BIMA MEMO No. 43
Polarization Switching for the BIMA array
M.C.H.Wright, 15-Dec-95
The BIMA array receivers each have a single linear polarization.
Polarization Switching is required in order to obtain polarization
observations. This memo explores efficient switching cycles.
For linear polarization measurements we wish to sample LL, LR, RL, and RR,
where L and R designate the sense of circular polarization for a pair of
antennas. The interferometer response, F, to polarized emission is
F(LL) = F(I) + F(V)
F(RR) = F(I) - F(V)
F(LR) = F(Q) + jF(U)
F(RL) = F(Q) - jF(U)
where I, Q, U, and V are the Stokes parameters (e.g. Fomalont & Wright, 1974).
If there is no circular polarization, V=0, and either LL or RR measure the
total intensity I. Both LR and RL are required to measure the linear
polarization since LR and RL uv-data are not Hermitian.
With dual polarization receivers and 4 x nants(nants-1)/2
correlators it would be possible to obtain all 4 combinations of L and R
simultaneously for all baselines in an array with nants antennas.
Although it is possible in principal to use the
4 spectral windows in the BIMA correlator to sample the 4 polarizations,
with a single polarization receiver we must time multiplex the polarization
observations. The array is currently being outfitted so that each antenna
can be switched between L and R circular polarization. Since
the mechanical switching takes a few seconds, we seek efficient
switching cycles. To sample all combinations on all baselines would
take 2**nants combinations of L and R on each antenna. This is not
compatible with good uv sampling, so we must compromise.
One possibility, suggested by John Lugten, is to use Walsh functions to
switch the polarizations. A Walsh function of length 16 provides
orthogonal series for switching the polarization on up to 15 antennas.
A complete polarization cycle takes 16 integrations, which may be too
long for good uv sampling on long baselines. With the 9-antenna array
we can use higher Walsh sequencies, which cycle through all polarization
on most baselines in fewer than 16 integrations, as noted by John.
Walsh function switching takes no account of how many baselines simultaneously
observe total intensity, which may be a problem for calibration. We could add
cycles to simultaneously observe total intensity, although this makes the
complete cycle longer.
Another possibility is to switch the polarizations in groups with a common
polarization. With Ng groups there are 2**Ng combinations, of polarizations,
but the number of baselines which sample all 4 polarizations is reduced
as Ng gets smaller. For the current 9-antenna array, 3 groups of 3 antennas
provides a reasonable compromise as illustrated in Table 1.
Table 1. Polarization switching for 9-antenna array
cycle antennas in group baselines in group
A B C AB AC BC A B C
3 3 3 9 9 9 3 3 3
1 L L L LL LL LL LL LL LL
2 R L L RL RL LL RR LL LL
3 L R L LR LL RL LL RR LL
4 L L R LL LR LR LL LL RR
5 R R L RR RL RL RR RR LL
6 R L R RL RR LR RR LL RR
7 L R R LR LR RR LL RR RR
8 R R R RR RR RR RR RR RR
Examining Table 1, we see that the 27 baselines in the groups AB, AC and BC
sample all 4 polarization pairs for an equal amount of time, and the other
9 baselines within the groups A, B, and C sample LL and RR for equal times.
Any antenna can be chosen as a reference antenna for calibration since
cycle 1 and 8 respond to the total intensity (Stokes I) simultaneously
for all baselines.
We can trade off polarization observations for more integration time
in total intensity, or vice-versa. For example, using only cycles 3 - 6
provides 18 baselines in groups AB and AC with full polarization, 9
baselines in group BC sample only LR and RL, and 9 baselines within
the groups A, B and C sample only LL and RR. In this case the reference
antenna for calibration must be chosen from group A, and the calibration
of all antennas cannot be made simultaneously. Adding cycle 1 or 8 to
3 - 6 provides simultaneous calibration of all antennas; alternatively
using cycles 2 - 7 gives more integration time to RL and LR.
The choice of antennas within each group should be made with consideration
to which baselines are required with full polarization, and which with only
total intensity. Multiple tracks would allow permutations of antennas
within each group. Other permutations are possible.
The calibration problem was investigated using the Miriad task UVGEN to
generate polarization switched uv-data for the Walsh cycle of length 16,
and for the cycle of length 8 in table 1. We used the BIMA a-array
and a 20% polarized point source model with 30 degrees of phase noise.
The model uv-data were imaged and combined to produce I, Q, U, and V images.
Note that both real and imaginary parts of the LR and RL data must be imaged.
The task WALPOL was used to generate Walsh patterns. The shortest calibration
interval which could be used with the Walsh cycle was 1.6 times the integration
interval if both LL and RR were used to calibrate the data. With this
calibration interval, the amplitude of both unpolarized and polarized
components was about 88% of the model amplitude, corresponding to 28 degrees
of residual phase noise after the calibration. The resulting images
were almost identical for both switching patterns using a point source model.
For the cycle of length 8, we can use a calibration interval equal to the
integration interval. In this case the LL and RR images are recovered almost
perfectly, and the LR and RL images are improved to 95% of the model amplitude.
For more complex models, different results will be obtained, depending on the
sample interval and the source structure. A model with 5 compact components
with 20% polarization and 30 degrees phase noise also worked well with either
switching pattern. Models with more complex source structures, or higher phase
noise which requires self-calibration, work better with a short polarization
cycle giving better uv-sampling
of RR and/or LL. If the polarization structure is also complex, then
good uv-sampling of LR and RL is also required, and multiple uv-tracks are
required with either switching pattern. The choice of switching pattern
will probably be dictated by practical constraints of switching time,
a-priori knowledge of source structure, and available observing time.
This memo is respectfully
submitted for consideration by those who may be mathematically inclined.
References
Fomalont,E.B, & Wright,M.C.H, 1974, in Galactic and Extragalactic Radio
Astronomy, Springer-Verlag, New York, eds. G.L.Verschuur & K.I.Kellermann.