# HFI cold optics

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## Cold optics

(Lamarre)

Mechanical and thermal structure of the HFI Focal Plane Unit

### Horns,lenses

#### Feed horns beams (sub-system level)

In order to meet with straylight, beam shapes and filtering requirements, a design using feedhorn coupled detectors has been chosen, with a triple horn configuration (see Figure 3.1.1). A detailed description of the HFI optical design and beam performances is given in Maffei et al.(2009) and Ade et al.(2009).

Optics of the HFI Focal plane unit. The back-to-back horn (front and back horns) is coupling the incoming radiation from the telescope to a detector horn which is then coupling the radiation to the bolometer. The filters determining the spectral bands are located in between the two horn assemblies. A lens is refocussing the radiation from the back horn to the detector

#### Single-moded horns patterns

The spectral and geometrical properties of the horns have been characterized individually. The measured beam pattern of a typical front horn is compared with the prediction from the design (figure 3.1.2). The fit is excellent down to very low levels, which validates the logics that prevailed for characterizing the horns: modelling and optimizing the horns before implementation. Validating the model with a complete measurement of the beam intensity patterns.

For the single-moded horns, a method has been developed to use the measured intensities, together with the phase information from modeling to derive “worst-case” horn beam patterns that can be used in GRASP simulations of the telescope beams (phase information is mandatory). Worst-case beam patterns have been computed for all single-moded HFI horn types. A .pdf file (IAS-FN-WCB-001-03022009) detailing the algorithm can be found on the optics ftp site at ctwg1.planck.fr, in the following directory: /File_Exchange_Box_GOPT/Files_from_FN/Worst_case_beams”

As an example, in Figure 3.1.3, we show the difference in encircled powers at constant isolevel intervals for the HFI 143_1a channel main beam using both the model and worst-case horns.

Typical measurement/theory comparison for the back-to-back horn co-pol beam pattern at 100GHz. Dots : model. Color : measurements (Off axis angle in degrees). Noise is limiting accuracy under -35 dB
Difference in power encircled within 3dB curves from peak value. Main beam encloses at least 99.5 % (model) and 99.4% (worst-case) of power

#### Ideal Multi-moded horns simulations

In the high frequency 545GHz and 857GHz Planck pixels both the back-to-back (BTB) horn and the detector horn have overmoded waveguide filters. The waveguide-horn structures are modelled using the scattering matrix approach used for the single-moded-CMB-channel horns, with the inclusion of modes of azimuthal orders, n = 0, 1, 2, 3 and 4. We assume reciprocity, and that the cavity plus cavity horn and filter behaves like a black-body: since the waveguide filter in the detector horn is wider than that of the BTB, it is reasonable to assume that any mode that can propagate through the BTB filter also propagates through the detector horn filter from the cavity. Thus, as far as the beam patterns are concerned the BTB is effectively coupled to a black body cavity also and all modes are excited at the back end of the BTB horn. Thus, all waveguide modes are equally excited in power but are also independent of each other so there is no phase relationship between them. Many of these modes may contribute independently (i.e. incoherently) to the beam on the sky. All coherent aperture fields have to be independently propagated through the PLANCK telescope.

From the S21 transmission scattering matrix for the whole BTB we recover the group of true independent hybrid fields that are transmitted by the waveguide filter and horn at the aperture of the BTB. These hybrid fields propagate independently through the telescope and onto the sky at a single frequency. The beam pattern for the band is obtained by adding the filter transmission weighted coherent fields in quadrature across the band (Microwave Horns and Feeds A. David Olver, Institution of Electrical Engineers) (“Shaped Corrugated horns for Cosmic Microwave Background Anisotropy Measurements” B. Maffei, P.A.R. Ade, C.E. Tucker, E. Wakui, R.J. Wylde, J.A. Murphy, R.M. Colgan, Int Jour Infrared & Millimeter waves, 21, (12) 2023-2033, December 2000).

Far field patterns of horns and comparison with test data:

The far-field patterns of the horns (which illuminate the Planck mirrors) have been simulated and are shown in figure 3.1.4a for a few spot frequencies across the 857 GHz band. Note that the edge taper is approx -30dB at 25 degrees as required at the centre of the band. Superimposed is the broadband measurement made at Cardiff (see below) which clearly looks narrower than the majority of the spot frequency measurements and requires explanation. The measured far field beam patterns across the band are narrower than the predicted far field beams right across the band. The simulated beams are too wide suggesting missing higher order modes, either due to attenuation between the cavity and the BTB, or to the experimental setup. Away from the band centre the theoretical beams also appear to be too wide, indicating missing modes (see figure 3.1.4b).

Grouping contributions according to azimuthal order and overlaying the measurement data to investigate missing field distributions, modes of order 2 and higher appear to be absent. The beam containing modes of azimuthal orders n = 0 and 1 appears to fit the measured beams reasonably. Adding modes with azimuthal order n = 2 gives a simulated beam that is too wide, suggesting that these modes are absent from the measured BTB front aperture field.

no legend provided
Model / Measurement comparison for the Far field pattern of the 857GHz_Horn ; left (a) : model at spot frequencies, superimposed with the broadband test data; right (b): measurements made at Cardiff for broadband and spot frequencies with the BTB horn both outside and inside the test dewar window (the inside case being more representative of the true Planck pixel)

### Spectral Characterization of the HFI Detector Focal Plane

(Spencer) This section is comprised of excerpts from the HFI - Spectral Calibration report of the IAS measurements, v3.02, an internal report describing detailed pre-flight ground based measurements of the HFI focal plane. The experimental setup, data collection, and related data processing are described. The official version of the HFI detector spectral transmission profiles is available within the HFI instrument model and the RIMO files in the Planck Legacy archive [ref.] . This data is comprised of broadband Fourier transform spectrometer (FTS) measurements conducted with the HFI focal plane assembly in a ground-based test cryostat, and includes a waveguide model for the low frequency spectral region, and component-level filter spectra for the remaining out of band spectral regions. Specific attention is given to in-band and near-band spectral regions surrounding CO rotational transitions in order to support the CO extraction component separation effort [ref.] . The spectral transmission profiles are evaluated with parameters such as cut-on, cut-off, centre frequency, effective frequency (including spectral index), and band-width, all provided in this analysis. Further evaluation yields band-average spectra ( [ref.] ) and unit conversion / colour correction coefficients ( [ref.] ) and software routines to generate additional unit conversion and colour correction coefficients ( [ref.] ).

The main goal of the spectral transmission tests of the HFI instrument is to measure the spectral response of all HFI detectors to a known source of EM radiation individually. This was determined by measuring the interferometric output of all detection channels for radiation propagated through a continuously scanned polarising Fourier transform spectrometer (FTS). The required accuracy to which the spectral transmission is to be recovered is 1$\%$. It is important to note that the absolute spectral calibration cannot be achieved solely from the analysis of the FTS data because of uncertainties in the coupling efficiency of the FTS source through the FTS, input optics, and integrating sphere. The relative FTS measurements must be combined with the optical efficiency tests which used internal black-body sources (EFF Test – see §EFF). A reference bolometer located in an intermediate integrating sphere (accepting $2\pi$ sr of incident radiation) within the FTS test setup was used to ratio the HFI detector spectra against to determine the throughput-normalized relative transmission spectra for each HFI detector. Data were collected over a series of pre-flight test campaigns, and processed/analyzed using standard Fourier data processing techniques.

There were two significant additional tests that are used in the derivation of the HFI detector spectral transmission profiles beyond the scope of the IAS HFI FTS measurements. Additional filter measurements were recorded in the Astronomical Instrumentation Group (AIG) test facility at Cardiff during filter stack production. These measurements are used to extend the IAS FTS spectral measurements beyond the HFI spectral passband. The EFF tests are used to obtain optical efficiency parameters for each detector. These parameters, when combined with the respective normalized spectral transmission profiles, provide an estimate of the absolute spectral transmission. The EFF tests are discussed in greater detail by A. Catalano ( [ref.] ). The filter measurements are described in §filter below.

#### Filter Measurements

Prior to the IAS measurements with the integrated HFI detectors and filter stacks, FTS measurements of the individual filters comprising the filter stacks for each band were conducted at Cardiff. As will be discussed in §[sec:OOB], the independent measure of the filter stack transmission is used for a portion of the HFI detector spectral transmission for regions of the spectrum where it is deemed to be of better quality than the IAS FTS measurements (i.e. for frequencies outside of the band edge filter cut-off(s)). The filter stacks for each of the frequency bands are comprised of 5 filters. There is an additional low frequency cut-on filter for the 545 and 857 GHz bands as the waveguide cut-on is too low for these multi-moded channels. Figures [fig:Filt100] – [fig:Filt857] show the individual filter transmission measurements as well as the combined filter stack product.

The uncertainty on the combined filter transmission measurement is determined as follows. Let $f_i(\sigma)$ represent the individual filter transmission. The standard deviation of all values of $f_i(\sigma)$ below a threshold of $0.001$ is used as an approximate uncertainty for each individual filter measurement, i.e. $\varsigma_{i}$. The individual uncertainty estimates are combined to provide an estimate of the combined filter transmission spectral uncertainty through standard error propagation. For the combined filter transmission represented as $F(\sigma)$, the associated uncertainty estimate is given as

$\label{eq:Ferror} \varsigma_F(\sigma) = \displaystyle\sqrt{[F(\sigma)]^2\Sigma_i [(\frac{\varsigma_i}{f_i(\sigma)})^2] } ~.$

#### Data Processing and Fourier Transformation

<1-- [sec:DP] -->

Having identified the data sections that are of interest, here follows the data processing sequence used to obtain the resulting spectra. The processing steps taken are as follows:

1. Selection and extraction of time sampled data sets
2. Conversion of time sampled data to arrays of OPD sampled interferogram data sets
3. Fourier transformation and averaging of interferogram data sets
4. Discrimination of poor quality spectra by standard deviation comparison (see §[sec:FTavg])
5. Division of detector spectra by reference bolometer spectra to obtain normalized spectral transmission profiles
6. Combination of relative transmission spectra with filter measurements and waveguide models
7. Determination of optical efficiency through evaluation of normalized spectral transmission in the context of the EFF experiments
8. Addition of over-sampled data into spectrum for the CO transition regions (see §[sec:CO])
9. Identification of common frequency sampling per channel and interpolation of spectra onto the common sampling

The bolometer signal is stored within the initial database in several formats: raw ADU, signal voltage, resistance, current, temperature, total power, electrical power, and radiant power, all of which may be exported as a function of sample time. A bolometer model was applied to the raw data to produce interferograms in units of absorbed optical power. This both reduced the in-band effects of detector nonlinearities, and allowed another comparison of optical efficiency to complement the EFF tests. The location of Zero Optical Path Difference (ZPD) is estimated for each interferogram, and the interferogram boundaries are determined as the mid-points between subsequent interferograms, less a small number of buffer points ($\sim$10) to ensure that the extracted interferograms include regions associated with the FTS stage travel having constant velocity while excluding the acceleration regions. Visual verification of the extracted interferograms is performed prior to subsequent processing and averaging to ensure that each ZPD was identified correctly, and to remove any low-quality interferograms. The overlap of the extracted interferograms is also verified visually. Once interferograms have been extracted from each data set, the overall spectral resolution is evaluated and an evenly sampled Optical Path Difference (OPD) grid onto which each interferogram in the combined data is then interpolated is generated. This ensures that each individual interferogram is sampled at ZPD and that each spectrum has identical frequency sampling and can thus be averaged together. An example of a combined interferogram data set is illustrated in Figure [fig:CombinedIFGMex] where the central portions of the recorded interferograms are shown. A similar plot for each HFI detector is included in the Appendices (see §[sec:allIFGMs]). The MPD value for each detector, and corresponding spectral resolution, is shown in Fig. [fig:MPD].

In preparation for Fourier transformation, a low-order polynomial baseline removal is performed on the individual interferograms. Consequently, no information can be recovered from the spectrum below $\sim 0.1~\mbox{cm}^{-1}$, but this is of no concern as this region of the spectrum is replaced by a waveguide fit in the final data product (see §[sec:WG]). An average interferogram is determined and used to identify glitches for removal from the interferogram data. Following glitch removal, individual interferograms undergo standard Fourier data processing. The modified Norton-Beer 1.5 apodization function ( [ref.] ) has been selected to be used for the final spectral transmission profile data set as it represents a good compromise between the desired ILS sidelobe reduction and improved S/N with marginal reduction in spectral resolution. Data averaging is then performed in the spectral domain. The uncertainty for every spectral data point is determined statistically through the standard deviation at a given frequency. A check for poor quality spectra is performed by comparing the overall standard deviation including and excluding any given spectrum. An example of the individual spectra and uncertainty for bc00 is shown in Fig. [fig:CombinedSpecEx], along with the corresponding S/N of the reference bolometer spectrum; similar plots for all of the detectors are shown in §[sec:allSpecs]. An estimate of the spectral S/N for each detector is obtained using the average spectrum and its statistical uncertainty, averaged across the in-band region of the spectrum. The averaged spectra are then normalized and divided by a (normalized) reference bolometer spectrum (see §[sec:setup] & §[sec:Bref]) to obtain the HFI detector relative spectral response.

 Band (GHz) bc # Ifgm. # Spec. MPD (cm) ILS$_{\mbox{FWHM}}$ (cm$^{-1}$) avg. S/N 100 00 96 93 29.648639 0.020355066 104.70555 100 01 96 95 29.648639 0.020355066 203.34226 100 20 96 96 29.649801 0.020354268 271.38262 100 21 95 94 29.650962 0.020353471 264.65385 100 40 96 94 29.652124 0.020352674 432.40921 100 41 96 96 29.652124 0.020352674 962.98064 100 80 95 95 29.650962 0.020353471 262.71036 100 81 96 96 29.650962 0.020353471 227.35782 143 02 96 95 29.652124 0.020352674 434.93136 143 03 96 96 29.652124 0.020352674 458.86741 143 10 95 95 29.652124 0.020352674 565.21311 143 30 92 92 29.653286 0.020351876 424.53824 143 31 92 92 29.652124 0.020352674 447.50343 143 42 92 92 29.652124 0.020352674 536.23423 143 50 96 96 29.652124 0.020352674 441.68983 143 51 96 96 29.652124 0.020352674 476.22684 143 60 96 96 29.652124 0.020352674 517.43269 143 70 95 94 29.652124 0.020352674 326.06542 143 82 96 96 29.652124 0.020352674 454.23293 143 83 96 96 29.652124 0.020352674 439.50612 217 04 96 96 29.652124 0.020352674 625.92713 217 11 96 96 29.652124 0.020352674 580.32904 217 12 67 67 29.663741 0.020344703 549.24264 217 22 96 96 29.652124 0.020352674 550.65242 217 43 95 95 29.652124 0.020352674 616.57750 217 44 96 96 29.652124 0.020352674 543.78288 217 52 95 94 29.652124 0.020352674 684.91039 217 61 92 92 29.652124 0.020352674 594.70296 217 62 95 95 29.652124 0.020352674 579.69385 217 71 96 96 29.652124 0.020352674 631.30065 217 72 96 96 29.652124 0.020352674 687.10844 217 84 92 92 29.652124 0.020352674 636.36790 353 05 55 55 29.651916 0.020352816 354.15815 353 13 57 57 29.651916 0.020352816 319.09858 353 23 55 55 29.651916 0.020352816 218.05772 353 24 64 64 29.651916 0.020352816 236.44464 353 32 56 56 29.651916 0.020352816 238.40018 353 33 55 55 29.651916 0.020352816 219.60443 353 45 64 64 29.653078 0.020352019 246.97889 353 53 57 57 29.651916 0.020352816 225.27621 353 54 57 57 29.651916 0.020352816 228.47058 353 63 57 56 29.651916 0.020352816 140.16991 353 64 55 55 29.651916 0.020352816 158.85037 353 85 57 57 29.653078 0.020352019 246.18460 545 14 63 63 29.654240 0.020351221 448.67624 545 34 64 64 29.654240 0.020351221 481.87111 545 55 57 57 29.654240 0.020351221 328.07835 545 73 57 57 29.654240 0.020351221 447.49901 857 25 64 64 29.653078 0.020352019 501.25106 857 35 64 64 29.654240 0.020351221 567.55125 857 65 42 42 29.653078 0.020352019 482.41906 857 74 64 64 29.654240 0.020351221 491.89133

#### Reference Bolometer Spectra

The reference bolometer spectra are obtained in a fashion similar to that used for the HFI detectors. Wherever possible, the same data processing is applied to the reference bolometer data as was applied to the HFI detector data, including optical filtering, scan speed, scan length, source intensity, apodization, phase correction, etc. Table [tab:Brefstats] contains the reference bolometer data set properties corresponding those listed for the HFI detectors in Table [tab:Igstats]. Figures [fig:BrefL] - [fig:Bref857] illustrate the resultant spectra and S/N from the reference bolometer data sets. Figure [fig:SNall] compares the approximate S/N of the average spectrum for each detector against the reference bolometer average spectrum S/N over the same spectral region.

 Band (GHz) # Ifgm. # Spec. MPD (cm) ILS$_{\mbox{FWHM}}$ (cm$^{-1}$) avg. S/N 100 164 164 29.644698 0.020357772 11.478057 143 164 164 29.644698 0.020357772 37.361317 217 164 164 29.644698 0.020357772 126.14280 353 24 24 29.652306 0.020352549 136.06159 545 24 24 29.652306 0.020352549 314.58093 857 24 24 29.652306 0.020352549 388.23050

#### Out-of-Band Spectral Transmission Content

The HFI detector spectral transmission profiles have been extended beyond the optical pass-band of the detectors. This is done by using a combination of a waveguide model and external filter measurements for the out-of-band regions of the detector spectral response. An uncertainty estimate for these additional spectral regions is also provided, however, it should be noted that the spectral uncertainty for the waveguide and filter spectra are determined indirectly (as described above/below). There is a transition from IAS FTS data to filter data for every band edge which is defined by an optical filter. For the 100, 143, 217, and 353 GHz bands this is the high frequency cut-off band edge. For the 545 and 857 GHz bands a separate filter is used to define each of the high and low frequency band edges. For the spectral regions outside of the HFI detector optical bands, first the IAS FTS data is used to qualitatively verify that there are no spectral leaks or features, and then the external filter measurements are grafted into the ratioed spectra where they better represent the relative spectral transmission.

The normalized ratioed spectrum and filter are both scaled by the optical efficiency (see §[sec:EFF]). For the 100 - 353 bands, the filter spectrum is also scaled by $\lambda^2$ to account for single-moded throughput normalization. This frequency scaling results in a more accurate in- and near-band match between the two sets of spectra, at the cost of less accuracy at much higher frequencies where any transmission will not be single-moded. As ransmission at higher frequencies is significantly reduced, this trade-off is acceptable. The 545 and 857 GHz bands have their filter spectra without the additional frequency scaling as this is not correct for multi-moded propagation. Both lower and upper frequency thresholds, $\nu_{l}$, and $\nu_{u}$, are defined for each band, below and above which the IAS-FTS spectra and/or the Cardiff composite filter spectra must be used, respectively. This is done to avoid introducing any detector nonlinearity residuals into the final spectral transmission data products. The region between these two points is defined as the transition region; within this region the amplitudes and slopes of the IAS and filter spectra are used to determine the spectral cross-oveer/transition point. Additional checks are performed to ensure that non-physical data processing artefacts are not introduced into the spectral transition region. Additionally, a similar technique is used, with decreasing frequency instead of increasing, for the 545 and 857 GHz bands with a filter-induced frequency cut-on. Figure [fig:OOBex] illustrates an example of both the FTS and filter spectra used in extending the transmission profiles beyond the HFI optical bands. Similar plots for every detector are shown in §[sec:stitch].

#### Waveguide Model

A waveguide model is used to provide the data for the lowest frequency portion of the HFI detector spectral transmission. For the 100, 143, 217, and 353 GHz bands the waveguide model is transitioned (with increasing frequency) to the FTS ratioed spectra directly. There is an intermediate transition to the filter data, and then the ratioed spectra for the 545 and 857 GHz bands. For each detector, the waveguide transmission, $W(\sigma)$, is given by the following relation

$\label{eq:WGmodel} W(\sigma) = \left\{ \begin{array}{lcl} \exp{\left[ -\displaystyle\left(\sqrt{(c_{nm}/r_{w})^2 - (2\pi\sigma)^2}\right) \left( l_w \right) \right]} & , & \mbox{ for }\sigma \leq c_{nm}/(2\pi r_w) \\ 1 & , & \mbox{ for }\sigma \gt c_{nm}/(2\pi r_w) \end{array}\right.~,$

where $\sigma$ is the frequency in cm$^{-1}$, $r_w$ is the waveguide radius in cm, $l_w$ is the waveguide length in cm, and $c_{nm}$ is a waveguide specific constant; 1.841 for the TE$_{11}$/TM$_{11}$ (HE$_{11}$) hybrid mode in this case . Table [tab:WG] lists the waveguide radii and lengths resultant from the waveguide model fit to the ratioed spectra. As all of the feedhorns for a given band are meant to be identical (i.e. within mechanical manufacturing tolerances) the uncertainty of the waveguide transmission is estimated statistically using all of the waveguide models for each band. I.e. for $n$ bands, the uncertainty at each spectral data point is determined by the standard deviation of $n$ transmission values at that frequency. As a result of each detector in a given band having a unique cut-on frequency, this method begins to over-estimate the uncertainty for frequencies approaching the cut-on; for regions very near the waveguide cut-on, the uncertainty is extrapolated from the ratioed spectrum as a more accurate representation.

 Band (GHz) bc Det. r$_{\mbox{w}}$ (mm) r$_{\mbox{w}}$ (mm) 100 00 1a 1.039705 1000 100 01 1b 1.038040 12.5813 100 20 2a 1.038375 14.3000 100 21 2b 1.041035 12.3750 100 40 3a 1.042370 12.9250 100 41 3b 1.042370 12.5125 100 80 4a 1.033050 13.2000 100 81 4b 1.033050 12.6500 143 02 1a 0.740345 9.4500 143 03 1b 0.737550 9.5625 143 10 5 0.736155 9.4500 143 30 2a 0.740345 9.4500 143 31 2b 0.739880 9.2250 143 42 6 0.741275 9.4500 143 50 3a 0.741275 9.4500 143 51 3b 0.741275 9.4500 143 60 7 0.729635 9.7875 143 70 8 0.739415 9.4500 143 82 4a 0.739415 9.4500 143 83 4b 0.738480 9.4500 217 04 1 0.4740750 7.8000 217 11 5a 0.4814250 7.8000 217 12 5b 0.4817310 8.0000 217 22 2 0.4749935 7.8000 217 43 6a 0.4832625 8.0000 217 44 6b 0.4832625 8.0000 217 52 3 0.4753000 8.0000 217 61 7a 0.4820375 8.0000 217 62 7b 0.4820375 8.0000 217 71 8a 0.4838750 8.0000 217 72 8b 0.4838750 1000 217 84 4 0.4740750 7.6000 353 05 1 0.2921850 6.47500 353 13 2 14300 7.00000 353 23 3a 0.2891650 7.70000 353 24 3b 3535 7.52500 353 32 4a 0.2869000 7.35000 353 33 4b 0.2870890 7.39375 353 45 7 0.2876550 6.12500 353 53 5a 0.2936950 7.35000 353 54 5b 0.2936950 7.61250 353 63 6a 0.2929400 6.73750 353 64 6b 0.2925625 6.62500 353 85 8 0.2889765 6.73750 545 14 1 1000 3.600 545 34 2 0.3865875 3.525 545 55 3 0.3800065 3.525 545 73 4 0.3787875 3.525 857 25 1 0.2985 2.4 857 35 2 0.2985 2.4 857 65 3 0.2985 2.4 857 74 4 0.2985 2.4

#### Optical Efficiency

The EFF tests involved exposing the HFI detectors to a known blackbody source and observing the response. Sufficient details for the HFI detector spectral transmission profiles are provided here while full details of the EFF experiments and results are provided in a separate technical report . A blackbody source internal to the Saturne cryostat was set to a variety of temperatures ($\sim$1 – 6 K) and the bolometer detector response was recorded. A bolometer model was applied to the recorded response in order to obtain the radiative optical power absorbed by the detector, in units of W, i.e. $P_{\mbox{abs}}(T_i)$ where $T_i$ represents the blackbody source temperature. Using the measured source temperature, the theoretical radiative optical power incident on the detector is also calculated using the Planck function. The ratio of the received power and the theoretical power provides the optical efficiency term. To remove any offsets in the measurement, a ratio of differences between unique temperature settings is used. The measured absorbed optical power difference is given by

$\label{eq:EFFdPabs} \Delta P_{\mbox{abs}} = P_{\mbox{abs}}(T_j) - P_{\mbox{abs}}(T_i) ~,$

where $T_j$ and $T_i$ represent two unique source temperature settings. The theoretical incident power is determined using the HFI detector spectral transmission profiles. Let $\tau(\nu)$ represent the normalized detector transmission spectrum (i.e. it has been ratioed and had the waveguide model and filter data appropriately grafted). The spectral transmission is scaled for $\lambda^2$ throughput and then re-normalized as follows

$\label{eq:EFFrenorm} \tau'(\nu) = \mbox{Norm}\left[ \tau(\nu) \left( \frac{\nu}{c}\right)^2 \right]~,$

where $\mbox{Norm}\left(f(x)\right)$ is the division of $f(x)$ by its maximum value, and $c$ is the speed of light. The normalized spectral transmission is then used with the Planck function at the temperature setting to determine the theoretical power, $P_{\mbox{th}}(T_i)$ , as follows

$\label{eq:EFFPth} P_{\mbox{th}}(T_i) = \displaystyle \int_{\nu_1}^{\nu_2}{ \left\{ \tau'(\nu) \left[ \displaystyle \frac{2 h \nu^3}{c^2 (\exp{(\frac{h\nu}{k T_i})} - 1)} \right] \left( \displaystyle \frac{c^2}{\nu^2}n_m \right) d\nu \right\}} ~,$

where $h$ is the Planck constant, $k$ is the Boltzmann constant, the integration limits are given by $\nu_1$ and $\nu_2$, and $n_m$ is the expected mode content of the frequency band. Table [tab:modes] lists the $n_m$ values used for each band . In this case the integration is performed over the range $\nu \in [67~\mbox{GHz},1142~\mbox{GHz}]$. The difference between the theoretical power loading is given by

$\label{eq:EFFdPth} \Delta P_{\mbox{th}} = P_{\mbox{th}}(T_j) - P_{\mbox{th}}(T_i) ~,$

which allows the optical efficiency term to be determined as follows

$\label{eq:EFFoptEff} \epsilon = \displaystyle \frac{\Delta P_{\mbox{abs}}}{\Delta P_{\mbox{th}}} ~.$

Thus, if $\epsilon \tau'(\nu)$ were used in Equation [eq:EFFPth] in place of $\tau'(\nu)$, the resultant optical efficiency would be unity, indicating that the transmission losses have already been taken into account.

The uncertainty estimate of the optical efficiency is statistically based on the results from the multiple temperature settings used in the EFF test sequences.

 Band (GHz) 100 143 217 353 545 857 $n_m$ 1 1 1 1 3.4 8.3

#### CO interpolation

The IAS FTS/Saturne data taken with the HFI detectors is limited to a spectral resolution of $\sim0.017~\mbox{cm}^{-1}~(\sim0.5~\mbox{GHz})$ by the mechanical travel of the FTS translation stage. This corresponds to an unapodized FTS ILS FWHM of $\sim0.020~\mbox{cm}^{-1}~(\sim0.61~\mbox{GHz})$. The previous iteration of the spectral transmission (IMO ver. 2_28), in addition to a triangular apodization kernel, has a factor of 3 sub-sampled spectral resolution to improve the signal-to-noise ratio (S/N) of the output spectra. In light of the CO contribution to the 100 GHz spectra, the spectral transmission has been reprocessed at the original full spectral resolution with a different apodization function which provides better noise reduction and less spectral resolution degradation, as discussed above.

In order to provide an improved estimate of the spectral transmission near the CO features, an interpolation of the spectra by a factor of $\sim$10 has been performed. This over-sampling was accomplished by zero-padding the FTS interferograms prior to Fourier transformation , and subsequently incorporated into the Nyquist-sampled spectral data near the CO transitions (see Table [tab:CO]). Although the data are presented at higher resolution, the resolution of independent data points is not improved, i.e. the ILS line-width remains the same. A flag column has been added to the spectral transmission profile data files to indicate whether a given data point originates from the actual data, or is a result of the ILS-based interpolation. The region of the over-sampled, interpolated, data has been extended to also include other CO isotopes; the COJ1-0 – J9-8 transitions should be oversampled for CO, C$^{13}$O, CO$^{17}$, and CO$^{18}$. The original data points within the over-sampled region have been preserved (i.e. every tenth data point – the data point that is not an interpolated one – is flagged with a zero rather than a one), as is indicated by the data flag, so a flag filter on the data will restore the independent data points easily. Figure [fig:COflag] illustrates the regions where the over-sampling has been incorporated into the spectral transmission profiles. An example of the over-sampled spectra is shown in Figure [fig:COzoom100] for the 100 GHz detectors. Examples for the other detector bands are shown in §[sec:COappend]

 Band (GHz) CO transition ($J_{Upper}$ - $J_{Lower}$) $\nu_o$ (GHz) over-sampled region (GHz) 100 1 – 0 115.2712018 109.67 – 115.39 143 1 – 0 115.2712018 109.67 – 115.39 143 2 – 1 230.5380000 219.34 – 230.77 217 2 – 1 230.5380000 219.34 – 230.77 353 3 – 2 345.7959899 329.00 – 346.15 545 4 – 3 461.0407682 438.64 – 461.51 545 5 – 4 576.2679305 548.28 – 576.85 857 6 – 5 691.4730763 657.89 – 692.17 857 7 – 6 806.6518060 767.48 – 807.46 857 8 – 7 921.7997000 877.04 – 922.73 857 9 – 8 1036.9123930 986.57 – 1037.95

353 || 05 || 1 || 3 – 2 || 0.4487 $\pm$ 0.0036
 BC Det. CO transition ($J_{\mbox{upper}} - J_{\mbox{lower}}$) Transmission 100 00 1a 1 – 0 0.1495 $\pm$ 0.0054 100 01 1b 1 – 0 0.1812 $\pm$ 0.0052 100 20 2a 1 – 0 0.2577 $\pm$ 0.0119 100 21 2b 1 – 0 0.1944 $\pm$ 0.0086 100 40 3a 1 – 0 0.2798 $\pm$ 0.0133 100 41 3b 1 – 0 0.1660 $\pm$ 0.0093 100 80 4a 1 – 0 0.2538 $\pm$ 0.0114 100 81 4b 1 – 0 2173 $\pm$ 0.0105 217 04 1 2 – 1 0.3781 $\pm$ 0.0026 217 11 5a 2 – 1 0.4530 $\pm$ 0.0029 217 12 5b 2 – 1 0.4174 $\pm$ 0.0030 217 22 2 2 – 1 0.3773 $\pm$ 0.0026 217 43 6a 2 – 1 0.3149 $\pm$ 0.0023 217 44 6b 2 – 1 0.3609 $\pm$ 0.0030 217 52 3 2 – 1 0.3924 $\pm$ 0.0025 217 61 7a 2 – 1 0.3428 $\pm$ 0.0022 217 62 7b 2 – 1 0.2770 $\pm$ 0.0017 217 71 8a 2 – 1 0.4623 $\pm$ 0.0031 217 72 8b 2 – 1 0.4340 $\pm$ 0.0031 217 84 4 2 – 1 0.3506 $\pm$ 0.0025 353 13 2 3 – 2 0.5461 $\pm$ 0.0044 353 23 3a 3 – 2 0.3443 $\pm$ 0.0030 353 24 3b 3 – 2 0.4706 $\pm$ 0.0037 353 32 4a 3 – 2 0.3099 $\pm$ 0.0024 353 33 4b 3 – 2 0.2801 $\pm$ 0.0027 353 45 7 3 – 2 0.2923 $\pm$ 0.0022 353 53 5a 3 – 2 0.3150 $\pm$ 0.0024 353 54 5b 3 – 2 0.3181 $\pm$ 0.0023 353 63 6a 3 – 2 0.2059 $\pm$ 0.0014 353 64 6b 3 – 2 0.2113 $\pm$ 0.0017 353 85 8 3 – 2 0.3509 $\pm$ 0.0028 545 14 1 4 – 3 0.0747 $\pm$ 0.0003 545 34 2 4 – 3 0.0731 $\pm$ 0.0003 545 55 3 4 – 3 0.0521 $\pm$ 0.0002 545 73 4 4 – 3 0.0473 $\pm$ 0.0002 545 14 1 5 – 4 0.3306 $\pm$ 0.0012 545 34 2 5 – 4 0.3183 $\pm$ 0.0011 545 55 3 5 – 4 0.2428 $\pm$ 0.0009 545 73 4 5 – 4 0.2597 $\pm$ 0.0009 857 25 1 6 – 5 0.0280 $\pm$ 0.0001 857 35 2 6 – 5 0.0241 $\pm$ 0.0001 857 65 3 6 – 5 0.0292 $\pm$ 0.0001 857 74 4 6 – 5 0.0159 $\pm$ 0.0001 857 25 1 7 – 6 0.1636 $\pm$ 0.0005 857 35 2 7 – 6 0.1427 $\pm$ 0.0004 857 65 3 7 – 6 2176 $\pm$ 0.0007 857 74 4 7 – 6 0.1168 $\pm$ 0.0003 857 25 1 8 – 7 0.2554 $\pm$ 0.0009 857 35 2 8 – 7 0.2218 $\pm$ 0.0007 857 65 3 8 – 7 0.2744 $\pm$ 0.0009 857 74 4 8 – 7 0.1119 $\pm$ 0.0004 857 25 1 9 – 8 0.0053 $\pm$ 0.0000 857 35 2 9 – 8 0.0060 $\pm$ 0.0000 857 65 3 9 – 8 0.0085 $\pm$ 0.0000 857 74 4 9 – 8 0.0001 $\pm$ 0.0000

#### Spectral Response Conclusions and Conformity With Requirements

The two defining requirements indicated in the HFI calibration plan are the acquisition of the spectral transmission of the single pixels with a prescribed accuracy and spectral resolution. The desired accuracy is 3% for the low frequency channels (CMB – 100, 143, and 217 GHz) and 1% for the high frequency channels (353, 545, 857 GHz). The spectral resolution requirement is for a resolution superior to 0.1 cm$^{-1}$.

The spectral resolution requirement has been exceeded by more than a factor of five. It is also possible to degrade the spectral resolution to the 0.1 cm$^{-1}$ requirement to gain an improvement in the S/N.

No quantitative number is present on the document regarding the blocking of high frequency (near IR, visible, UV) radiation outside the range of the instrument. Checks in order to quantify the rejection have been performed at subsystem level and estimates of the out-of-band transmission profiles have been incorporated into the data products. The high level of out-of-band signal attenuation is verified by in-flight observations as demonstrated in the Spectral Response paper ([ref.]).

Considering statistical fluctuations in the determination of the spectra, these goals have been achieved. There are, however, caveats regarding the nature of error bars when dealing with frequency space. The nature of uncertainties in spectra determination is less obvious than when dealing with timestream data. Systematic effects produced from instrumental setup, but also by data reduction can in some cases exceed the actual statistical oscillation in the determination of the final spectra. This is the case for the high-frequency data for instance where the statistical fluctuation of the different determinations of the spectra in some case are better than 1 part in $10^3$.

A second caveat regards the method of data analysis of the calibration test data, for which the ratio with the reference bolometer data (of which the relative error is a function of frequency) introduces an error that increase with wavelength. With the spectral resolution of the data provided being much higher than the stated 0.1 cm$^{-1}$, the transmission accuracy requirement is not met for the 100 GHz detectors. It is possible to degrade the spectral resolution to the 0.1cm$^{-1}$ level to allow the accuracy to achieve the required level, but the higher spectral resolution data has been provided to better assist with the CO contamination removal from the 100 GHz signal.

Figures containing the full spectral response of each HFI detector, and band-average spectra, are provided in the Appendix [ref.]. Details of the generation of band-average spectra, unit conversion coefficients, and colour correction coefficients are provided in the Data Processing sections of the Explanatory Supplement. IDL scripts have been provided alongside the PLA to allow users to generate unit conversion and colour correction coefficients; these are described in the PLA section [ref.].

(Planck) High Frequency Instrument

Back To Back HFI horns

Cosmic Microwave background

LFI Waveguide

Instrument Line Shape

Full-Width-at-Half-Maximum

Planck Legacy Archive