Beams LFI

Overview

LFI is observing the sky with 11 pairs of beams associated with the 22 pseudo-correlation radiometers. Each beam of the radiometer pair (Radiometer Chain Assembly - RCA) is named as LFIXXM or LFIXXS. XX is the RCA number ranging from 18 to 28; M and S are the two polarization namely main-arm and side-arm of the Orthomode transducers #darcangelo2009b (see also LFI naming convention).

Figure 1. A sketch of the Planck LFI field of view in the (u,v) plane is shown. The polarization direction on the sky are highlighted by the colored arrows. The M-polarization is shown in green and the S-polarization in red. Main beam shapes are shown for completness and they are not representative of flight beams.

Main Beams and Focal Plane calibration

As the focal plane calibration we refer to the determination of the beam pointing parameters in the nominal Line of Sight (LOS) frame through main beam measurments using Jupiter transits. the parametes that characterise the beam pointing are the following:

• THETA_UV ($\theta_{uv}$)
• PHI_UV ($\phi_{uv}$)

They are calculated starting from u,v coordinates derived form the beam reconstruction algorithm as

$\theta_{uv} = \arcsin(u^2+v^2)$

$\phi_{uv} = \arctan(v/u)$

Two additional angles are used to characterize the beams in the RIMO:

• PSI_UV ($\psi_{uv}$)
• PSI_POL ($\psi_{pol}$)

$\psi_{uv}$ and $\psi_{pol}$ are not derived from measurements but they are extimated form optical simulations. They are the quantities that represent the polarization direction of each beam, in the following approximation: the M- and S- beams of the same RCA point at the same direction on the sky.

The main beams are characterised by 2 method:

• elliptical (or bivariate) gaussian fit as in #[planck2011-1-6] with modification explained in #[planck2013-p02d]. This method is used to determine
  • the beam centre
  • the average full width half maximum defined as $\sqrt{FWHM_{max}\cdot FWHM_{min}}$
  • the beam ellipticity defined as $FWHM_{max}\over{FWHM_{min}}$
  • the beam tilting, $\psi_{ell}$, with respect the u-axis.

• Electromagnetic simulation (using GRASP Physical Optics code) by appropriately tuning the Radio Frequency Flight Model (RFFM) #tauber2010b. The Radio Frequency Tuned Model, called RFTM, was implemented to fit the beam data with electromagnetic model. It is derived as follow:
  • the Focal plane unit electromagnetic model has shifted by 3.5mm toward the secondary mirror;
  • All the simulated beams where monochromatic, i.e calculated at a single frequency called Optical Center Frequency (OCF). For the RFTM model the OCF has been chosen at $28.0$, $44.0$, $70.0$. In fact the optical and radiometer bandshapes as reported in #zonca2009 demonstrates that for the 30 GHz channel, the radiometer responses are better described by a central frequency closer to 28 GHz with respect to the nominal one, whereas for the other two frequency channels the OCF is close to the nominal one.
  • each feed horn phase centre has been moved alogn horn axis to optimize the match between simulations and data. The optimization was obtained by minimizing the variance according to the following definition: If $B_s[u,v]$ is the peak-normalized scanning beam matrix (for semplicity we use here $(u,v)$ coordinates also as indexes of the beam matrix) and $B_o[u,v]$ is the smeared peak-normalized simulated GRASP beam, the variance, $\sigma$, can be evaluated for each beam:

\begin{eqnarray} \label{eqsigma} \sigma &=& {\sum_{u,v}{(f[u,v] - \overline{f})^2}}\cdot {1\over N} \\ f[u,v]&=&w[u,v] \cdot (B^{dB}_s(u,v)- B^{dB}_o(u,v)) \\ w[u,v]&=&\sqrt{T[u,v]} \end{eqnarray} where also $T[u,v]$ is the temperature or the scanning beam not normalized to peak, and $N$ is the number of points considered in the comparison so that the number of point in the (u,v) plane. The parameter $\sigma$, as already said, is the variance of the difference between two beams weighted by the measured beam itself. For each beam the variance has computed computed down to $-15$dB from beam peak to avoid bias due to noise and background. The comparison between the simulated RFTM beams and the data are reported in #planck2013-p02d

Effective beams

TBW

Window Functions

TBW

Sidelobes

There is no direct measurements of sidelobes for LFI. The sidelobe pattern for LFI was been simulated using GRASP9 Multi-reflector GTD. We used the RFTM electromagnetic model. Seven beams for each radiometer have been computed in spherical polar cuts with a step of 0.5 degrees both in theta and phi. The beams have been computed in the same frames used for the main beams. The intermediate beam region (theta < 5 degrees) has been replaced with null values.

• In the computation we considered:
  • the direct field from the feed
  • the 1st order contributions: Bd, Br, Pd, Pr, Sd, Sr, Fr
  • the 2nd order contributions SrPd and SdPd

where B=buffle', P=primary reflector, S=secondary reflector, F=Focal Plane Unit Box. and where d=diffraction, r=reflection. For example Br, means that we considered in the calculation the reflection on the telescope baffle system.

A refinement of the sidelobes model will be considered in a future release, taking into account more contributions together with Physical Optics models.

References

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1. References

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