the matching of phased-array antenna beams in the aperture plane is described in detail in O’Brient et al. 2012.
86
By rotating the beam splitter relative to the focal plane, the tilt axes of the calibrator rotate with respect to the polarization axes of the detectors, thus creating a periodic polarized signal. As the hardware for the measurement was inherited fromBicep1, we refer the reader to Yuki Takahashi’s thesis (2010) for a complete description of the calibrator, appropriately deemed the “Yuki-cal.”
The procedure for acquiring polarization angle measurements using the dielectric sheet calibrator is as follows: the apparatus is installed in place of the forebaffle directly above the window of the telescope. Because a substantial fraction of the beam is transmitted to the sky, data are acquired in only the best of weather. A film thickness and index is chosen to provide the requisite signal-to-noise while avoiding gain instability. The detectors are then biased onto either the Ti or Al transitions.4 Before acquiring the calibration scan, the telescope is “dipped” in elevation to provide an unpolarized signal modulation, from which a relative gain correction between A and B is derived. Scans are acquired by counter-rotating in DK and AZ. The counter-rotation fixes the beam location on the sky, while the calibrator (attached toAZ axis but not theDK axis) rotates about the boresight.
Extracting polarization angles from this measurement requires a full model of the projected axes of polarization sensitivity onto the dielectric sheet. This model is presented in detail in Appendix B. This is based on a similar model presented in Takahashi 2010. Unlike the Takahashi model, we assume at the outset that the deviation of the polarization axes of the detector pair from nominal can be represented by a single angle↵, which is equivalent to assuming perfect orthogonality between the detectors. This is a reasonable approximation: Measurements of the cross-polar beam in the far field constrain cross-polar response to<0.005. If we assume a simple sinusoidal model, this implies that the axes must be co-aligned to <0.28degrees. This simplifying assumption allows us to fit a single polarization angle to the pair-difference signal. By assuming orthogonality, we can reject the common-mode atmospheric fluctuations and enormously improve the polarization angle fits. Fitting the response of each detector separately requires stability in the common-mode signal beyond what we were able to achieve during the austral summers.
The model relies on a few externally measured quantities: the tilt of the dielectric sheet, the sheet material properties (including the index of refraction and thickness), and the ray angles of each of the detectors. The tilt of the sheet was measured with respect to gravity with a digital level before and after each scan (represented by t in the appendix, and close to 45 degrees). The agreement between the beginning and end of the scan was typically<0.03degrees. The lateral tilt across the surface of the dielectric sheet (represented by in the appendix) was also measured with a digital level meter. Between installations of the calibrator, the value of was observed to change by as much as 1.5 degrees, but during a measurement was repeatably measured to<0.02degrees.
The sheet thickness was measured in the lab, while the index of refraction was taken from external
4In the end, the data acquired on the Al transition proved less susceptible to gain compression than the Ti data, though both data sets were used in analysis.
100 50 0 50 100 150 200 250 0.5
0 0.5
DK [degrees]
Idi↵/I
Detector pair 1
DK-averaged data Model
Figure 3.16: Dielectric sheet calibrator pair-difference data and fitted model for Detector Pair 1. The detectors are first relatively calibrated offof the atmosphere by performing a dip in elevation, then differenced to construct the pair-difference irradiance, Idiff. The pair-difference signal, normalized by the brightness difference between the warm absorber and the sky, I (a free parameter in the model), is extremely well-matched by the model of the polarized signal from the dielectric sheet, the details of which can be found in Appendix B.
sources. The fit angle↵is a weak function of both the index and the sheet thickness, and thus does not present significant uncertainties. The ray angles of the detectors are taken from the detector centroid fits in the far field, as described in Section 3.1.3. With these external inputs accurately measured, the model leaves only two free parameters. The first is ↵, which is the angle of the A and B polarization axes relative to DK. The second free parameter is I, the amplitude of the signal, which is proportional to the difference in temperature between the absorptive lining and the sky temperature at zenith.
Polarization angles are derived from a total of 5 independent measurements separated in time.
These were acquired in August 2010, November 2010, March 2011, November 2011, and December 2012. The first three measurements used a 2 mil thick mylar film while the final two measurements were taken with a thinner 1 mil sheet. To combine the measurement results, weights were derived from the inverse variance of the residual after subtracting the fitted model, thus down-weighting noisy or poorly modeled data. Using this weighted combination, we find a best-fit global rotation of the polarization angles = 0.18degrees. This is in excellent agreement with an overall rotation of the focal plane inferred from beam centroids, which was fitted using CMB data to be equal to 0.17 degrees. Among the five scans, the maximum deviation from the mean was +0.1 degrees, but this scan also contained the least weight. We therefore quote a conservative estimate of the 1- uncertainty in the global rotation of the polarization angles to be0.1/p
5 = 0.04degrees. This uncertainty is a significant improvement over Bicep1, which achieved a polarization orientation
88
0 20 40 60 80 100 120 140 160 180 200 220 240 1
0.5 0 0.5 1
Detector pair number
↵[degrees]
Figure 3.17: Bicep2 measured polarization angle deviation with associated measurement uncer- tainty. The angle↵corresponds to the deviation of the polarization axes from nominal, as defined in Equation B.15. These measurements are consistent with pointing center fits, which measure a global rotation of the focal plane of -0.18 degrees fromDK. Error bars are equal to the square root of the weighted variance.
uncertainty of < 0.7 degrees (Takahashi et al. 2010). The dominant uncertainty in Bicep1 was an apparent overall offset of 1.0 degree between calibration measurements taken in 2006 and in 2007-2008 that was never fully understood. Bicep2observed much tighter consistency over repeated independent measurements, thus achieving a much tighter global constraint.
In addition to measuring the global rotation of the polarization axes, we can similarly assess the per-detector scatter and uncertainty. We repeatably measure scatter in the polarization angles across the focal plane: The standard deviation taken across all operational detector pairs (referred to as “really good lights”) is 0.14 degrees. We estimate the median 1- per-detector polarization uncertainty to be0.08degrees. This is calculated as the square root of the weighted variance, where, as before, the weights are derived from the fit residuals.