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arrangement was also constrained by the sky sampling requirements(Ade et al. 2010). Because of the focal surface curvature,the optical requirements summarised in Table 1 and theconstraints due to shadowing, two different horn designs wererequired per frequency band (100, 143 and 217 GHz channels)depending on their location within the focal plane. The high frequencychannels are located close to the centre of the focal planewhere aberration effects are less pronounced.A&A 520, A12 (2010)4. Horn design and beam optimisationSection 2 gives the reasons behind our detection assembly conceptand our decision to use cold optics based on corrugatedfeedhorns. The front horns are coupled directly to the telescope.They define the resolution, the antenna beam shape (main beamand some of the sidelobes) and the spillover of the instrument.Thus, a careful design and an accurate characterisation of eachfeedhorn are crucial. This section describes the theoretical modellinginvolved in the horn design leading to the focal instrumentshown in Figs. 4 and 5.Wealsopresentsomeresultsofthehornbeam measurement campaign together with an analysis of thehorn performances.Fig. 3. Section view of one of Planck-HFI back-to-back horns. Theclose-up on the waveguide shows the sequence of cylindrical sectionsof alternating radii r 0 and r 1 .4.1. Effects to take into considerationThe back and detector horns couple the radiation from the fronthorn to the detector through the spectral filters. It is essentiallytrue for the single-mode channels (100, 143, 217 and 353 GHz −CMB channels) that for a given frequency, the filters at this locationwill not impact the beam shape although they do affectthe detection assembly transmission. However, the coupling betweenthe back and detector horns as well as the filter transmissionwill affect the optical efficiency of the detection assemblynon-uniformly with frequency (Ade et al. 2010). Since the beamshape of the horn depends on the specific frequency within thespectral band of the channel (see Sect. 4.6), this frequency dependenttransmission will not only affect the sensitivity but alsothe overall beam shape of the detection assemblies (i.e. for thebroad-band beam the beam shape is integrated over the wholebandwidth of the channel).On the other hand, for the high frequency multi-mode horns(545 and 857 GHz dedicated to high frequency foregroundsremoval) the balance between the modes that can propagatethrough the waveguide filter determines the beam illuminatingthe telescope and thus the resolution, spillover and edge taper.This balance depends ultimately on how these modes couple tothe detector. The beam shapes will then be affected by the detailednature of the coupling to the detector cavity via the hornwaveguidefilter relay system, resulting in a beam modification.The beam will vary to a larger extent compared to single-modechannels. The filter transmission as well as the gap between theback-to-back horn and the detector horn, due to the filter stackthickness located in-between, will affect each mode transmissionto the detector in a different way.Fig. 4. Top:HFIfocalplanelayout.Emptycirclesrepresentthelocationof the front horn of each total intensity pixel. The crosses within thecircles give the axis orientation of the dual-polarised detectors a and bof the PSBs. Bottom:top-viewoftheflightmodel4Kfocalplanearray.4.2. Computation of corrugated horn beam patternsTwo main modelling techniques can be used for horn beam predictions.With the finite model analysis method, the horn and itssurrounding is divided into small cells and Maxwell equationsare solved within and at the boundaries of each cell. Softwarepackages such as Ansoft HFSS (www.ansoft.com)orCSTmicrowavestudio (www.cst.com) arebasedonthistechnique.While accurate, they are usually very time consuming to run andneed fairly high computation capabilities.After cross-checks between various modelling techniquesand validation with experimental data, we decided to use asecond technique which has been used for many years in the fieldof radio and microwave technology development: the modalmatchingtechnique.Page 4 of 15
B. Maffei et al.: Planck pre-launch status: HFI beam expectations from the optical optimisation of the focal planeFig. 6. Typical back-to-back horn beam pattern at 100 GHz. The 2 copolarisationbeams (model and experimental data) are shown (E and Hplanes) as well as the model of the cross-polarisation beam.Fig. 5. Side-view pictures of the 4 K cold plate showing both curvaturesof the focal surface.In the modal-matching model, a corrugated horn, such asthose employed by HFI, is regarded as a sequence of cylindricalwaveguide sections with alternating radii as shown in Fig. 3.For each horn segment the natural modes of propagationare the usual TE n,l and TM n,l modes of a cylindricalwaveguide (Olver et al. 1994; Murphy et al 2001; Colgan 2001;Gleeson 2004). For each value of n > 0andl, twoorthogonalmodal fields exist independently because of cylindrical symmetry.In the single-mode corrugated horn antennas, the waveguidesection is designed to filter out all but the TE 11 and TM 11 modesto propagate between the wider sections of the horn.The term single-mode is somewhat misleading, but can beunderstood within the context of the hybrid mode model. In thismodel, it is assumed that there are a sufficient number of corrugationsalong the guide walls to mimic a continuous surfaceimpedance. This assumption is valid if there are several corrugationsper wavelength and if there is no abrupt change inthe waveguide profile: four corrugations per free space wavelengthare usually sufficient (Clarricoats & Olver 1984). Withinthis framework, each hybrid mode propagating in the horn canbe considered as a combination of TE and TM scattering matrixmodes (Gleeson 2004; Noviello 2008). For instance, the combinationof the TE 11 and TM 11 scattering matrix modes is calledan HE 11 hybrid mode. The HE 11 fundamental hybrid mode isthen the only one actually propagating along the full length of asingle-mode horn, although other modes, such as the EH 11 ,willalso propagate, but will then subsequently get cut off (evanescentmode).Several channels of Planck-HFI are also dual-polarised (100,143, 217 and 353 GHz). In order to limit the cold optics crosspolarisation,the resulting beam needs to be the same for anypolarisation of the incoming radiation.Ahornradiationbeampatterngivesthevariationoftheelectromagneticfield amplitude (or intensity in this case, as bolometersare power sensitive detectors) as a function of the polar angleθ respectively to the boresight (horn axis of symmetry). φ isthe azimuthal angle relating to a cut of the beam. By convention,the E field is aligned along the axis for which φ = 0deg,whilethe H field is associated with φ = 90 deg. For this reason a cut ofthe beam in the φ = 0degplaneisreferredastheH-plane cut (orH plane) while a cut in the φ = 90 deg plane will be the E-planecut (E plane). The difference between these two cuts will givean indication of the cross-polarisation level of the horn. Figure 6shows the two cuts (both measured and simulated) and also thesimulated cross-polarisation beam of the horn.It has been shown that the easiest way to achieve lowlevel of cross-polarisation is to use single-mode corrugatedhorns selecting only the HE 11 fundamental hybrid mode(Clarricoats & Olver 1984).In the case of multi-mode horns, a larger number of higherorder modes above the TE 11 or the TM 11 modes can propagate.These give rise effectively to hybrid modes of order higherthan the HE 11 mode discussed above for the single-mode case.These hybrid fields can be derived using the same scattering matrixmode-matching techniques used for single-mode horns (exceptthat we include modes of all necessary azimuthal orders,n, notjustthosewithn = 1). The number of modes propagatingthrough the waveguide filters ultimately needs to be adjustedto produce the correct illumination beam on the telescope.These modes are independent of each other, so there is no phaserelationship between them. Thus, these modes will contributeindependently (i.e. incoherently) to the beam on the sky. In asingle-mode horn only one pair of orthogonal coherent fieldsmakes it through the waveguide filter to the horn aperture. Inamulti-modehornmanyindependentpairsoforthogonalcoherentfields will be present. Each of these has to be independentlypropagated through the Planck telescope to form the independentsum, taking into account a weighting factor for each modecorresponding to the coupling efficiency through the cold opticschain.4.3. Waveguide definitionAcircularcorrugatedwaveguideisdescribedbyitsradiusandby the depth of the corrugations giving two parameters r 0 andr 1 (Fig. 3). These two radii will define the cut-on frequency ofthe modes that can propagate. In the case of the single-modechannels, we want to transmit only the fundamental HE 11 hybridmode through the detection assembly, which has a verylow cross-polarisation and well defined Gaussian beam. On theother hand, in order to obtain a high sensitivity we need a largebandwidth. The maximum bandwidth with only the fundamentalmode transmitted is about 33%. Above this a second mode willbe transmitted compromising the polarisation purity. The highPage 5 of 15
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A&A 520, E1 (2010)DOI: 10.1051/0004
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ABSTRACTThe European Space Agency
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A&A 520, A1 (2010)Fig. 2. An artist
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A&A 520, A1 (2010)Table 4. Summary
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A&A 520, A1 (2010)three frequency c
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A&A 520, A1 (2010)flux limit of the
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A&A 520, A1 (2010)- University of C
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A&A 520, A1 (2010)57 Instituto de A
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A&A 520, A3 (2010)Ω ch 2τn s0.130
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3.3.1. SpecificationsThe main requi
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-30-40-30-6-3-20A&A 520, A7 (2010)0
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