Planck 2013 results. IV. Low Frequency Instrument beams and window
  functions
  This paper presents the characterization of the in-flight beams, the beam
window functions and the associated uncertainties for the Planck Low Frequency
Instrument (LFI). Knowledge of the beam profiles is necessary for determining
the transfer function to go from the observed to the actual sky anisotropy
power spectrum. The main beam distortions affect the beam window function,
complicating the reconstruction of the anisotropy power spectrum at high
multipoles, whereas the sidelobes affect the low and intermediate multipoles.
The in-flight assessment of the LFI main beams relies on the measurements
performed during Jupiter observations. By stacking the data from multiple
Jupiter transits, the main beam profiles are measured down to -20 dB at 30 and
44 GHz, and down to -25 dB at 70 GHz. The main beam solid angles are determined
to better than 0.2% at each LFI frequency band. The Planck pre-launch optical
model is conveniently tuned to characterize the main beams independently of any
noise effects. This approach provides an optical model whose beams fully
reproduce the measurements in the main beam region, but also allows a
description of the beams at power levels lower than can be achieved by the
Jupiter measurements themselves. The agreement between the simulated beams and
the measured beams is better than 1% at each LFI frequency band. The simulated
beams are used for the computation of the window functions for the effective
beams. The error budget for the window functions is estimated from both main
beam and sidelobe contributions, and accounts for the radiometer bandshapes.
The total uncertainties in the effective beam window functions are: 2% and 1.2%
at 30 and 44 GHz, respectively (at $\ell \approx 600$), and 0.7% at 70 GHz (at
$\ell \approx 1000$).
