BIMA memo 48. Calibrating and Imaging Polarization Switched Data Melvyn Wright, 01-AUG-96 This memo defines the algebra and outlines the steps used for calibrating polarization switched data, and making polarization images. A tutorial csh script is available in \$MIRBIN/polcal. Definitions ----------- I Stokes I, total intensity. Ip linear polarized intensity. psi polarization position angle Q Stokes Q = Ip * cos(2*psi) U Stokes U = Ip * sin(2*psi) V Stokes V, circular polarized intensity. chi Parallactic angle m,n baseline from antenna, m, to antenna, n p,q polarizations, either L,R circular, or X,Y linear. D(p,m) Leakage into polarization, p, on antenna, m At Hat Creek we measure ----------------------- RR = I + V LL = I - V LR = Ip * expi(2*psi) * expi(-2*chi) + I * (D(L,m) + conjg(D(R,n)) RL = Ip * expi(-2*psi) * expi(2*chi) + I * (D(R,m) + conjg(D(L,n)) Assuming that Stokes V is small, we calibrate using both LL and RR with a long enough calibration interval to include LR and RL data. For Alt-Az antennas, the phase of a linearly polarized source rotates with the parallactic angle, whereas the leakage terms are constant. Plotting the LR and RL uv-data imaginary versus real over a wide range of parallactic angle shows the linear polarization as circles offset by the polarization leakage. Average to get source polarization. ----------------------------------- We can remove the parallactic angle variation using UVCAL options=parang LR = LR * expi(2*chi) RL = RL * expi(-2*chi) and average the output to reduce the effect of the polarization leakage. LR = Ip * expi(2*psi) + I * (D(L,m) + conjg(D(R,n)) * expi(2*chi) RL = Ip * expi(-2*psi) + I * (D(R,m) + conjg(D(L,n)) * expi(-2*chi) For an unresolved source, averaged over a range of parallactic angle, = Ip * expi(2*psi) = Ip * expi(-2*psi) Determine the polarization leakage. ----------------------------------- To estimate the polarization leakage, we subtract the source polarization from the calibrated data using UVCAL polcal=Ip * expi(2*psi) and average in time to get the polarization leakage for each baseline. Finding the polarization leakage for each antenna. -------------------------------------------------- The task ANTPOL lists RL/(0.5*(RR+LL)) and RL/(0.5*(RR+LL)) for each baseline. The average polarization leakage is about 6% for each baseline, or 3% for each antenna at 86 GHz. We can fit the leakage for each antenna using the task GPCAL. Although GPCAL was designed for simultaneous measurements of YX,XY,YY,XX, we can average our polarization switched data so that each averaged interval contains LR,RL,LL,RR data. We then change the polarization codes from LR,RL,LL,RR to YX,XY,YY,XX using UVCAL polcode=-4 and fit antenna-based instrumental polarization. GPCAL also fits the antenna gains and XY phases, which in our case are the phase differences between L and R polarizations. These gains can be applied to the data, if needed. The fitted antenna-based leakage terms are relative to the leakage into the reference antenna. Subtracting the instrumental polarization using a source model. --------------------------------------------------------------- In order to subtract the polarization leakage we must have an estimate of the total intensity, I, for each time interval and baseline. With polarization switching we do not have simultaneous measurements of LR, RL, LL and RR on each baseline, but can estimate I from a source model, and subtract the polarization leakage using UVMODEL options=polcal. The polarization leakage is given as a table of four values for each antenna, being the real and imaginary parts for leakage into L and R polarizations respectively. The table can be extracted from the history of the GPCAL fit. Iterate if needed to get better estimates for source and instrumental polarization. Making polarization images. --------------------------- After calibrating the uv-data and correcting for the polarization leakage, images can be made and deconvolved using the usual Miriad tasks. We can make images of the Stokes I, Q, U, and V from the uv-data using the relations: RR = I + V LL = I - V LR = Q + jU RL = Q - jU If Stokes V is small, then both LL and RR data can be used together to make an I image. Alternatively LL and RR data can be imaged separately and the sum and difference used to make I and V images. With polarization switching the sampling is different for LL and RR data. Better sampling and deconvolution may be obtained using LL and RR data together. The deconvolved I image can be used as the source model for subtracting the instrumental polarization from LR and RL data. The LR and RL data are not Hermitian, as assumed by the INVERT task. The Q image is obtained in the usual way by imaging LR or RL data. We must use both the select and stokes keywords to select either LR or RL data. The U image is obtained using INVERT options=imaginary to image the conjugate of the uv-data. Thus we obtain an estimate of Q and U from the LR data, and an estimate of Q and U from the RL data. The sampling is different for LR and RL data. The synthesised beam is obtained with the Q image, and can be used to deconvolve both Q and U images. A better deconvolution may be obtained by averaging the Q and U images and beams from the LR and RL data. An estimate of the errors can be obtained by examining the differences. The Q, U, and I images can be combined into Ip and psi images using either MATHS or IMPOL tasks, and plotted using CGDISP or IMPLOT.