their relative phase.
In reality, the incoming signal is elliptically polarized due to the
properties of the source,
imperfections in the reflector, the feeds, and
the polarizer. Measurements of continuum sources (Turlo et al.
1985) or pulsars (Stinebring et al. 1984; Xilouris
1991) can estimate those imperfections introduced by the
receiving system which tend to be stable over long time periods.
To derive full polarization information, a correlation device
is often employed which produces power outputs proportional
to the Stokes parameters
defined as follows (see Saikia & Salter 1988):
.
The polarization state of the incident radiation is degraded
further by phase and gain imbalances
introduced by
the various amplification stages between the two channels
following the polarizer as well
as by the correlation devices used to produce the Stokes parameters.
The gain imbalances are time dependent, and for high quality
polarimetry, they must be continuously monitored and dynamically
corrected for. In the following sections,
our approach to the problem of dynamic calibration is presented
for both the adding and
multiplying polarimeters which we used.
The adding polarimeter is a passive device which performs analog phase-correct additions of the two incoming signals before their detection. This type of polarimeter is used when further analog signal processing (i.e. via a filter bank) follows.
Figure 1: Block diagram of the 40-MHz adding polarimeter
at Effelsberg.
The RCH and LHC signals are split into three pairs. Each
two pairs, appropriately delayed in phase are added by the quadrature
and 
hybrids, resulting in the denoted signal outputs
The adding polarimeter
consists of a network of 
and hybrids
which perform phase-correct additions with wide-band response
and minimal loss. It is this quality of wide-band
analog response that has made them the best approach
for high-frequency wide-bandwidth polarimetry.
Generally, an adding polarimeter can provide
six analog outputs (see Fig. 1 (click here)).
The data acquisition system at Effelsberg PUB86,
is capable of synchronously detecting only 4 inputs.
We chose to record the LHC and RHC components separately
and used these values to furnish the total intensity because it is
straightforward to convert the measured deflections into flux densities.
To make an absolute flux-density calibration,
one compares the noise-source deflections with those of standard
continuum sources.
The voltages provided by each output of the adding polarimeter
are shown in Fig. 1 (click here).
Following square-law detection,
the power detected in
each of the four chosen
outputs is given by:
where the 
denotes the complex conjugate of
a vector. The Stokes parameters can then be
derived as:
and
Prior to subtraction, a scaling factor is necessary between the total power I and the outputs of channels 3 and 4 in order to produce the linear components Q and U. Such scaling is necessary because of the gain differences between the paths of channels 3 and 4, and the total power I.
The quadrature and hybrids have been measured
to introduce with great repeatability their nominal gain and
phase delays in the circular components of the propagating
signals. Other active components in the propagation
path such as amplifiers, and phase shifters, as well as any piece
of wave guide, introduce differential gains and phases between
the two circular components. These differences are reflected in
the Stokes parameters as spurious polarization and therefore
they have to be detected and accounted for.
To detect and monitor the stability of these gains, a noise signal is
fired synchronously to the pulsar period into the receiver horn
prior to the polarizer. The injection scheme is such
that most of this calibration
signal propagates towards the receiver and not the reflector.
To get a correlated calibration signal, the power from a single
diode is split and fed into the feed horn.
A visual conception of the gain factors involved in the
system is presented in the following block diagram
in Fig. 2 (click here).
Figure 2: Different gain-factors of the system
The gains along the propagation path can be identified in
three different groups.
Following Rankin et al. (1975),
the gains introduced in the path between the reflector and the
entrance to the polarimeter
are symbolized as AR and AL.
The gains between the inputs and the
outputs of the polarimeter are symbolized as
F1, F2, F13,F14,F23 and
F24.
The gains introduced by the
variable attenuators, the dedisperser, and the analog to digital detectors
are symbolized as G1 to G4.
The various gains will
influence the measured power in the output of each channel as:
with R and L as the two circular components which
represent real signals.
The main variations will occur in the A- and G-gains. The
polarimeter-gains (F) were found to be
relatively constant due to the passive components in the signal path.
In particular,
the products 
and 
should be equal.
If this is the case, then
we can set 
. Further, we assume that 
and 
.
Thus, the system of Eqs. (9 (click here)) can be simplified as:
The remaining five gains (
) can now be calculated
when the following
assumptions are made for the calibration noise-signal:
(a) the noise-signal is 100% linearly polarized, which requires the
intensity in channels 1 and 2 to be equal and (b) the
PPA of the signal is set at 
. Deviations from this would result in
unwanted terms inserted in the calculations.
The signals in channels 1 and 2 have to be calibrated to the same deflection
levels.
In channel 3 (=I-Q), no signal is expected because
when 
. In channel 4 (=I-U), we expect the
maximum signal because, 
when .
This is accomplished by introducing the necessary delays via a
"delay-box''. This is a unit that allows extra electrical length
in the signal path
so that the calibration signal as monitored on an oscilloscope display
following the multiplying polarimeter has all the power in the U
channel while there is virtually no power in the Q channel.
This is
checked over the entire bandwidth and the phases are
adjusted in such a way as to put as few zero crossings
in the Q channel as possible thereby, ensuring that the polarimeter
operates within
the white-light fringe regime. This calibration is performed
before each run. For the two year period of
these polarization experiments, the phase delays that
we had to introduce were very stable with time. Changes were only
encountered when
a component in the propagation path
was deliberately changed or malfunctioning.
With 
to 
representing the power
of the calibration-signal in the four channels, we
obtain the four gain-factors:
where 
should be close to unity.
The 
and 
give the size of the statistical noise,
measured over a
baseline in the off-pulse region of channels 3 and 4.
To obtain g3 and , one has to normalize
channel 3 to channel 4. Therefore, channel 3 is scaled to the same rms
level as
channel 4. The factor of 2 in channels 3 and 4 is introduced because
the total intensity of
the noise-signal is measured in channel 4, whereas
only half of the noise signal is measured in channels 1 and 2.
The calibration signal is given in units of half of the noise-signal.
The Stokes-Parameters are then calculated as follows:
We dynamically monitor
the gain values for each pulsar
as the calibration signal is synchronously introduced to the pulsar
period at the first 50 bins of each pulsar period. These gains
were found to be very stable in their ratios for long
observing runs. Furthermore,
when different ratios were detected this indicated a malfunction
of the system.
The multiplying polarimeter is easier to calibrate than the adding
polarimeter.
A multiplying polarimeter works on the principle
of correlating the input signals. If the two circular components
carry radiation that is linearly polarized,
the signals will be 100% correlated.
In one of the branches, a phase shift is induced which enables
the estimation of all four Stokes parameters. The response
of this device is linear
for the bandwidths of interest.
The outputs of the multiplying polarimeter are as follows:
and therefore, the Stokes parameters are given by:
Figure 3: Block diagram of the multiplying polarimeter.
A three-way power splitter feeds the RHC and LHC
signals into the input
of broadband mixers. In one of the branches, a phase delay
is introduced, to allow for all Stokes parameters to be derived
It is again helpful to consider the different gain-factors (see Fig. 4 (click here)) introduced by the observing set-up.
Figure 4: Different gain-factors of the system
The calibration is much easier
in this case because the signal is detected directly so
there are no vector additions involved (Eq. 15 (click here)).
Using the same definitions for the gains as for the adding
polarimeter, the gain propagation along the signal path is described
as follows:
Because there are only multiplicative gains involved,
the set of Eq. (15 (click here)) can be simplified
without further assumptions by introducing Eq. (3 (click here)) as follows:
To derive the gain-factors,
a noise-signal is used in the same way as with the adding
polarimeter. However, here we only need to assume that
the noise-signal is
100% linearly polarized. The PPA of the signal can then be calculated
from channels 3 and 4 as:
The following gain-factors are then derived:
Finally, the Stokes parameters can be calculated and normalized to
units of half the noise-signal:
The linear (L) and circular (V) polarized intensity and PPA
can then be calculated as described in Sect. (3 (click here)).
The error of the PPA was calculated as:
The error of the polarized linear and circular intensities is
estimated by the variance calculated in their off-pulse position
of the very nearly Gaussian fluctuations in the
Stokes parameters Q, U and V respectively.
Typically, each sub-integration lasted for a few hundred pulses.
Due to the lengthy integrations and the low gain of the telescope
at high frequencies, it is not necessary to correct for bright
pulses which could increase the overall
system temperature 
. The statistics of the linearly
polarized intensity are directly derived from the average
baseline level of the array containing the linear polarization.
Following the instrumental correction procedure, polarization
profiles of the same source from different epochs were sometimes
added to produce high signal-to-noise profiles that we used for our
fitting purposes. To enable this adding,
the parallactic angle of each integration was referenced to zero. This
transformation was performed on the PPA curve following every
instrumental calibration.
observations made with the adding polarimeter.
Differential gains, cross-coupling and depolarization will
transform the incident polarization state
to the observed 
.
The origin of these effects is discussed by
Stinebring et al. (1984) and Xilouris (1991),
while an evaluation of the instrumental polarization performance
of Effelsberg radio telescope at 1.7 GHz was presented by
Xilouris (1991). Typically, most of the feeds used have
an elliptical rather than a circular response. This means
that the
axial ratio 
of the LHC and RHC polarized
components of the antenna response is not equal, but
very close to, unity.
Even if the response is elliptical rather than
circular, the feeds can be orthogonal to one another
resulting in the antenna fully responding to an arbitrarily
polarized wave. Assuming an orthogonally responding antenna,
and following
proper gain calibration procedures,
the cross-coupling will have
the net effect of mixing the Stokes parameters. Under the assumption
that the amplitude of this effect 
is small
(
1), the Stokes parameters are modified to
first order as:
where 
and are the cross-coupling phase and
amplitude. Here,
represents the angle between the effective dipole axis
of the antenna
and the electric vector of the incident polarization state
, while 
is the true PPA
and n the parallactic angle.
In most pulsars, the circularly polarized component is small
compared to the linear polarization. Assuming
to be small which is usually the case for most
radio telescopes, the effect of cross-coupling on the
observed polarization state can be further simplified as:
When measurements of the incident polarization state
are conducted over a sufficient range of parallactic angle n,
or equivalently, effective parallactic angle 
,
the ratio of 
will exhibit a sinusoidal behavior. The cross
coupling parameters and are readily obtained
from the amplitude and phase of a cosine wave fit to the data.
Two well studied and bright pulsars, PSR B1929+10 and PSR B0355+54,
were chosen as calibrators and were observed for a small
range of parallactic
angle during each observing run. Those pulsars apart from
exhibiting standard polarization profiles, involve a number of
different polarization states with strong linear polarization
and small circular, with the associated PPA
well defined. This provides a good estimate of the
ratio as a function of .
The fitted parameters and were used to
correct the L and V waveforms as well as the pulsar intrinsic
polarization position angle curve by solving the system
of Eqs. (21 (click here)) for the true polarization states.
The results from a two year monitoring of the polarization properties
for all of the receivers that we used are presented in Table 1 (click here).
The cross-coupling amplitude was
less than 10
and quite stable within the time intervals that
the instrumental set-up remained unchanged.
A known source of depolarization is Faraday rotation which occurs
both in the galactic interstellar medium and in the Earth's ionosphere.
These effects combine with unequal phase delays
between the two polarization channels of the instrumental setup
to effectively depolarize the incident polarization state.
In all cases, this effect results from different
propagation velocities between the
left-hand and right-hand circularly polarized components.
The net effect of either
true or instrumental Faraday rotation is that the position
angle of the linear polarization becomes frequency dependent.
Across some finite bandwidth, the dispersion of the position angle
results in a net loss of linear polarization.
Both the shape and the width of the IF filters that are used to
form gain corrected Stokes parameters play an important role
in a quantitative determination of the depolarization. Following
the analysis of Rankin et al. (1975) for square filters,
the largest amounts
of depolarization estimated for the highest rotation
measure pulsars in our sample at 1.4 GHz is between 3 to 8
with the rest of the sources less than 1.
At frequencies higher than 1.4 GHz, this
effect was negligible. The Faraday-induced
position angle rotation across the bandwidths used was
between 16
and 32
for the high RM pulsars in our sample.
Most of the RMs in our sample were low and in these cases
less than 
of rotation were calculated.
The instrumentally-induced position angle rotation across the bandwidths
used is proportional to the difference in electrical path length
between the two circular channels. Severe depolarization would
occur if the time delay exceeded the reciprocal of the observing
bandwidth. For the 1.4 GHz setup, a difference
in the electrical path of 7.5 m
would be required to cause severe instrumental
depolarization.
]