Planck 2013 results. VII. HFI time response and beams
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Warsaw University Observatory, Aleje Ujazdowskie 4, 00478
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Poland
Received:
21
March
2013
Accepted:
10
December
2013
This paper characterizes the effective beams, the effective beam window functions and the associated errors for the Planck High Frequency Instrument (HFI) detectors. The effective beam is theangular response including the effect of the optics, detectors, data processing and the scan strategy. The window function is the representation of this beam in the harmonic domain which is required to recover an unbiased measurement of the cosmic microwave background angular power spectrum. The HFI is a scanning instrument and its effective beams are the convolution of: a) the optical response of the telescope and feeds; b) the processing of the timeordered data and deconvolution of the bolometric and electronic transfer function; and c) the merging of several surveys to produce maps. The time response transfer functions are measured using observations of Jupiter and Saturn and by minimizing survey difference residuals. The scanning beam is the postdeconvolution angular response of the instrument, and is characterized with observations of Mars. The main beam solid angles are determined to better than 0.5% at each HFI frequency band. Observations of Jupiter and Saturn limit near sidelobes (within 5°) to about 0.1% of the total solid angle. Time response residuals remain as long tails in the scanning beams, but contribute less than 0.1% of the total solid angle. The bias and uncertainty in the beam products are estimated using ensembles of simulated planet observations that include the impact of instrumental noise and known systematic effects. The correlation structure of these ensembles is welldescribed by five error eigenmodes that are subdominant to sample variance and instrumental noise in the harmonic domain. A suite of consistency tests provide confidence that the error model represents a sufficient description of the data. The total error in the effective beam window functions is below 1% at 100 GHz up to multipole ℓ ~ 1500, and below 0.5% at 143 and 217 GHz up to ℓ ~ 2000.
Key words: cosmic background radiation / cosmology: observations / instrumentation: detectors / surveys
© ESO, 2014
1. Introduction
This paper, one of a set associated with the 2013 release of data from the Planck^{1} mission (Planck Collaboration I 2014), describes the impact of the optical system, detector response, analogue and digital filtering and the scan strategy on the determination of the High Frequency Instrument (HFI) angular power spectra. An accurate understanding of this response, and the corresponding errors, is necessary in order to extract astrophysical and cosmological information from cosmic microwave background (CMB) data (Hill et al. 2009; Nolta et al. 2009; Huffenberger et al. 2010).
Bolometers, such as those used in the HFI on Planck, are phononmediated thermal detectors with finite response time to changes in the absorbed optical power. The observations are affected by attenuation and the phase shift of the signal resulting from a detector’s thermal response, as well as the analogue and digital filtering in the associated electronics.
In the small signal regime (appropriate for CMB fluctuations and Galactic emission), the receiver response can be well approximated by a complex Fourier domain transfer function, termed the time response. The timeordered data (also referred to as timeordered information, or TOI) are approximately deconvolved by the time response function prior to calibration and mapmaking (for recent examples of CMB observations with similar semiconductor bolometers see Tristram et al. 2005; Masi et al. 2006).
Ideally, the deconvolved TOIs represent the true sky signal convolved with the optical response of the telescope (or physical beam) and filtered by the TOI processing. The combination of time domain processing and physical beam convolution is, in practice, degenerate, due to the nature of the scan strategy; the Planck spacecraft scans at 1 rpm, with variations less than 0.1% (Planck Collaboration I 2011). The beams reconstructed from the deconvolved planetary observations are referred to as the scanning beam.
These deconvolved data are then projected into a pixelized map, as discussed in Sect. 6. of Planck Collaboration VI (2014). To a good approximation, the effect of the mapmaking algorithm is to average the beam over the observed locations in a given pixel. This average is referred to as the effective beam, which will vary from pixel to pixel across the sky. The mapmaking procedure implicitly ignores any smearing of the input TOI; no attempt is made to deconvolve the optical beam and any remaining electronic time response. Thus, any further use of the resulting maps must take the effective beam into account.
To obtain an unbiased estimate of the angular power spectrum of the CMB, one must determine the impact of this effective beam pattern on a measurement of an isotropic Gaussian random signal in multipole (ℓ) space. The filtering effect is well approximated by a multiplicative effective beam window function, derived by coupling the scan history with the scanning beam profile, which is used to relate the angular power spectrum of the map to that of the underlying sky (Hivon et al. 2002).
The solar and orbital motion of Planck with respect to the surface of last scattering provides a 60s periodic signal in the time ordered data that is used as a primary calibrator (Planck Collaboration V 2014; Planck Collaboration VIII 2014). This normalizes the window function at a multipole ℓ = 1; the effective beam window function is required to transfer this calibration to smaller angular scales.
These successive products (the scanning beam, the effective beam and the effective beam window function) must be accompanied by an account of their errors, which are characterized through ensembles of dedicated simulations of the planetary observations. The error on the effective beam window function is found to be subdominant to other errors in the cosmological parameter analysis (Planck Collaboration XVI 2014).
The scanning beam is thus measured with onorbit planetary data, coupling the response of the optical system to the deconvolved time response function and additional filters in the TOI processing. As shown in the following, the main effect of residual deconvolution errors is a bias in the beam window function of order 10^{4} at ℓ > 100, due to a residual tail in the beam along the scan direction.
The scanning beam can be further separated into the following components;

1.
The main beam, which is defined as extending to30′ from the beam centroid.

2.
The near sidelobes, which extend between 30′ and 5°. These are typically features below −30 dB, and include the optical effects of phase errors, consisting of both random and periodic surface errors and residuals due to the imperfect deconvolution of the time response.

3.
The far sidelobes, which extend beyond 5°. These features are dominated by spillover: power coupling from the sky to the feed antennas directly, or via a single reflection around the mirrors and baffles. The minimum in the beam response between the main beam and direct spillover of the feed over the top of the secondary mirror falls at roughly 5°, making such a division natural (Tauber et al. 2010).
This paper describes the main beam and the near sidelobes, the resulting effective beam patterns on the sky and the effective beam window function used for the measurement of angular power spectra, along with their errors. The effects of the far sidelobes are mainly discussed in Planck Collaboration VI (2014) and Planck Collaboration XIV (2014), and their effects on calibration described in Planck Collaboration X (2014). A companion paper (Planck Collaboration IV 2014) computes the effective beams and window functions for Planck’s Low Frequency Instrument (LFI). Despite using very different methods which depend more strongly on optical modelling, the two instruments produce compatible power spectra, providing a crosscheck of both approaches (Planck Collaboration XXXI 2014).
The signaltonoise ratio of Jupiter observations with HFI allows the measurement of near sidelobes to a noise floor of − 45(−55) dB relative to the forward gain at 100 (857) GHz. The HFI analysis does not rely on a physical optics model to constrain the behaviour of the beam in this regime.
Onorbit measurements of the Planck HFI scanning beam and temporal transfer function have been previously reported in Planck HFI Core Team (2011a) for the Early Release Compact Source Catalog (ERCSC; Planck Collaboration VII 2011) and early science from Planck. Section 2 presents an improved model for the time response and explains how its parameters are derived using onorbit data, and how it is deconvolved from the data. Figure 1 provides a flowchart of the determination of the time response parameters. Figure 2 is the flowchart of the steps that lead from planet measurements to effective beams, effective beam window functions and assessment of uncertainties. Section 3 describes how the scanning beams are reconstructed from planet observations. Section 3.4 specifically describes the effects of long time scale residuals in the data due to imperfect deconvolution of the time response. Section 4 describes the propagation of the scanning beams to effective beams and effective beam window functions using the scanning history of Planck. Section 5 describes the techniques used to propagate statistical and systematic errors and to check the consistency of the beams and window functions. Section 6 describes the final error budget and the eigenmode decomposition of the errors in the effective beam window function.
Fig. 1 Data flow for fitting time response model parameters. 

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Fig. 2 Data flow for determining the HFI beams. 

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2. Time response
In Planck’s early data release, a 10parameter model TF10 describes the time response of the bolometer/electronics system (Planck HFI Core Team 2011a). This section describes an improved model of the time response based on the HFI readout electronics schematics which more accurately reproduces phase shifts in the system close to the Nyquist frequency. The improved model also provides more degrees of freedom for the bolometer’s thermal response in order to describe more accurately the low frequency response of the bolometer.
2.1. Model
The new model is named LFER4 (Low Frequency Excess Response with four time constants) and consists of an analytic model of the HFI readout electronics (Lamarre et al. 2010) and four thermal time constants and associated amplitudes for the bolometer: (1)where TF(ω) represents the full time response as a function of angular frequency ω. The time response of the bolometer alone is modelled by (2)and H′(ω;S_{phase},τ_{stray}) is the analytic model of the electronics transfer function (whose detailed equations and parameters are given in Appendix A) with two parameters.
The overall normalization of the transfer function is forced to be 1.0 at the signal frequency of the dipole, leaving a total of 9 free parameters for each bolometer.
The sum of singlepole lowpass filters represents a lumpedelement thermal model with four elements. The thermal model underlying the temporal transfer function is described elsewhere (Spencer et al., in prep.); this work adopts an empirical approach to correcting the data.
The two parameters of H′ mainly affect the high frequency portion of the time response. S_{phase} represents the phase difference between the bias and the lockin summation, and is fixed in the model as a readout electronics setting. The second parameter τ_{stray} is the time constant of the bolometer resistance and the parasitic capacitance of the wiring and is measured independently during the checkout and performance verification (CPV) phase of the mission prior to the sky survey. All resistance and capacitance values of the readout electronics chain are fixed at values from the circuit diagram.
The inflight data are used to determine the remaining seven free parameters. Low frequency parameters are constrained by minimizing the difference between the first and second survey maps (Sect. 2.2), while planetary observations are used to constrain those parameters governing the high frequency portion of the time response (Sect. 2.3).
The fastest thermal time constants in the LFER4 model roughly correspond to the time constants measured during prelaunch tests of the bolometers (Holmes et al. 2008; Pajot et al. 2010). The slower time constants contribute lower frequency response at the several percent level. The time constants in the LFER4 model are not exactly identical to those measured on glitches (Planck Collaboration X 2014) due to additional filtering applied by the deglitching. A future publication (Spencer et al., in prep.) will relate the time constants and amplitudes to the thermal properties of the bolometer and module.
2.2. Fitting slow time constants with galaxy residuals
As Planck scans across the Galactic plane, the low frequency time response of HFI spreads Galactic signal away from the plane. Surveys 1 and 2 consist of roughly six months of data each and cover nearly the same sky, scanned at almost opposite angles. The difference between maps made with data from the two individual surveys highlights the effect since the Galactic power is spread in different directions in the two surveys, creating symmetric positive and negative residuals in the difference map. The LFER4 parameters are varied to minimize the difference between these surveys. For most bolometers, the fit is limited to the slowest time constant and its associated amplitude, and in others the fit is extended to the two slowest time constants and associated amplitudes.
Other systematic effects can confuse the measurement of LFER4 parameters by creating a similar positive/negative residual pattern in the survey difference. The philosophy employed here is to minimize the survey residuals fitting only for LFER4 parameters, but to use simulations and data selections to test the dependency of the results with other systematics.
2.2.1. Survey difference method 1
Two techniques are used to perform the fit. The first method is based on map resampling in the time domain, using the pointing to generate synthetic TOI. The synthetic TOI of each survey is compared with the synthetic TOI of both surveys combined. Before the production of these synthetic TOI, the maps are smoothed to 30′. Given the fact that in consecutive surveys the scan direction is nearly opposite, the survey 2 TOI is very similar to the timereversal of the survey 1 TOI. This symmetry is assumed to be exact, ensuring that time reversal is equivalent to taking the complex conjugate in the frequency domain. Since the sky signal has been convolved by the true transfer function, TF_{true}(ω), and deconvolved by the estimated – not fully correct – transfer function TF_{0}(ω), the single survey resampled timestream is (3)where ℱ is the Fourier transform operator, s is the true sky map observed at time t′, and d_{1} is the synthetic TOI. This can be written as, (4)having defined δTF(ω) = TF_{true}(ω)/TF_{0}(ω) as the corrective factor which should be applied to the estimated transfer function. The synthetic TOI of survey 2, d_{2}, is similarly defined. For the TOI of both surveys combined, the average of the two maps is used.
Using the time reversal property, the synthetic TOI d(t) of the full sky map (survey 1 and survey 2 combined) may be written as In the Fourier domain, the ratio of the TOIs is Since the real part of the transfer function at low frequency is close to 1, the imaginary part of the ratio of the synthetic beams is equal to the imaginary part of the corrective factor, δTF(ω): The LFER4 model of the transfer function is fitted to this measure of the imaginary part of the correction to infer the parameters of the true transfer function, in the low frequency regime, typically below a few Hz.
2.2.2. Survey difference method 2
The second method looks at onedimensional slices through the Galactic plane for each survey independently. A sky signal slice is obtained by resampling a sky map for a single survey and a single bolometer. The slices are taken along the scan direction and the sky signal is averaged over 5° in longitude. Only the sky region close to the Galactic plane is considered (10° above and below the Galactic plane and longitudes between −40° and 60°). The slice is convolved with the ratio of the transfer function used to create the map and a new LFER4 transfer function with trial parameter values. New sky survey maps are obtained by reprojecting the slices into pixelized maps. As for the previous method, parameters of the lowfrequency part of the LFER4 transfer functions are chosen so that they minimize the residuals in the difference between surveys 1 and 2.
Fig. 3 Survey 2 minus Survey 1 residual close to the Galactic centre before (upper) and after (lower) fitting and deconvolving the low frequency part of the time response for bolometer 3532. Remaining residuals are dominated by gain difference between the surveys due to ADC nonlinearity (Planck Collaboration VIII 2014) and artifacts of the different scanning directions (beam asymmetry) and pixel coverage survey to survey. 

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In practice, the first method is used for the 100–353 GHz bolometers, and at 545 GHz and 857 GHz the second method is used, being better adapted to the maps having the most structure in the Galactic difference residuals. Figure 3 shows an example of the residual remaining in a HEALPix^{2} map (Górski et al. 2005) of the survey difference, after fitting the long time response, showing reduced asymmetry in the Galactic plane.
2.2.3. Survey difference systematics
In the survey difference solution for the time response, any systematic effect that creates a residual in the survey difference can be confused with a time response effect, in particular affecting the low frequency time response. This section identifies a number of residualproducing systematics and quantifies the resulting bias in the transfer function. These residualproducing systematics include far sidelobes, zodiacal light, pointing offset uncertainty, gain drifts, main beam asymmetry and polarized sky signal.
Due to the very high signaltonoise ratio of Galactic signal at submillimetre wavelengths, far sidelobe pickup of the Galactic plane is detected in the 545 GHz and 857 GHz channels. A physical optics model of the far sidelobe pickup is used to estimate the signal from the Galactic centre. The optical depth of the zodiacal dust cloud along the line of sight varies as Planck orbits the Sun, leaving a residual in the survey difference (Planck Collaboration XIV 2014). A model of the zodiacal dust emission is subtracted in the reconstruction of the time response. The reconstruction of the time response is then repeated without subtracting the models; these do not significantly affect the result (Fig. 4).
As a probe of the effect of far sidelobes on the time constant determination, the pipeline is run on a survey difference map obtained from the sidelobe model alone for each of the 100, 143, 217, and 353 GHz channels. The method did not find a long time constant, as the sidelobe effect on the survey map is very different from the time constant effect.
Fig. 4 68%, 95%, and 99% likelihood contours for the long time constant τ_{3} and associated amplitude a_{3} for a 545 GHz bolometer (5454) with (black) and without (red) zodiacal emission and far sidelobe removal. The square and cross indicate the maximum likelihood values. 

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A systematic offset in the pixel pointing creates a residual in the survey difference. The pointing solution reduces the pointing error to a few arcseconds RMS in both the coscan and crossscan direction. With the 6° s^{1} scanning speed, this error corresponds to frequencies greater than 1 kHz, far from the range affected by the long time constant.
Gain variability can also bias the estimation; due to nonlinearity in the analogue to digital converters (ADC), the HFI responsivity to the sky signal varies at a few tenths of a percent throughout the mission. As a probe of this effect, gaincorrected data for the 100, 143 and 217 GHz bolometers are used to reconstruct the long time response. This has a negligible impact on the fitted parameters. Residual gain errors tend to leave monopolar residuals that are not coupled to the long time constants in the fitting procedure.
Some residual is expected in the survey difference maps because the asymmetric beam scans the sky at different angles in the two surveys. This is especially an issue at 545 GHz where the beam is substantially asymmetric. To quantify the amplitude, simulated survey difference maps are generated using a realistic asymmetric beam model convolved with the Planck Sky Model (Delabrouille et al. 2013); the residuals in the Galactic plane at 545 GHz are dominated by the main beam asymmetry (see Fig. 5).
Fig. 5 Survey 1 minus survey 2 residual for the 545 GHz bolometers, averaged from Galactic longitude 0 through 20°. The black curves show the Planck data, and red is a simulation. 

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Polarization sensitive bolometers (PSBs) show an additional residual in the survey difference maps because of polarized sky signal observed at slightly different crossing angles in the two surveys. For the PSBs, the low frequency time response is therefore determined using different levels of polarization masking. These studies do not suggest the presence of any significant level of bias from polarization, but only higher noise with wider masking. As an additional check on the contribution of residual polarization to the low frequency response, the survey difference of the sum of the two arms of each PSB pair is input to the pipeline. This sum is not sensitive to polarization, and no bias is found in the determination of the time response.
Fig. 6 Gridded Jupiter data for bolometer 1433b before and after fitting for LFER4 parameters. The best fit Gaussian is subtracted from each plot to emphasize residuals. Residuals in the main beam show the deviation from a Gaussian shape, captured in the representations of the scanning beam, as described in Appendices B and C. 

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2.3. Fitting fast time constants with planet crossings
Filtering of the TOIs and errors in the deconvolution kernel results in ringing along the scan direction. The planets Mars, Jupiter and Saturn are high signaltonoise sources that can be used to minimize this smearing by adjusting the parameters of the LFER4 model. See Sect. 3.1 for a description of the planet data.
In solving for the high frequency portion of the time response, the beam profile is forced to be compact. The optical beam is modelled as a spline function on a twodimensional grid (see Appendix C for details) and the LFER4 parameters are fit by forcing conditions on the resulting beam shape (Fig. 6).
The planet data are first deconvolved with a time response model derived from prelaunch data, to recover an initial estimate of the beam profile. Jupiter is used for the 100, 143 and 217 GHz channels, while Saturn is used for higher frequencies (see Sect. 3.4 for a discussion of the nonlinearity of Jupiter at higher frequencies).
Rather than deconvolve the planet data, the model parameters are determined in the forward sense. Since the beam is decomposed into Bspline functions, this basis is convolved with the temporal transfer function to retrieve the coefficients for each basis function using the planet data. These coefficients are applied to the original deconvolved Bspline functions to recover an estimate of the optical beam. The convolution is made in the Fourier domain by resampling the Bspline function onto a timeline with a sample separation corresponding to 4.̋5. The typical knot separation length of the basis function is between 1′ and 2′.
In the Fourier domain, the convolution of the temporal transfer function with the planet signal is (5)where ℬ^{0}(ω) is the Fourier transform of the slice through the peak planet signal in the scan direction b^{0}(t), which is obtained by deconvolving planet data using a fiducial estimate, TF^{0}(ω) of the transfer function. The slice b^{0}(t) is then symmetrized about the origin (defined by the location of the maximum of the Bspline representation)to obtain b^{sym}(t), and its Fourier transform ℬ^{sym}(ω). This operation aims to recover a beam that, by construction, is symmetric, within the limits allowed by the model of the temporal transfer function, (6)Here ℬ^{sym}(ω) is entirely defined by and TF^{0}(ω) and the new estimate of the time response TF^{∗}(ω) is derived from Eq. (6). This function is parameterized in terms of the three shortest time constants (τ_{1}, τ_{2}, τ_{3}) and their associated amplitudes (a_{1},a_{2},a_{3}).
2.4. Stationarity of the time response
The time response of each HFI detector/readout channel is a function of the cryogenic temperature of the bolometers and the ambienttemperature components of the readout electronics. Both cryogenic and ambient temperatures change throughout the mission as the Galactic particle flux and the Planck spacecraft solar distance are modulated. The seasonal consistency of the scanning beam sets an upper limit on changes in the time response through the mission, shown below in Sect. 5.2.1.
2.5. Deconvolution of the data
The time response transfer function is deconvolved from the data and not included as part of the scanning beam, because the low frequency response of the bolometer would give an extended scanning beam, stretching many degrees from the main lobe along the scan direction.
Since the time response amplitude decreases as a function of frequency, the deconvolution operation increases the noise at high frequency to an unacceptably high fraction of the RMS. During the deconvolution stage of the TOI processing, a phaseless lowpass filter is applied in order to suppress the high frequency noise and keep pixel aliasing at a manageable level.
In the early data release, a finite impulse response lowpass filter was used for this purpose (Planck HFI Core Team 2011b). In the 2013 cosmology data release, the lowpass filter is further tuned for the 100, 143, 217 and 353 GHz channels to reduce filter ripple produced by the lowpass filters used in the earlyrelease data. The filter is now implemented in the Fourier domain, with a kernel consisting of the product of a Gaussian and a squared cosine, (7)where (8)and (9)Here f_{max} = f_{c} + k(f_{samp}/2 − f_{c}) and f_{samp} is the sampling frequency of the data. The parameters of the filter are the same for all bolometers in the bands 100–353 GHz: f_{Gauss} = 65 Hz; f_{c} = 80 Hz; and k = 0.9. To first order, this filter widens the scanning beam along the scan in an equivalent way to convolving the optical beam with a Gaussian with fullwidth at halfmaximum (FWHM) of 2.́07. The filter introduces some rippling along the scan direction at the −40 dB level at 217 and 353 GHz, where the beams allow harmonic signal content close to the filter edge. The rippling is captured by the Bspline beam representation (see Fig. 11).
The 3point finite impulse response filter is still used for the 545 and 857 GHz channels^{3}. This extends the scanning beams along the scan direction more than the Gaussiancosine Fourier filter (the 545/857 GHz filter time scale corresponds to 3.́0 on the sky), but has the advantage of reducing rippling and signal aliased from above the Nyquist frequency.
The deconvolution kernel multiplied by the data in the Fourier domain is the product of the lowpass filter with the inverse of the bolometer/electronics time response, (10)Figure 7 shows a comparison of the deconvolution functions resulting from the LFER4 model and from the TF10 model used in the ERCSC data. The differences between the two models appear mainly in the phase at high frequency, mostly above the signal frequency corresponding to the beam size. Although the phase of LFER4 is a more accurate description of the system, in practice replacing LFER4 with TF10 had little effect on the data, because of the lowpass filter applied at the time of deconvolution and the empirical determination of an overall sample offset in the pointing reconstruction.
In the HFI data processing (Planck Collaboration VI 2014), data chunks of length 2^{19}(≈5 × 10^{5}) samples are Fourier transformed at a time, overlapping by half with the subsequent chunk to avoid edge effects.
Fig. 7 Phase and amplitude as a function of signal frequency of the deconvolution function of bolometer 2171. The solid black curve is the LFER4 model, while the dashed red curve shows the TF10 model used in the earlier Planck papers. The vertical dotted line marks the signal frequency corresponding to the half power point of the average effective beam. 

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3. Scanning beams
The filtering of the TOI and the accuracy of the deconvolution kernel affect the angular response of the HFI detectors. An accurate estimate of the scanning beam, resulting from the filtering of the physical (optical) beam by these time domain convolutions, is required to relate the angular power spectra of the maps to that of the underlying sky. As described in Planck HFI Core Team (2011a) and Planck HFI Core Team (2011b), the HFI scanning beam profiles are measured using the planetary observations. HFI uses two flat sky representations of the twodimensional scanning beams, one using GaussHermite polynomials, and another using Bspline functions.
Three selections of planetary data are used to derive estimates of different aspects of the beam:

1.
the first two observing seasons of Mars (main beam, andwindow functions);

2.
all available seasons of Mars [3], Jupiter [5] and Saturn [4] (near sidelobe); and

3.
all five Jupiter observations (residual time response).
The effective beam window functions used in the CMB analysis are ultimately derived from the first of these. In each case, the statistical properties of the beam representations and the choice of planetary data are studied using ensembles of simulated planet observations (Sect. 5).
While the signaltonoise ratio of the Jupiter and Saturn data is significantly greater than the Mars data, at this stage of the analysis a reconstruction bias results in the main beams recovered from simulated Jupiter and Saturn observations that is not present in the simulations of Mars. Additionally, the nonlinear response of some HFI detectors to the Jupiter signal (Planck HFI Core Team 2011a) makes the normalization of the planet peak response uncertain at the few percent level (see Sect. 3.4). Therefore the main beam model is established using Mars data, while observations of Jupiter and Saturn are used only to estimate the near sidelobes and residuals in the deconvolution of the transfer function.
The Bspline representation of the joint Mars observations is used as input for the calculation of the effective beam and the effective beam window function. Simulations have shown the Bspline representation to better capture the features outside of the main lobe, due to the necessarily finite order of the GaussHermite decomposition. However, the GaussHermite model is used for other systematics checks, including the consistency of the planets and observing seasons.
Because the Jupiter and Saturn data allow measurement of the beam response below −45 dB from the peak, there is no need to rely on a model of the telescope to determine the main beam or near sidelobe structure.
3.1. Planetary data handling
The JPL Horizons package^{4} (Giorgini et al. 1996) is programmed with Planck’s orbit to calculate the positions of the planets. Table 1 shows the dates when the planets were within 2° of the centre of the focal plane. By the end of HFI operations Mars was observed three times, Saturn four times, and Jupiter five.
The planets Jupiter, Saturn and Mars are among the brightest compact objects seen by Planck HFI; the signal amplitude affects the data handling in a number of ways. Moving solar system objects are flagged and removed from the TOI in the standard processing pipeline. Specialized processing for the planet data is required, with two major differences from the nominal processing (see Planck Collaboration VI 2014 for details).
Observation seasons of the planets observed by Planck: date range and position in Galactic coordinates.
While the pickup from the ^{4}HeJT cooler is removed from the data as usual, pointing periods containing very bright sources such as the planets cannot be used to reliably estimate the line amplitude. Instead, during the planet observations, these are extrapolated from neighbouring pointing periods.
Fig. 8 Window functions of the planetary discs of Jupiter, Saturn, and Mars, equivalent to the bias in the inferred effective beam window function if the beam is reconstructed from observations of one of these planets alone. The labels show the corresponding angular radius of each disc. 

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To better detect glitches near the extremely bright planet crossings, an estimate of the planet signal is subtracted from the data prior to glitch detection. Glitch template subtraction is performed on the signalsubtracted timeline in the same way as during nominal observations.
In order to remove the (quasistationary) astrophysical background from the planetary data, a bilinear interpolation of the frequency averaged map is subtracted. The full mission map is used for the fiveseason Jupiter data, while the nominal survey sky maps are used in the processing of the other planetary data.
The planets are oblate spheroids, and appear as ellipses slightly extended along the direction of the ecliptic. The Planck planet range and Planck sublatitude calculated by JPL Horizons are used in combination with the polar and equatorial radii of the planets reported by Horizons to compute the angular size and ellipticity of each planet. During HFI observations, the mean angular radii of Jupiter, Saturn and Mars are 20.̋44, 8.̋542, and 4.̋111, respectively. The ratio of the equatorial to polar radii are 1.069, 1.109 and 1.006, respectively.
The finite planetary disc size increases the apparent size of the scanning beam and biases the inferred effective beam window function. The filtering in multipole space of a circular disc of angular radius R can be written as B_{disc}(ℓ) = 2J_{1}(ℓR) /(ℓR), where J_{1}(x) is the Bessel function of order 1. Figure 8 shows the for the three planetary discs. In practice, the effects of the disc size are mitigated by merging observations from the three brightest planets. The effects of the large Jupiter disc size are greatest where the spatial gradient of the beam is greatest, between the –3 to −10 dB contours of the beam. By excluding the Jupiter observations above − 10 dB, the disc size smearing is reduced, and setting a −20 dB threshold results in a bias in the window function below 10^{3} at all multipoles.
Planck observes Saturn at a range of ring inclination angles: 6.̊03, 2.̊45, 12.̊6 and 9.̊4 for seasons 1, 2, 3 and 4, respectively. While emission from the solid angle of the ring increases the effective planetary disc area, the brightness temperature of the ring tends to be much less than the planetary disc temperature for the Planck bands. For example Weiland et al. (2011) fit a single 90 GHz ring brightness temperature 14% that of the Saturn disc. In our beam reconstruction the average of the first three Planck observations of Saturn is used, and even in the extreme limit where the ring brightness temperature is the same and only Saturn data are used, the multipole space correction is 2 × 10^{3} at ℓ = 3000; this correction is ignored.
The elliptical shape of the planetary disc gives a further bias in the inferred window function, depending on whether the long axis of the beam is aligned with or perpendicular to the long axis of the planet. In this case the planetary disc is approximated as an elliptical Gaussian with σ = 0.5R. The worst case is a 10^{3} effect in the case of 100 GHz beams measured on Jupiter at ℓ = 3000 (where the 100 GHz window function is vanishingly small). At 143 and 217 GHz, the 10^{3} level is reached only at ℓ = 4000. The effect is negligible in the range of Planck’s sensitivity. With Saturn and Mars, the ellipticity effects are <10^{4} and <10^{6}, respectively, at all multipoles; the ellipticity of the planetary discs has a negligible effect on the estimate of the scanning beam.
3.2. Main beam model
Two representations are used to describe the (two dimensional) main beam; a GaussHermite basis (described in Appendix B following Huffenberger et al. 2010 and Planck HFI Core Team 2011a) and a Bspline basis (Appendix C).
The GaussHermite (GH) and Bspline bases have very different characteristics. The GH representation uses a relatively small number of parameters and, in practice, amounts to a perturbative expansion about a Gaussian shape. The Bspline is quite general, using many more degrees of freedom to fit the data on a defined grid, with the spline controlling the behaviour in between. The bias and correlation structure of the noise of these two representations are characterized using Monte Carlo simulations of the planetary data which include all the details of the beamprocessing pipeline used on the data (these simulations are described more completely in Sect. 5). In each case, the parameters of the representation are derived directly from the timeordered data, without recourse to a pixelized map.
Figure 9 shows the Bspline scanning beams for the entire HFI focal plane, as reconstructed from the Jupiter and Saturn data. Figure 10 shows the radially binned, frequency averaged beam profile for the HFI channels, comparing the Bspline representation of the Mars data (solid black lines) with the combined Mars, Jupiter and Saturn data (filled and open points, and the blue dashed line). The Bspline maps are apodized with a Gaussian at a radius beyond the signaltonoise floor of the Mars data: 13.́4, 13.́0, 11.́4, 12.́9, 17.́8, 17.́8 on average at 100, 143, 217, 353, 545,and 857 GHz respectively.
Fig. 9 Bspline scanning beams reconstructed from Mars, Saturn, and Jupiter seasons 1, 2 and 3 data for near sidelobe studies. The beams are plotted in logarithmic contours of − 3, − 10, − 20 and − 30 dB from the peak. PSB pairs are indicated with the a bolometer in black and the b bolometer in blue. 

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Fig. 10 Azimuthally and bandaveraged main beam profiles (black solid curve) derived from the Bspline representation of the first and second Mars observations compared to that derived from a combination of Mars, Jupiter and Saturn observations (filled and open markers represent positive and negative data respectively). The red dashed line is defined as the joint envelope of the main beam and near sidelobe dataset, the integral of which represents the maximal solid angle that is compatible with these data. A nominal near lobe model, provided as a reasonable extrapolation of the data below the noise floor, is shown as the blue dashed line. The fractional increase in solid angle, relative to the Marsalone derived beam profile, is displayed in each panel. The black dotted line shows the GRASP physical optics model averaged over a subset of detectors that have been simulated (100–353 GHz). The data show a clear excess in power over the model at 143, 217 and 353 GHz that is consistent with a spectrum of surface errors on scales between 2 and 12 cm, with an RMS of order 10 μm. Table 2 contains an estimate of the fraction of the solid angle in the near sidelobes that is not captured in the Bspline representation. For clarity, the figure extends only to 45′. In all cases the solid angle is derived from the profile extending out to 5°. Due to the high signaltonoise of the Jupiter data (− 40 to − 55 dB, depending on the frequency), and the rapidly falling response of the beam, the solid angle estimates are insensitive to the limit of integration. 

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The azimuthally averaged beam window function, , from each of these models is compared to the known input beam model. At 100, 143 and 217 GHz the two methods perform comparably, with the Bspline having slightly smaller bias and variance; at the higher frequencies the Bspline performs demonstrably better, especially at 545 and 857 GHz, as expected due to the highly nonGaussian shape of these multimode detectors.
3.3. Near sidelobes
While the HFI beams are Gaussian at the −25 dB level, nontrivial structure in the beam is captured in the data at lower power. There are two distinct components to the near lobe response: a discrete pattern of secondary lobes evident at frequencies of 217 GHz and above; and a diffuse shoulder consistent with phase errors in the aperture plane.
The Planck reflectors suffer from printthrough of the honeycomb structure that supports the carbonfiber face sheets. The size of the deformation has been measured during thermal testing to be less than 20 μm (Tauber et al. 2010). While small in amplitude, the strict periodicity of this pattern results in a correspondingly periodic structure in the near lobes, seen clearly in Fig. 12, which is slightly larger than predicted based on the prelaunch measurements. A simple grating equation describes the angular positions of the resulting contributions to the near sidelobes, (11)where θ_{n} is the angular position of the order n lobe from the central beam peak, λ is the wavelength of the radiation, d is the grating periodicity and Y is a factor that describes the position of each reflector along the optical path, with Y = 1.00 for the primary reflector and Y = 1.80 for the secondary reflector. Three possible periodicities (19.6 mm, 30 mm, 52 mm) in the honeycomb array dominate the Planck dimpling pattern for the 857 GHz detectors, though only those for the 52 mm periodicity can be seen for the 545 GHz and 353 GHz detectors. For the highest frequency detectors, only the weaker lobes due to the 19.6 mm and 30 mm periodicities are seen outside the 40′ main beam model, but they contribute at most (0.050 ± 0.008)% to the integrated beam solid angle. A forthcoming publication (Oxborrow et al., in prep.) will present an indepth study of the mirror surface.
The second component is a beam shoulder becoming significant near the −30 dB contour, and extending to 2–4 FWHM from the beam centre. This shoulder is consistent with scattering from random surface errors on the primary and secondary reflectors, and is reasonably welldescribed by a spectrum of surface errors with correlation lengths ranging from 2 to 12 cm, with an RMS of order 10 μm (Ruze 1966). The contribution of each of these components is included in the radially binned profiles shown in Fig. 10.
While the Bspline parameterization captures both the main beam and near sidelobe structure, the extended features must be separately included in the GaussHermite beam representation. Figure 11 shows contour plots of a Bspline beam using Mars, Jupiter and Saturn data at each frequency.
The Bspline representation of the scanning beam used to compute the window function includes only that portion of the near sidelobe structure that falls within the signaltonoise of the Mars data; the Jupiter and Saturn data provide an estimate of the beam solid angle that is neglected in the scanning beam product. The near sidelobe solid angle, and the resulting window function error, are sensitive to the details of the analysis, including the sky subtraction, offset removal and masking of inscan ringing at the partpermillion level. Although a comprehensive study of these effects in the Jupiter and Saturn data are underway, a conservative, and model independent upper limit is obtained by taking the envelope of the noise floor to define the maximum solid angle allowed by the data (the red dashed line in Fig. 10). A reasonable estimate of the true solid angle in the near lobes can be obtained by extrapolating the data below the noise floor (the blue dashed line in Fig. 10). By either measure, the grating lobes and diffuse shoulder account for a small fraction of the total beam solid angle; for the 100, 143 and 217 GHz channels this contribution represents less than 0.15% of the total solid angle (see Table 2). The amplitude of the impact on the window function is estimated by comparing the Legendre transform of the maximal envelope of the Jupiter and Saturn data with that of the nominal Mars derived scanning beam. The result is shown as the family of green curves in Fig. 19. Because the Monte Carlo ensembles that are used to derive the error envelope neglect this near sidelobe structure in the beam that is input to the simulations, the window function error amplitudes have been scaled to accomodate the upper limit defined by the noise floor of the Jupiter and Saturn data.
Fig. 11 One scanning beam at each HFI frequency (1003b, 1436, 2171, 3537, 5451, and 8573). Contours are in dB from the peak in steps of −5 dB. The lowest contours are set at −30 dB, −35 dB, −40 dB, −45 dB, −45 dB, −45 dB at 100 GHz, 143 GHz, 217 GHz, 353 GHz, 545 GHz, and 857 GHz, respectively. 

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Scanning beam solid angle (Ω_{SB}) error budget, showing the bias and fractional error due to the residual time response (ΔΩ_{τ}), near sidelobes (ΔΩ_{NSL}) and solid angle colour correction (ΔΩ_{CC}).
Fig. 12 Gridded data from all five seasons of Jupiter. The colour scale shows the absolute value of the peak signal. 

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3.4. Residual time response
The Planck spacecraft spin rate is constant to 0.1%, making the time response of the electronics and the bolometer degenerate with the angular response of the optical system.
The Bspline beam model extends ±20′ from the centre of the beam. An error in the time response on fast timescales will thus be exactly compensated by the scanning beam. However, errors in the time response beyond the limit of the scanning beam reconstruction will not be accounted for, and will bias the recovered beam window function.
To look for residual long tails due to incomplete deconvolution of the time response, all five seasons of Jupiter observations are binned into a 2D grid of 2′ pixel size extending 6° from the planet (Fig. 12). These are backgroundsubtracted using the Planck maps and stacked by fitting a Gaussian to estimate the peak amplitude and centroid of each observation. The data are binned as a function of pointing relative to the planet centre.
While a static nonlinearity correction is included in the TOI processing, partial ADC saturation and dynamical nonlinearity bias the normalization of these tails by underestimating the Jupiter peak signal. An estimate of the nonlinear correction is derived by fitting a gridded map of all three Mars observations to a map of all five Jupiter observations for each detector. A signal reduction at the peak of Jupiter is ruled out at the 1% level at 100 and 143 GHz, but detected at higher frequencies. Relative to Mars, the peak Jupiter signal is reduced by nonlinearity on average by 3 ± 3%, 12 ± 3%, 12 ± 4% and 65 ± 20% at 217, 353, 545 and 857 GHz, respectively. The tail normalizations are scaled by these factors.
An example of the residual tail is shown in Fig. 13. As well as a tail following the planet due to imperfect deconvolution of the time response, there is also a tail preceding the planet crossing; this is due to the lowpass filter applied in the Fourier domain. The residual beam tails have amplitudes typically at the level of 10^{4} of the peak but extend several degrees from the centre of the beam. The response beyond 6° on the sky is consistent with noise for all detectors.
The signal to noise of the tail measurement greater than 6° from Jupiter sets a modelindependent limit on the knowledge of the time response at signal frequencies from 0.016 Hz to 0.1 Hz at <10^{4}.
These stacked Jupiter data are integrated to determine the expected bias in total beam solid angle from this remaining uncorrected time response (see Table 2). The mean integral values are typically a few times 10^{4} of the main beam solid angle, an order of magnitude lower than the error in the beam solid angle due to noise and other systematics.
The residual scanning beam tails can also bias the effective beam window function. The spherical harmonic transform of the residual that is not included in the model for the main scanning beam is computed, , where is the multipole space representation of the main scanning beam model and is the multipole space representation of the long tail model. In all cases, the m = 0 (symmetric) mode of the ratio of to dominates higher order modes by at least a factor of 1000, meaning that the coupling to the scan strategy is negligible and the bias in the effective beam window function can be approximated by (12)The main effect on the effective beam window function is that at low ℓ, the bias δW_{ℓ} approaches a value of twice the fractional contribution to the total solid angle. When the window function is normalized to unity at the dipole frequency, the effect is a roughly constant bias in the window function at a level of a few × 10^{4} for ℓ > 100. The contribution of the residual tail to the window function is neglected in the error budget.
Fig. 13 A slice through stacked Jupiter data for bolometer 1436, illustrating residual long time response after deconvolution. The vertical dotted line shows the extent of the scanning beam map (plotted in blue). 

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3.5. Far sidelobes
The far sidelobes (FSL) are defined as the response of the instrument at angles more than 5° from the main beam centroid. Tauber et al. (2010) describes the prelaunch measurements and predictions of the far sidelobe response using physical optics models. Figure 14 compares the measured beam profile of detector 3531 with the FSL physical optics model. The way the FSL are treated in the dipole calibration and in the scanning beam model affects the effective beam window function, and care is needed to check whether the offaxis response could bias the window function at ℓ > 40 (angular scales 5°) relative to ℓ< 40. To the extent that the physical optics simulations correctly predict the far sidelobe response (which is shown to be roughly the case in the survey difference maps), they produce effects negligible in the effective beam window function of HFI. Appendix D presents the details of this calculation.
Fig. 14 Azimuthally averaged profiles of measured beams of channel 3531 compared to the azimuthal average of the far sidelobe physical optics model. 

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As a check of the quality of the physical optics model of the far sidelobes, Planck Collaboration XIV (2014) attempt a template fit of the physical optics model to the survey difference maps in combination with a zodiacal light model. The template fits are presented in Table 4 of Planck Collaboration XIV (2014). The FSL signature is clearly detected at 857 GHz at a level much higher than predicted. As these channels are multimoded (Maffei et al. 2010), the differences are not that surprising; it is very difficult to perform the calculations necessary for the prediction. In addition, the specifications for the horn fabrication were quite demanding, and small variations, though still within the mechanical tolerances, could give large variations in the amount of spillover.
For the lower frequency, singlemoded channels, there is no clear detection of primary (PR) spillover (i.e., pickup from close to the spacecraft spin axis; see Appendix D). While the significant negative values of the best fit template amplitude may indicate some lowlevel, largescale systematic, there seems to be nothing with the distinctive signature of PR spillover at frequencies between 100 and 353 GHz. These values indicate that the PR spillover values in Table 2 of Tauber et al. (2010) may be overestimated.
For the direct contribution of the secondary (SR) spillover, the situation is similar at 353 GHz, but at 217 and 143 GHz there is a 3σ detection at about the level expected, while at 100 GHz the value is about 2.5 times higher than expected, though the signaltonoise level of the detection is less than 2σ. The baffle contribution to the SR spillover appears higher than predicted, though Planck Collaboration XIV (2014) note that the baffle spillover is difficult to model and the diffuse signal is easily contaminated by other residuals in the survey difference.
Mean values of effective beam parameters for each HFI frequency.
4. Effective beams
Unlike WMAP (Jarosik et al. 2011), for large portions of the sky the scan strategy of Planck does not azimuthally symmetrize the effect of the beams on the CMB map. Treating the beams as azimuthally symmetric leads to a flawed power spectrum reconstruction. To remedy this, the effective beam takes the coupling between the azimuthal asymmetry of the beam and the uneven distribution of scanning angles across the sky into account.
The effective beam is computed for each HFI frequency scanning beam and scan history in real space using the FEBeCoP (Mitra et al. 2011) method, as in Planck’s early release (Planck HFI Core Team 2011b). A companion paper describes the details of the application of FEBeCoP to Planck data (Rocha et al., in prep.).
FEBeCoP calculates the effective beam at a position in the sky by computing the real space average of the scanning beam over all crossings angles of that sky position. The observed temperature sky is a convolution of the true sky T and the effective beam ℬ, (13)where Ω_{pix} is the solid angle of a pixel, and the effective beam can be written in terms of the pointing matrix A_{ti} and the scanning beam as (14)where t is the timeordered data sample index and i is the pixel index. A_{ti} is 1 if the pointing direction falls in pixel number i, else it is 0, p_{t} represents the pointing direction of the timeordered data sample, and is the centre of pixel number j, where the scanning beam is being evaluated (if the pointing direction falls within the cutoff radius).
The sky variation of the effective beam solid angle and the ellipticity of the bestfit Gaussian to the effective beam at HEALPix N_{side} = 16 pixel centres are shown for 100 GHz in Fig. 15. The effective beam is less elliptical near the ecliptic poles, where more scanning angles symmetrize the beam.
The mean and RMS variation of the effective beam solid angle across the sky for each HFI map are presented in Table 3.
Fig. 15 Effective beam solid angle (upper panel) and the bestfit Gaussian ellipticity (lower panel) of the 100 GHz effective beam across the sky in Galactic coordinates. 

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4.1. Effective beam window functions
The multipole space window function of one (or two) observed map(s) is defined, in the absence of instrumental noise and other systematics, as the ratio of the ensemble averaged auto or crosspower spectrum of the map(s) to the true theoretical sky power spectrum (15)It must account for the azimuthal asymmetry of the scan history and the beam profile. This is done in the HFI data processing pipeline using the harmonic space method Quickbeam (described in Appendix E.2), which allows a quick determination of the nominal effective beam window functions and of their Monte Carlo based error eigenmodes (Sect. 6) for all auto and crossspectra pairs of HFI detectors, the error determination being computationally intractable with FEBeCop.
In the FEBeCoP approach, many (approximately 1000) random realizations of the CMB sky are generated starting from a given fiducial power spectrum . For each beam model, the maps are convolved with the precomputed effective beams, and the pseudopower spectra of the resulting maps are computed and corrected by the mode coupling kernel matrix M (Hivon et al. 2002) for a given sky mask: The Monte Carlo average of kernelcorrected power spectra compared to the input power spectrum then gives the effective beam window function (Eq. (15), with replacing ).
In addition, another harmonic space method (FICSBell; Appendix E.1) was also tested and all three methods give consistent results for the nominal window functions at 100–353 GHz (see Fig. E.1).
For two different maps obtained with different detectors or combination of detectors X and Y, because of the optical beam noncircularity and Planck’s scanning strategy, the crossbeam window function is not the geometrical mean of the respective autobeam window functions, i.e., (16)as illustrated in Fig. 16, while of course W^{XY} = W^{YX} for any X and Y.
The effective beam window functions for the 2013 maps are shown in Fig. 17. Variations in the effective beam window function from place to place on the sky are significant; the published window functions have been appropriately weighted for the analysis of the nominal mission on the full sky. Analyses requiring effective beam data for more restricted data ranges or for particular regions of the sky should refer to the specialized tools provided in Planck Collaboration (2013).
5. Uncertainties and robustness
Ensembles of simulated planetary observations are used to propagate noise and other systematic effects in the timeordered data into errors in the beam reconstruction. These simulations are also used to estimate bias in the reconstruction of the beam by comparing the ensemble average beam with the input beam.
To assess the completeness of the statistical and systematic noise model, the consistency of the beam reconstruction derived from different planetary observations is measured against corresponding Monte Carlo ensembles. The MCderived beam statistics, including both the bias and the correlation structure of the errors, have been found to be somewhat sensitive to the near lobe structure that is included in the beam used as an input to the simulations. An investigation of this effect is ongoing.
In addition, several sources of systematic error that can potentially impact the beam determination, but are not included in the current simulation pipeline, have been investigated. The most significant of these are the beam colour correction and distortion due to ADC nonlinearity. These errors are estimated separately and found to be small in comparison to the error bars estimated in the eigenmode analysis described in Sect. 6.
Fig. 16 Ratios of the crossbeam window functions to the product of the respective autobeam window functions B_{a}(ℓ)B_{b}(ℓ) for maps at 100, 143, 217 and 353 GHz, illustrating Eq. (16). 

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Fig. 17 Effective beam window functions (solid lines) for each HFI frequency. The shaded region shows the ± 1σ error envelope. Dashed lines show the effective beam window function for Gaussian beams with FWHM 9.́65, 7.́25, 4.́99, 4.́82, 4.́68, and 4.́33 for 100, 143, 217, 353, 545 and 857 GHz, respectively. 

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5.1. Simulation of planet observations
The bias and uncertainty in the scanning beams are determined using ensembles of simulated planet observations. The simulation pipeline uses the LevelS Planck simulation code (Reinecke et al. 2006) to generate simulations of the first two observations of Mars and the first three observations of Jupiter and Saturn for each bolometer. This pipeline is used to generate 100 simulations for each bolometer at 353 GHz, 545 GHz and 857 GHz, 200 at 143 GHz and 217 GHz, and 400 at 100 GHz. Each simulation is reconstructed into a beam model using the identical procedure and software as for the real data.
The components of the simulations are as follows.

1.
The scanning beams used as input to the simulations are thesame Mars beams used to calculate the effective beam windowfunction. As shown in Appendix B, thereconstruction bias depends more on the beam representationthan the exact input beam used in a simulation.

2.
The Planck spacecraft pointing and the Horizons ephemerides are used to calculate the pointing relative to the planets for the simulation. An additional random 2.̋5 RMS per sample pointing error is added in both the inscan and crossscan directions with a power spectral density given by the pointing error estimate, consistent with the estimated error in the spacecraft pointing (see Sect. 4, and Fig. 7 of Planck Collaboration VI 2014). Error in the beam centroid determination is not simulated.

3.
Detector noise is generated in the timeline by a random realization of Gaussian noise with a power spectral density as reported in Planck Collaboration VI (2014).
For a small number of the simulations, cosmic ray glitches are added to the simulation with the energy spectrum and rate measured in the data (Planck Collaboration X 2014), and the deglitching procedure from the standard pipeline is applied to detect samples contaminated by glitch transients and to subtract the long tails (Planck Collaboration VI 2014).
Each simulation in the ensemble provides an estimate of the scanning beam, which is then used as input to Quickbeam (Appendix E.2) to estimate an effective beam window function (EBWF). The resulting ensemble of EBWFs are then used to compute the bias and errors on the EBWF derived from the data; the mean of the ensemble provides a measure of the reconstruction bias, and the distribution of the ensemble gives the uncertainty. The bulk of the RMS of the ensemble can be captured with a small number of eigenmodes (Sect. 6).
This procedure allows for the direct determination of the EBWF and associated errors for each of the detector crosscorrelations input to the angular power spectrum likelihood, described in Planck Collaboration XV (2014).
5.2. Absolute consistency: comparison of systematics against simulations
5.2.1. Seasonal consistency
One important test of the consistency of the scanning beam measurement is the stability over time, as measured in different seasons and on different objects over the course of the mission. The difference of the scanning beam window function between the first two observations of Mars is shown in Fig. 18. The residuals are well within the estimated uncertainty that is derived from the simulation ensemble for 100–353 GHz. Due to the sparse crossscan sampling, the Bspline beam representation does not converge for a single planetary observation; for this test the GaussHermite (GH) formalism is used.
Fig. 18 Seasonal consistency of the beams at four HFI frequencies.Thick blue lines show the difference of the Mars season 1 and Mars season 2 window functions normalized to the bandaverage full window function. The grey shaded band is the ±1σ error for Mars 1 and the blue shaded band is the ±1σ error for Mars 2. 

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5.2.2. Time variability of planet flux density
Mars is known to have diurnal variability at HFI frequencies due to the nonuniformity of the surface albedo^{5}. A single detector scans Mars on time scales of a few hours, during which the brightness temperature of Mars can vary by several percent. Additionally, the Mars planetary disc area varies by several percent during an observation as it moves relative to Planck. To assess the impact of Mars variability on the scanning beam determination, GH representations of the beam are derived for each bolometer, with and without a model for the diurnal variability of Mars. Using χ^{2} tests for goodness of fit, these results are indistinguishable.
5.2.3. Beam colour correction
The scanning beams are measured using planets, whose spectral energy distribution (SED) roughly follows the RayleighJeans spectrum. However, the beam window functions from these measurements are used to correct the angular power spectra of CMB anisotropies, which is characterized by a very different spectrum, one that is falling in power as a function of frequency relative to a RayleighJeans spectrum across the HFI bands.
Fig. 19 An estimate of known biases in the effective beam window function compared to the RMS statistical error including the additional factor 2.7 (grey shaded region) for each HFI frequency band. Green is the bias due to near sidelobes, blue is the colourcorrection bias, and magenta shows an upper limit on the effect of ADC nonlinearity. 

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Physical optics simulations, using the GRASP^{6} software, are used to investigate this effect. For each HFI channel from 100–353 GHz, monochromatic simulations are generated at five frequencies across each band using the prelaunch telescope model (Maffei et al. 2010; Tauber et al. 2010). The solid angle of these simulated beams varies across the band, due to diffraction and focussing effects. For 100–217 GHz, the beam size comes to a minimum near the band centre, while at 353 GHz the beam size rises with frequency.
No attempt is made to colourcorrect the planetderived window functions, because a telescope solution has not yet been determined that agrees with the measured solid angles. Spot checks at 143 and 353 GHz with a defocussed telescope model improve agreement between the data and the model, but does not change the trend of solid angle with frequency across the band. The investigation of these effects suggests that numerical models can bound the uncertainty, but cannot reliably predict a bias. RayleighJeansweighted average and CMB anisotropyweighted average beams are produced and used to compute an effective beam window function (blue curves in Fig. 19). The largest bias results at 353 GHz. For the frequencies 100, 143, and 217 GHz, however, the band average effect is less than 0.1% across the entire multipole range used in the CMB likelihood.
The beam solid angle also varies as a function of source SED; the GRASP simulations constrain the size of the beam solid angle colour correction from a ν^{2} source SED (like the planets) to the IRAS convention ν^{1}. At 545 and 857 GHz, the GRASP models of Murphy et al. (2010) provide a measure of the effect, which is found to be significant at 353 GHz, but not at the other frequencies, at the level of a few tenths of a percent (Table 2).
5.2.4. ADC nonlinearity
Nonlinearity in the ADC in the HFI readout electronics mainly manifests itself as an apparent gain drift (Planck Collaboration VI 2014; Planck Collaboration VIII 2014), however, the measured beam shapes are also biased by the effect.
Correcting the ADC nonlinearity relies on a model that predicts which ADC codes contribute to each data sample. The presence of large signal gradients, such as a planet, or pickup from the ^{4}HeJT cooler, make modelling difficult. A model for correcting every detector is still under development.
A model of the behaviour of the ADCs is used to apply nonlinearity to simulated Mars observations. The Bspline scanning beams reconstructed from these simulations predict biases that are typically under 2% at ℓ ≤ 2000, less than the RMS error (see the magenta curves in Fig. 19 for simulated bias of the 100 through 353 GHz channels).
Using the brighter planets in future Planck main beam models will tend to reduce the effect, as the higher signal amplitudes sample a broader range of the ADC, tending to average down the resulting bias (Mather et al. 1993).
5.2.5. Pointing errors
While the simulated planet observations account for random pointing errors, pointing drifts remain at the level of a few arcseconds per pointing period in the coscan and crossscan directions common to all detectors (Planck Collaboration I 2014; Planck Collaboration VI 2014). These have the effect of widening the beam. Even in the worst case, if remaining errors are Gaussiandistributed with a 2.̋5 residual, the window function is 0.2% biased at multipole ℓ = 3000. Since this is a very small effect relative to the other beam errors, it is considered negligible.
5.2.6. Glitches
The HFI data are affected by transient signals due to interactions with high energy particles. The planet simulation tools allow for the addition of a random population of glitches to simulated Mars observations with a realistic rate and energy spectrum (Planck Collaboration X 2014), testing the performance of the deglitching algorithm near planet signals and the effects of undetected low energy glitches in the channel with the highest glitch rate at each frequency band.
The RMS of the noise increases by at most 20% in the bolometer with the highest glitch rate, but more typically less than 10%, and the reconstruction bias changes by a negligible amount. The effects are small enough that they are not included in the error budget.
5.3. Relative consistency: window function ratios vs. spectral ratios
The CMB sky itself allows an additional, and statistically powerful, check on the relative consistency (but not the absolute accuracy) of the effective beam window functions within a frequency band. See Fig. 33, and the associated discussion in Sect. 7.3 of Planck Collaboration VI (2014), showing the selfconsistency of the window functions at a level better than 0.4%, across the full range of multipoles employed in cosmological parameter estimation.
6. Total error budget
The distribution of beam solid angles reconstructed from simulated planet reconstructions provides the statistical error budget for the beam solid angles (Table 2). Other systematic effects are small in comparison.
The uncertainty in the effective beam window function is determined with the distribution of window functions computed from the simulated beams. These errors are highly correlated in multipole space, and we find that they can be fully described by a small number of eigenmodes and their covariance matrix (Appendices G.1 and G.2), which can be retrieved, together with the nominal beam window functions, from Planck Collaboration (2013) and the Planck Legacy Archive^{7}. These simulations also allow us to determine the bias induced by the beam reconstruction pipeline, and correct the final beam window functions from it (Appendix G.3).
Studies have been performed to probe the impact of assumptions regarding the near sidelobe response of the beam. These studies suggest the potential that the Monte Carlo ensembles used to derive the error eigenmodes may not capture aspects of the bias and the correlation structure of the errors. A visual summary of the total error budget, including the impact of the near sidelobe response, is provided in Fig. 19. For the initial data release a conservative scaling of the MCderived beam errors, a factor of 2.7 in power, is applied in order to accommodate the maximum extent of the bias that is allowable, given the signaltonoise ratio of the Jupiter and Saturn data.
7. Conclusion
A combination of Jupiter and Saturn observations, survey difference maps, and checkout and performance verification (CPV) phase data is used to derive the bolometer/readout electronics time response transfer function. This transfer function is deconvolved from the HFI timeordered data. Residuals can be detected as long tails, stretching up to 6° from compact sources, but are at a level less than 10^{4} of the beam peak, and so are insignificant for the scanning beams.
The effective beams and window functions are estimated for all HFI detectors using a scanning beam derived from a combination of the first two seasons of Mars observations. The effective beam products account for the partial symmetrization of the scanning beam which results from the scan strategy. The FEBeCoP algorithm produces effective beams in real space, while the Quickbeam method is used to produce the effective beam window functions and errors in harmonic space.
The final error budget for each effective beam window function (both auto and crossbeams) is welldescribed by five eigenmodes for each beam and a correlation matrix of eigenvalues. The error parameters are estimated with an ensemble of simulated Mars observations. The simulations are based on random realizations of noise and pointing errors.
No significant time variation of the scanning beam or the time response is found. Crosspower window functions are also produced that describe the beam filtering effects in crosscorrelations between independent HFI maps.
The high signaltonoise ratio of the Jupiter and Saturn data limit the contribution of the near sidelobe response (between 30′ and 5° of the beam centroid) to the total beam solid angle to less than 0.2% at 100 to 217 GHz. This portion of the beam, which is beyond the signaltonoise of the Mars data, is not fully represented in the beam simulations. The Monte Carlo derived error amplitudes are scaled by a factor of 2.7 to account for the contribution of the near sidelobe response to the beam window function.
Far sidelobes contribute negligibly to the window function, but may potentially impact the primary calibration and result in pickup from bright Galactic sources. Improved physical optics models for the far sidelobes will be forthcoming for future releases.
The impact of nonlinearity in the ADC, the sensitivity of beam shape to the spectral intensity of the source, and residual cosmic ray transients are found to be insignificant sources of systematic error.
Knowledge of the beam window functions allows Planck HFI to extrapolate the dipole calibration over more than three orders of magnitude in angular scale. For the current data release, beam errors are subdominant to noise, foreground marginalization and sample variance in cosmological parameter estimation. Future Planck data releases will fully exploit all of the available planetary data and create widefield beam maps, allowing an even more precise, and accurate, measurement of the main beams and near sidelobes.
Planck is a project of the European Space Agency – ESA – with instruments provided by two scientific Consortia funded by ESA member states (in particular the lead countries: France and Italy) with contributions from NASA (USA), and telescope reflectors provided in a collaboration between ESA and a scientific Consortium led and funded by Denmark.
TICRA, http://www.ticra.com
Acknowledgments
The development of Planck has been supported by: ESA; CNES and CNRS/INSUIN2P3INP (France); ASI, CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MICINN, JA and RES (Spain); Tekes, AoF and CSC (Finland); DLR and MPG (Germany); CSA (Canada); DTU Space (Denmark); SER/SSO (Switzerland); RCN (Norway); SFI (Ireland); FCT/MCTES (Portugal); and PRACE (EU). A description of the Planck Collaboration and a list of its members, including the technical or scientific activities in which they have been involved, can be found at http://www.sciops.esa.int/index.php?project=planck&page=Planck_Collaboration.
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Appendix A: Electronics model
HFI electronics filter sequence; here s = iω.
Parameters for LFER4 model that are deconvolved from the data.
This Appendix derives the effect of the HFI electronics filtering on the TOI. If the input signal (power) on a bolometer is (A.1)the bolometer physical impedance can be written as: (A.2)where ω is the angular frequency of the signal and F(ω) is the complex intrinsic bolometer transfer function. For HFI the bolometer transfer function is modelled as the sum of 4 single pole low pass filters, (A.3)The modulation of the signal is done with a square wave, written here as a composition of sine waves of decreasing amplitude, (A.4)where the Euler relation sinx = (e^{ix}−e^{−ix}) /2i is used, and ω_{r} is the angular frequency of the square wave. The modulation frequency is f_{mod} = ω_{r}/2π and was set to f_{mod} = 90.18759 Hz in flight. This signal is then filtered by the complex electronic transfer function H(ω). Setting and gives (A.5)This signal is then sampled at high frequency (2f_{mod}N_{samp}). N_{samp} is a parameter of the HFI electronics corresponding to the number of high frequency samples in each modulation semiperiod. In order to obtain an output signal sampled every π/ω_{r} seconds, the waveform is integrated on a semiperiod, as done in the HFI readout. To also include a time shift Δt, the integral is calculated between nπ/ω_{r} + Δt and (n + 1)π/ω_{r} + Δt (with T = 2π/ω_{r} period of the modulation). The time shift Δt is encoded in the HFI electronics by the parameter S_{phase}, with the relation Δt = S_{phase}/N_{samp}/f_{mod}.
After integration, a sample of a bolometer at time t_{n} can be written as (A.6)where (A.7)The output signal in Eq. (A.6) can be demodulated (thus removing the (− 1)^{n}) and compared to the input signal in Eq. (A.1). The overall transfer function is composed of the bolometer transfer function and the effective electronics transfer function, H′(ω), (A.8)The shape of H(ω) is obtained by combining low and highpass filters with Sallen Key topologies (Sallen & Key 1955), with their respective time constants, and accounting also for the stray capacitance low pass filter given by the bolometer impedance combined with the stray capacitance of the cables. The sequence of filters that define the electronic bandpass function H(ω) = h_{0} ∗ h_{1} ∗ h_{2} ∗ h_{3} ∗ h_{4} ∗ h_{5} are listed in Table A.1, with parameters listed in Table A.2.
Appendix B: GaussHermite beams
HFI’s scanning beams are described by an elliptical Gaussian shape to an accuracy of 2–5% in solid angle. A GaussHermite representation of the beam uses Hermite polynomials to provide higherorder corrections to a Gaussian model (Huffenberger et al. 2010). There are fewer degrees of freedom than in a gridded beam map, allowing higher signaltonoise on each mode. However, because the order of the decomposition is truncated, in practice the description is ill suited to a description of features much beyond the extent of the main beam. Larger scale features of the beam, including a beam shoulder (Ruze 1966) or the effect of the printthrough of the backing structure (grating lobes), must be included separately.
The initial twodimensional Gaussian is parametrized as (B.1)where A is an overall amplitude, (x_{1},x_{2}) are twodimensional Cartesian coordinates (the pointing is projected to a flat sky), are the coordinates of the beam centre, and the correlation matrix is given by (B.2)Hence, the Gaussian model parameters are fitted. These can also be expressed in terms of the ellipticity, e (defined here as the ratio between the major and minor axes), and rotation angle α, of the Gaussian ellipse.
The coefficients to the GaussHermite polynomials multiplying the elliptical Gaussian are then fitted. The basis functions are defined as (B.3)where H_{n}(x) is the ordern Hermite polynomial and the primed coordinates x^{′} rotates into a system aligned with the axes of the Gaussian and scaled to the major and minor axes σ_{i} (i.e., to the principle axes of the correlation matrix Σ). Having determined the elliptical Gaussian separately, the subsequent GaussHermite polynomial fit is a generalized leastsquares procedure, solvable by the usual matrix techniques, under the assumption of white noise. Because the full data model includes further effects such as pointing error, glitch effects, etc, a full error analysis requires a broad suite of simulations, described in Sect. 5.
Appendix C: Bspline beams
In this model of the scanning beam, a twodimensional Bspline surface is fit to the timeordered planet data. A smoothing criterion is applied during the fit to minimize the effects of high spatial frequency variations due to noise. This representation of the beam is superior to a simple twodimensional binning of the data in its ability to capture large signal gradients.
Bsplines are a linear combination of polynomials of degree k and order k + 1. They are defined by the location of their control points (called knots) of which there are 5 for 3rd degree polynomials.
The kdegree Bspline built using the knots {λ_{i},...,λ_{i + k + 1}} (de Boor 1972; Cox 1972) is given by (C.1)with recursion relations (C.2)where the index l runs from 1 through the Bspline degree k. The Bspline knots {λ_{i},...,zλ_{i + k + 1}} are located on a regularly spaced grid in the detector coordinate frame. At the edge of the reconstructed beam map area, 4 coincident knots are added to avoid vanishing basis functions, allowing a unique decomposition.
In the solution to the Bspline coefficients P_{i,l}(x), a smoothing criterion is introduced as a constraint on the sum of the derivatives of the beam at each knot, motivated by the assumption that the true beam does not contain very high spatial frequencies and prevents noise from biasing the reconstruction at spatial frequencies smaller than the smoothing scale.
A smoothing criterion η (Dierckx 1993) is related to the sum of the order k derivative of the beam model (P^{k}) evaluated at the knot locations λ_{i}: (C.3)where g is the total number of distinct knots and λ_{i +} and λ_{i −} are the left and right derivative of the beam model evaluated at the knot location. The smoothing criterion is introduced as an extra term in the score function ζ in the leastsquares minimization of the beam map with respect to the data, (C.4)where δ is the usual squared difference between the data points y_{r} and the model P(x_{r}) at pointings x_{r}, (C.5)and p is a relative weighting factor for the smoothing criterion. The knot spacing and the smoothing criterion weight p are determined separately for each frequency band based on physical optics simulations and the coverage of the two Mars transits given by the scanning strategy (see Table C.1). Simulated planet observations (see Sect. 5) show that the choice of Bspline knot spacing and smoothing parameters do not significantly bias the beam reconstruction.
Summary of Bspline knot spacing and smoothing criterion weight p used in the beam reconstruction.
Appendix D: Far sidelobe effects on the effective beam window function
The far sidelobes (FSL) are defined as the response of the instrument at angles >5° from the main beam centroid. The FSL affect both 1. the gain calibration of the instrument with the dipole, and 2. how the calibration is transfered to higher multipoles with the effective beam window function. Here we discuss the interplay of these effects.
The FSL can be separated into three main components (see Fig. 5 of Tauber et al. 2010):

1.
Primary Reflector Spillover (PR Spillover) is the response ofthe instrument to radiation from just above the primary mirror thatreflects off the secondary mirror and arrives at the feed horns. Thisresponse is nearly aligned with the spin axis of the spacecraft, andtherefore scans very little of the sky on 1 min timescales.

2.
Secondary Reflector Direct Spillover(SR Direct Spillover, or SRD) is the response from directly above the secondary mirror. The sidelobe peaks roughly 10° from the main beam, and as such scans the sky in nearly the same way as the main beam as the satellite rotates. The azimuthal extent is roughly 30°.

3.
Secondary Reflector Baffle Spillover (SR Baffle Spillover, or SRB) is response from radiation reflecting off the baffles. This is difficult to model, being diffuse radiation reflecting off the poorly known inner baffle surfaces. It is spread over a large fraction of the sky.
The HFI dipole calibration is performed assuming a deltafunction (pencil) beam (Planck Collaboration VIII 2014). This leads to a bias in the calibration described by the ratio of the dipole convolved with the fullsky beam to the dipole convolved by a pencil beam, where is the estimated gain, g is the true gain, P is the true fullsky beam, P^{pencil} is the assumed pencil beam, and D is the dipole. The true fullsky beam is taken to consist of three portions, the main beam P_{main} (within 20′ of the centroid), the near sidelobes P_{NSL} (between 20′ and 5°), and the far sidelobes P_{FSL} (response further than 5° from the centroid). The quantity is defined as the beam within 5° of the centroid (the main lobe and near sidelobes). The bias in the calibration can be rewritten as (D.1)where f_{FSL} is the integral of the far sidelobe response relative to the full beam integral. The first term, 1 − f_{FSL}, is due to the loss in response of the main lobe and near sidelobes to far sidelobes, while the second term represents the coupling of the dipole into the sidelobes.
For each pointing period the convolution of the pencil beam with the dipole is approximated by (D.2)where θ_{main} is the angle between the main beam centroid and the spin axis, and d_{pencil} is the dipole amplitude that would be observed with a pencil beam. The components of the far sidelobes enter into Eq. (D.1) differently, and assuming the majority of the response to the dipole in each component comes from one direction on the sky, can be approximated as where ϵ_{PR}, ϵ_{SRD}, and ϵ_{SRB} are the fraction of the total solid angle in the PR spillover, the SR direct spillover, and the SR baffle spillover, respectively, and θ_{PR}, θ_{SRD}, θ_{SRB} are the angles between the spin axis and the peak of the PR spillover response, the SR direct spillover response, and the SR baffle spillover response respectively. Equation (D.1) then simplifies to (D.3)From Tauber et al. (2010), θ_{pencil} ≃ 85°, θ_{PR} ≃ 10°, θ_{SRD} ≃ 75°, and θ_{SRB} ≃ 45° making the PR spillover and SR baffle spillover negligible. Equation (D.3) reduces further to (D.4)SR direct spillover scans the dipole, but with a slightly different amplitude, since it is offset by 10° from the main lobe. The PR spillover does not modulate the dipole; aligned with the spin axis, the PR spillover contributes a nearly constant signal during each stable pointing period. The predicted values of the solid angle fractions are f_{FSL} = 3.3 × 10^{3} and ϵ_{SRD} = 1.9 × 10^{3} at 100 GHz (Tauber et al. 2010), making the measured gain .
However, the effect of the gain bias on the angular power spectrum is further reduced by corrections to the effective beam window function due to the FSL. Not including the FSL in the beam model tends to bias the window function at very low multipoles relative to high multipoles. Considering the measured sky signal at small scales as compared to the true sky T_{sky} one has where S is the signal measured by the bolometer. Solving for the true sky signal gives Considering that P_{MNSL} ⊗ T_{sky} ≃ (1 − f_{FSL})T_{sky}, Now also considering that P_{MNSL} ⊗ D ≃ (1 − f_{FSL})P^{pencil} ⊗ D, the true sky signal can be written as or (D.5)where the secondorder correction term is dropped, and the following factors are defined: and Here φ_{sky} represents the relative pickup of anisotropy in the sidelobe beam as compared to the main beam and near sidelobes. The response of the FSL to CMB anisotropy is negligible, but Galactic response may not be. The PR spillover contributes only a constant value per pointing period, because it is not modulated with the scan. Only the SR direct spillover enters into the formula. Since the SR direct spillover is nearly aligned with the main beam, Galactic signal is not picked up in the CMB anisotropy except close to the Galactic plane. So for foregroundclean regions of the sky, φ_{sky} ≪ f_{FSL}.
The quantity φ_{D} is the bias due to the ratio of dipole response in the far sidelobes to dipole response in the main beam. Again the PR spillover contributes only an offset per pointing period, which is removed by the polkapix destriping algorithm (Tristram et al. 2011; Planck Collaboration VIII 2014), so we are left with the SR direct spillover and SR baffle spillover. Applying the same approximation as in Eq. (D.2) gives and Eq. (D.5) becomes (D.6)This result implies that FSL bias of the effective beam window function tends to cancel the gain bias (Eq. (D.4)), and the response to CMB anisotropy is unaffected to first order.
Simulations of the sky scanned with the far sidelobe physical optics model (Planck Collaboration XIV 2014; Planck Collaboration VI 2014) confirm this.
Appendix E: Harmonic space computation of the effective beam window function
Two harmonic space techniques (FICSBell and Quickbeam), developed independently but closely related in their formalism, have been used to compute the effective beam window functions. They provide a valuable crosscheck of the pixelbased results obtained with FEBeCoP (Fig. 2) and their low computational requirements allow fast calculation of the window function errors through the processing of Monte Carlo simulations of planet observations.
Appendix E.1: FICSBell
FICSBell is a harmonic space method for computing the effective beam window function directly from the scanning beam and the scan history. The steps of this method are as follows.

1.
The statistics of the orientation of each detector d within each map pixel p is computed first, andonly once for a given observation campaign:(E.1)where ψ_{j} is the orientation of the detector with respect to the local meridian during the measurement j occurring in the direction r_{p}. Note that the s = 0 moment is simply the hit count map. The orientation hit moments are computed up to degree s = 4. At the same time, the first two moments of the distribution of samples within each pixel (i.e., the centre of mass and moments of inertia) are computed and stored.

2.
The scanning beam map of each detector d is transformed into spherical harmonics: (E.2)where B_{d}(r) is the beam map centred on the north pole, and the Y_{ℓs}(r) are the spherical harmonic basis functions. Higher s indices describe higher degrees of departure from azimuthal symmetry and, for HFI beams, the coefficients are decreasing functions of s at most multipoles considered. It also appears that, for ℓ< 3000, the coefficients with s  > 4 account for much less than 1% of the beam solid angle. Spot checks where window functions are computed with s ≤ 6 show a difference of less than 10^{4} for ℓ< 2000 at 100 GHz and for ℓ< 3000 at 143 and 217 GHz. For these reasons, only modes with s ≤ 4 are considered in the present analysis. ArmitageCaplan & Wandelt (2009) reached a similar conclusion in their deconvolution of LFI beams.

3.
For a given CMB sky realization t, described by its spherical harmonics coefficients a_{ℓm} = ^{∫}drt(r)Y_{ℓm}(r), the coefficients computed above are used to generate sspin weighted maps, (E.3)as well as the first and second derivatives, using standard HEALPix tools.

4.
The spinweighted maps and orientation hit moments of the same order s are combined for all detectors involved, to provide an “observed” map (E.4)Similarly the local spatial derivatives are combined with the location hit moments to describe the effect of the nonideal sampling of each pixel (see Appendix F). In this combination, the respective number of hits of each detector in each pixel is considered, as well as the detector weighting (generally proportional to the inverse noise variance).

5.
The power spectrum of this map can then be computed, and compared to the input CMB power spectrum to estimate the effective beam window function over the whole sky, or over a given region of the sky.
Monte Carlo (MC) simulations in which the sky realizations are changed can be performed by repeating steps 3, 4 and 5. The impact of beam model uncertainties can be studied by including step 2 in the MC simulations.
Appendix E.2: Quickbeam
By decomposing the scanning beam into harmonic coefficients B_{ℓm}, each TOI sample can be modelled as (neglecting the contribution from instrumental noise, which is independent of beam asymmetry) (E.5)where the TOI samples are indexed by i, and is the underlying sky signal. The spin spherical harmonic rotates the scanning beam to the pointing location (θ,φ), while the e^{− isαi} factor gives it the correct orientation. Equation (E.5)may be evaluated using techniques developed for convolution in Wandelt & Gorski (2001) and Prezeau & Reinecke (2010), although manipulating a TOIsized object is necessarily slow.
On the small angular scales comparable to the size of the beam, it is a good approximation to assume that the procedure of mapmaking from TOI samples is essentially a process of binning: (E.6)where H(p) is the total number of hits in pixel .
Given a normalized, rescaled harmonic transform of the beam B_{ℓm}, sky multipoles and a scan history object given by (E.7)where the sum is over all hits j of pixel p at location , and α_{j} is the scan angle for observation j. The harmonic transform of this scanstrategy object is given by (E.8)The beamconvolved observation is then given by (E.9)Taking the ensemble average of the pseudoC_{ℓ} power spectrum of these T_{lm} gives (Hanson et al. 2010; Hinshaw et al. 2007) (E.10)where (E.11)is a crosspower spectrum of scan history objects. Note that the w(n,s) used here can also incorporate a position dependent weighting to optimize the pseudoC_{ℓ} estimate, such as inversenoise or a mask, the equations are unchanged. Writing the pseudoC_{ℓ} in position space (following Dvorkin & Smith 2009) with Wignerd matrices gives (E.12)This integral can be implemented exactly using GaussLegendre quadrature, at a cost of . For simplicity, the equations here are written for the autospectrum of a single detector, but the generalization to a map made by adding several detectors with different weightings is straightforward. The cost to compute all of the necessary terms exactly in that case becomes .
Fig. E.1 Difference between effective beam window functions computed with a real space method (FEBeCoP) and a harmonic space method (Quickbeam). The shaded region denotes the RMS error at each mutipole. 

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On the flat sky, beam convolution is multiplication in Fourier space by a beam rotated onto the scan direction. Multiple hits with different scan directions are incorporated by averaging (as the scan history objects above encapsulate).
A scan strategy which is fairly smooth across the sky is nearly equivalent to observing many independent flatsky patches at high L. There is a fairly good approximation to the beam convolved pseudopower spectrum which is essentially a flatsky approximation. In the limit that L ≫ ℓ_{1}, with C_{ℓ2} and B_{ℓ2} is a slowlyvarying function in ℓ_{2}, and using the equality (E.13)the pseudoC_{ℓ} sum above can be approximated as (E.14)where the average ⟨⟩_{p} is taken over the full sky. It is illustrative to consider two limits of this equation. Firstly, for a “raster” scan strategy in which each pixel is observed with the same direction:(E.15)and the predicted transfer function is just the power spectrum of the beam. Secondly, for a “bestcase” scan strategy, in which each pixel is observed many times with many different orientation angles, (E.16)and the transfer function is the azimuthally symmetric part of the beam. Note that this is for a fullsky observation; in the presence of a mask, the average above produces an f_{sky} factor, as expected but neglects the coupling between L multipoles (which can be calculated with the more complete equations above).
Appendix F: Pixelization artefacts
PlanckHFI maps are produced at HEALPix resolution 11 (N_{side} = 2048), corresponding to pixels with a typical dimension of 1.́7. With the resolution comparable to the spacing between scanning rings (Planck Collaboration I 2011) there is an uneven distribution of hits within pixels, introducing a complication in the analysis and interpretation of the Planck maps. A sample of the Planck distribution of sample hits within pixels is illustrated in Fig. F.
Fig. F.1 Illustration of TOI samples near the Galactic plane (grey dots), overplotted on a simulated CMB realization which has been convolved with a Gaussian 7′ FWHM beam and pixelized at (N_{side} = 2048). Associated scanning rings (grey lines) as well as centres of mass for the hit distribution (black arrows) are also plotted. 

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The three effective beam codes may also be used to simulate the effect of pixelization on the observed sky: LevelS/TotalConvoler/Conviqt (Reinecke et al. 2006; Wandelt & Gorski 2001; Prezeau & Reinecke 2010); FEBeCoP (Mitra et al. 2011; and FICSBell (Appendix E).
For the measurement of CMB fluctuations, the effects of pixelization may be studied analytically. On the small scales relevant to pixelization, the observed CMB is smooth, both due to physical damping and the convolution of the instrumental beam. Taylor expanding the CMB temperature about a pixel centre to second order, the typical gradient amplitude is given by (F.1)Here the approximate value is calculated for a ΛCDM cosmology with a 7′ FWHM Gaussian beam. The typical curvature of the observed temperature, on the other hand is given by (F.2)On the scales relevant to the maximum displacement from the centre of a 1.7′ pixel, the maximum displacement is of order 1′(3 × 10^{4}rad)), and so the gradient term tends to dominate, although the curvature term is still nonnegligible. For each observation of a pixel, the displacement from the pixel centre can be denoted as d = d_{θ} + id_{φ}. The average over all hits within a pixel gives an overall deflection vector for a pixel centre located at denoted as . This represents the centre of mass of the hit distribution; Fig. F shows these average deflections using black arrows. The deflection field may be decomposed into spin1 spherical harmonics as (F.3)With a secondorder Taylor expansion of the CMB temperature about each pixel centre, it is then possible to calculate the average pseudoC_{ℓ} power spectrum of the pixelized sky. This is given by (F.4)where R^{d} = ⟨d^{2} ⟩ /2 is half the meansquared deflection magnitude (averaged over hits within a pixel, as well as over pixels), is the sum of the gradient and curl power spectra of d_{ℓm}, and is the gradient spectrum minus the curl spectrum. The R^{d} term describes a smearing of the observed sky due to pixelization. For uniform pixel coverage of N_{side} = 2048 pixels ⟨d^{2} ⟩ ^{1/2} = (2R^{d})^{1/2} = 0.725′, while, for the hit distribution of Planck frequency maps, R^{d} is within 0.2% of this value for CMB channels, and 0.4% for all channels. This term is therefore accurately described by the HEALPix pixel window function, which is derived under the assumption of uniform pixel coverage, and the resulting relative error on the beam window function is at most 4 × 10^{4} for ℓ ≤ 3000.
The effect of pixelization is degenerate with that of gravitational lensing of the CMB, with the difference that it: (1) acts on the beamconvolved sky, rather than the actual sky; and (2) produces a curlmode deflection field as well as a gradient mode. This is discussed further in Planck Collaboration XVII (2014), where the subpixel deflection field constitutes a potential source of bias for the measured Planck lensing potential. Indeed, Eq. (F.4) is just a slightly modified version of the usual first order CMB lensing power spectrum (Hu 2000), Lewis & Challinor (2006) to accommodate curl modes.
A useful approximation to Eq. (F.4) which is derived in the unrealistic limit that the deflection vectors are uncorrelated between pixels, but in practice gives a good description of the power induced by the pixelization, is that the couples the CMB gradient into a source of noise with an effective level given by (F.5)where the average is taken over all pixels and R^{T} is half the meansquared power in the CMB gradient: (F.6)For frequencycombined maps, is typically on the order of 0.́1, and so the induced noise σ^{N} is approximately 2 μK′. This is small compared to the instrumental contribution, although it does not disappear when taking crossspectra, depending on the coherence of the hit distributions of the two maps in the crossspectra.
Appendix G: Beam window function error
Fig. G.1 Beam window function error modes. Panels a), b) and c) show the first six eigenmodes defined in Eq. (G.11) for respectively the effective autobeam 100ds1 and 143ds2 and the effective crossbeam 2171 × 2172. The first five modes used in the current beam error modelling are shown as solid coloured curves, while the 6th mode (the first one to be ignored) appears as yellow dashes. The grey dashes show the ±1 σ envelope obtained by adding the first five modes in quadrature, while the solid grey curve is the ±1 σ envelope estimated from the simulations (therefore including all eigenmodes). The black dashes show the relative difference between and B_{mean}(ℓ) defined in Eqs. (G.6) and (G.5), respectively. In panel d), for a selection of effective beams, the coloured symbols show (d_{k}/d_{1})^{2} where d_{k} is the kth eigenvalue of the diagonal matrix D (Eq. (G.12)), while the coloured dashes show the error made on the quadratic sum of the eigenvalues by truncating it to n_{modes}: . The vertical dashes show the current n_{modes} = 5 value. 

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Appendix G.1: Error eigenmodes
Fig. G.2 Effect of ℓ truncation on beam error modes for frequencyaveraged beam window functions at 100, 143, 217 GHz and 143 × 217. For clarity, only the three leading modes are shown, respectively, in blue, green and red, while the solid grey line shows the 1 σ level, obtained by adding all modes in quadrature. Dotted lines are the original eigenmodes computed on a wide ℓrange. Solid lines are the eigenmodes computed directly from the MC simulations on the truncated ℓrange used for cosmological analysis. Dashed lines show the eigenmodes computed on the truncated ℓrange with Eqs. (G.16) and (G.17), starting from the first five eigenmodes for the wide range. In all four cases considered, the first leading mode on the truncated range, which dominates the error budget, is perfectly reconstructed out of the information available, while the second leading mode is well estimated in all cases except for 217 GHz. 

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Consider two individual detectors or detection units (weighted sum of detectors) X and Y for which sky maps are available. Here X and Y can be the same or different. Putting aside the instrumental noise and other contingencies for the time being, the cross (or auto) angular power spectrum measured of the observed maps is on average related to the real input signal through (G.1)where is the effective beam window function. Note that because of the optical beam noncircularity and Planck scanning strategy, (G.2)as illustrated in Fig. 16, while W^{XY} = W^{YX} for any X and Y. It also appears that in the ℓ range of interest, W^{XY}(ℓ) ≥ 0; therefore , analogous to what is usually done for simple (circular) beam models. In what follows, the XY pair superscript is dropped except when they are required for clarity. In most cases, scientific analyses will be conducted on a best guess C_{est}(ℓ) of the sky power spectrum, in which the empirical C_{obs}(ℓ) is corrected from a nominal effective window W_{eff,nom}(ℓ)(G.3)therefore, on average, (G.4)The ratio B_{eff,true}(ℓ) /B_{eff,nom}(ℓ) which determines the uncertainties on the angular power spectrum originating from our beam knowledge is studied using the planet transit MC simulations described in Sect. 5. The scanning beam map determined with the BSpline code described in Appendix C on each of these simulations is turned into an effective beam window function W_{i}(ℓ) for i = 1,2,...,n_{MC} (where n_{MC} = 100 is the number of MC simulations) using the Quickbeam formalism described in Appendix E.2.
Defining the means B_{mean}(ℓ) and W_{mean}(ℓ) as (G.5)and, (G.6)one can build the matrix of the deviations around the mean (G.7)where the factor f_{s} has been applied. As discussed in Sect. 6, f_{s} = 2.7. This scaling factor is applied throughout the rest of discussion and is included in the delivered products and plotted error modes.
Sect. G.3 contains a discussion of how the MC average B_{mean} and nominal beam B_{eff,nom} are related and focus here on the dispersion around the mean.
Since the relative dispersion of the simulated W_{i}(ℓ) generally is very small (less than 1%), then W_{mean}(ℓ) ≃ B_{mean}(ℓ)^{2} to a very good approximation (as illustrated in Fig. G.1) and the matrix Δ is well approximated by (G.8)The quantity B(ℓ) was preferred over W(ℓ) in order to remain consistent with the usual description of the beam in linear map space, instead of quadratic space.
Fig. G.3 Distribution of the eigenmodes determined from MC simulations, for all HFI detector sets, for the first n_{modes} = 5 modes, compared to a Gaussian distribution of zero mean and unit variance. 

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The statistical properties of the MC based beam window functions can be summarized in the covariance of the deviations Δ, defined as (G.9)where Δ is a matrix with n_{MC} rows and ℓ_{max} + 1 columns, and C is a square symmetric positive semidefinite matrix with ℓ_{max} + 1 rows and columns. It can be diagonalized into (G.10)where D is a diagonal matrix, with at most n_{MC} positive eigenvalues, and the eigenmodes matrix (G.11)of the beam uncertainty. Alternatively, one can perform a singular value decomposition (SVD) of Δ, which reads where M is an orthogonal n_{MC} × n_{MC} matrix (i.e., M^{T}·M = M·M^{T} = I_{nMC}), D is a diagonal matrix with n_{MC} nonnegative eigenvalues of decreasing amplitude, and V is a matrix with ℓ_{max} + 1 rows whose n_{MC} columns are orthonormal vectors (i.e., V^{T}V = I_{nMC}), with I_{nMC} being the identity matrix of size n_{MC} × n_{MC}. The diagonalization of the covariance matrix C (Eq. (G.10)) has a numerical complexity scaling like , while the SVD of Δ (Eq. (G.12)) scales like . Since n_{MC} ≪ ℓ_{max} the latter approach was prefered because it is much faster, and it naturally provides the mixing matrix M. Equation (G.12) has a degeneracy on the sign of M and V, which was lifted by constraining the vectors of V (and therefore E) to all be positive at ℓ = 200.
It appears that most of the statistical content of Δ or C is described by the first few modes n_{modes} with the largest eigenvalues, in which case the E matrix is truncated to its first n_{modes} rows with little loss of information. For HFI, n_{modes} = 5 is chosen, as illustrated in Fig. G.1.
The statistics of the elements of the mixing matrix M, and therefore of the MC measured beam window function fluctuations, is shown in Fig. G.3 to be very close to Gaussian, justifying the current analysis in terms of a covariance matrix.
The beam uncertainty model therefore is where g is a n_{modes} element vector of independant Gaussian variates of zero mean and unit variance and e_{k}(ℓ) is the kth row of E.
The SVD decomposition of the beam uncertainty was performed for the ℓ range [ ℓ_{min},ℓ_{max} ] with ℓ_{min} = 0 and ℓ_{max} depending on the frequencies of the two detectors involved in the beam considered. Currently ℓ_{max} = 2500 when the lowest frequency is 100 GHz, ℓ_{max} = 3000 when that frequency is 143 GHz, and ℓ_{max} = 4000 at higher frequency.
If orthogonal error eigenmodes are needed for the range , with and , the provided E must first be truncated to the new range to give the E_{t} matrix with n_{modes} rows and columns, and then singular value decomposed into (G.16)where the new set of orthogonal eigenmodes is (G.17)This is illustrated in Fig. G.2, where the eigenmodes are truncated to the frequency dependent ℓ ranges used in the PlanckC(ℓ) likelihood (Planck Collaboration XV 2014).
Appendix G.2: Eigenmode covariance
The previous approach, dedicated to the study of the uncertainty on the beam window function associated with a single pair of detectors (or maps), can be generalized to the simultaneous characterization of any number of pairs. For instance, for the three disjoint pairs, a = { UV }, b = { XY } and c = { ZT }, one writes (G.18)and the covariance matrix reads (G.19)where M^{a,b,c} is a square symmetric matrix with 3n_{modes} rows, and if one denotes L^{a,b,c} its Cholesky “root,” such that M^{a,b,c} = L^{a,b,c}·L^{a,b,cT}, then where g is the 3n_{modes} element vector of independent Gaussian deviates and is the same for Eqs. (G.20) to (G.22).
The PlanckHFI Reduced Instrument Model (RIMO) available at Planck Legacy Archive^{8} and described in Planck Collaboration (2013) contains the correlation matrix M^{a,b,c,d,...}, where a,b,c,d... each are a different element of the set of pairs that can be built out of the detection units available. So, for n_{d} detection units, the number of pairs will be n_{d}(n_{d} + 1)/2 and the correlation matrix will be square and symmetric, with the value “1” on its diagonal and have n_{modes}n_{d}(n_{d} + 1)/2 rows and as many columns. The n_{modes} relative to the same detector pair form adjacent rows and columns in the matrix, and the order of appearance of the pairs in the matrix is specified in the header of the FITS extension containing the matrix. The nominal B_{ℓ} and eigenmodes E are provided for each pair in a different extension.
The results above were obtained in the basis of eigenmodes provided, if one wants to obtain a beam correlated model on a different ℓrange, with the eigenmodes E′ defined in Eq. (G.17), then the covariance matrix becomes (G.23)with (G.24)where the R′ matrices are obtained from the SVD in Eq. (G.16).
Appendix G.3: Monte Carlo bias
As discussed in Sect. 3.2, the Monte Carlo average of the BSpline reconstructed effective beam window function B_{mean}(ℓ) can be different from the effective beam that would be obtained directly out of the simulation input beam map B_{eff,in}(ℓ), introduces a bias (G.25)which is interpreted as a consequence of the imperfect sampling of the beam map by the planet, the effect of the instrumental noise and pointing uncertainty and the smoothing feature of the Bspline functions. It is found that ε_{bias}(ℓ = 500) ≤ 2 × 10^{4} and ε_{bias}(ℓ = 1000) < 5 × 10^{4} for all effective beams, making it smaller than the relative dispersion on the beam window function described above. It is assumed that the beam reconstruction on the real data suffers from the same bias, and its beam window function B_{eff,raw}(ℓ) is corrected for this to provide the nominal beam In doing so, ε_{bias} is assumed to be perfectly determined by the simulations, and no error is associated with this correction.
All Tables
Observation seasons of the planets observed by Planck: date range and position in Galactic coordinates.
Scanning beam solid angle (Ω_{SB}) error budget, showing the bias and fractional error due to the residual time response (ΔΩ_{τ}), near sidelobes (ΔΩ_{NSL}) and solid angle colour correction (ΔΩ_{CC}).
Summary of Bspline knot spacing and smoothing criterion weight p used in the beam reconstruction.
All Figures
Fig. 1 Data flow for fitting time response model parameters. 

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In the text 
Fig. 2 Data flow for determining the HFI beams. 

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In the text 
Fig. 3 Survey 2 minus Survey 1 residual close to the Galactic centre before (upper) and after (lower) fitting and deconvolving the low frequency part of the time response for bolometer 3532. Remaining residuals are dominated by gain difference between the surveys due to ADC nonlinearity (Planck Collaboration VIII 2014) and artifacts of the different scanning directions (beam asymmetry) and pixel coverage survey to survey. 

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In the text 
Fig. 4 68%, 95%, and 99% likelihood contours for the long time constant τ_{3} and associated amplitude a_{3} for a 545 GHz bolometer (5454) with (black) and without (red) zodiacal emission and far sidelobe removal. The square and cross indicate the maximum likelihood values. 

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In the text 
Fig. 5 Survey 1 minus survey 2 residual for the 545 GHz bolometers, averaged from Galactic longitude 0 through 20°. The black curves show the Planck data, and red is a simulation. 

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In the text 
Fig. 6 Gridded Jupiter data for bolometer 1433b before and after fitting for LFER4 parameters. The best fit Gaussian is subtracted from each plot to emphasize residuals. Residuals in the main beam show the deviation from a Gaussian shape, captured in the representations of the scanning beam, as described in Appendices B and C. 

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In the text 
Fig. 7 Phase and amplitude as a function of signal frequency of the deconvolution function of bolometer 2171. The solid black curve is the LFER4 model, while the dashed red curve shows the TF10 model used in the earlier Planck papers. The vertical dotted line marks the signal frequency corresponding to the half power point of the average effective beam. 

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In the text 
Fig. 8 Window functions of the planetary discs of Jupiter, Saturn, and Mars, equivalent to the bias in the inferred effective beam window function if the beam is reconstructed from observations of one of these planets alone. The labels show the corresponding angular radius of each disc. 

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In the text 
Fig. 9 Bspline scanning beams reconstructed from Mars, Saturn, and Jupiter seasons 1, 2 and 3 data for near sidelobe studies. The beams are plotted in logarithmic contours of − 3, − 10, − 20 and − 30 dB from the peak. PSB pairs are indicated with the a bolometer in black and the b bolometer in blue. 

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In the text 
Fig. 10 Azimuthally and bandaveraged main beam profiles (black solid curve) derived from the Bspline representation of the first and second Mars observations compared to that derived from a combination of Mars, Jupiter and Saturn observations (filled and open markers represent positive and negative data respectively). The red dashed line is defined as the joint envelope of the main beam and near sidelobe dataset, the integral of which represents the maximal solid angle that is compatible with these data. A nominal near lobe model, provided as a reasonable extrapolation of the data below the noise floor, is shown as the blue dashed line. The fractional increase in solid angle, relative to the Marsalone derived beam profile, is displayed in each panel. The black dotted line shows the GRASP physical optics model averaged over a subset of detectors that have been simulated (100–353 GHz). The data show a clear excess in power over the model at 143, 217 and 353 GHz that is consistent with a spectrum of surface errors on scales between 2 and 12 cm, with an RMS of order 10 μm. Table 2 contains an estimate of the fraction of the solid angle in the near sidelobes that is not captured in the Bspline representation. For clarity, the figure extends only to 45′. In all cases the solid angle is derived from the profile extending out to 5°. Due to the high signaltonoise of the Jupiter data (− 40 to − 55 dB, depending on the frequency), and the rapidly falling response of the beam, the solid angle estimates are insensitive to the limit of integration. 

Open with DEXTER  
In the text 
Fig. 11 One scanning beam at each HFI frequency (1003b, 1436, 2171, 3537, 5451, and 8573). Contours are in dB from the peak in steps of −5 dB. The lowest contours are set at −30 dB, −35 dB, −40 dB, −45 dB, −45 dB, −45 dB at 100 GHz, 143 GHz, 217 GHz, 353 GHz, 545 GHz, and 857 GHz, respectively. 

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In the text 
Fig. 12 Gridded data from all five seasons of Jupiter. The colour scale shows the absolute value of the peak signal. 

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In the text 
Fig. 13 A slice through stacked Jupiter data for bolometer 1436, illustrating residual long time response after deconvolution. The vertical dotted line shows the extent of the scanning beam map (plotted in blue). 

Open with DEXTER  
In the text 
Fig. 14 Azimuthally averaged profiles of measured beams of channel 3531 compared to the azimuthal average of the far sidelobe physical optics model. 

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In the text 
Fig. 15 Effective beam solid angle (upper panel) and the bestfit Gaussian ellipticity (lower panel) of the 100 GHz effective beam across the sky in Galactic coordinates. 

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In the text 
Fig. 16 Ratios of the crossbeam window functions to the product of the respective autobeam window functions B_{a}(ℓ)B_{b}(ℓ) for maps at 100, 143, 217 and 353 GHz, illustrating Eq. (16). 

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In the text 
Fig. 17 Effective beam window functions (solid lines) for each HFI frequency. The shaded region shows the ± 1σ error envelope. Dashed lines show the effective beam window function for Gaussian beams with FWHM 9.́65, 7.́25, 4.́99, 4.́82, 4.́68, and 4.́33 for 100, 143, 217, 353, 545 and 857 GHz, respectively. 

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In the text 
Fig. 18 Seasonal consistency of the beams at four HFI frequencies.Thick blue lines show the difference of the Mars season 1 and Mars season 2 window functions normalized to the bandaverage full window function. The grey shaded band is the ±1σ error for Mars 1 and the blue shaded band is the ±1σ error for Mars 2. 

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In the text 
Fig. 19 An estimate of known biases in the effective beam window function compared to the RMS statistical error including the additional factor 2.7 (grey shaded region) for each HFI frequency band. Green is the bias due to near sidelobes, blue is the colourcorrection bias, and magenta shows an upper limit on the effect of ADC nonlinearity. 

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In the text 
Fig. E.1 Difference between effective beam window functions computed with a real space method (FEBeCoP) and a harmonic space method (Quickbeam). The shaded region denotes the RMS error at each mutipole. 

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In the text 
Fig. F.1 Illustration of TOI samples near the Galactic plane (grey dots), overplotted on a simulated CMB realization which has been convolved with a Gaussian 7′ FWHM beam and pixelized at (N_{side} = 2048). Associated scanning rings (grey lines) as well as centres of mass for the hit distribution (black arrows) are also plotted. 

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In the text 
Fig. G.1 Beam window function error modes. Panels a), b) and c) show the first six eigenmodes defined in Eq. (G.11) for respectively the effective autobeam 100ds1 and 143ds2 and the effective crossbeam 2171 × 2172. The first five modes used in the current beam error modelling are shown as solid coloured curves, while the 6th mode (the first one to be ignored) appears as yellow dashes. The grey dashes show the ±1 σ envelope obtained by adding the first five modes in quadrature, while the solid grey curve is the ±1 σ envelope estimated from the simulations (therefore including all eigenmodes). The black dashes show the relative difference between and B_{mean}(ℓ) defined in Eqs. (G.6) and (G.5), respectively. In panel d), for a selection of effective beams, the coloured symbols show (d_{k}/d_{1})^{2} where d_{k} is the kth eigenvalue of the diagonal matrix D (Eq. (G.12)), while the coloured dashes show the error made on the quadratic sum of the eigenvalues by truncating it to n_{modes}: . The vertical dashes show the current n_{modes} = 5 value. 

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In the text 
Fig. G.2 Effect of ℓ truncation on beam error modes for frequencyaveraged beam window functions at 100, 143, 217 GHz and 143 × 217. For clarity, only the three leading modes are shown, respectively, in blue, green and red, while the solid grey line shows the 1 σ level, obtained by adding all modes in quadrature. Dotted lines are the original eigenmodes computed on a wide ℓrange. Solid lines are the eigenmodes computed directly from the MC simulations on the truncated ℓrange used for cosmological analysis. Dashed lines show the eigenmodes computed on the truncated ℓrange with Eqs. (G.16) and (G.17), starting from the first five eigenmodes for the wide range. In all four cases considered, the first leading mode on the truncated range, which dominates the error budget, is perfectly reconstructed out of the information available, while the second leading mode is well estimated in all cases except for 217 GHz. 

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In the text 
Fig. G.3 Distribution of the eigenmodes determined from MC simulations, for all HFI detector sets, for the first n_{modes} = 5 modes, compared to a Gaussian distribution of zero mean and unit variance. 

Open with DEXTER  
In the text 