Beam Window Functions

Beam window functions have computed with the Febecop Pipeline (as described there), and the QuickPol pipeline (see Hivon et al, 2017, and the Planck 2016 Likelihood paper).

The beam window function relates, over the full sky or over a masked sky, the angular power spectrum measured (in the absence of noise) on a map produced by a set of detectors [math]C^{XX}_\text{map}(\ell)[/math], to the true underlying sky angular power spectrum [math]C^{XX}_\text{sky}(\ell)[/math] (assumed to have isotropic statistical properties, as is the case for the CMB).

QuickPol effective beam window products

They are available in three forms:

Beam window functions

For unpolarized detectors (HFI SWB)

The temperature beam window function [math]b_{T}(\ell),[/math] is such that
[math]C^{TT}_\text{map}(\ell)\, = \, b_{T}^2(\ell) \, w_\text{pix}^2(\ell) \, C^{TT}_\text{sky}(\ell)[/math]
where [math]w_\text{pix}[/math] is the pixel window function, parameterized by the HEALPix resolution parameter [math]N_\text{side}[/math] (=2048 for Planck HFI maps).

For polarized detectors (HFI PSB)

The temperature and polarization beam window functions [math]b_{T}(\ell), b_{E}(\ell), b_{B}(\ell),[/math] are such that
[math]C^{XX}_\text{map}(\ell)\, = \, b_{X}^2(\ell) \, w_\text{pix}^2(\ell) \, C^{XX}_\text{sky}(\ell)[/math]
for X=T, E or B.

They are provided for each multipole [math] 0 \le \ell \le \ell_\text{max} = 4000[/math], in FITS format files compatible with HEALPix tools (such as synfast, smoothing, syn_alm_cxx, smoothing_cxx), as well as with PolSpice.

Beam matrices for polarized detectors

The beam matrices [math]W_{XY,X'Y'}(\ell)[/math] are such that
[math]C_{XY}^\text{map}(\ell) \, = \, \sum_{X',Y'} W_{XY,X'Y'}(\ell) \, w_\text{pix}^2(\ell) \, C_{X'Y'}^\text{sky}(\ell)[/math]
for X,Y,X',Y'= T, E or B, and where [math]w_\text{pix}[/math] is defined above.

They are provided in FITS files, containing 4 extensions each:

1. first one, named 'TT', contains the 9 fields: 'TT_2_TT', 'TT_2_EE', 'TT_2_BB', 'TT_2_TE', 'TT_2_TB', 'TT_2_EB', 'TT_2_ET', 'TT_2_BT', 'TT_2_BE'
   describing the ℓ-dependent leakage template of TT towards TT, EE, BB, ... respectively.
   TT_2_TT is the usual [math]W_{TT}(\ell) = b_T(\ell)^2,[/math] with [math]b_T(\ell=0)=1.[/math]
2. second extension, named 'EE', contains the 9 fields 'EE_2_TT', 'EE_2_EE', 'EE_2_BB', ... for leakage of EE towards TT, EE, BB, ...
   EE_2_EE is the usual [math]W_{EE}(\ell) = b_E(\ell)^2.[/math]
3. 3rd extension: 'BB' with 'BB_2_TT', ...
4. 4th extension: 'TE' with 'TE_2_TT', ...

Beware: there is no extension #5 nor 6, corresponding to TB and EB, since these terms are unlikely to be major sources of contamination for the other spectra.
They are provided for each multipole [math] 0 \le \ell \le \ell_\text{max} = 4000[/math].

The measured [math]C^*(\ell)[/math] are then related to the sky ones [math]C(\ell)[/math] via (ignoring the pixel window function [math]w_\text{pix}^2(\ell)[/math])
C^TT*(ℓ) = C^TT(ℓ) TT_2_TT(ℓ) + C^EE(ℓ) EE_2_TT(ℓ) + C^BB(ℓ) BB_2_TT(ℓ) + C^TE(ℓ) TE_2_TT(ℓ)
C^EE*(ℓ) = C^TT(ℓ) TT_2_EE(ℓ) + C^EE(ℓ) EE_2_EE(ℓ) + C^BB(ℓ) BB_2_EE(ℓ) + C^TE(ℓ) TE_2_EE(ℓ)
C^BB*(ℓ) = C^TT(ℓ) TT_2_BB(ℓ) + C^EE(ℓ) EE_2_BB(ℓ) + C^BB(ℓ) BB_2_BB(ℓ) + C^TE(ℓ) TE_2_BB(ℓ)
C^TE*(ℓ) = C^TT(ℓ) TT_2_TE(ℓ) + C^EE(ℓ) EE_2_TE(ℓ) + C^BB(ℓ) BB_2_TE(ℓ) + C^TE(ℓ) TE_2_TE(ℓ)
C^ET*(ℓ) = C^TT(ℓ) TT_2_ET(ℓ) + C^EE(ℓ) EE_2_ET(ℓ) + C^BB(ℓ) BB_2_ET(ℓ) + C^TE(ℓ) TE_2_ET(ℓ)

• To read these FITS file in IDL or python, see [[1]]