SERVAL Methodology
Under Construction
The Visibility Integral
\[
\begin{align}
\mathcal{V}^{ij\nu}(\phi) = \int \diff\Omega~
A^{i\nu} A^{j\nu*}(\dirvec)~
\exp{\left(-2\pi i~\uvvec^{i-j}\cdot\dirvec\right)}
T^{\nu}(\dirvec, \phi)
\end{align}
\]
Where,
- \(A^{i\nu} A^{j\nu*}(\dirvec)\) is the primary beam term.
- \(\exp{\left(-2\pi i~\uvvec^{i-j}\cdot\dirvec\right)}\) is the baseline term
- \(T^{\nu}(\dirvec, \phi)\) is the sky term which rotates with \(\phi\) relative to the other terms.
We use \(i\) and \(j\) to indicate indices for individual antennas. \(nu\) designates the central frequency of a frequency channel and \(\uvvec\) is a vector
representation of the baseline in units of wavelength. The notation \(i-j\) indicates this is the baseline corresponding to the pair of antennas \(i\) and \(j\).
This may be shared with many pairs of antennas with difference indices. \(\dirvec\) indicates a unit vector on the sphere and the integral \(\diff\Omega\) is
carried out all such directions on the sphere although this is, in practice, often limited by a horizon term in the primary beam.
Spherical Harmonic Expansions
We will write functions on the sphere in terms of their (band-limited) spherical harmonics,
\[
\begin{align}
f(\dirvec) = \sum_{\ell=0}^{l_{\rm max}} \sum_{m=-\min{\left(m_{\rm max}, \ell\right)}}^{\min{\left(m_{\rm max}, \ell\right)}} a_{\ell m}Y_{\ell m}(\dirvec).
\end{align}
\]
Here \(\ell_{\rm max}\) and \(m_{\rm max}\) are imposed band-limits. The inverse transform is
\[
\begin{align}
a_{\ell m} = \int \diff\Omega~ f(\dirvec) Y^*_{\ell m}(\dirvec).
\end{align}
\]
SERVAL uses the orthonormal normalisation convention with the Condon-Shotley phase term for spherical harmonic expansions.
From orthonormality we then have,
\[
\begin{align}
\int \diff\Omega~ Y_{\ell_1 m_1}(\dirvec) Y^*_{\ell_2 m_2}(\dirvec) = \delta_{\ell_1\ell_2}\delta_{m_1m_2}.
\end{align}
\]
Additionally the symmetry relation,
\[
\begin{align}
Y^*_{\ell, m}(\dirvec) = (-1)^{m}Y_{\ell, -m}(\dirvec).
\end{align}
\]
This last relation can be used to show that, for real functions, the negative \(m\) harmonics can be constructed by conjugating the positive \(m\) harmonics. All but the sky terms that need to be computed in terms of their spherical harmonic expansions are potentially complex. Therefore, SERVAL does not make broad use of the \(\pm m\) symmetry in spherical harmonic expansions as is common in many codes.
We also make use of the spherical harmonic product rule,
\[
\begin{align}
&Y_{\ell_1 m_1}(\dirvec) Y_{\ell_2 m_2}(\dirvec) \\
&= \sum_{\ell m} \left(\frac{(2\ell_1+1)(2\ell_2+1)(2\ell+1)}{4 \pi}\right)^\frac{1}{2}
\begin{pmatrix}
\ell_1&\ell_2&\ell_3 \\
0&0&0
\end{pmatrix}
\begin{pmatrix}
\ell_1&\ell_2&\ell \\
m_1&m_2&m
\end{pmatrix}
Y^*_{\ell m}(\dirvec) \\
&= \sum_{\ell m} \mathcal{G}^{\ell_1~\ell_2~\ell}_{m_1 m_2 m} Y^*_{\ell m}(\dirvec).
\end{align}
\]
Here the bracketed terms are Wigner-3j symbols, this particular product of which corresponds to the Gaunt coefficients, \(\mathcal{G}\). We also summarise the \(\ell\) and \(m\) sum for brevity as will be done often in these notes.
Coordinate Systems and Rotations
Using Integral of Two Spherical Harmonics (Standard m-mode)
\[
\begin{align}
\mathcal{V}^{ij\nu}(\phi) &= \int d\Omega~
\left[A^{i\nu} A^{j\nu*}(\dirvec)~
\exp{\left(-2\pi i~\uvvec^{i-j}\cdot\dirvec\right)}\right]
\left[T^{\nu}(\dirvec, \phi)\right] \\
\mathcal{V}^{ij\nu}(\phi) &= \int d\Omega~
\left[\sum_{{\ell_{B} m_{B}}} B^{ij\nu}_{\ell_{B} m_{B}}~Y^*_{\ell_{B} m_{B}}(\dirvec_{\rm TIRS})\right]~
\left[\sum_{\ell_a m_a} a^{\nu}_{\ell_a m_a} Y_{\ell_a m_a}(\dirvec_{\rm CIRS})\right] \\
&=
\sum_{{\ell_{B} m_{B}} {\ell_a m_a}} B^{ij\nu}_{\ell_{B} m_{B}} a^{\nu}_{\ell_a m_a} \int d\Omega~Y^*_{\ell_{B} m_{B}}(\dirvec_{\rm TIRS})Y_{\ell_a m_a}(\dirvec_{\rm CIRS}) \\
&= \sum_{{\ell_{B} m_{B}} {\ell_a m_a}} B^{ij\nu}_{\ell_{B} m_{B}} a^{\nu}_{\ell_a m_a} \int d\Omega~Y^*_{\ell_{B} m_{B}}(\dirvec_{\rm TIRS})Y_{\ell_a m_a}(\dirvec_{\rm TIRS}) e^{im_a\phi} \\
&= \sum_{{\ell_{B} m_B} {\ell_a m_a}} B^{ij\nu}_{\ell_{B} m_{B}} a^{\nu}_{\ell_a m_a} \delta_{\ell_{B} \ell_{a}}\delta_{m_{B} m_{a}} e^{i m_a\phi} \\
&= \sum_{\ell m} B^{ij\nu}_{\ell m} a^{\nu}_{\ell m} e^{im\phi} \\
\mathcal{V}^{ij\nu}_{m^\prime} &= \int \frac{\diff\phi}{2\pi} \sum_{\ell m} B^{ij\nu}_{\ell m} a^{\nu}_{\ell m} e^{i(m - m^\prime)\phi} \\
&= \sum_{\ell m} B^{ij\nu}_{\ell m} a^{\nu}_{\ell m} \delta_{m m^\prime} \\
\mathcal{V}^{ij\nu}_m &= \sum_{\ell } B^{ij\nu}_{\ell m} a^{\nu}_{\ell m}\\
\end{align}
\]
\[
\begin{align}
\mathcal{V}^{ij\nu}(\phi) = \int d\Omega~
\left[A^{i\nu} A^{j\nu*}(\dirvec)\right]~
\left[\exp{\left(-2\pi i~\uvvec^{i-j}\cdot\dirvec\right)}\right]
\left[T^{\nu}(\dirvec, \phi)\right]
\end{align}
\]
Using Integral of Three Spherical Harmonics (SERVAL)
\[
\begin{align}
\mathcal{V}^{ij\nu}(\phi) &= \int d\Omega~
\left[\sum_{{\ell_{p} m_{p}}} p^{ij\nu}_{\ell_{p} m_{p}}~Y_{\ell_{p} m_{p}}(\dirvec_{\rm TIRS})\right]~
\left[\sum_{\ell_u m_u} u^{i-j,\nu}_{\ell_u m_u}~Y_{\ell_u m_u}(\dirvec_{\rm TIRS})\right]
\left[\sum_{\ell_a m_a} a^{\nu}_{\ell_a m_a} Y_{\ell_a m_a}(\dirvec_{\rm CIRS})\right] \\
&= \sum_{\ell_{pua}, m_{pua}} p^{ij\nu}_{\ell_{p} m_{p}} u^{i-j,\nu}_{\ell_u m_u} a^{\nu}_{\ell_a m_a} \int d\Omega~Y_{\ell_{p} m_{p}}(\dirvec_{\rm TIRS}) Y_{\ell_{u} m_{u}}(\dirvec_{\rm TIRS}) Y_{\ell_a m_a}(\dirvec_{\rm TIRS}) e^{i m_a \phi} \\
&= \sum_{\ell_{pua}, m_{pua}} p^{ij\nu}_{\ell_{p} m_{p}} u^{i-j,\nu}_{\ell_u m_u} a^{\nu}_{\ell_a m_a} e^{i m_a \phi} \mathcal{G}^{~\ell_p~~\ell_u~\ell_a}_{m_p~m_u~m_a} \\
\mathcal{V}^{ij\nu}_m &= \sum_{\ell_{pua}, m_{pu}} p^{ij\nu}_{\ell_{p} m_{p}} u^{i-j,\nu}_{\ell_u m_u} a^{\nu}_{\ell_a m}\mathcal{G}^{~\ell_p~~\ell_u~\ell_a}_{m_p~m_u~m} \\
\mathcal{V}^{ij\nu}_m &= \sum_{\ell_{p}, m_p} P^{i-j,\nu}_{l_p m_p m} p^{ij,\nu}_{\ell_{p} m_{p}} \\
\end{align}
\]
Using Integral of Four Spherical Harmonics (SERVAL Voltage Beams)
Handling Polarisation
Appendix: The Gaunt Coefficients
See more info on these mathematical objects at: DLMF 34.3
Appendix: Wigner 3j Symbols