Beams

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 (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 Planck-Early-V with modification explained in Planck-2013-IV. 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) Planck-PreLaunch-XII. 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 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 Planck-2013-IV.

Effective beams

The effective beam is the average of all scanning beams pointing at a certain direction within a given pixel of the sky map for a given scan strategy. It takes into account the coupling between azimuthal asymmetry of the beam and the uneven distribution of scanning angles across the sky. It captures the complete information about the difference between the true and observed image of the sky. They are, by definition, the objects whose convolution with the true CMB sky produce the observed sky map.

The full algebra involving the effective beams for temperature and polarisation was presented elsewhere. Here we summarise the main results. The observed temperature sky [math]\widetilde{\mathbf{T}} [/math] is a convolution of the true sky [math]\mathbf{T} [/math] and the effective beam [math]\mathbf{B}[/math]:

[math] \widetilde{\mathbf{T}} \ = \ \Delta\Omega \, \mathbf{B} \cdot \mathbf{T}, \label{eq:a0} [/math]

where

[math] B_{ij} \ = \ \left( \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \right) / \left({\sum_t A_{ti}} \right) \, , \label{eq:EBT2} [/math]

[math]t[/math] is time samples, [math]A_{ti}[/math] is [math]1[/math] if the pointing direction falls in pixel number [math]i[/math], else it is [math]0[/math], [math]\mathbf{p}_t[/math] represents the exact pointing direction (not approximated by the pixel centre location), and [math]\hat{\mathbf{r}}_j[/math] is the centre of the pixel number [math]j[/math], where the scanbeam [math]b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t)[/math] is being evaluated (if the pointing direction falls within the cut-off radius of [math]\sim 2.5 \times[/math] FWHM.

The algebra is a bit more involved for polarised detectors. The observed stokes parameters at a pixel [math]i[/math], [math](\widetilde{I}, \widetilde{Q}, \widetilde{U})_i[/math], are related to the true stokes parameters [math](I, Q, U)_i[/math], by the following relation:

[math] ( \widetilde{I} \quad \widetilde{Q} \quad \widetilde{U})_i^T \ = \ \Delta\Omega \sum_j \mathbf{B}_{ij} \cdot (I \quad Q \quad U)_j^T, \label{eq:a1} [/math]

where the polarised effective beam matrix

[math] \mathbf{B}_{ij} \ = \ \left[ \sum_t A_{tp} \mathbf{w}_t \mathbf{w}^T_t \right]^{-1} \sum_t A_{ti} \, b(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) \, \mathbf{w}_t \mathbf{W}^T(\hat{\mathbf{n}}_j,\hat{\mathbf{p}}_t) \, , \label{eq:a2} [/math]

and [math]\mathbf{w}_t [/math]and [math]\mathbf{W}(\hat{\mathbf{r}}_j, \hat{\mathbf{p}}_t) [/math] are the the polarisation weight vectors, as defined in \cite{mitra2010}.

The task is to compute [math]B_{ij}[/math] for temperature only beams and the [math]3 \times 3[/math] matrices [math]\mathbf{B}_{ij}[/math] for each pixel [math]i[/math], at every neighbouring pixel [math]j[/math] that fall within the cut-off radius around the the center of the [math]i^\text{th}[/math] pixel.

The effective beam is computed by stacking within a small field around each pixel of the HEALPix sky map. Due to the particular features of Planck scanning strategy coupled to the beam asymmetries in the focal plane, and data processing of the bolometer and radiometer TOIs, the resulting Planck effective beams vary over the sky.

FEBeCoP, given information on Planck scanning beams and detector pointing during a mission period of interest, provides the pixelized stamps of both the Effective Beam, EB, and the Point Spread Function, PSF, at all positions of the HEALPix-formatted map pixel centres.

Production process

The methodology for computing effective beams for a scanning CMB experiment like Planck was presented elsewhere.

FEBeCoP, or Fast Effective Beam Convolution in Pixel space, is an approach to representing and computing effective beams (including both intrinsic beam shapes and the effects of scanning) that comprises the following steps:

• identify the individual detectors' instantaneous optical response function (presently we use elliptical Gaussian fits of Planck beams from observations of planets; eventually, an arbitrary mathematical representation of the beam can be used on input)
• follow exactly the Planck scanning, and project the intrinsic beam on the sky at each actual sampling position
• project instantaneous beams onto the pixelized map over a small region (typically <2.5 FWHM diameter)
• add up all beams that cross the same pixel and its vicinity over the observing period of interest
• create a data object of all beams pointed at all N'_pix_' directions of pixels in the map at a resolution at which this precomputation was executed (dimension N'_pix_' x a few hundred)
• use the resulting beam object for very fast convolution of all sky signals with the effective optical response of the observing mission

Computation of the effective beams at each pixel for every detector is a challenging task for high resolution experiments. FEBeCoP is an efficient algorithm and implementation which enabled us to compute the pixel based effective beams using moderate computational resources. The algorithm used different mathematical and computational techniques to bring down the computation cost to a practical level, whereby several estimations of the effective beams were possible for all Planck detectors for different scanbeam models and different lengths of datasets.

Pixel Ordered Detector Angles (PODA)

The main challenge in computing the effective beams is to go through the trillion samples, which gets severely limited by I/O. In the first stage, for a given dataset, ordered lists of pointing angles for each pixels---the Pixel Ordered Detector Angles (PODA) are made. This is an one-time process for each dataset. We used computers with large memory and used tedious memory management bookkeeping to make this step efficient.

effBeam

The effBeam part makes use of the precomputed PODA and unsynchronized reading from the disk to compute the beam. Here we tried to made sure that no repetition occurs in evaluating a trigonometric quantity.

One important reason for separating the two steps is that they use different schemes of parallel computing. The PODA part requires parallelisation over time-order-data samples, while the effBeam part requires distribution of pixels among different computers.

Computational Cost

The whole computation of the effective beams has been performed at the NERSC Supercomputing Center. In the table below it isn displayed the computation cost on NERSC for nominal mission both in terms of CPU hrs and in Human time.

Computational cost for PODA, Effective Beam and single map convolution.The cost in Human time is computed using an arbitrary number of nodes/core on Carver or Hopper NERSC Supercomputers

Channel	030	044	070
PODA/Detector Computation time (CPU hrs)	85	100	250
PODA/Detector Computation time (Human minutes)	7	10	20
Beam/Channel Computation time (CPU hrs)	900	2000	2300
Beam/Channel Computation time (Human hrs)	0.5	0.8	1
Convolution Computation time (CPU hr)	1	1.2	1.3
Convolution Computation time (Human sec)	1	1	1
Effective Beam Size (GB)	173	123	28

The computation cost, especially for PODA and Convolution, is heavily limited by the I/O capacity of the disc and so it depends on the overall usage of the cluster done by other users.

Inputs

In order to fix the convention of presentation of the scanning and effective beams, we show the classic view of the Planck focal plane as seen by the incoming CMB photon. The scan direction is marked, and the toward the center of the focal plane is at the 85 deg angle w.r.t spin axis pointing upward in the picture.

Planck Focal Plane

The Focal Plane DataBase (FPDB)

The FPDB contains information on each detector, e.g., the orientation of the polarisation axis, different weight factors, (see the instrument RIMOs):

• The LFI RIMO

The scanning strategy

The scanning strategy, the three pointing angle for each detector for each sample: Detector pointings for the nominal mission covers about 15 months of observation from Operational Day (OD) 91 to OD 563 covering 3 surveys and half.

The scanbeam

The scanbeam modeled for each detector through the observation of planets. Which was assumed to be constant over the whole mission, though FEBeCoP could be used for a few sets of scanbeams too.

• LFI: GRASP scanning beam - the scanning beams used are based on Radio Frequency Tuned Model (RFTM) smeared to simulate the in-flight optical response.

(see the instrument RIMOs)

• The LFI RIMO

Beam cutoff radii

• N times the geometric mean of FWHM of all detectors in a channel, where N=2.5 for all LFI frequency channels.

Map resolution for the derived beam data object

• [math]N_{side} = 1024 [/math] for all LFI frequency channels.

Comparison of the images of compact sources observed by Planck with FEBeCoP products

We show here a comparison of the FEBeCoP derived effective beams, and associated point spread functions,PSF (the transpose of the beam matrix), to the actual images of a few compact sources observed by Planck, for 30GHz frequency channel, as an example. We show below a few panels of source images organized as follows:

• Row #1- DX9 images of four ERCSC objects with their galactic (l,b) coordinates shown under the color bar
• Row #2- linear scale FEBeCoP PSFs computed using input scanning beams, Grasp Beams, GB, for LFI and B-Spline beams,BS, Mars12 apodized for the CMB channels and the BS Mars12 for the sub-mm channels, for HFI (see section Inputs below).
• Row #3- log scale of #2; PSF iso-contours shown in solid line, elliptical Gaussian fit iso-contours shown in broken line

30GHz

Histograms of the effective beam parameters

Here we present histograms of the three fit parameters - beam FWHM, ellipticity, and orientation with respect to the local meridian and of the beam solid angle. The shy is sampled (pretty sparsely) at 768 directions which were chosen as HEALpix nside=8 pixel centers for LFI to uniformly sample the sky.

Where beam solid angle is estimated according to the definition: 4pi* sum(effbeam)/max(effbeam) ie [math] 4 \pi \sum(B_{ij}) / max(B_{ij}) [/math]

Histograms for LFI effective beam parameters

Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian

• The discontinuities at the Healpix domain edges in the maps are a visual artifact due to the interplay of the discretized effective beam and the Healpix pixel grid.

ellipticity - 30GHz

beam solid angle (relative variations wrt scanning beam - 30GHz

Statistics of the effective beams computed using FEBeCoP

We tabulate the simple statistics of FWHM, ellipticity (e), orientation ([math] \psi[/math]) and beam solid angle, ([math] \Omega [/math]), for a sample of 768 directions on the sky for LFI data. Statistics shown in the Table are derived from the histograms shown above.

• The derived beam parameters are representative of the DPC NSIDE 1024 healpix maps (they include the pixel window function).
• The reported FWHM_eff are derived from the beam solid angles, under a Gaussian approximation. These are best used for flux determination while the the Gaussian fits to the effective beam maps are more suited for source identification.

Statistics of the FEBeCoP Effective Beams Computed with the BS Mars12 apodized for the CMB channels and oversampled

frequency	mean(fwhm) [arcmin]	sd(fwhm) [arcmin]	mean(e)	sd(e)	mean([math] \psi[/math]) [degree]	sd([math] \psi[/math]) [degree]	mean([math] \Omega [/math]) [arcmin[math]^{2}[/math]]	sd([math] \Omega [/math]) [arcmin[math]^{2}[/math]]	FWHM_eff [arcmin]
030	32.239	0.013	1.320	0.031	-0.304	55.349	1189.513	0.842	32.34
044	27.005	0.552	1.034	0.033	0.059	53.767	832.946	31.774	27.12
070	13.252	0.033	1.223	0.026	0.587	55.066	200.742	1.027	13.31

Beam solid angles for the PCCS

• [math]\Omega_{eff}[/math] - is the mean beam solid angle of the effective beam, where beam solid angle is estimated according to the definition: [math] 4 \pi*sum(effective_{beam})/max(effective_{beam})[/math] , i.e. as an integral over the full extent of the effective beam, i.e. [math] 4 \pi \sum(B_{ij}) / max(B_{ij}) [/math].

• from [math]\Omega_{eff}[/math] we estimate the [math]fwhm_{eff}[/math], under a Gaussian approximation - these are tabulated above
  • [math]\Omega^{(1)}_{eff}[/math] is the beam solid angle estimated up to a radius equal to one [math]fwhm_{eff}[/math] and [math]\Omega^{(2)}_{eff}[/math] up to a radius equal to twice the [math]fwhm_{eff}[/math].
    • These were estimated according to the procedure followed in the aperture photometry code for the PCCS: if the pixel centre does not lie within the given radius it is not included (so inclusive=0 in query disc).

Band averaged beam solid angles

Band	[math]\Omega_{eff}[/math][arcmin[math]^{2}[/math]]	spatial variation [arcmin[math]^{2}[/math]]	[math]\Omega^{(1)}_{eff}[/math] [arcmin[math]^{2}[/math]]	spatial variation-1 [arcmin[math]^{2}[/math]]	[math]\Omega^{(2)}_{eff}[/math] [arcmin[math]^{2}[/math]]	spatial variation-2 [arcmin[math]^{2}[/math]]
30	1189.513	0.842	1116.494	2.274	1188.945	0.847
44	832.946	31.774	758.684	29.701	832.168	31.811
70	200.742	1.027	186.260	2.300	200.591	1.027

Monte Carlo simulations

FEBeCoP software enables fast, full-sky convolutions of the sky signals with the Effective beams in pixel domain. Hence, a large number of Monte Carlo simulations of the sky signal maps map convolved with realistically rendered, spatially varying, asymmetric Planck beams can be easily generated. We performed the following steps:

• generate the effective beams with FEBeCoP for all frequencies for Nominal Mission data
• generate 100 realizations of maps from a fiducial CMB power spectrum
• convolve each one of these maps with the effective beams using FEBeCoP
• estimate the average of the Power Spectrum of each convolved realization, [math]C_\ell^{out}[/math], and 1 sigma errors

As FEBeCoP enables fast convolutions of the input signal sky with the effective beam, thousands of simulations are generated. These Monte Carlo simulations of the signal (might it be CMB or a foreground (e.g. dust)) sky along with LevelS+Madam noise simulations were used widely for the analysis of Planck data. A suite of simulations were rendered during the mission tagged as Full Focalplane simulations.

Window Functions

The Transfer Function or the Beam Window Function [math] W_\ell [/math] relates the true angular power spectra [math]C_\ell [/math] with the observed angular power spectra [math]\widetilde{C}_\ell [/math]:

[math] W_\ell= \widetilde{C}_\ell / C_\ell \label{eqn:wl1}[/math]

Note that, the window function can contain a pixel window function (depending on the definition) and it is {\em not the angular power spectra of the scanbeams}, though, in principle, one may be able to connect them though fairly complicated algebra.

The window functions are estimated by performing Monte-Carlo simulations. We generate several random realisations of the CMB sky starting from a given fiducial [math] C_\ell [/math], convolve the maps with the pre-computed effective beams, compute the convolved power spectra [math] C^\text{conv}_\ell [/math], divide by the power spectra of the unconvolved map [math]C^\text{in}_\ell [/math] and average over their ratio. Thus, the estimated window function

[math] W^{est}_\ell = \lt C^{conv}_\ell / C^{in}_\ell \gt \label{eqn:wl2}[/math]

For subtle reasons, we perform a more rigorous estimation of the window function by comparing [math] C^{conv}_\ell[/math] with convolved power spectra of the input maps convolved with a symmetric Gaussian beam of comparable (but need not be exact) size and then scaling the estimated window function accordingly.

Beam window functions are provided in the RIMO.

Beam Window functions, Wl, for LFI channels

Beam Window functions, [math]W_\ell [/math], for LFI channels

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.

The image of the LFI27-M sidelobes is created as RGB picture where the red channel is the 27 GHz (f0), the green channel is the 30 GHz (f3), and the blue channel is the 33 GHz (f6). Because of the combined map does not show any wide white region, the sidelobe pattern change with frequency, as expected.