# L2-CalSat: A Calibration Satellite for Ultra-Sensitive CMB Polarization Space Missions
## 3 Calibration Mission Impact over Systematic Errors
### Beam Patterns  
Important polarization systematics can be produced by the leakage from intensity to polarization signals, due to non-idealities in the beam patterns of the telescopes, if they are not well understood and accounted for. Beam patterns are a 2D representation of the angular response of the telescope. Precise knowledge of them is crucial to reconstruct the sky maps and extract accurate scientific results. As in the previous cases, beam irregularities can produce TB, EB and especially BB spurious signals. They arise from different effects related to the optics, the filters, the focal plane, and the readout electronics and can be quantified and taken into account properly, only if characterised during the operation of the telescope. Figure 4: BB signals produced by random pointing errors of 10 (green), 5 (blue) and 1 (red) arc-minute. Figure 5: TB (**a**) and EB (**b**) signals produced by random pointing errors of 10 (green), 5 (blue) and 1 (red) arc-minute. The best method to measure beam patterns is to observe a well-characterized point-source in the far-field of the telescope. Ideal beam patterns should be symmetric and should not affect the polarized properties of the incoming light. However, they often exhibit different properties for different detectors and their shape can be affected by ellipticity and other irregularities. These effects introduce polarization systematics and polarization leakage, including non-zero TB and EB signals, and, more importantly, BB bias. Sometimes beam features can be taken into account with optical models, but some complex features must be quantified using an experimental beam calibration. Examples include features caused by stray light from internal reflections or scattering on optical elements, mirror surface errors, diffraction, transmission elements emission, and non-uniformities. Beam window functions are also affected by focal plane systematics, such as feed-horn asymmetries or cross-polar response, detector time constants, and detector orientation. Coupling with sinuous antennas and lenslets may introduce cross-polarization and beam ellipticity as well. Sinuous antennas have a periodic, frequency-dependent polarization angle defined by the antenna geometry. In order to accurately characterise the intensity beam, the [PERSON] (1948); [PERSON] (1949) CMB space mission showed that with several years of observations of a typical celestial source as Jupiter, a NF level of about \(-50\) dB respect to the peak amplitude of the planet emission can be achieved, taking into account the overall contribution of the cosmological bands detectors. Although this level of accuracy can be slightly improved combining the data from other planets as Mars, Saturn, Uranus and Neptune, in the case of future ultra-sensitive space missions, it may result insufficient to characterise far side-lobes (FSL) placed at \(10^{\circ}\) or more from the beam center, specially in polarization. To illustrate this, Figure 6 shows the BB signals (a) produced by the FSL of two non-ideal beam models (b) with FWHM = 30 arcmin and for a frequency of 140 GHz. The beam coloured in red has been modeled following the trend of the ones shown in [PERSON] (1948), while the beam coloured in blue has been modeled from the red one but presenting 10 dB lower side lobes from \(10^{\circ}\) onwards. In this case, the spurious signal is produced by FSL placed at more than \(20^{\circ}\) from the beam center, which would remain non-characterised using Jupiter. It has been calculated subtracting the signal of an auxiliary beam that is equal to the red one until \(20^{\circ}\), but from this angle it is equivalent to a Gaussian one (see the dashed red line in Figure 6b). In the case of the beam coloured in blue the spurious signal is produced by FSL placed at more than \(10^{\circ}\) from the beam center and it is calculated, equivalently to the red one, using an auxiliary beam that behaves as a Gaussian one from \(10^{\circ}\) onwards.