HFI optical efficiency

The HFI optical efficiency 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 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 (about 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_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

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

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 τ(ν) represent the normalized detector transmission spectrum (i.e., it has been taken as a ratio and had the waveguide model and filter data appropriately attached). The spectral transmission is scaled for λ² 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 Norm[f(x)] 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_th(T_i), 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 h is the Planck constant, k is the Boltzmann constant, the integration limits are given by ν₁ and ν₂, and n_m is the expected mode content of the frequency band. The mode table lists the n_m values used for each band. In this case the integration is performed over the range ν = 67 to 1142 GHz. 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 ετ'(ν) were used in the power equation in place of τ'(ν), 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 optical efficiency test sequences.

Mode content for the HFI detector bands.

Band [GHz]	100	143	217	353	545	857
nm	1	1	1	1	3.4	8.3