Receiver requirements

For technical reasons, the ALMA receivers will apparently be nominally sensitive to orthogonal linear polarizations. Let the signal from an astronomical object measured at the two orthogonal directions be X and Y. Then one may derive Stokes parameters as follows: Q = X - Y, $U = XYcos\phi$, $V = XYsin\phi$, where $\phi$ is the phase difference between the X and Y components of the signal. Systematic errors in Stokes U and V will be only linearly sensitive to the gain calibration of X and Y since X and Y occur as a product, but will be very sensitive to the phase calibration of the two receivers, something that is necessary for interferometry anyway. This discussion assumes that the two receivers are truly orthogonal, but if they are not the difference from a $90^\circ$ phase difference can be calibrated. But Stokes Q will be extremely sensitive to the gain uncertainty between X and Y; a gain error of 1% means a 100% error in a 1% polarization signal. Maintaining the difference between the gains of the two orthogonal receivers to better than -30 dB in order to achieve 0.1% polarization fidelity would be extremely difficult.

One way to overcome this problem is to insert a phase rotation device into the optical path such that the receivers detect opposite circular polarizations, R and L. This pushes the extreme sensitivity to the gain uncertainty into Stokes V = L - R, but means that both Stokes Q and U are derived from products of R and L, with the phase uncertainty rather than gain uncertainty dominating. Although this approach has been considered by the receiver group, it has major disadvantages. The phase rotation device (e.g., a quarter-wave plate) would introduce mechanical complications, would increase noise temperatures, and would yield circular polarization only over a relative small bandwidth.

One alternative solution for deriving accurate measurements of linear polarization with linearly polarized feeds is to take advantage of the fact that the position angle of the feeds will rotate on the sky due to the alt-azimuth mounting of the antennas. In practice, this is a poor solution, since it yields highly variable polarization fidelity over the uv plane. For snapshots, there will be little rotation. For sources transiting near zenith, rotation of position angle will be very rapid, with very little integration time for Stokes Q and U in many uv cells.

A somewhat related approach (suggested by Johan Hamaker; see Astron. & Ap. Suppl., 143, 515, 2000) is to have the orthogonal feeds at mixed orientations on various antennas. Thus, if on half of the antennas the linearly polarized feeds were rotated by 45$^\circ$ with respect to the other half, half (say) of the antennas may at any one instant derive Stokes Q from X - Y, while the other half would derive Q from $XYcos\phi$. During source tracking, position angle rotation would change this such that the situation would be more complicated, but the mathematics of inferring the four Stokes parameters from this situation is well known. It does mean that in many uv cells the derivation of Q (say) would be dominated by X - Y; without excellent receiver gain calibration the fidelity of the Stokes parameters would be highly variable across the uv plane. Nonetheless, this approach would make it possible to make polarization maps of high fidelity even without accurate receiver gain stability or calibration, with loss of some sensitivity due to having to assign low weights to individual Stokes parameter visibilities derived primarily from X - Y information.

If both receiver gain and phase can be very accurately calibrated, it should be possible to derive all four Stokes parameters from orthogonal linear feeds with common orientation, with acceptable loss of fidelity in the Stokes parameter derived from X - Y. An example of a careful investigation of deriving Stokes parameters with such a system is the study by Heiles (Arecibo Observatory Technical Memo 99-01, available on the NAIC web pages). Heiles employed a Mueller matrix formalism of the polarization properties of the Arecibo system and measured the Mueller matrix elements. Off-diagonal matrix elements describe undesirable polarization properties, such as leakage of Stokes I into Stokes Q ($m_{\rm QI}$), but if these off-diagonal terms can be accurately determined, accurate Stokes parameters can be derived. It of course helps that these terms be as small as possible, which is the requirement for a good hardware design and implementation. The Arecibo L-band wide receiver system has nominally orthogonal linear feeds whose signals are combined in the Arecibo correlator to produce the four Stokes parameters. The system has a correlated calibration source used to calibrate the gains and phases of the two receivers. Although the Arecibo system is a single dish rather than an interferometer, considerations for polarization work are similar, and are the same when single dish mapping with ALMA antennas, which will be essential for much polarization science, is considered. Heiles showed that with a system carefully engineered to produce good uncalibrated polarization results and with a stable calibration source available, excellent results may be obtained. He found that the error in determining the $m_{\rm QI}$ matrix term (with Q being derived from the difference X - Y) was only twice that in the $m_{\rm UI}$ term (with U being derived from the product).

A design for a bandpass calibration system presented by D. Emerson at the present ASAC meeting could also be used for polarization calibration of the receiver. This would consist of an amplitude and phase stable signal that would be broadcast into the receiver, so one could essentially continuously calibrate the gain and phase difference between the two nominally orthogonal receptors on each antenna. Based on the Heiles memo of his calibration of the Arecibo system using a similar calibration scheme, this system should make it possible with ALMA to derive the four Stokes parameters with sufficient accuracy.

Either, mixed orientation orthogonal feeds or the signal injection design, seems capable of ensuring accurate polarization calibration and both should be studied in the future.