane
of polarisation when passing through a medium with a magnetic field $B$.
Faraday rotation of the plane of polarisation of an electromagnetic wave
traveling through a region with a non-zero magnetic field and free electron
density.

While the effect of Faraday rotation is to rotate only the phases and ideally
should not result in depolarisation. However, since the radio instruments have
a certain spectral resolution and the Faraday rotation is slightly different
at different frequencies, the finite spectral resolution of the instrument
causes a depolarisation of the polarised signal. However, this will not result
in the reversal of sign of the polarisation. Thus Faraday rotation can be
ruled out as a possible cause of reversal of Stokes Q or U in GMRT bands 3 and
4. Faraday rotation rotates the plane of polarisation of the linearly
polarised emission. Hence, it can not cause the sign change of Stokes V.

###  5.2 Role of parallactic angle

For the case of an alt-azimuth mounted telescope, the feed dipoles rotate in
the plane of the sky, and hence to the plane of polarisation of the incoming
radiation. This rotation of the feed to sky coordinate is characterized by
parallactic angle. Correction of parallactic angle is necessary to decipher
the correct polarisation of radiation. In figure 13, we plot the raw beam data
taken for a pulsar B0740-28 on multiple days. The data plotted are without
correcting for parallactic angle. One can see without parallactic angle
correction, Stokes profiles are not to be believed. Parallactic angles need to
be calculated for each time and corrected for.

Figure 13: GMRT phased array observations of pulsar B0740-28 at different
epochs in 2013, at different parallactic angles. The left panels show band 3
data and the right panels show band 4 data (courtesy Yashwant Gupta). Please
note that these data are taken in USB mode and no correction or calibration
has been applied to the data.

###  5.3 Necessity of polarisation calibration

The signals received in the two polarisation channels go through the receiver
chain which may have unequal gains. Gain calibration is required to correct
for these effects. We aim to determine if, under certain circumstances, lack
of polarisation calibration can cause the results seen in bands 3 and 4, as
mentioned in the Section 4.5.

In full polarimetric calibration, the EM signal is fully treated as a vector.
For a single baseline of two antennas ‘j’ and ‘k’, the incoming signal in an
$XYZ$ coordinate system can be written as a column vector

$e_{+}=\begin{pmatrix}e_{x}\\\ e_{y}\end{pmatrix}$ (30)

The $e_{+}$ vector can be written on the circular basis

$e_{\odot}=Ce_{+}=\frac{1}{\sqrt{2}}\begin{pmatrix}i&i\\\
1&-i\end{pmatrix}\begin{pmatrix}e_{x}\\\
e_{y}\end{pmatrix}=\begin{pmatrix}e_{R}\\\ e_{L}\end{pmatrix}$ (31)

Then the correlation between these two signals can be written in the form of a
fully coherence matrix

$\mathbf{E_{jk}}=\mathbf{e_{j}e_{k}^{\dagger}}=\begin{pmatrix}e_{jR}\\\
e_{jL}\end{pmatrix}\begin{pmatrix}e^{*}_{kR}&e^{*}_{kL}\end{pmatrix}=\begin{pmatrix}e_{jR}e^{*}_{kR}&e_{jR}e^{*}_{kL}\\\
e_{jL}e^{*}_{kR}&e_{jL}e^{*}_{kL}\end{pmatrix}$ (32)

If the coordinate system is chosen according to IAU/IEEE convention the
coherency matrix can be written in terms of Stokes parameters

$\mathbf{E_{jk}}=\frac{1}{2}\begin{pmatrix}I+V&Q+iU\\\ Q-iU&I-V\end{pmatrix}$
(33)

But the true source signal is always modified by the atmospheric and
instrumental effects. The transformation of the signal by the instrument can
be represented by $2\times 2$ Jones matrices as, $\mathbf{w_{j}=J_{j}e_{j}}$,
where $\mathbf{w_{j}}$ is the output signal from the single antenna chain.
Then the output coherency matrix will be

$\mathbf{W_{jk}=w_{j}w_{k}^{\dagger}=J_{j}E_{jk}J_{k}^{\dagger}}$ (34)

In polarimetric calibration, one wants to find out these Jones matrices for
each antenna.

Now Jones matrix can be decomposed in the following form

$\mathbf{J_{i}=G_{i}B_{i}D_{i}}$

where $\mathbf{G_{i},\,\,B_{i}}$ are the time dependent antenna gains and
bandpass, respectively, and

$\mathbf{D_{i}}=\begin{pmatrix}1&d_{R}\\\ d_{L}&1\end{pmatrix}$

is the leakage matrix representing the polarisation leakage from one
polarisation channel to another polarisation channel.

Using unpolarised calibrators these factors can easily be solved within an
unknown unitary transformation between the corrected coherency matrix and true
source coherency matrix: $\mathbf{E^{\prime}_{jk}}$ as
$\mathbf{E^{\prime}_{jk}=XE_{jk}X^{\dagger}}$ where $\mathbf{X}$ is the
unitary matrix (Hamaker, 2000). To calibrate this unitary ambiguity we need
polarised calibrator sources.

The unitary transformation can represent two things:

  * •

Reflection of the vector

  * •

Rotation of the vector

In all interferometric analysis software only the rotational property is
considered and not the reflection property.

The rotation matrix represents the rotation of a polarisation vector
$\mathbf{P}=(Q,U,V)$ in the 3D polvector space and can be decomposed into
rotations about three perpendicular axes.

In circular basis it is decomposed as,

$\mathbf{X=X_{q}(\phi)X_{u}(\epsilon)X_{v}(\theta)}=\begin{pmatrix}e^{i\phi}&0\\\
0&e^{-i\phi}\end{pmatrix}\begin{pmatrix}\cos\epsilon&i\sin\epsilon\\\
i\sin\epsilon&\cos\epsilon\end{pmatrix}\begin{pmatrix}e^{i\theta}&0\\\
0&e^{-i\theta}\end{pmatrix}$ (35)

  * •

$\mathbf{X_{q}(\phi)}$ represents the phase difference between the R and L
polarisation channel.

  * •

$\mathbf{X_{u}(\epsilon)}$ represents the ellipticity matrix, which will cause
leakage from stokes Q to stokes U.

  * •

$\mathbf{X_{u}(\epsilon)}$ represents the geometrical rotational matrix, which
includes Faraday rotation, parallactic rotation, and any absolute physical
rotation of the feed itself. In circular basis it takes the form
$\left(\begin{smallmatrix}e^{i\theta}&0\\\
0&e^{-i\theta}\end{smallmatrix}\right)$. This will also cause a change in the
polarisation angle.

  * •

An important property of rotation matrices to note here is that the
determinant of a rotation matrix is $+1$.

The important property of the reflection matrix is that the determinant of a
reflection matrix is $-1$. Although both rotation and reflection matrices are
unitary, this property distinguishes the reflection matrix from the rotation
matrix.

These properties can be used to understand the possible reason causing an
incomprehensible result from the initial test described in Section 4.5. We
will attempt this in both band 3 and band 4.

####  5.3.1 Possible reason of uGMRT band 3 and 4 initial test results

In uGMRT band 3, the observations showed that the Stokes V and Stokes U change
sign, but stokes Q remained the same. This means that a true source coherence
matrix $\mathbf{B_{3}}=\left(\begin{smallmatrix}I+V&Q+iU\\\
Q-iU&I-V\end{smallmatrix}\right)$ is transformed to

$\mathbf{B^{\prime}_{3}}=\begin{pmatrix}I-V&Q-iU\\\
Q+iU&I+V\end{pmatrix}=\mathbf{X}\begin{pmatrix}I+V&Q+iU\\\
Q-iU&I-V\end{pmatrix}\mathbf{X^{\dagger}}$ (36)

The required unitary matrix to carry out the above operation is

$\mathbf{X}=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$

The determinant of this unitary matrix is $-1$. This means it is an effect of
reflection, not rotation. Thus the calibration software package cannot
calibrate this effect.

It was suggested above that such a flip in band 3 could be due to the swapping
of R and L. This can also be seen if we write the coherency matrix in R and L
form.

$\mathbf{XB_{3}X^{\dagger}}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}\begin{pmatrix}RR^{*}&RL^{*}&\\\
LR^{*}&LL^{*}\end{pmatrix}\begin{pmatrix}0&1\\\
1&0\end{pmatrix}=\begin{pmatrix}LL^{*}&LR^{*}&\\\ RL^{*}&RR^{*}\end{pmatrix}$
(37)

There is a possibility that other rotational effects are present. If we do not
calibrate these rotational effects, it may also change the values of the
Stokes parameters, but cannot explain the simultaneous sign change of stokes U
and V.

In band 4, the signs of Stokes V and Q were found to be swapped but the Stokes
U sign remained the same. It can be easily shown that there exists no unitary
matrix on a circular basis to explain this. But on a linear basis, there
exists a unitary transformation, which can be caused by $X$ and $Y$ channel
swapping.

####  5.3.2 A coherent framework explaining bands 3 and 4 results

An important question is whether both band 3 and band 4 polarisation swaps can
be described in a single coherent framework or not. There are rotational
effects, which we did not calibrate for the initial tests. We need to first
take care of those before justifying the problem.

Let us first assume that R and L are swapped in both bands 3 and 4. Then in
this swapped coordinate system, the coherency matrices for bands 3 and 4 can
be related as

$\mathbf{B^{\prime}_{4}=X^{\prime}B^{\prime}_{3}X^{\prime\dagger}}$ (38)

which becomes

$\mathbf{\begin{pmatrix}I-V&-Q+iU\\\
-Q-iU&I+V\end{pmatrix}=X^{\prime}\begin{pmatrix}I-V&Q-iU\\\
Q+iU&I+V\end{pmatrix}X^{\prime\dagger}}$ (39)

From this expression the corresponding unitary matrix is

$\mathbf{X^{\prime}}=\begin{pmatrix}e^{i\pi/2}&0\\\
0&e^{-i\pi/2}\end{pmatrix}$

. The determinant of this matrix is $+1$, thus it is a rotational matrix. This
rotational effect can be calibrated using the calibration software. Then both
band 3 and band 4 conventions will be the same and can be explained by only
the R and L swap.

In this exercise above, an assumption has been made that the polarisation
properties of the source do not change between bands 3 and 4. But in reality
the source polarisation can be different in different bands. In that case the
true rotation angle could be different than $\pi/2$. While polarisation
fraction may vary, there is no physical mechanism that can cause reflection in
the polarisation vector 3D polvector space.

The conclusion of the above mathematical analysis is as follows:

  * •

The sign changes between bands 3 and 4 cannot be explained by any kind of
rotation, like Faraday rotation and parallactic rotation, absolute geometric
rotation of the feed, and cross-phase between two polarisation channels.

  * •

The change in the Stokes sign can only be explained by reflection, which
cannot be calibrated by any software package because the reflection effect is
never considered in the calibration framework. This reflection effect can
simply arise from the swapping of the dipoles with respect to the IAU/IEEE
convention.

  * •

The difference of the Stokes Q and U signs between band 3 and band 4 can be
explained by rotation. If we calibrate the rotational effects then both band 3
and band 4 Stokes signs can be explained in a single coherent framework, that
is R and L polarisation channels are swapped. However, these are theoretical
results and they need to be confirmed with observations, as we show in later
sections.

To summarise, both band 3 and band 4 results can be explained in a single
coherent scenario involving R & L swap (reflection effect), and uncorrected
rotational effect in the case of band 4.

##  6 New Sets of Experiments

###  6.1 Tracking polarisation from the feed to fibre optic system

The main aim here is to determine how H and V are positioned as this will
determine which channel is RCP and LCP. The first step was to track the
polarisation from the feeds to the fibre optic system.

####  6.1.1 Determine the GMRT Azimuth 0 direction

Figure 14: GMRT antenna positioned at Azimuth $0^{o}$ and Elevation at
$18^{o}$ from the control room. The Az $0^{o}$ of the antenna points in the
South direction.

As a first step, we aimed to see whether the Azimuth $0^{o}$ of a GMRT antenna
is towards the North or South? For this exercise, a GMRT antenna was
positioned at Azimuth $0^{o}$ and Elevation at $18^{o}$ from the control room.
The servo reading was checked from the antenna base and confirmed the antenna
position. We checked the direction to which the antenna is pointing using a
GPS receiver (Figure 14). We confirmed that Az $0^{o}$ of the antenna points
towards the South direction. This is different than the regular convention
which is always towards the North, which will cause in swap in the angle which
is corrected in the GMRT. This was also verified at the antenna site and
obtained the same result.

####  6.1.2 Identify feed rotating axis and the direction of feed rotation

The next step was to determine the direction of the feed rotating axis and
feed rotation. For this test, the antenna was positioned at $Az=0^{o}$ and
$EL=90^{o}$ (Figure 15). We find that the feed moves along North-South or
South-North direction i.e along the elevation gear of the antenna. The feed
rotating axis is positioned along the East-West direction i.e perpendicular to
the elevation gear of the antenna

The Horizontal dipole element (H) is placed along the East-West Direction
(Python assembly (West) is along this axis which is taken as a reference to
mount the Horizontal dipole element). The Vertical dipole element (V) is
mounted along the North-South direction (Fig 15).

To test these further, we also created a dipole element and put it on the
ground, and directly let the feed get it. The horizontal component goes to V
and the vertical goes to H (Figure 16).

Figure 15:  Experiment to identify the feed rotating axis and the direction of
feed rotation. The GMRT antenna is positioned at $Az=0^{o}$ and $EL=90^{o}$.
The Horizontal dipole element is along East-West Direction and the Vertical
dipole element is mounted in a North-South direction.  Figure 16: label

Since IAU/IEEE convention defines North to be $X$ and East to be $Y$, we can
conclude that the Vertical element receives $X$ component of the electric
field $E_{x}$. The Horizontal polarisation receives $Y$ component of the
electric field $E_{y}$. The Same convention is followed to orient the dipole
elements for band 2, band 3, and band 4.

####  6.1.3 Lab test to determine H and V mapping with Channels 1 and 2

So far we have determined that the H represents $Y$ and V represents $X$. We
know that as per IAU/IEEE convention, radiation is RCP if $X$ leads $Y$ and
LCP if vice versa (Figure 4). Now we have to identify the characteristics of
GMRT channels 1 and 2 at the receiver end.

As we saw, each GMRT antenna has V and H polarisers, 90 degrees in phase. The
polariser at bands 2, 3, and 4 converts linear polarisation appearing in H and
V, to circular polarisation appearing in Channel 1 and 2 (Fig 17).

Figure 17: The sketch diagram shows the functional diagram of the polariser of
all bands, which converts the linear polarisation to circular polarisation i.e
as Channel 1 and Channel 2.

To determine this mapping, first, the signal was fed at V input of the
polariser, and H input was terminated with a 50 $\Omega$ resister. V is then
split into two beams and the output is recorded in channels 1 and 2, both
amplitude and phase. The result shows that while the signal directly comes to
Channel 1, it goes to channel 2 with a 90 degrees phase shift. When the same
experiment is done with H, channel 2 receives 0 phase shift and channel 1
receives 90 degree phase shift. This has been tracked throughout the
electronic chain till the fibre output. Our results from these experiments are
summarised in Table 1.

Hence we can conclude that,

$\begin{split}\mathrm{Channel\ 1}=V(0\,\rm deg)+H(90\,\rm deg)=V+iH\\\
\mathrm{Channel\ 2}=H(0\,\rm deg)+V(90\,\rm deg)=H+iV\\\ \end{split}$ (40)

Since previous experiment showed $V\equiv X$ and $H\equiv Y$, hence,

$\begin{split}\mathrm{Channel\ 1}=X+iY\\\ \mathrm{Channel\ 2}=Y+iX\\\
\end{split}$ (41)

Thus channel 1 is RCP and channel 2 is LCP. This is a very important result,
which shows that the GMRT convention of channel 1 being $RR^{*}$ and channel 2
being $LL^{*}$ is correct, and there is no swap from the feed to at least till
the optical fibre.

Table 1: Experiment to determine mapping of V and H to channels 1 and 2  Signal input | Channel 1 | Channel 2
---|---|---
V | 0 deg | 90 deg
H | 90 deg | 0 deg

###  6.2 Tracking the polarisation from optical fibre to user end

In the GMRT, the signal flow from the front end to the correlator through many
electronic blocks, parallel in two separate electronic chains. There are
chances of two polarisation signals to interchange their electronic path at a
couple of places. These two chains are called channel 1 (130 MHz polarisation
channel in the legacy GMRT) and channel 2 (175 MHz polarisation in the legacy
GMRT) and further, the autocorrelation of these polarisations are named RR∗
and LL∗ respectively. It may be possible that the incoming R & L signals have
flipped in the receiver chain in which case definitions of Stokes V would have
changed its meaning. We performed the following tests to check the conventions
of Stokes parameters.

Figure 18: u/GMRT signal flow from the optical fibre to the user end. Cable
lines come from optical fibre to GWB input. From GWB input, the electric cable
goes to the hardware. One signal goes to the interferometer, and another to
the beamformer. In correlator room, Channel 1 is on the right side of the
correlator facing the wall in the correlator room.

Test 1:
In the first test, the signals from the noise generator were fed to V and H
dipole elements with a 90 degree phase shift. The signal for the injection is
generated in the lab using a noise generator and can be split into two by a
power splitter and quadrature hybrid generating two polarisations, say Pol 1
and Pol 2 (90-degree phase shift with respect to Pol 1). The test will not
determine the RCP versus LCP nature of the two polarisations but will
determine if there is a swap or not from optical fibre to the user end. We
inject this signal at the analog backend input as well as input to the optical
fibre at the antenna base. No swap of the signal was found.

Test 2:
The second test was a mixed setup test where one channel was fed with the
output from the antenna and the second from the power generated by the noise
generator. Here also, the aim was to look for any kind of swap of cables. For
the antenna signal, we pointed the antenna to a calibrator and used the
received signal. Several combinations were checked. Data were recorded in
visibility and beam modes simultaneously. Below we show various combinations
which were used. As mentioned above Pol 1 and Pol 2 refer to noise-generated
signals.

  1. 1.

GWB CH-1 was fed with Pol 1 input and GWB CH-2 was connected to the antenna.

  2. 2.

GWB CH-1 was fed with Pol 2 input and GWB CH-2 was connected to the antenna.

  3. 3.

GWB CH-2 was fed with Pol 1 input and GWB CH-1 was connected to the antenna.

  4. 4.

GWB CH-2 was fed with Pol 2 input and GWB CH-1 was connected to the antenna.

  5. 5.

GWB CH-1 was fed with Pol 1 input and GWB CH-2 was fed with Pol 2 input.

  6. 6.

GWB CH-1 was fed with Pol 2 input and GWB CH-2 was fed with Pol 1 input.

The output was checked at various monitoring levels and using different tools,
which are:

  1. 1.

bandmon: Realtime bandshape monitoring of visibility data.

  2. 2.

tax: Offline data checking of visibility data.

  3. 3.

CASA Offline data processing package to monitor visibility (in FITS) +
imaging.

  4. 4.

gptool: Full polar beam data analysis package.

Following are our findings:

  * •

We found no swap of polarisation anywhere between the optical fibre to the
back end in test 1.

  * •

When Pol 1 frequency signal is injected, CH-1 is showing higher signals at all
four levels listed above.

  * •

With Pol 2 frequency signal injected, CH-2 is showing signals at all four
levels listed above.

  * •

There is no swap of the signal path from optical fibre output to GWB output,
to beam data, to visibility data to FITS data.

  * •

Pol 1 represents channel 1, Pol 2 represents channel 2.

Test 3:
Another test was done by Analog Backend team666courtesy Mr. Abhijit Dhende &
Ms. Sweta Gupta to see if two polarisation signals are exchanged in their path
using GWB & GSB.

Experiment 1:

  1. 1.

Signal coming from RF chan 1 input was disconnected (at point A in Figure 18).
Findings: GWB Bandmon Pol1 showed power drop, and the fringe vanished.

  2. 2.

Signal coming from RF chan 2 input disconnected (at point B in Figure 18)
Findings: GWB Bandmon Pol2 showed power drop, and the fringe vanished.

Experiment 2:

  1. 1.

RF Terminated & Single frequency injected at input GAB/baseband (at point C in
Figure 18)
Findings: GWB showed in bandmon a single line in the band at the desired
location for Pol 1.

  2. 2.

RF Terminated & Single frequency injected at input GAB/baseband (at point D in
Figure 18)
Findings: GWB showed in bandmon a single line in the band at the desired
location for Pol 2.

Hence this proves that there is no swap in the signal path.

###  6.3 Observational test results using astronomical source

Now we explore the effect of observational settings and calibration in the
actual astronomical data. Several test observations were carried out to
understand polarisation convention by analyzing beam as well as
interferometric data (Table 2). The target pulsars were chosen such that their
polarisation properties are well known with significant amount of both linear
and circular polarisation. Our chosen targets are PSR B0740- 28, PSR B0138+57
and PSR B1702-19.

Band 3 & band 4 observations were carried in default sideband mode for basic
tests, i.e. band 4 in USB (550 – 750 MHz), band 3 in LSB (500 – 300 MHz).
Additionally, we carried observations in the reversed side-band mode in a few
observations. Data were recorded in full polar mode with 2048 channel in all
observations. Band 3 & band 4 observations are recorded in 200 MHz bandwidth
mode some band 4 observations were carried out in 100 MHz bandwidth mode).
Time resolution (sampling interval) differ from one observation to another,
which was chosen based on pulsar period, available disk space and on-pulse
duty cycle. We now discuss beam data results in band 4 and all interferometric
data analysis and results. The details of beamformer data and other bands’
results will be discussed by Sanjay Kudale in a separate technical report.

####  6.3.1 Beamformer observation (pulsar mode observation)

Beamformer observation has an advantage over imaging analysis that it produce
high time resolution Stokes profile of the pulsar and swing of polarisation
position angle of linear polarisation across pulse (PPA), which can be
directly compared with the pulsar profiles obtained from EPN database. Beam
data were recorded with the GWB which offers the beam products as $RR^{*}$,
$Im(RL^{*})$, $LL^{*}$ and $Re(RL^{*})$. To check data quality first analysis
was done with gptool. After data was found to be good, it was further
processed with analysis process described in Kudale & Gupta, 2008 (NCRA :
internal technical report). The pipeline is modified and details of the
modifications are subject to another GMRT technical report by Sanjay Kudale.

Table 2: Details of all the test observations

Sr | Date of | Target | Freq | Side | Starting | BW | Samp | Stokes Q, U, V
---|---|---|---|---|---|---|---|---
| Observation |  | Band | Band | Freq, MHz | MHz |  $\mu$sec | IAU/IEEE
1 | 19 Mar 2020 | B1702-19 | B4 | USB | 550 | 200 | 327.68 |  $-$23%, $-$10%, +15%
2 | 20 May 2020 | B1702-19 | B3 | LSB | 500 | 200 | 327.68 |  $+$8%, $-$19%, +26%
3 | 13 Feb 2021 | B1702-19 | B4 | USB | 550 | 200 | 163.84 |  $-$23%, $-$10%, +15%
$\\**$ 4 | 04 Dec 2021 | B1702-19 | B3 | LSB | 500 | 200 | 327.68 |  $+$8%, $-$19%, +26%
4 | 12 Feb 2021 | B0136+57 | B4 | USB | 550 | 200 | 163.84 |  $-$27%, $-$64%, $-$16%
5 | 09 Mar 2021 | B0136+57 | B3 | LSB | 500 | 200 | 163.84 |  $-$62%, $-$26%, $-$5%
6 | 19 Mar 2021 | B0136+57 | B4 | USB | 550 | 100 | 327.68 |  $-$27%, $-$64%, $-$16%
7 | 19 Mar 2021 | B0136+57 | B4 | LSB | 650 | 100 | 327.68 |  $-$27%, $-$64%, $-$16%
8 | 31 May 2021 | B0136+57 | B3 | LSB | 500 | 200 | 163.84 |  $-$62%, $-$26%, $-$5%
9 | 24 Mar 2021 | B0740-28 | B4 | USB | 550 | 100 | 163.84 | +59%, +88%, +5%
10 | 24 Mar 2021 | B0740-28 | B4 | LSB | 650 | 100 | 163.84 | +59%, +88%, +5%
11 | 24 Mar 2021 | B0740-28 | B3 | USB | 400 | 100 | 163.84 |  $-$63%, +34%, +6%
12 | 24 Mar 2021 | B0740-28 | B3 | LSB | 500 | 100 | 163.84 |  $-$63%, +34%, +6%
$\\**$ 13 | 2 May 2021 | B0740-28 | B4 | LSB | 650 | 100 | 327.68 | +59%, +88%, +5%
$\\**$ 14 | 4 May 2021 | B0740-28 | B4 | USB | 550 | 100 | 163.84 | +59%, +88%, +5%
$\\**$ 15 | 4 May 2021 | B0740-28 | B4 | LSB | 650 | 100 | 163.84 | +59%, +88%, +5%

$\\**$ These observations were done with RF SWAP switch.

Band 4 observations:
Band 4 observations were carried out in many modes, including USB, LSB as well
as RF swap model. To record in LSB and USB mode of uGMRT, we could record only
100 MHz bandwidth. LSB & USB observations were separated by very short time
(about an hour). A few observations were also carried with RF swap applied to
check the effect of exchanging R-voltage and L-voltage on final profiles.

LSB vs USB observations:

To check how the Stokes profile vary from one frequency to other and from
lower sideband to upper sideband we observed pulsar B0740-28 with both
sidebands with recording bandwidth to be 100 MHz. All these observations were
carried over just 3.5 hours so that there is no major change any environment
which can cause profile change. During analysis PPA is made zero at the peak
in total intensity (and linear polarisation) pulse profile. We show band 4
profiles in (Figure 19), along with 610 MHz EPN profile corrected for IAU/IEEE
convention.

For given frequency band profiles of LSB & USB are exactly opposite for
Stokes-U, Stokes-V and for the PPA. These results are consistent for GWB as
well as GSB. The profiles obtained by these are showing match band3 & band4
for LSB and band3 & band4 for USB separately (Figure 19 and Figure
LABEL:fig:b3lsbusb).

Figure 19: Top panel: The uGMRT Stokes profile of PSR B0740-28 observed at
band 4 in USB mode (left) and LSB mode (right). Bottom panel: Same as top left
two panels but for GSB 610 MHz data.

The same trend can also be seen in the pulsar B0136+57 (Fig 20).

Figure 20: The uGMRT Stokes profile of PSR B0136+57 observed at band 4 in USB
mode (left) and LSB mode (middle), along with EPN profile on the right. Stokes
U indeed seems to invert in USB vs LSB.

RF swap observation:
As we mentioned in section 2, the GMRT antennas have RF swap switch at the end
of the common box, which when set in swap mode, interchange the signal path
and interchange R and L. We carried two observations of PSR B0740-28 to check
if R & L channels are exchanged how it would affect the final full Stokes
profiles. In figure 21, we show USB and LSB profiles for regular data, data
with RF swap and EPN data.

Figure 21: Left panel shows USB band 4 data for PSR B0740-28 . Middle panel
shows the RF SWAP data for the same pulsar and right are the EPN profile. One
can see swapping ’R’ and ’L’ in the USB mode data, the results match with the
EPN profile.

The result of above beam observations in different settings are the following:

  * •

In USB observations, after PPA correction, Stokes $U$ and $V$ remain mismatch
from that of EPN profiles.

  * •

In LSB mode, Stokes $U$ and PA are inverted than that of USB data.

  * •

For observations made in RF swap mode, in USB mode, all Stokes profiles match
with that of EPN data.

  * •

For observations made in RF swap mode, in LSB mode, all Stokes profiles except
Stokes $U$ and PA match with that of EPN data.

####  6.3.2 Interferometric data

We now focus on the results of the interferometric data. For interferometric
observations, we have used the pulse phase averaged values of the flux
densities and polarisation fractions obtained from EPN database for each of
the target pulsars. EPN database followed PSR/IEEE convention. We have
converted those values in IAU/IEEE convention.

Polarisation calibration has been performed before deriving the Stokes
information. Calibration has been performed in CASA. We have calibrated the
following effects as a part of instrumental polarisation calibration:

  1. 1.

Instrumental bandpass and flux density scale: Using the flux density
calibrator.

  2. 2.

Time variable instrumental gain: Phase calibrator close to the target source.

  3. 3.

Cross-hand delay: Cross-hand delay causes a phase ramp with frequency between
R and L polarisation channels. Cross-hand delay can be calibrated using a
polarised calibrator. We have used the strong polarisation angle calibrator
for this purpose.

  4. 4.

Polarisation leakage: Polarisation leakage has been calibrated using the
unpolarised leakage calibrator. In one epoch, 21 May 2020, we have used the
phase calibrator 1822-096 for leakage calibration, because no unpolarised
calibrator observation was available for this epoch. We have verified that
1822-096 does not have strong polarisation, hence can be used as a unpolarised
calibrator at uGMRT frequencies.

  5. 5.

Cross-hand phase and absolute polarisation angle: In circular basis, the
cross-hand phase (phase difference between R-L polarisation channels) and
absolute polarisation angle calibration are degenerate. Both of them are
calibrated together using the polarisation angle calibrator 3C 138 or 3C 286.

Below we discuss the data analysis procedure.

Flagging :
We have performed standard flagging of the bad antennas and bad channels. At
the initial stage, strong RFIs are flagged using CASA task flagdata using its
tfcrop mode. We also performed flagging in between calibration processes using
rflag mode of flagdata task. Since central square baselines are more prone to
RFI and some of our data is also affected by the issue with central square
baselines777Central Square Issues, we have flagged all central square
baselines from the beginning of the data analysis.

Model import : For polarisation calibration, one needs a full Stokes model of
the calibrator. The standard models available in CASA are the Stokes I model
only. Thus those models have not been used for the polarisation angle
calibrators; 3C 286 and 3C 138. For phase calibrators and instrumental leakage
calibrators, the standard Stokes I model has been used. The polarisation model
of 3C 286 and 3C 138 available in Perley & Butler (2008), have been used here.
The CASA task setjy has been used in $standard=manual$ mode to import the
polarisation models.

3C 286 is a very good polarisation angle calibrator and has polarisation
fraction $\sim 9$%, and the polarisation angle is almost constant over our
observing frequency. 3C 138 on the other hand has $\sim$6% linear
polarisation, and the polarisation angle varies with frequency. Thus two avoid
bandwidth depolarisation (particularly for 3C138), calibration has been
performed at the high-frequency resolution ($\sim 97$ kHz). The flux density
of 3C 138 is a little lower compared to 3C 138, thus large integration time
has been used for 3C 138.

Calibration steps:

  * •

Step 1: First we performed calibration of the flux density calibrator. We
choose a small chunk of good channels ($\sim 5-10$ MHz) and perform a phase-
only gain calibration to calculate the phase variation over the entire scan of
the flux calibrator. We used gaincal task only over the selected channels.

  * •

Step 2: We then used this phase solutions and performed the geometric delay
calibration using gaincal task with gaintype=‘k’ and bandpass calibration over
the entire band using bandpass task. We obtained a normalized bandpass
solution.

  * •

Step 3: Using the delay and bandpass solution, we have now estimated amplitude
and phase variations of the flux density calibrator and phase calibrator.
After these three steps, we performed some level of flagging on the residual
visibilities and repeat these steps until we found that there are no
significant RFI present.

  * •

Step 4: When we converged on the previous steps, we first apply the gain and
bandpass solutions on the unpolarised leakage calibrator and on the
polarisation angle calibrators. After applying the solutions, we performed a
flagging using tfcrop mode of flagdata.

  * •

Step 5: Then cross-hand delay is estimated from the polarised calibrator
sources. In this case, we used the polarisation angle calibrator source (3C
286 or 3C 138). The cross-hand delays are estimated using the entire observing
band using CASA task gaincal with gaintype=‘KCROSS’. Step 6: After applying
the cross-hand phase solutions, instrumental leakages are estimated using the
un-polarised calibrator source. Leakages are estimated for every frequency
channel separately. We have used CASA task polcal with the poltype=‘Df’ and
with solint=‘ınf’. This calibrates the leakage using un-polarised source per
frequency channels.

  * •

Step 7: After applying the correction for the instrumental leakages, the
cross-hand phases are estimated for every frequency channel using the
polarisation angle calibrator. This step performs the absolute polarisation
angle calibration. CASA task polcal has been used for this purpose with
poltype=‘Xf’, which denotes the polarisation angle calibration per frequency
channels.

Once all the calibrations are done, the calibration solutions are applied to
the target data. The corrected target data is then flagged for RFIs. Since the
calibration method used in CASA is only valid for the low-level instrumental
leakages (< 10%), the calibration solutions are flagged when the instrumental
leakage for an antenna at a certain frequency is more than 10%. After flagging
these high leakage frequency channels per antenna, Steps 1 to Step 7 are
repeated from scratch. When this is done, several self-calibration iterations
are performed to further improve the image dynamic range.

  * •

Step 8: Self-calibration is performed using separate sky models for both the
co-polar visibilities (RR and LL). This is done to make sure that if there is
a strong circularly polarised source present in the field, the circular
polarisation fraction remains the same with the self-calibration iteration.
Ideally, one should perform self-calibration until the dynamic range is
conserved. To reduce the computational burden, three rounds of phase-only and
four rounds of amplitude-phase self-calibration have been performed. A single
flagging is performed after the first amplitude-phase self-calibration round.

Final Imaging: Once the self-calibration is done, the final spectral cube of
Stokes parameters is made. At different epochs, depending on the integration
time, spectral cubes are made at smaller or larger frequency resolutions. Only
Stokes Q and U spectral cubes are made and for Stokes V (circular
polarisation) a continuum image is made over the entire band. The spectral
slices where the source is not detected either in Stokes Q or U are removed
from further analysis.

RM fit: RM fit was performed on each separately using Stokes Q and Stokes U
spectral cube. PSR B1702-19 has low RM ($RM=-19$ rad m-2) with sufficient
circular polarisation, however, the linear polarisation fraction is low. PSR
B0136+57 has high RM ($RM=-94$ rad m-2), with large fraction linear
polarisation but low circular polarisation. Thus it is a good source for the
sign check of Stokes Q and Stokes U.

Below are our results:

PSR B0136+57, band 4 USB

  * •

Pulse averaged source polarisation properties (IAU/IEEE): RM $=-94$ rad/m2,
Stokes Q = -27%, U = -64%, V = -16%

  * •

Observations date and settings: 12 Feb 2021; Band 4, LO I: 550 MHz, USB

  * •

Polarisation leakage and position angle calibration was done using 3C147 and
3C138 respectively.

  * •

Post calibration and post RM fit results:

    1. 1.

RMfit = $-90$ rad/m2

    2. 2.

$2\chi_{0}$ ($\chi_{0}$ = Intrinsic position angle) = 123 degree (Figure 22).

    3. 3.

Stokes Q $=-$ve, Stokes $U=+$ve, Stokes $V=+$ve.

    4. 4.

Result: Stokes U and V flipped, and Stokes Q remained the same.

Figure 22: RM fit to PSR B0136+57, band 4 USB data on 12 Feb 2021.

PSR B1702-19, band 4 USB

  * •

Pulse averaged source polarisation properties (IAU/IEEE): RM $=-19$ rad/m2, Q
= -23%, U = -10%, V = +15%

  * •

Observations date and settings: 13 Feb 2021; Band 4, LO I : 550 MHz LO, USB

  * •

Polarisation leakage and position angle calibration was done using 2355+498
and 3C 286 respectively.

  * •

Post calibration and post RM fit results:

    1. 1.

RMfit = $-14$ rad/m2

    2. 2.

$2\chi_{0}$ ($\chi_{0}$ = Intrinsic position angle) = 124 degree (Figure 23).

    3. 3.

Stokes Q $=-$ve, Stokes $U=+$ve, Stokes $V=-$ve.

    4. 4.

Result: Stokes U and V flipped, and Stokes Q remained the same.

Figure 23: RM fit to PSR B1702-19, band 4 USB data on 13 Feb 2021.

PSR B0136+57, band 4 LSB

  * •

Pulse averaged source polarisation properties (IAU/IEEE): RM $=-94$ rad/m2,
Stokes Q = -27%, U = -64%, V = -16%

  * •

Observations date and settings: 19 Mar 2021; Band 4, LO I : 650 MHz LO, LSB

  * •

Polarisation leakage and position angle calibration was done using 3C48 and
3C138 respectively.

  * •

Post calibration and post RM fit results:

    1. 1.

RMfit = $-90$ rad/m2

    2. 2.

$2\chi_{0}$ ($\chi_{0}$ = Intrinsic position angle) = 102 degree (Figure 24).

    3. 3.

Stokes Q $=-$ve, Stokes $U=+$ve, Stokes $V=+$ve.

    4. 4.

Result: Stokes U and V flipped, and Stokes Q remained the same.

Figure 24: RM fit to PSR B0136+57, band 4 LSB on 19 Mar 2021.

PSR B0136+57, band 3 LSB

  * •

Pulse averaged source polarisation properties (IAU/IEEE): RM $=-94$ rad/m2,
Stokes Q = -62%, U = -26%, V = -5%

  * •

Observations date and settings: 31 May 2021; Band 3, LO I : 500 MHz LO, LSB,
200 MHz BW

  * •

Polarisation leakage and position angle calibration was done using 3C 48 and
3C 138 respectively.

  * •

Post calibration and post RM fit results:

    1. 1.

RMfit = $-99$ rad/m2

    2. 2.

$2\chi_{0}$ ($\chi_{0}$ = Intrinsic position angle) = 117 degree (Figure 25).

    3. 3.

Stokes Q $=-$ve, Stokes $U=+$ve, Stokes $V=+$ve.

    4. 4.

Result: Stokes U and V flipped, and Stokes Q remained the same.

Figure 25: RM fit to PSR B0136+57, band 3 LSB data on 31 May 2021.

Hence we performed experiments on two PSRs in four different epochs and using
USB and LSB at both band 3 and band 4. In all three experiments, results are
similar, after performing polarisation calibration and RM correction. Stokes
$Q$ sign remained the same, and Stokes $U$ and $V$ signs flipped. There is no
difference between the results from the data taken using LSB and USB in
interferometric mode after the proper full polarisation calibration. This
result is consistent with a swap of $R$ and $L$, as discussed in Section
5.3.2.

Figure 26: A right-handed helical antenna as per IEEE convention. Return loss
performance of the helical antenna. The antenna tunes well within uGMRT band
4.

##  7 Understanding the effect of the prime focus feed

So far, all the results have shown that GMRT data can be explained by the swap
of $R$ and $L$ except beamformed data in LSB, where polarisation calibration
is not done). The swap of $R$ and $L$ is equivalent to the effect of GMRT
feeds being the prime focus, i.e. the radiation from the astronomical source
falls on the antenna and gets reflected when reaches the feed and in that
process, RCP and LCP are interchanged due to reflection. To explicitly check
for these effects, we needed to carry out tests with artificial sources of
known polarisation. We built an RCP and an LCP helical antenna in the lab. We
also carried out some tests with the satellites of known polarisation.

###  7.1 Engineering tests using helical antenna at band 3 and band 4

A helical antenna is in the shape of a corkscrew that produces radiation along
the axis of the helix antenna. The helical antenna can produce circularly
polarised fields. In November 2020 an axial mode helical antenna was built in
the lab (Figure 26). In a design of a helical antenna, the radiation is fed
from below via the wire in the image. The direction of the coil determines the
sense of circular polarisation. As per this image, according to IEEE
standards, this is a right-hand circularly polarised helical antenna.

Figure 27: Top Left: The test setup with the RHCP helical antenna as per IEEE
convention. Top right: Results of band 3 set up. The yellow curve is channel 1
and the blue curve is channel 2. Bottom left and right: Left one is channel 1
and the right one is channel 2. Higher power seen in channel 1

The antenna was designed for 585 MHz and it was found to be tuning well
between $480-700$ MHz, i.e. in the uGMRT band 4 (Figure 26). In this test, we
brought the GMRT antenna to the lowest possible elevation and focused the band
4 feed to the ground (Figure 27). The RCP helical antenna was placed on the
ground in the line of sight of the feed. Note that the GMRT parabolic dish is
not used here, the radiation directly goes to the feed, and there is no
reflection of the radiation from the dish. We radiated the EM signal at 610
MHz, and we have seen a strong signal of -22.8 dB in channel 1 and a weaker
signature of -47.9 dB in channel 2. The power difference between the two
channels is 26 dB, which is quite significant. This ensures that the channel 1
is RCP when the dish is not involved in the experiment.

Since the helical antenna was also mildly tuning to 400 MHz, we carried the
same test in band 3 as well. In Figure 27, channel 1 (yellow curve) receives
higher power at $-27.2$ dB, and channel 2 receives a lower power at $-37.5$
dB. This also ensures that for band 3 as well, channel 1 is RCP when the dish
is not involved in the experiment.

We also built an LCP helical antenna and carried out the same test. In this
case, channel 2 showed higher power in both band 3 and band 4. This
engineering test ensures channel 1 is RCP and channel 2 is LCP, when dish is
not involved in the experiment.

###  7.2 Engineering test using dipole antenna at band 5

We also attempted to test channels 1 and 2 for band 5 for which the
polarisation channels are linear. For this, we used a newly designed cross-
dipole feed. The setup is the same as that for the helical antenna. We first
brought the antenna dish to the lowest elevation. We then pointed band 5 feed
towards the ground and put a linearly polarised horn antenna in the line of
sight on the ground.

Table 3: Test results with the cross dipole feeds at band 5  Radiated power | FE output Channel 1 | FE output Channel 2
---|---|---
(–40 dBm) polarisation | (dBm) | (dBm)
Vertical | $-18.1$ | $-38.2$
Horizontal | $-38.8$ | $-16.9$

When we radiate vertical polarisation via the vertical probe of the horn
antenna and terminate the horizontal polarisation by placing 50 $\Omega$
resister, channel 1 gets higher power. Now we rotate the element and radiate
through the horizontal probe and terminate the vertical component by putting
50 $\Omega$ resister, the channel 2 gets higher power (Table 3). This means
channel 1 is X and channel 2 is Y.

In both tests, the effect of the reflection due to the GMRT antenna is avoided
since the feed is directly receiving radiation. These tests also verified that
Channel 1 and Channel 2 are labeled corrected from the feed to the final
output.

##  8 Tests using satellites

A satellite is an artificial spacecraft for communication that relays and
amplifies telecommunications signals with a transponder. It can be considered
as a middleman between a transmitter and a receiver. The satellites are used
for television, telephone, radio, internet, and military applications. In
satellite telecommunication, a downlink (the link from a satellite down to one
or more ground stations or receivers), and an uplink (the link from a ground
station up to a satellite) are used.

Many of these satellites are circularly polarised and they work at the GMRT
frequency range. To understand the effect of the antenna dish, we chose
circularly polarised satellites with downlink ranges between bands 2 and 3
frequencies888Thanks to Ankur, Sanjeet, Pravin, Imran, Gaurav, Technical
support staff, NCRA Workshop, and Control Room. In addition, we carried out
some tests in band 5 as well.

Figure 28: Description of satellites used for various tests.

###  8.1 Tests at band 2

For band 2, we used weather satellites, which are polar-orbiting, non-
geosynchronous, environmental satellites such as NOAA, and Russian Meteor
satellites. They have a downlink frequency of 137 to 138 MHz and are right-
handed circularly polarised (downlink polarisation)999A downlink is a link
from a satellite down to one or more ground stations or receivers, and an
uplink is a link from a ground station up to a satellite. . Skynet-5B as well
as Sicral 1B satellites were used for band 2 tests.

Skynet -5B satellite uses downlink frequency in the 240-270 MHz band. However,
this band is visible in band-2 at the furthest right side of our 120- 250MHz
band and hence can be used for band 2 testing. The satellite is RCP (the
downlink signal) as per IAU/IEEE convention.

Sicral 1B is an Italian military communications satellite. It is equipped with
a communications package that includes three frequency bands, the extreme-high
frequency (EHF), the ultra-high frequency (UHF), and super-high frequency
(SHF) transponders. The satellite is located at 37.0 degrees east latitude.
The uplink and downlink frequencies are 292 - 317 MHz and 243 - 270 MHz,
respectively. This is also RCP as per IAU/IEEE convention. The satellite has
17 25-kHz UHF channels and 21 5-KHz communication channels.

These tests were done with the GMRT antennas C11, E03, and S02. The satellites
were tracked and the data were recorded at the receiver room and looked at the
fibre optics receiver output monitoring port. We note that channel 2 shows
high power reception with $\sim-20$ dB power and channel 1 power is down by
around 20–27 dB at a level of $-40$ dB. Hence for this RCP satellite, channel
2 receives higher power (Figure 29).

Figure 29: Satellite tests at band 2. The top panel is the tests with C11
antenna, the middle panel with E3, and the bottom panel with S2

###  8.2 Test at band 3

The test was carried out by Sicral 1B and Sicral 2 satellites. We did the test
with C10 antenna and the data were recorded at 252 and 267 MHz frequencies. In
both bands, channel 2 received higher power with -21.7 and -18.2 dB power at
both frequencies, respectively. Channel 1 received lower power between -40 and
-45 dB causing the RF isolation of 18-27 dB, confirming that for this RCP
satellite, channel 2 receives higher power (Figure 30).

Figure 30: Satellite tests at band 3 at frequencies 252.342 and 267.236 MHz.
Channel 2 receives higher power in both cases. label

###  8.3 Tests at band 5

For band 5, we used GPS satellites. These were IRNSS and BEIDOU satellites.
IRNSS 6 used was at the Geostationary orbit at 32.5 degrees east longitudes.
The downlink frequency of the satellite is 1176.45-1189 MHz. BEIDOU is a G6
satellite, which is 80 degrees east in the geostationary orbit with downlink
frequencies of 1207.14 MHz, 1268.52 MHz, 1561.98 MHz, and 1589.742 MHz. Both
are RCP satellites as per IAU/IEEE convention. Since GMRT band 5 feeds are
dipolar, we should not receive a power difference. This is indeed the case as
can be seen in figure and table in 31, both channels receive equal power.

Figure 31: Tests with IRNSS-1F and BEIDOU-G6 in band 5. Both channels receive
equal power.

###  8.4 Tests with and without the GMRT antenna

Figure 32: Tests with the satellite with the feed in focus (i.e. default
setting with radiation first falling onto the antenna and then reflected and
received by the feed) and feed out of focus (i.e. feed directly receiving the
radiation from the satellite).

Tests with helical antenna were without the reflection from the antenna dish
as the feeds directly received the radiation. The tests showed channel 1 to be
RCP and channel 2 to be LCP. To reconfirm the effect of dish, we carried out
tests with feed in focus (i.e. default setting with radiation first falling
onto the antenna dish and then reflected and received by the feed) and feed
out of focus (i.e. feed directly receiving the radiation from the satellite).
We used Sicral 1B and Skynet 5B for band 2, Sicral 1B, Sicral 2, and MUOS 4
for band 3. All of these satellites are RCP in nature. As seen in Table 32,
with feed in focus, in all settings, channel 2 receives higher power. However,
channel 1 receives higher power when the feed is directly receiving the
radiation.

This result suggests that channel 1 receives higher power when the RCP
radiation directly falls on the feed. However, when it is reflected from the
antenna, it reverses its polarisation and channel 2 receives higher power.
This experiment concludes that the reflection at the dish causes a reversal in
polarisation between $R$ and $L$, which is consistent with the results from
astronomical observation as described in Section 6.3.

##  9 Conclusions

This brings us to the end of this study. Below are our main takeaway points.

  1. 1.

GMRT antenna Azimuth 0 points to South. The H dipole is parallel to the feed-
rotating axis which is along the East-West direction, V dipole is
perpendicular to the feed-rotating axis, which is along the North-South
direction. Hence as per IAU/IEEE convention, H is equivalent to the Y
component of the electric field and V is equivalent to the X component of the
electric field.

  2. 2.

Engineering tests have shown that channel 1 receives V polarisation without
any phase delay and channel 2 receives V polarisation with 90 degrees phase
delay. Similarly, channel 2 receives H polarisation without any phase delay
and channel 1 receives H polarisation with 90 degrees phase delay. This means
channel 1 $=V+iH$ and channel 2 $=H+iV$. Translating it into the cartesian
coordinate system, Channel 1 is $X+iY$ and channel 2 is $Y+iX$, which
according to IAU/IEEE convention are RCP (North component ($X$) of the
electric vector leading East component $Y$) and LCP (East component ($Y$) of
the electric vector leading North component $X$), respectively.

  3. 3.

The above results are also confirmed by the tests performed using the helical
antenna. When this directly radiates towards the feeds, channel 1 receives
higher power for the right circularly polarised helical antenna, and channel 2
receives higher power for the left circularly polarised helical antenna. This
confirms that the Channel 1 is RCP ($R$) and channel 2 is LCP ($L$) from feed
to the final data products.

  4. 4.

The observations have revealed that when the data are fully polarised
calibrated (flux and phase calibration, bandpass calibration, and polarisation
calibration) for pulsars with well-established polarisation properties with
known Stokes $I$, $Q$, $U$, $V$, the GMRT Stokes $U$ and $V$ always have a
mismatch. This can be explained if $R$ and $L$ are swapped. This solves the
incomprehensible result obtained from the initial test (Section 4).

  5. 5.

The helical antenna tests and the lab tests did not involve the reflection of
the GMRT antenna. The observations involved reflection of the antenna and
indicated a swap of $R$ and $L$, which is equivalent to ignoring the effect of
reflection, which converts RCP to LCP and vice versa.

  6. 6.

To incorporate the effect of reflection by the GMRT antenna, we carried out
tests with satellites of known polarisation with feed in focus and feed out of
focus. In bands 2 and 3, for RCP satellites, with feed in focus (i.e.
radiation first falling on the antenna dish and then reflecting to reach the
feed), channel 2 showed higher power and channel 1 showed lower power. With
feeds directly receiving the radiation, channel 1 showed higher power for RCP
satellites. This is consistent with observations, which were showing channel 2
to receive higher power (which is done with feed in focus). This test again
establishes the polarisation reversal due to the reflection by the antenna
dish.

  7. 7.

All of these results (tests with astronomical source in interferometric mode,
tests using helical antennas, tests using satellites keeping the feed in focus
to the dish and directly to the satellite) are consistent with each other.

The study indeed proves that GMRT channels 1 and 2 are true $R$ and $L$.
However, since GMRT is a prime focus instrument, the reflection due to the
dish reverses the sense of polarisation and converts RCP into LCP and vice
versa. This has not been taken into account and because of that reason, a
right circularly polarised source behaves like a left circularly polarised
source upon reflection and produces higher power in channel 2.

  8. 8.

Since astronomical observations are always done with feed in focus ((i.e.
radiation first falling on the antenna dish and then reflected to reach the
feed), one needs to reassign channel 1 to $L$ and channel 2 to $R$ for all
circular feeds, i.e. bands 2, 3 and 4 (however, see next section for caveats
in band 2) to obtain correct polarisation of the source.

##  10 Caveats and future studies

  1. 1.

While there is no difference in the interferometric data whether it is taken
in the USB mode vs LSB mode, and consistent with the engineering tests, the
beamformed data does not show this result. In beam data, Stokes $U$ (thus
position angle) are consistently inverted between USB vs LSB data. This needs
to be understood. As of now, parallactic angle corrected beam data in the USB
mode with $R$ and $L$ swapped will give the correct results. One thing that
needs to be noted here is that like the interferometric analysis, no robust
polarisation calibration is performed for beamformed data.

  2. 2.

The above tests have been done only with bands 3 and 4. The uGMRT band 2
polarisation calibration is still in progress. Once that is done, the
observational tests have to be done in bands 2 as well101010Preeti Kharb and
the team is working on the band 2 calibration. However, the engineering and
satellite tests have shown that the band 2 behavior is the same as bands 3 and
4. Hence even without the observational confirmation, one can successfully
apply our results.

  3. 3.

We have no conclusive tests for band 5 which have dipolar feeds. These tests
have to be carried out separately and determined for band 5. Hence the results
in band 5 remain inconclusive. Engineering tests have shown that band 5
channels 1 and 2 are $X$ and $Y$. The calculations show that the reflection
effect due to GMRT being a prime focus instrument must result in $X$ and $Y$
becoming $X$ and $-Y$. Its effects on Stokes parameters need to be tested via
astronomical observations and polarisation calibrations.

Our main conclusion is GMRT Stokes V and U signs need to be reversed by the
observer to make them consistent with the IAU/IEEE convention. The cause of
this is R & L swap incurred by the reflection suffered by sky signals at the
antenna dish before entering the feed at the prime focus. This objective can
be achieved by reassigning channel 1 to $L$ and channel 2 to $R$ for all
circular feeds, i.e. bands 2, 3, and 4. The study remains inconclusive for
band 5, which has linear feeds.

## Acknowledgements

We thank the staff of the GMRT that made these observations possible. The GMRT
is run by the National Centre for Radio Astrophysics of the Tata Institute of
Fundamental Research. We thank Yashwant Gupta for his useful insights. B.D.
thanks, Minhajur Rahaman for suggesting the pulsar B1702–19 to use as the
polarisation standard. B.D. and P.C. thank Jayaram N. Chengalur for useful
discussions. S.K. thanks Kadaladi Pavankumar for helping with beam
polarisation data. P.C. thanks, Dipanjan Mitra for clarifying some important
polarisation concepts. We thank the GMRT engineering team for help on multiple
occasions. We especially acknowledge Ankur, Pravin Raybole, Sanjeet Rai, Imran
Khan, Gaurav Parikh and Vilas Bhalerao for the engineering tests carried out
in this study. This research has used NASA’s Astrophysics Data system.

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