Beam convolution with polarized beams

Giuseppe Puglisi, Davide Maino, Reijo Keskitalo - 10/20/2021

We report on the possible implementation of beam convolution within the libconviqt algorithm, in present of polarised sky and accounting for the polarisation properties of the beam. This is a severe issue in presence of an experiment using an half-wave plate (HWP) to modulate the polarization signal. Although this device is very useful to enlarge the sampling of the polarisation angles, it actually introduces a further complication for the beam convolution problem. Indeed the HWP produces a rotation of the beam which is no more fixed in the sky. Convolution methods working in harmonic space usually create data-cubes where the convolution result is stored for each $(\theta ,\varphi ,\psi )$$(\theta,\phi,\psi)$ position and orientation in the sky (with a suitable grids in the three dimensions). The actual value of the convolved signal for a generic pointing direction and orientation is obtained via interpolation of the data-cube. However, the HWP adds an extra-dimension of the data-cube and the required resolution and number of detectors of modern experiments make the problem almost unfeasible.

To overcome this we propose here an approach already proposed in the literature (Duivenvorden et al. 2018 ).

toast.ops.SimWeightedConviqt

This routine is the only routine to date implemented in TOAST that allows to perform beam convolution in presence of HWP modulator. See the TOAST operator here.

We stress that this routine is based on the following approximation :

We assume that the same perfect co-polar beam from total intensity is adopted also for the polarisation components of the beam, i.e. the $Q$$Q$ and $U$$U$ Stokes parameters of the beam

/!\ Notice that in this way we totally lose any information on the simulated co-polar components of the beam

The three beams, $I,Q$$I, Q$ and $U$$U$ are then expanded into harmonic coefficients and convolved with the input sky. The convolution can be schematized as three consecutive convolutions:

• convolution of the $I$$I$ beam with the unpolarised sky $a_{\ell m}$$a_{\ell m}$
• convolution of the perfect $Q$$Q$ beam with the polarised sky weighted by $\cos (2{\psi }_{\mathrm{pol}})$$\mathrm{cos}(2\psi_{\mathrm{pol}})$ where ${\psi }_{\mathrm{pol}}$$\psi_{\mathrm{pol}}$ accounts for the intrinsic polarisation orientation and for the HWP
• convolution of the perfect $U$$U$ beam with the polarised sky weighted by $\sin (2{\psi }_{\mathrm{pol}})$$\mathrm{sin}(2\psi_{\mathrm{pol}})$

We adopted this routine for the convolution runs performed for recent LiteBird beam simulations, i.e. $4\pi$$4\pi$, main beam, near and far side-lobes.

New procedure: toast.ops.SimTEBConviqt

We propose below a new procedure to perform the convolution in presence of the HWP.

CMB temperature and polarization (TQU) maps can be decomposed into scalar fields with the usual spherical harmonic functions $Y_{\ell m}(\hat{n})$$Y_{\ell m}(\hat{n})$ and the spin-2 ${}_{\pm 2}{Y}_{\ell m}(\hat{n})$${}_{\pm2}Y_{\ell m}(\hat{n})$on the celestial sphere, as

$T(\hat{n})=\sum _{\ell m}{a}_{\ell m}^T{Y}_{\ell m}(\hat{n})$$T(\hat{n}) = \sum_{\ell m} a^T_{\ell m} Y_{\ell m}(\hat{n})$

$ (Q\pm iU)(\hat{n}) = \sum_{\ell m}(a_{\ell m}^E \pm i a_{\ell m}^B)\,_{\pm 2}Y_{\ell m}(\hat{n}) $

Similarly we can decompose the beam function. We can have a pure total-intensity $T$$T$ beam given by the $I$$I$ Stokes parameter and a ``polarised beam'' combining $Q$$Q$ and $U$$U$ components of the beam in the same way: $b_{\ell m}^P=b_{\ell m}^E+i{b}_{\ell m}^B$$b_{\ell m}^P = b_{\ell m}^E+ib_{\ell m}^B$ .

The beam convolved polarization signal is therefore given by :

$\begin{array}{ccc}(\tilde{Q}\pm i\tilde{U})(\hat{n})& =& [\sum _{\ell ms}(b_{\ell s}^E+i{b}_{\ell s}^B)\cdot (a_{\ell m}^E+i{a}_{\ell m}^B)]\times (-1{)}^m{D}_{-m,s}^{\ell }\\ & =& [\sum _{\ell m}{b}_{\ell m}^P(a_{\ell m}^E\pm i{a}_{\ell m}^B)]\times (-1{)}^m{D}_{-m,s}^{\ell }\end{array}$$(\tilde{Q}\pm i\tilde{U})(\hat{n}) &=& \left[ \sum_{\ell m s}(b_{\ell s}^E+ib_{\ell s}^B)\cdot (a_{\ell m}^E + i a_{\ell m}^B) \right] \times (-1)^m D^{\ell}_{-m,s} \nonumber \\ & = &\left[ \sum_{\ell m}b_{\ell m}^P(a_{\ell m}^E \pm i a_{\ell m}^B)\, \right] \times (-1)^m D^{\ell}_{-m,s}$

With $D_{-m,s}^{\ell }$$D^{\ell}_{-m,s}$ being the $(2\ell +1)\times (2\ell +1)$$(2\ell +1)\times(2\ell+1)$ Wigner $D$$D$-matrices.

Following this reasoning, we can perform the convolution using three separated steps:

1. convolution for I: $b_{\ell m}^T\ast a_{\ell m}^T$$b_{\ell m}^T * a_{\ell m}^T$ ;
2. convolution for Q: $b_{\ell m}^P\ast a_{\ell m}^E$$b_{\ell m}^P * a_{\ell m}^E$
3. convolution for U: $b_{\ell m}^P\ast a_{\ell m}^B$$b_{\ell m}^P * a_{\ell m}^B$

This routine is currently implemented in TOAST here and there is an open pull request to be fully integrated within the package.

Validation results

To validate the outlined procedure, we performed a runs with TOAST2.3 for a LiteBIRD-like satellite observing at 100 GHz with 104 detectors. We perform the convolution by means of the two routines (described above) onto signal encoding only Galactic emission with the far side lobe component of beam (simulated with GRASP) extracted at different cuts in the $\theta$$\theta$ radial coordinate. We consider 3 side lobe cuts, i.e. ${\theta }_c=10,20,30$$\theta_c= 10,20,30$deg .

Figure1 Side lobe convolutions obtained with toast.ops.SimWeightedConviqt by considering 3 different cuts ${\theta }_c=10,20,30$$\theta_c= 10,20,30$deg going from left to right panels. (top) side-lobe pick up for intensity signal, (bottom) for polarization amplitude.

Figure2 Side lobe convolutions obtained with toast.ops.SimTEBConviqt by considering 3 different cuts ${\theta }_c=10,20,30$$\theta_c= 10,20,30$deg going from left to right panels. (top) side-lobe pick up for intensity signal, (bottom) for polarization amplitude.

Results are shown in Fig.1 and 2. Notice that the difference in the side-lobe pick up is clearly visible in polarized amplitude whereas for total intensity we don't notice any difference. This is clearly expected as the convolution with the two procedures in intensity are performed in the exact same way.

Moreover, the larger residuals in polarization obtained in Fig.1 are clearly due to the fact that the assumption of adopting the intensity beam for the polarized component of the signal results in an over estimation of the side-lobe pick up.