Formalism

The case of single detector

SBM addresses the map-making problem in spin space. By this approach, binnning of time-ordered data (TOD) will be replaced to a convolution of the map in spin space.

The signal field \(S\) (SignalFields) is defined as a function of the detector’s crossing angle \(\psi\) and the HWP angle \(\phi\). The real space scan field \(h\) is also a function of \(\Omega\), \(\psi\), and \(\phi\). The signal detected by a detector within a sky pixel of spherical coordinates \(\Omega=(\theta, \varphi)\) is given by

\[S^{d}(\Omega,\psi,\phi)=h(\Omega,\psi,\phi)S(\Omega,\psi,\phi).\]

Since the signal field is expanded to a two dimensional field given by \(\psi\) and \(\phi\), we consider corresponding scan field \(h\) as

\[h(\Omega,\psi, \phi)= \frac{4 \pi^{2}}{N_{\rm hits}(\Omega)}\sum_{j}\delta(\psi -\psi_{j})\delta(\phi-\phi_{j}).\]

Now we consider Fourier transform to bring the signal field to spin space. Defining \(n\) and \(m\) as the spin moment that is the variable conjugate to the angle \(\psi\) and \(\phi\), the transformation \((\psi,\phi)\to(n,m)\) is given by

\[\begin{split}S^{d}(\Omega,\psi,\phi) = \sum_{n,m}{}_{n,m}\tilde{S}^{d}(\Omega)e^{i n\psi}e^{i m\phi}, \\ {}_{n,m}\tilde{S}^{d}(\Omega) = \sum_{n'=-\infty}^{\infty}\sum_{m'=-\infty}^{\infty}{}_{\Delta n,\Delta m}\tilde{h}(\Omega){}_{n',m'}\tilde{S}(\Omega),\end{split}\]

where we introduce \(\Delta n = n-n'\) and \(\Delta m = m-m'\). \({}_{n,m}\tilde{S}^{d}\) can be obtained SignalFields.get_coupled_field(). And define the two dimensional orientation function, \({}_{\Delta n,\Delta m}\tilde{h}\) by Fourier transform of the real space scan field as

\[\begin{split}{}_{n,m}\tilde{h}(\Omega) &= \frac{1}{4\pi^{2}}\int d\psi \int d\phi h(\Omega,\psi,\phi)e^{-i n\psi}e^{-i m\phi} \\ &= \frac{1}{N_{\rm hits}}\sum_{j}e^{-i(n\psi_j + m \phi_j)}.\end{split}\]

This is waht we refer to as the cross-link which can be obtained by ScanFields.get_xlink().

The case of multiple detectors

Nowadays, the CMB experiment usually has multiple detectors about \(10^{3}\) to \(10^{4}\) to take statistics. Here we consider the implementation of multiple detectors in the map-making procedure. This can be simply described by modifying several quantities in the previous section. We introduce the detector index \(\mu\) and total number of detectors \(N_{\rm dets}\), then the total number of hits per pixel \(N_{\rm tot}\) is given by

\[N_{\rm tot}(\Omega) = \sum_{\mu}N_{\rm hits}^{(\mu)}(\Omega),\]

where \(N_{\rm hits}^{(\mu)}\) is the number of hits of the \(\mu^{\rm th}\) detector in the sky pixel \(\Omega\). The orientation function given by total number of observations, \({}_{\Delta n,\Delta m}\tilde{h}_{\rm tot}\), is

\[{}_{n,m}h_{\rm tot}(\Omega) = \frac{1}{N_{\rm tot}(\Omega)}\sum_{\mu}{}_{n,m}\tilde{h}^{(\mu)}(\Omega)N_{\rm hits}^{(\mu)}(\Omega).\]

Here, we define orthogonal pair detector which is named as \(\texttt{T}\) and \(\texttt{B}\) that stands for Top and Bottom detectors. These detectors observe the same direction though different crossing angle \(\psi\), let us denote the crossing angle of the \(\texttt{T}\) and \(\texttt{B}\) detectors as \(\psi^{\texttt{T}}\) and \(\psi^{\texttt{B}}\), respectively. Then, the orientation function of the \(\texttt{T}\) can be exchanged to that of the \(\texttt{B}\) by the following relation

\[{}_{n,m}\tilde{h}^{(\texttt{B})} = {}_{n,m}\tilde{h}^{(\texttt{T})}e^{i n \frac{\pi}{2}}.\]

The detected signal in spin space per detector is given by

\[{}_{n,m}{S^{d}}^{(\mu)}(\Omega) = \sum_{n'=-\infty}^{\infty}\sum_{m'=-\infty}^{\infty}{}_{n-n',m-m'}\tilde{h}^{(\mu)}(\Omega) {}_{n',m'}\tilde{S}^{(\mu)}(\Omega).\]