# HFI cold optics

## Contents

- 1 Horns, lenses and filters
- 2 Spectral response
- 3 References

## Horns, lenses and filters[edit]

The cold optics is described in Planck-PreLaunch-I^{[1]}}.

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 below).
A detailed description of the HFI optical design and beam performances is given in Planck-PreLaunch-V^{[2]}.

### Horn design and patterns[edit]

The cosmological channels (100 GHz, 143 GHz, 217 GHz, 353 GHz) have been designed to detect a single mode mode with one or two polarizations, for Polarization Sensitive Bolometers or Spider Web Bolometers respectively. The spectral and geometrical properties of the horns have been characterized individually. A typical example for a single-moded horn is given in the figure below.

The fit is excellent down to very low levels, which validates, at least for single-moded horns, the logics that prevailed for ctheir development: Modelling and optimizing the horns before implementation and then 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 in order to estimate an upper limit of the spill over. As an example, in the figure below, we show the difference in encircled powers at isolevel intervals for the HFI 143_1a channel main beam using both the model and worst-case horns.

In the high frequency 545 GHz and 857 GHz Planck pixels both the back-to-back (BTB) horns and the detector horn have overmoded waveguide filters ^{[3]}. The far-field patterns of the horns (which illuminate the Planck mirrors) have been simulated and are shown in the two figures below for a few spot frequencies across the 857 GHz band. Note that the edge taper is approx -30 dB at 25 degrees as required at the centre of the band. Superimposed is the broadband measurement 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.

## Spectral response[edit]

The measurement of the spectral response of HFI is fully described in Planck-2013-IX^{[4]}. 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. 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 (Planck-2013-XIII^{[5]} and Planck-2013-XII^{[6]}. 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 and unit conversion / colour correction coefficients, Planck-2013-IX^{[4]} and software routines to generate additional unit conversion and colour correction coefficients (i.e., the Unit conversion and Color correction (UcCC) routines).

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 (see section HFI optical efficiency). 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[edit]

Two significant additional tests were used in the derivation of the HFI detector spectral transmission profiles beyond the scope of the integrated HFI FTS measurements. Additional filter measurements were recorded at component level during filter stack production. These measurements are used to extend the integrated HFI FTS spectral measurements beyond the HFI spectral passband. The other experiments, herein the EFF tests, are used to obtain optical efficiency parameters for each detector, and thus an estimate of the absolute spectral transmission. 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 in ^{[7]}. The filter measurements and the optical efficiency experiments are described below.

#### Filter measurements[edit]

Prior to the integrated HFI measurements with the integrated HFI detectors and filter stacks, FTS measurements of the individual filters comprising the filter stacks for each band were conducted. As discussed in this section, 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 integrated HFI 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. The figures below 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)."

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]

As described below, the composite filter spectra illustrated above form part of the official HFI detector spectral response, specifically for a portion of the out-of-band spectral region. The band-average spectra, containing this data as a constituent, is provided in the RIMO files (see here). The RIMO files will also include the detector level spectra for future Planck data releases.

#### Optical efficiency[edit]

The optical efficiency tests, hereafter the EFF tests, were conducted to allow an estimate of the overall optical efficiency of the HFI detectors. These tests involved exposing the HFI detectors to a known blackbody source and observing the response, and comparing this to a theoretical model in combination with the detector spectral response data. The optical efficiency parameter derived for each detector is meant as a multiplicative factor to be combined with the normalized transmission spectra (spectra whose maxima are set to one). This product is an estimate of the absolute spectral transmission of a given HFI detector. The detector optical efficiency parameters are included within the header of the detector spectra in the RIMO files (here).

Further details regarding the derivation of the optical efficiencies are provided here.

### Data processing and Fourier transformation[edit]

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:

- Selection and extraction of time sampled data sets
- Conversion of time sampled data to arrays of Optical Path Difference (OPD) sampled interferogram data sets
- Fourier transformation and averaging of interferogram data sets
- Discrimination of poor quality spectra by standard deviation comparison (see HFI spectral response data processing)
- Division of detector spectra by reference bolometer spectra to obtain normalized spectral transmission profiles
- Combination of relative transmission spectra with filter measurements and waveguide models
- Determination of optical efficiency through evaluation of normalized spectral transmission in the context of the EFF experiments
- Addition of over-sampled data into spectrum for the CO transition regions (see HFI CO response)
- Identification of common frequency sampling per channel and interpolation of spectra onto the common sampling

Further details regarding the interferogram processing are provided here.

### Reference bolometer spectra[edit]

The reference bolometer is assumed to be spectrally flat across each HFI band to within 1%. Its 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. The table included in HFI reference bolometer contains the reference bolometer data set properties corresponding to those listed for the HFI detectors in the table included in HFI spectral response data processing. Figures in HFI reference bolometer illustrate the resultant spectra and S/N from the reference bolometer data sets and compare the approximate S/N of the average spectrum for each detector against the reference bolometer average spectrum S/N over the same spectral region.

### Out-of-band spectral transmission content[edit]

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 integrated HFI 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 integrated HFI 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 (HFI detector spectrum divided by the reference bolometer spectrum) spectrum and filter are both scaled by the optical efficiency (see HFI optical efficiency). 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 integrated HFI FTS spectra and/or the 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 integrated HFI and filter spectra are used to determine the spectral cross-over/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. The figure illustrates an example of both the FTS and filter spectra used in extending the transmission profiles beyond the HFI optical bands.

### Waveguide model[edit]

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 . The table in HFI detector feedhorn model_parameters 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.

### CO interpolation[edit]

The integrated HFI 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 Instrument Line Shape(ILS) FWHM of [math]\sim0.020~\mbox{cm}^{-1}~(\sim0.61~\mbox{GHz})[/math].

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 the CO table). 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, [math]^{13}[/math]CO, C[math]^{17}[/math]O, and C[math]^{18}[/math]O. 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. The figure below 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 the figure below for the 100 GHz detectors.

Details of the HFI CO data products are available in the CO emission maps section.

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 |

**Regions of HFI spectral transmission profiles near CO transitions.**

### Spectral response conclusions and conformity with requirements[edit]

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 Planck-2013-IX^{[4]}.

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 Unit conversion and Color correction section.

Further details regarding the use of the spectral response of the HFI detectors are found in the spectral response section.

## References[edit]

- ↑
**Planck pre-launch status: The optical architecture of the HFI**, P. A. R. Ade, G. Savini, R. Sudiwala, C. Tucker, A. Catalano, S. Church, R. Colgan, F. X. Desert, E. Gleeson, W. C. Jones, J.-M. Lamarre, A. Lange, Y. Longval, B. Maffei, J. A. Murphy, F. Noviello, F. Pajot, J.-L. Puget, I. Ristorcelli, A. Woodcraft, V. Yurchenko, A&A,**520**, A11+, (2010). - ↑
**Planck pre-launch status: HFI beam expectations from the optical optimisation of the focal plane**, B. Maffei, F. Noviello, J. A. Murphy, P. A. R. Ade, J.-M. Lamarre, F. R. Bouchet, J. Brossard, A. Catalano, R. Colgan, R. Gispert, E. Gleeson, C. V. Haynes, W. C. Jones, A. E. Lange, Y. Longval, I. McAuley, F. Pajot, T. Peacocke, G. Pisano, J.-L. Puget, I. Ristorcelli, G. Savini, R. Sudiwala, R. J. Wylde, V. Yurchenko, A&A,**520**, A12+, (2010). - ↑
**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, International Journal of Infrared and Millimeter Waves,**21**, 2023-2033, (2000). - ↑
^{4.0}^{4.1}^{4.2}**Planck 2013 results: HFI spectral response**, Planck Collaboration 2013 IX, A&A, in press, (2014). - ↑
**Planck 2013 results: Galactic CO emission as seen by Planck**, Planck Collaboration XIII, A&A, in press, (2014). - ↑
**Planck 2013 results: Component separation**, Planck Collaboration XII, A&A, in press, (2014). - ↑
**Développement de modèles numériques del'Instrument Haute Fréquence (HFI) de Planck nécessaires à son exploitation**, A. Catalano, (PhD Thesis), École doctorale Astronomie et Astrophysique d'Ile de France, (2008).

(Planck) High Frequency Instrument

Back To Back HFI horns

Instrument Line Shape

Full-Width-at-Half-Maximum

Cosmic Microwave background

Planck Legacy Archive