HFI cold optics & spectral response

(Lamarre)

Mechanical and thermal structure of the HFI Focal Plane Unit

Horns,lenses and filters

links to Peter's paper

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 response

(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[math]\%[/math]. 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 [math]2\pi[/math] 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.

Additional Experiments

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.

• Combined (bottom) and individual (top) filter transmission measurements for the 5 filters within the HFI band filter stacks (6 for 545 and 857 GHz).
• 

  100GHz

• 

  143GHz

• 

  217GHz

• 

  353GHz

• 

  545GHz

• 

  857GHz

The uncertainty on the combined filter transmission measurement is determined as follows. Let [math]f_i(\sigma)[/math] represent the individual filter transmission. The standard deviation of all values of [math]f_i(\sigma)[/math] below a threshold of [math]0.001[/math] is used as an approximate uncertainty for each individual filter measurement, i.e. [math]\varsigma_{i}[/math]. 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 [math]F(\sigma)[/math], the associated uncertainty estimate is given as

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

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 ([math]\sim[/math]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 [math]\sim 0.1~\mbox{cm}^{-1}[/math], 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.

Properties of HFI Detector Spectra

Band (GHz)	bc	# Ifgm.	# Spec.	MPD (cm)	ILS[math]_{\mbox{FWHM}}[/math] (cm[math]^{-1}[/math])	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.

Properties of Ref. Bolometer Spectra

Band (GHz)	# Ifgm.	# Spec.	MPD (cm)	ILS[math]_{\mbox{FWHM}}[/math] (cm[math]^{-1}[/math])	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

• Reference bolometer spectra and S/N.
• 

  100 - 217 GHz

• 

  353 - 857 GHz

• 

  all bands

• Reference bolometer spectra (top) and S/N (bottom) for each of the HFI bands (vertical bars represent band edges).
• 

  100GHz

• 

  143GHz

• 

  217GHz

• 

  353GHz

• 

  545GHz

• 

  857GHz

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 [math]\lambda^2[/math] 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, [math]\nu_{l}[/math], and [math]\nu_{u}[/math], 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, [math]W(\sigma)[/math], is given by the following relation

[math]\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.~,[/math]

where [math]\sigma[/math] is the frequency in cm[math]^{-1}[/math], [math]r_w[/math] is the waveguide radius in cm, [math]l_w[/math] is the waveguide length in cm, and [math]c_{nm}[/math] is a waveguide specific constant; 1.841 for the TE[math]_{11}[/math]/TM[math]_{11}[/math] (HE[math]_{11}[/math]) 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 [math]n[/math] bands, the uncertainty at each spectral data point is determined by the standard deviation of [math]n[/math] 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.

Waveguide model parameters for the HFI detectors.

Band (GHz)	bc	Det.	r[math]_{\mbox{w}}[/math] (mm)	r[math]_{\mbox{w}}[/math] (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 ([math]\sim[/math]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. [math]P_{\mbox{abs}}(T_i)[/math] where [math]T_i[/math] 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

[math]\label{eq:EFFdPabs} \Delta P_{\mbox{abs}} = P_{\mbox{abs}}(T_j) - P_{\mbox{abs}}(T_i) ~,[/math]

where [math]T_j[/math] and [math]T_i[/math] represent two unique source temperature settings. The theoretical incident power is determined using the HFI detector spectral transmission profiles. Let [math]\tau(\nu)[/math] 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 [math]\lambda^2[/math] throughput and then re-normalized as follows

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

where [math]\mbox{Norm}\left(f(x)\right)[/math] is the division of [math]f(x)[/math] by its maximum value, and [math]c[/math] 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, [math]P_{\mbox{th}}(T_i)[/math] , as follows

[math]\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\}} ~,[/math]

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

[math]\label{eq:EFFdPth} \Delta P_{\mbox{th}} = P_{\mbox{th}}(T_j) - P_{\mbox{th}}(T_i) ~,[/math]

which allows the optical efficiency term to be determined as follows

[math]\label{eq:EFFoptEff} \epsilon = \displaystyle \frac{\Delta P_{\mbox{abs}}}{\Delta P_{\mbox{th}}} ~.[/math]

Thus, if [math]\epsilon \tau'(\nu)[/math] were used in Equation [eq:EFFPth] in place of [math]\tau'(\nu)[/math], 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.

Mode Content for the HFI detector bands.

Band (GHz)	100	143	217	353	545	857
[math]n_m[/math]	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 [math]\sim0.017~\mbox{cm}^{-1}~(\sim0.5~\mbox{GHz})[/math] by the mechanical travel of the FTS translation stage. This corresponds to an unapodized FTS ILS FWHM of [math]\sim0.020~\mbox{cm}^{-1}~(\sim0.61~\mbox{GHz})[/math]. 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 [math]\sim[/math]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[math]^{13}[/math]O, CO[math]^{17}[/math], and CO[math]^{18}[/math]. 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]

Regions of HFI spectral transmission profiles near CO transitions.

Band (GHz)	CO transition ([math]J_{Upper}[/math] - [math]J_{Lower}[/math])	[math]\nu_o[/math] (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 [math]\pm[/math] 0.0036

Spectral transmission for CO transitions for HFI detectors.

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

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[math]^{-1}[/math] 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 [math]10^3[/math].

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[math]^{-1}[/math], the transmission accuracy requirement is not met for the 100 GHz detectors. It is possible to degrade the spectral resolution to the 0.1cm[math]^{-1}[/math] 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.].