3.1. Quantifying geometric delay for FARSIDE

To see if the offsets of the two polarisations will have an effect on the uv-coverage and resulting Point Spread Functions (PSFs), we simulate visibilities using FARSIDE antenna positions (phase centres) of the spiral configuration, shown in Figure 3, The top panel of the Figure shows the complete extent of the array spanning from –6 to 6 km in the X and Y spatial directions. The two colours, black and green, indicate the two polarisations of the array. The bottom plot zooms into the array’s centre to highlight the offsets in the antenna positions. Following the notional design of FARSIDE discussed in Section 2, we take the offset between the orthogonal polarisations to be 50 m in both the X and Y directions. This is the minimum offset required to fit two dipoles, each with a half-arm length of 50 m, sequentially. For all the analyses in this paper, we assume the ideal offsets between the X and Y polarisation to be $\Delta x=50$ m and $\Delta y=50$ m.

The offset causes the sets of XY and YX baselines to fill different regions in visibility space compared to the corresponding XX and YY polarisations. Figure 5a and b show the simulated uv-coverages for XX/YY and XY/YX sets of baselines, respectively, at 2 MHz. Given the maximum baseline of the array is 12 km (Figure 4), the maximum uv-bin is $\pm$ 80 $\lambda$ (Figure 5a and b). The perturbed array (with offset) will fill in the uv-space better than the XX/YY case. This in turn results in a better PSF for the XY baselines as shown at the bottom of Figure 5e, where the sidelobes are lower with deeper nulls for the XY case baseline at zenith angles greater than the resolution of the array at each frequency.

Figure 4. Simulation of the dipole phase centres of the FARSIDE spiral arm array layout. Each arm has 32 pairs of dual-polarised dipoles. The green phase centres are offset from the black by 50 m in the X and Y directions. The top panel shows the top view of the complete layout spanning over 12 km in the X and Y extents. The bottom panel shows the inner $3\times3$ km of the layout and a closer look at the offsets between the X- and Y- dipoles in each antenna node.

Although offset polarisation effectively doubles the UV coverage density and improves cross-polarised PSF considerably, this model does not take into account the primary beam, which will affect the accuracy of polarisation measurements.

4.1. Review of the RIME formulation

Consider a simple interferometer with just two antennas represented by p and q, and each of these antennas has two orthogonal feeds (x and y) sensitive to the two polarisations of the incoming wave. The two orthogonal feeds of each antenna produce voltages proportional to the sum of the electric fields ( $e_{x}$ and $e_{y}$ ) for each polarisation at the location of the antenna. These voltages are products of the antenna properties and sky electric fields. Expressing the antenna properties in a Jones matrix ( $J_p$ ) for each antenna p, we can write the voltages ( $\overline{v}_p$ ) at a particular antenna as follows:

(1) \begin{equation} \overline{v}_p =\bigg[\begin{array}{c}v_{px} \\[-5pt]v_{py}\\\end{array}\bigg] = J_p \bigg[\begin{array}{c}e_{x} \\[-5pt]e_{y}\\\end{array}\bigg].\end{equation}

The beam of the dipole antenna has a pronounced gain pattern across the sky. Even an unpolarised sky appears polarised when the two polarisation feeds have unequal gains. In the case of offset phase centres, this polarised sky will also have additional phase offsets. Through simulation and analysis, we will show that these phase offsets will vary with frequency and antenna offset errors.

Figure 5. (a,b) Snapshot uv-coverage at 2 MHz of the four arm spiral array layout for zenith pointing. (a) uv-coverage for the XX and YY baselines and (b) shows uv-sampling for the XY baselines of the antenna pairs. (c,d) Normalised 2D Point Spread Functions (PSF) of the FARSIDE spiral arm layout with and without offset. (e) Azimuthally-averaged PSF versus elevation angle for the XX/YY and XY sets of baselines of the FARSIDE spiral arm layout plotted for three characteristic frequencies within the operating bandwidth.

The voltages from two antennas can be cross-correlated to produce a visibility matrix that depends on the sky coherency matrix (Born et al. Reference Born1999). For polarisation studies, as in our case, it is more convenient to express this in terms of the visibility vector and, equivalently, the coherency vector (Hamaker, Bregman, & Sault Reference Hamaker, Bregman and Sault1996). This is obtained by taking the outer product or the Kronecker product of the two input voltage vectors:

(2) \begin{align}V_{pq} & = <\overline{v}_p \otimes \overline{v}_q^* > \nonumber\\& =\bigg\langle J_p \bigg[\begin{array}{c} e_{x} \\[-3pt]e_{y}\\\end{array}\bigg] \otimes\bigg(J_q\bigg[\begin{array}{c} e_{x} \\[-3pt]e_{y}\\\end{array}\bigg]\bigg)^* \bigg\rangle , \end{align}

where $^*$ denotes element-by-element complex conjugate of the matrix and $<>$ indicates we calculate the average over time. Applying the property of the Kronecker product that holds true for any four matrices, $(A\otimes B) (C \otimes D) = AC \otimes BD$ , allowing us to write:

\begin{equation*}\begin{split}V_{pq} = \left\langle\begin{bmatrix}v_{px}v_{qx}^* \\v_{px}v_{qy}^* \\v_{py}v_{qx}^*\\v_{py}v_{qy}^*\end{bmatrix} \right\rangle = \big(J_p\otimes J_q^*\big) \left\langle\begin{bmatrix} e_{x}e_{x}^* \\e_{x}e_{y}^* \\e_{y}e_{x}^*\\e_{y}e_{y}^*\end{bmatrix} \right\rangle\end{split}.\end{equation*}

For a single unresolved source, if we take into account the phase delay between the antennas p and q, we can represent the above equation in terms of the source coherency vector $\langle[ e_{x}e_{x}^* \,\,e_{x}e_{y}^* \,\,e_{y}e_{x}^* \,\, e_{y}e_{y}^* ]^T\rangle = \mathcal{E}_{sky}$ since the source is spatially incoherent. For this, we need the baseline vector between the two antennas in Cartesian coordinates represented by u,v,w and as a function of wavelength;

\begin{equation*}\begin{split}\left\langle\left[V_{pq}\right] \right\rangle = \big(J_p\otimes J_q^*\big) \left\langle\begin{bmatrix}e_{x}e_{x}^* \\e_{x}e_{y}^* \\e_{y}e_{x}^*\\e_{y}e_{y}^*\end{bmatrix} \right\rangle \exp{({-}2\unicode{x03C0} i ( u l + v m + w n))},\end{split}\end{equation*}

where l, m, n are direction cosines that represent the coordinates of the source in the sky. To observe an extended region of the sky instead of a single source, we have to integrate the above equation over all the direction cosines of the sky (van Cittert Zernike theorem, Thompson, Moran, & Swenson 2001):

(3) \begin{equation}V_{pq} = \int \int \big(J_p\otimes J_q^*\big) \mathcal{E}_{{\mathrm{sky}}} \exp{({-}2\unicode{x03C0} i ( u l + v m + w n))} \frac{dl dm}{n} .\end{equation}

When imaging a small region ( $\theta < 30^\circ$ ) of the sky or making the ‘flat-sky’ approximation where w = 0 or where $l^2 + m^2 << 1$ and the n direction cosine ( $n=\sqrt{1-l^2 - m^2}$ ) evaluates to $\approx 1$ , the above equation simplifies to:

(4) \begin{equation}V_{pq} = \int \int \big(J_p\otimes J_q^*\big) \mathcal{E}_{{\mathrm{sky}}} \exp{({-}2\unicode{x03C0} i (u l + v m))} dl dm .\end{equation}

We note that unless $J_p$ is both diagonal and has, at any given point on the sphere, equal diagonal elements, there will be mixing or ‘leaking’ of different Stokes parameters together into each element of the visibility vector in a direction-dependent way (Geil, Gaensler, & Wyithe Reference Geil, Gaensler and Wyithe2011; Smirnov Reference Smirnov2011; Nunhokee et al. Reference Nunhokee2017; Asad et al. Reference Asad2016). We can expand $J_i$ into a product of matrices, each representing different antenna properties. In our analysis, we look at the beam and offset properties and analyse if and how these cause intermixing of the various polarisation components.

To conserve space, we will represent $\exp{({-}2\unicode{x03C0} i ( u l + v m))}$ as $K_{pq}$ .

4.2. Stokes polarimeter and Muller matrix

In all our calculations till now, we have represented the sky coherency vector in the Cartesian frame. The Stokes frame will give us the details of the source’s polarisation leaking into the polarisation components of the system. So, we will apply a coordinate transform to the above equation in the Cartesian system and transfer it to the Stokes system. We use the unitary transform matrix (S) on the linear operator J as: $S^{-1} \big(J_p\otimes J_q^*\big) S$ (Hamaker et al. Reference Hamaker, Bregman and Sault1996). The transformation matrix (S) used here (Equation 5) multiplied by a scaling factor of $\frac{1}{\sqrt{2}}$ is unitary such that $(\frac{1}{\sqrt{2}}S)^{-1} = \frac{1}{\sqrt{2}}S^\dagger$ ( $^\dagger$ represents the complex conjugate transpose). The term $S^{-1}(J_p \otimes J_q^*)S$ is the defined as the Muller matrix ( $M_{pq}$ ). It helps us quantify how the sky Stokes components are received by the XX, XY, YX, and YY components of the voltage vectors. And $\mathcal{E}^S_{\mathrm{sky}} = S^{-1}\mathcal{E}_{\mathrm{sky}}$ is the coherency vector of the sky in terms of the Stokes coordinate frame; $\mathcal{E}_{\mathrm{sky}}^S = (I\,Q\,U\,V)^T$ . We apply the coordinate transformation to Equation (4) using these definitions:

(5) \begin{equation}S = \begin{bmatrix}1 &\quad 1 &\quad 0 &\quad 0\\0 &\quad 0 &\quad 1 &\quad i \\0 &\quad 0 &\quad 1 &\quad -i \\1 &\quad -1 &\quad 0 &\quad 0 \\\end{bmatrix} \\[-10pt] ,\end{equation}

(6) \begin{align} S^{-1} V_{pq} = \int \int S^{-1}\big(J_p\otimes J_q^*\big)S \,\big(S^{-1}\mathcal{E}_{{\mathrm{sky}}}\big) K_{pq} dl dm, \nonumber\\\begin{bmatrix}1 &\quad 0 &\quad 0 &\quad 1\\1 &\quad 0 &\quad 0 &\quad -1 \\0 &\quad 1 &\quad 1 &\quad 0 \\0 &\quad -i &\quad i &\quad 0\\\end{bmatrix} \begin{bmatrix} v_{px}v_{qx}^* \\v_{px}v_{qy}^* \\v_{py}v_{qx}^*\\v_{py}v_{qy}^*\end{bmatrix} = \int \int S^{-1}\big(J_p\otimes J_q^*\big)S \,\mathcal{E}^s_{{\mathrm{sky}}} K_{pq} dl dm, \end{align}

(7) \begin{align}\begin{split}\begin{bmatrix}v_{px}v_{qx}^* + v_{py}v_{qy}^* \\ v_{px}v_{qx}^* - v_{py}v_{qy}^*\\ v_{px}v_{qy}^* + v_{py}v_{qx}^* \\ -iv_{px}v_{qy}^* +i v_{py}v_{qx}^* \end{bmatrix} = \begin{bmatrix}\mathcal{V}_I \\ \mathcal{V}_Q \\ \mathcal{V}_U \\ \mathcal{V}_V\end{bmatrix} &= \int \int S^{-1}\big(J_p\otimes J_q^*\big)S \,\mathcal{E}^s_{{\mathrm{sky}}} K_{pq} dl dm \\ & =\int \int M_{pq} \,\mathcal{E}^s_{{\mathrm{sky}}} K_{pq} dl dm \,. \end{split} \nonumber \\[-10pt] \end{align}

For example, to understand how all the Stokes components of the sky enter the measured or pseudo-Stokes ( $\mathcal{V}$ ), we look at just the first component of the $\mathcal{V}$ vector, which is

\begin{equation*}\begin{split} \mathcal{V}_I = \int \big(M00_{pq}I +M01_{pq}Q + & M02_{pq}U + M03_{pq}V\big) K_{pq} dl dm \,.\end{split}\end{equation*}

4.3. Methods of quantifying the polarisation performance of an array

We estimate and quantify the effects of the beam and dipole phase offsets by calculating the Muller matrices defined above. This calculation is carried out using Muller matrices for the representation of the Stokes leakages. The Muller matrices are calculated in the following manner:

First, we define the Jones matrix (J) for the effects of the array to be analysed. For the co-located case, we only use the direction-dependent beam of antenna node p. The electric beam patterns to determine $J = J_{beam}$ are obtained from the FEKO electromagnetic simulations. The simulation coordinates are represented by $\theta$ , which corresponds to the elevation angle with zero at zenith, and $\unicode{x03D5}$ , which corresponds to the azimuthal direction with zero along the excitation axis of the antenna.

\begin{align*}J_{\mathrm{beam}, p}(\hat s,\nu) = \begin{bmatrix}E_\theta^{px}(\hat s,\nu) &\quad E_{\unicode{x03D5}} ^{px} (\hat s,\nu)\\ E_\theta ^{py}(\hat s,\nu) &\quad E_{\unicode{x03D5}} ^{py}(\hat s,\nu)\end{bmatrix} ,\end{align*}

where $J_{beam}$ is a function of pointing direction $\hat s$ and frequency $\nu$ . As seen above, the beam Jones matrix is calculated using the electric fields of both the orthogonal dipoles. It is important to note that the orthogonal dipole beams are placed as two rows in the matrix as given in Equation (2).

For the offset phase of the array, in addition to $J_{beam}$ , we define another Jones matrix to capture the spatial offset between the phase centres. The offset between the orthogonal dipoles presents itself as an additional phase term that can be represented in the form of a Jones matrix:

\begin{align*} & \begin{bmatrix}\exp{({-}i \unicode{x03C8})} &\quad 0 \\ 0&\quad \exp{({-}i(\unicode{x03C8} + \Delta \unicode{x03C8}))} \end{bmatrix}\nonumber\\[5pt]& = \exp{({-}i \unicode{x03C8})} \begin{bmatrix}1 &\quad 0 \\0 &\quad \exp{({-}i \Delta \unicode{x03C8})} \end{bmatrix},\end{align*}

where $-i \unicode{x03C8} = -2\unicode{x03C0} i (u l + v m)$ is a function of ( $\hat s, \nu$ ) is taken into account by the $K_{pq}$ in Equation (4) and (6). So the Jones matrix due to the offset is given by:

(8) \begin{equation}J_\mathrm{offset} =\begin{bmatrix}1 &\quad 0 \\0&\quad \exp{({-}i \Delta \unicode{x03C8})} \end{bmatrix}, \end{equation}

where $\Delta \unicode{x03C8} = 2\unicode{x03C0} (\Delta u l + \Delta v m)$ captures the phase delay due to the Y-dipole being offset from the X-dipole. In this case, the total Jones matrix is now given as:

(9) \begin{equation}J = J_{\mathrm{offset}} \times J_{\mathrm{beam}}\,.\end{equation}

Next, we calculate the Muller matrices using the Jones matrix(ces) and the coordinate transform matrix S (Equation 6):

(10) \begin{equation} M_{pq} = S^{-1}(J_p \otimes J_q^*)S , \end{equation}

where we insert the appropriate J for the two cases: for the no-offset case, it would be just the $J_{Beam}$ ; for the offset phase case, the total J would be given by Equation (9).

Finally, we use Mueller matrix elements to calculate the fraction of Stokes sky ( $\mathcal{E}^S = [I,Q,U,V]$ ) captured by the instrumental Pseudo Stokes parameters

(11) \begin{equation}\begin{split} {\mathcal{V}_{I}} = M00_{pq}I + M01_{pq}Q + M02_{pq}U + M03_{pq}V ,\\{\mathcal{V}_Q} =M10_{pq}I + M11_{pq}Q + M12_{pq}U + M13_{pq}V ,\\{\mathcal{V}_{U}} = M20_{pq}I + M21_{pq}Q + M22_{pq}U + M23_{pq}V ,\\{\mathcal{V}_{V}} =M30_{pq}I + M31_{pq}Q + M32_{pq}U + M33_{pq}V\,. \end{split}\end{equation}

5.1. Co-located phase centres

The beam was simulated over a range of frequencies between 100 kHz and 40 MHz, and to quantify the polarisation leakages in detail, we selected one frequency in the band, 2 MHz, which is roughly at the multiplicative centre of the band and is also close to the resonant frequency of the dipole. With the generated electric field patterns, we now estimate the Stokes leakages by defining the Jones matrix, as shown in Step 1 of the methods section, and then using that to calculate the Muller matrix, as shown in Step 2. As a reference, the initial calculation is done for the case of co-located dipoles in the array, i.e. only the effect of the beam is considered. The Muller matrix evaluated using Equation (10) in Step2 will result in:

(12) \begin{equation}Jp \otimes Jq^* = \begin{bmatrix}E_{\theta}^{px} \overline{E_{\theta}^{qx}} & \quad E_{\theta}^{px} \overline{E_{\unicode{x03D5}}^{qx}} & \quad E_{\unicode{x03D5}}^{px} \overline{E_{\theta}^{qx}} & \quad E_{\unicode{x03D5}}^{px} \overline{E_{\unicode{x03D5}}^{qx}} \\[3pt]E_{\theta}^{px} \overline{E_{\theta}^{qy}} & \quad E_{\theta}^{px} \overline{E_{\unicode{x03D5}}^{qy}} & \quad E_{\unicode{x03D5}}^{px} \overline{E_{\theta}^{qy}} & \quad E_{\unicode{x03D5}}^{px} \overline{E_{\unicode{x03D5}}^{qy}} \\[3pt]E_{\theta}^{py} \overline{E_{\theta}^{qx}} & \quad E_{\theta}^{py} \overline{E_{\unicode{x03D5}}^{qx}}& \quad E_{\unicode{x03D5}}^{py} \overline{E_{\theta}^{qx}} & \quad E_{\unicode{x03D5}}^{py} \overline{E_{\unicode{x03D5}}^{qx}} \\[3pt]E_{\theta}^{py} \overline{E_{\theta}^{qy}} & \quad E_{\theta}^{py} \overline{E_{\unicode{x03D5}}^{qy}} & \quad E_{\unicode{x03D5}}^{py} \overline{E_{\theta}^{qy}} & \quad E_{\unicode{x03D5}}^{py} \overline{E_{\unicode{x03D5}}^{qy}}\end{bmatrix} .\end{equation}

We multiply this equation by the inverse of the unitary matrix, $S^{-1}$ :

(13) \begin{equation}S^{-1} (J_p \otimes J_q^*) = \frac{1}{2}\begin{bmatrix}E_{\theta}^{px} \overline{E_{\theta}^{qx}} +E_{\theta}^{py} \overline{E_{\theta}^{qy}}& \quad E_{\theta}^{px} \overline{E_{\unicode{x03D5}}^{qx}} +E_{\theta}^{py} \overline{E_{\unicode{x03D5}}^{qy}}& \quad E_{\unicode{x03D5}}^{px} \overline{E_{\theta}^{qx}} +E_{\unicode{x03D5}}^{py} \overline{E_{\theta}^{qy}}& \quad E_{\unicode{x03D5}}^{px} \overline{E_{\unicode{x03D5}}^{qx}} +E_{\theta}^{px} \overline{E_{\theta}^{qx}}\\E_{\theta}^{px} \overline{E_{\theta}^{qx}}-E_{\theta}^{py} \overline{E_{\theta}^{qy}}& \quad E_{\theta}^{px} \overline{E_{\unicode{x03D5}}^{qx}} -E_{\theta}^{py} \overline{E_{\unicode{x03D5}}^{qy}}& \quad E_{\unicode{x03D5}}^{px} \overline{E_{\theta}^{qx}} -E_{\unicode{x03D5}}^{py} \overline{E_{\theta}^{qy}}& \quad E_{\unicode{x03D5}}^{px} \overline{E_{\unicode{x03D5}}^{qx}} -E_{\unicode{x03D5}}^{py} \overline{E_{\unicode{x03D5}}^{qy}} \\E_{\theta}^{px} \overline{E_{\theta}^{qy}} +E_{\theta}^{py} \overline{E_{\theta}^{qx}}& \quad E_{\theta}^{px} \overline{E_{\unicode{x03D5}}^{qy}} +E_{\theta}^{py} \overline{E_{\unicode{x03D5}}^{qx}}& \quad E_{\unicode{x03D5}}^{px} \overline{E_{\theta}^{qy}} +E_{\unicode{x03D5}}^{py} \overline{E_{\theta}^{qx}}& \quad E_{\unicode{x03D5}}^{px} \overline{E_{\unicode{x03D5}}^{qy}} +E_{\unicode{x03D5}}^{py} \overline{E_{\unicode{x03D5}}^{qx}} \\-i E_{\theta}^{px} \overline{E_{\theta}^{qy}} +i E_{\theta}^{py} \overline{E_{\theta}^{qx}}& \quad -i E_{\theta}^{px} \overline{E_{\unicode{x03D5}}^{qy}} +i E_{\theta}^{py} \overline{E_{\unicode{x03D5}}^{qx}}& \quad -i E_{\unicode{x03D5}}^{px} \overline{E_{\theta}^{qy}} +i E_{\unicode{x03D5}}^{py} \overline{E_{\theta}^{qx}}& \quad -i E_{\unicode{x03D5}}^{px} \overline{E_{\unicode{x03D5}}^{qy}} +i E_{\unicode{x03D5}}^{py} \overline{E_{\unicode{x03D5}}^{qx}}\end{bmatrix}.\end{equation}

This is then multiplied by the unitary matrix S, so the total $S^{-1} (J_p \otimes J_q^*) S$ is:

(14) \begin{equation}{M_{pq} = \frac{1}{2} \begin{bmatrix}E_{\theta}^{px} \overline{E_{\theta}^{qx}} + E_{\theta}^{py} \overline{E_{\theta}^{qy}} + E_{\unicode{x03D5}}^{px} \overline{E_{\unicode{x03D5}}^{qx}}+ E_{\unicode{x03D5}}^{py} \overline{E_{\unicode{x03D5}}^{qy}}&E_{\theta}^{px} \overline{E_{\theta}^{qx}} + E_{\theta}^{py} \overline{E_{\theta}^{qy}} - E_{\unicode{x03D5}}^{px} \overline{E_{\unicode{x03D5}}^{qx}}- E_{\unicode{x03D5}}^{py} \overline{E_{\unicode{x03D5}}^{qy}}& E_{\theta}^{px} \overline{E_{\unicode{x03D5}}^{qx}} + E_{\theta}^{py} \overline{E_{\unicode{x03D5}}^{qy}}+ E_{\unicode{x03D5}}^{px} \overline{E_{\theta}^{qx}} + E_{\unicode{x03D5}}^{py} \overline{E_{\theta}^{qy}}& i E_{\theta}^{px} \overline{E_{\unicode{x03D5}}^{qx}} + i E_{\theta}^{py} \overline{E_{\unicode{x03D5}}^{qy}}-i E_{\unicode{x03D5}}^{px} \overline{E_{\theta}^{qx}} -i E_{\unicode{x03D5}}^{py} \overline{E_{\theta}^{qy}} \\E_{\theta}^{px} \overline{E_{\theta}^{qx}} - E_{\theta}^{py} \overline{E_{\theta}^{qy}} +E_{\unicode{x03D5}}^{px} \overline{E_{\unicode{x03D5}}^{qx}}- E_{\unicode{x03D5}}^{py} \overline{E_{\unicode{x03D5}}^{qy}} &E_{\theta}^{px} \overline{E_{\theta}^{qx}} - E_{\theta}^{py} \overline{E_{\theta}^{qy}} -E_{\unicode{x03D5}}^{px} \overline{E_{\unicode{x03D5}}^{qx}}+ E_{\unicode{x03D5}}^{py} \overline{E_{\unicode{x03D5}}^{qy}}& E_{\theta}^{px} \overline{E_{\unicode{x03D5}}^{qx}} - E_{\theta}^{py} \overline{E_{\unicode{x03D5}}^{qy}}+E_{\unicode{x03D5}}^{px} \overline{E_{\theta}^{qx}} - E_{\unicode{x03D5}}^{py} \overline{E_{\theta}^{qy}}& i E_{\theta}^{px} \overline{E_{\unicode{x03D5}}^{qx}} - i E_{\theta}^{py} \overline{E_{\unicode{x03D5}}^{qy}}-i E_{\unicode{x03D5}}^{px} \overline{E_{\theta}^{qx}} + i E_{\unicode{x03D5}}^{py} \overline{E_{\theta}^{qy}}\\E_{\theta}^{px} \overline{E_{\theta}^{qy}} + E_{\theta}^{py} \overline{E_{\theta}^{qx}}+ E_{\unicode{x03D5}}^{px} \overline{E_{\unicode{x03D5}}^{qy}} + E_{\unicode{x03D5}}^{py} \overline{E_{\unicode{x03D5}}^{qx}}&E_{\theta}^{px} \overline{E_{\theta}^{qy}} + E_{\theta}^{py} \overline{E_{\theta}^{qx}} -E_{\unicode{x03D5}}^{px} \overline{E_{\unicode{x03D5}}^{qy}} - E_{\unicode{x03D5}}^{py} \overline{E_{\unicode{x03D5}}^{qx}}& E_{\theta}^{px} \overline{E_{\unicode{x03D5}}^{qy}} + E_{\theta}^{py} \overline{E_{\unicode{x03D5}}^{qx}}+E_{\unicode{x03D5}}^{px} \overline{E_{\theta}^{qy}} + E_{\unicode{x03D5}}^{py} \overline{E_{\theta}^{qx}}& i E_{\theta}^{px} \overline{E_{\unicode{x03D5}}^{qy}} + i E_{\theta}^{py} \overline{E_{\unicode{x03D5}}^{qx}}-i E_{\unicode{x03D5}}^{px} \overline{E_{\theta}^{qy}} - i E_{\unicode{x03D5}}^{py} \overline{E_{\theta}^{qx}}\\-i E_{\theta}^{px} \overline{E_{\theta}^{qy}} + iE_{\theta}^{py} \overline{E_{\theta}^{qx}}-i E_{\unicode{x03D5}}^{px} \overline{E_{\unicode{x03D5}}^{qy}} +i E_{\unicode{x03D5}}^{py} \overline{E_{\unicode{x03D5}}^{qx}} &-i E_{\theta}^{px} \overline{E_{\theta}^{qy}} +i E_{\theta}^{py} \overline{E_{\theta}^{qx}}+i E_{\unicode{x03D5}}^{px} \overline{E_{\unicode{x03D5}}^{qy}} -i E_{\unicode{x03D5}}^{py} \overline{E_{\unicode{x03D5}}^{qx}}& -i E_{\theta}^{px} \overline{E_{\unicode{x03D5}}^{qy}} + i E_{\theta}^{py} \overline{E_{\unicode{x03D5}}^{qx}}-i E_{\unicode{x03D5}}^{px} \overline{E_{\theta}^{qy}} +i E_{\unicode{x03D5}}^{py} \overline{E_{\theta}^{qx}}&E_{\theta}^{px} \overline{E_{\unicode{x03D5}}^{qy}} -E_{\theta}^{py} \overline{E_{\unicode{x03D5}}^{qx}}- E_{\unicode{x03D5}}^{px} \overline{E_{\theta}^{qy}} + E_{\unicode{x03D5}}^{py} \overline{E_{\theta}^{qx}}\end{bmatrix}.}\end{equation}

The absolute values of the $M_{pq}$ at 2 MHz for the co-located case are plotted in Figure 7a to estimate the fractional leakages between the various Stokes components. The Muller matrices are projected in the $\theta/\unicode{x03D5}$ basis, and all of the dynamic ranges are normalised to the peak of $M_{00}$ , which is one at the zenith. The off-diagonal Muller matrix components would be zero for an ideal instrument with no polarisation leakage/mixing. The first column corresponds to the sky Stokes I coupling into the instrument’s all four Stokes components (I $\rightarrow \mathcal{V}^I, \mathcal{V}^Q,\mathcal{V}^U,\mathcal{V}^V$ ). Similarly, the second, third, and fourth columns capture the sky’s Stokes Q, U, and V components into all four pseudo-Stokes visibilities of the instrument. At low frequencies and large scales probed by many low-frequency interferometers, Stokes I of the sky is extremely bright compared to the other Stokes parameters. In early studies, only a few polarised point sources have been observed at frequencies below 300 MHz (Bernardi et al. Reference Bernardi2013; Asad et al. Reference Asad2016) and low-level diffused polarised emission, which peaks at 4K at 154 MHz (Lenc et al. Reference Lenc2016; Byrne et al. Reference Byrne2021). However, more recent papers by O’Sullivan et al. (Reference O’Sullivan2023), Van Eck et al. (Reference Van Eck2018) analysing the LOTS-DR2 LOFAR data (120–168 MHz) have detected more polarised point sources. But they also show that these are fewer in number compared to the NVSS survey (1.4 GHz) in a given sky area. In addition, O’Sullivan et al. (Reference O’Sullivan2023) find that the median degree of polarisation (1.8%) is about 3–10 times lower than those estimated with the NVSS survey. Applying a similar depolarisation scaling to FARSIDE frequencies, we can expect the degree of polarisation to be $< 1$ %. This is similar to the findings in Farnes et al. (Reference Farnes, Gaensler and Carretti2014), which showed evidence for systematic depolarisation of steep-spectrum point sources towards low frequencies, causing low polarisation fractions ( $\leq$ 1%) below 300 MHz. At FARSIDE frequencies, we would be primarily concerned about sky Stokes I leaking into other instrumental Stokes components. The Muller matrices indicate 20%, 4% and 0.075% fractional leakages of Stokes I into $\mathcal{V}_Q$ , $\mathcal{V}_U$ , and, $\mathcal{V}_V$ , respectively, for the co-located case.

Figure 7. Plots of Muller matrix elements for simulated co-located [top] and non-colocated [bottom] cross dipoles on regolith at 2 MHz as a function of elevation angle ( $\theta$ ) and azimuth angle ( $\unicode{x03D5}$ ). The absolute values of the elements are plotted. Colour bar scales are relative to the peak of M $_{00}$ (normalised to 1 at the zenith). The elements capture the fractional leakages of the sky Stokes components [I, Q, U, V] into the instrumental Psuedo-Stokes [ $\mathcal{V}_I$ , $\mathcal{V}_Q$ , $\mathcal{V}_U$ , $\mathcal{V}_V$ ]. For example, the first column corresponds to the sky Stokes I coupling into the instrument’s all four Stokes components ( $I \rightarrow \mathcal{V}^I, \mathcal{V}^Q,\mathcal{V}^U,\mathcal{V}^V$ ). See Equation (11) for a key to these matrices.

For the exoplanet science case, we are concerned about the Muller matrices $M_{30}$ , $M_{31}$ , and $M_{32}$ ; leakage of the sky components into the measured pseudo-Stokes $\mathcal{V}_V$ (bottom row of Figure 7). In this case, with no spatial offset and only direction-dependent beam effects, we find the leakage to be at least two orders magnitude lower compared to $M_{33}$ which is the instrumental pseudo-Stokes $\mathcal{V}_{V}$ capturing the sky Stokes V. For the hydrogen cosmology science case, we are interested in the sky Stokes components Q, U, and V leaking into the $\mathcal{V}_I$ . This is quantified by $M_{01}$ , $M_{02}$ , and $M_{03}$ and maximum percentages from each are 20%,7.5%, and 0.002%, respectively. So, the major contribution is from the Stokes Q leaking into the instrumental Stokes I. Due to the direction-dependent beam of the dipoles on the lunar regolith, we see considerable leakage of the polarised sky into the unpolarised power received.

5.2. Non-colocated phase centres

To study the polarisation leakages due to spatial offsets between the orthogonal dipole pairs of an antenna, we follow the same procedure as above and define a Jones matrix to account for the differential delay between the signal received by the X and Y dipoles (as shown in Section 4.3). The Muller matrix accounting for only this offset (excluding the antenna beam contribution) evaluates to:

(15) \begin{equation}J \otimes J^* = \left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c}1 & 0 & 0 & 0\\0 & \exp{(i \Delta \unicode{x03C8})} & 0 & 0 \\0 & 0 & \exp{({-}i \Delta \unicode{x03C8})} & 0 \\0 & 0 & 0 & 1\end{array}\right],\end{equation}

(16) \begin{equation}S^{-1}(J \otimes J^*) = \frac{1}{2}\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c}1 & 0 & 0 & 1\\1 & 0 & 0 & -1 \\0 &\exp{(i \Delta \unicode{x03C8})} & \exp{({-}i \Delta \unicode{x03C8})} & 0 \\0 &-i \exp{(i \Delta \unicode{x03C8})} & i \exp{({-}i \Delta \unicode{x03C8})} & 0\end{array}\right],\end{equation}

(17) \begin{equation}M_\mathrm{offset} = \left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c}1 & 0 & 0 & 0\\0 & 1 & 0 & 0 \\0 &0&\cos{\Delta \unicode{x03C8}} &-\sin{\Delta \unicode{x03C8}} \\0 &0& \sin{\Delta \unicode{x03C8}} & \cos{\Delta \unicode{x03C8}}\end{array}\right].\end{equation}

The case of only offset is seen to affect $ M_{22}, M_{23}, M_{32}\,\textrm{and}\,M_{33} $ , i.e. the coefficients that quantify the leakage from the sky Stokes U and V to the instrument’s Stokes $\mathcal{V}_U$ and $\mathcal{V}_V$ . This is seen in Equation (17) and Figure 8. In Figure 8, we plot the relevant Muller matrix coefficients for a few frequencies to assess how the offset effects change with frequency. Confirming what was obtained in the M $_{\textrm{offset}}$ equation, the $M_{22}$ and $M_{33}$ are the same and the $M_{23}$ and $M_{32}$ are the same in magnitude. As the frequency increases, the delay due to the offset, which depends on the ratio of the offset value to the observation wavelength, undergoes multiple cycles. In the case of 0.6 MHz (or 500 m wavelength), the offset is only 0.1 $\lambda$ in the u and v directions, and the exact number of cycles of variations between 0 and 1 can be understood by the total delay given as:

\begin{equation*} \cos\{\Delta \unicode{x03C8}\} = \cos \left\{2 \unicode{x03C0} \frac{50}{\lambda} (\sin \theta (\cos \unicode{x03D5} + \sin \unicode{x03D5}))\right\}.\end{equation*}

To obtain the number of cycles, we calculate the number of times $\cos\{\Delta \unicode{x03C8}\} $ evaluates to 1, i.e. what is the integer multiple of 2 $\unicode{x03C0}$ possible in the above equation for different wavelengths/frequencies. To do so, we need the $\theta$ and $\unicode{x03D5}$ at which the maximum value is possible, and substituting $\theta = 90^\circ$ and $\unicode{x03D5} = 45^\circ$ gives us that.

\begin{equation*}\cos\{\Delta \unicode{x03C8}\}_{\max} = \cos\left\{2 \unicode{x03C0}\frac{50}{\lambda} \sqrt{2}\right\}\,.\end{equation*}

Figure 8. Muller matrices that are non-zero when only the non-colocated phase centre effect is considered, excluding the beam pattern contribution. These indicate that direction-dependent intermixing of only Stokes U and V sky components occurs due to the spatial offset between the phase centres of the orthogonal dipoles. The effects are shown for three frequencies: 0.6, 2, and 10 MHz.

This implies it will have approximately 1/4 cycle for 0.6 MHz, one cycle for 2 MHz, and five cycles for 10 MHz.

Next, we combine the effects of offset and the direction-dependent beam so the joint Jones matrix is now defined as:

(18) \begin{equation}J_\mathrm{offset} \times J_\mathrm{beam} = \left[\begin{array}{c@{\quad}c}E_{\theta}^x & E_{\unicode{x03D5}}^x \\\exp{({-}i \Delta \unicode{x03C8})}E_{\theta}^y & \exp{({-}i \Delta \unicode{x03C8})} E_{\unicode{x03D5}}^y\end{array}\right].\end{equation}

Then the full $J \otimes J^*$ is calculated as above:

(19) \begin{align}&(J_\mathrm{offset} \times J_\mathrm{beam})\otimes (J_\mathrm{offset} \times J_\mathrm{beam})^* \nonumber\\[5pt]& = \left[\!\begin{array}{c@{\quad}c@{\quad}c@{\quad}c}E_{\theta}^x \overline{E_{\theta}^x}& E_{\theta}^x \overline{E_{\unicode{x03D5}}^x}& E_{\unicode{x03D5}}^x \overline{E_{\theta}^x}& E_{\unicode{x03D5}}^x \overline{E_{\unicode{x03D5}}^x} \\[4pt]\exp{(i \Delta \unicode{x03C8})} E_{\theta}^x \overline{E_{\theta}^y}& \exp{(i \Delta \unicode{x03C8})} E_{\theta}^x \overline{E_{\unicode{x03D5}}^y}& \exp{(i \Delta \unicode{x03C8})} E_{\unicode{x03D5}}^x\overline{E_{\theta}^y}& \exp{(i \Delta \unicode{x03C8})}E_{\unicode{x03D5}}^x \overline{E_{\unicode{x03D5}}^y} \\[4pt]\exp{({-}i \Delta \unicode{x03C8})}E_{\theta}^y \overline{E_{\theta}^x}& \exp{({-}i \Delta \unicode{x03C8})}E_{\theta}^y \overline{E_{\unicode{x03D5}}^x}& \exp{({-}i \Delta \unicode{x03C8})}E_{\unicode{x03D5}}^y \overline{E_{\theta}^x}& \exp{({-}i \Delta \unicode{x03C8})}E_{\unicode{x03D5}}^y \overline{E_{\unicode{x03D5}}^x} \\[4pt]E_{\theta}^y \overline{E_{\theta}^y}& E_{\theta}^y \overline{E_{\unicode{x03D5}}^y}& E_{\unicode{x03D5}}^y \overline{E_{\theta}^y}& E_{\unicode{x03D5}}^y \overline{E_{\unicode{x03D5}}^y} \end{array}\!\right].\end{align}

The joint Jones matrix of offset and beam is seen to have an additional delay factor multiplying the electric field patterns of the y-dipole (Equation 19). Thus, the resulting combined beam and offset Muller matrix affects every matrix element with an x- and a y-electric field component. Referring back to the Muller matrix (M $_{\mathrm{beam}}$ ), that would imply every element that determines the instrumental Psuedo-Stokes $\mathcal{V}_U$ and $\mathcal{V}_V$ . This is seen in the last two rows of the bottom plot in Figure 7, which shows the Muller matrices of the beam + offset at 2 MHz.

The offset does not add additional leakages or mixing in the collected Stokes I ( $\mathcal{V}_I$ ); thus, it does not directly affect 21 cm measurements. But it causes more mixing of other sky Stokes components into the acquired Stokes V ( $\mathcal{V}_V$ ). This affects the exoplanet science case where we want to capture the pure Stokes V from the ECM emission of the star’s type II bursts and the host-planet interaction. Specifically the max values of $M_{30}$ increases from 0.07% to 4%, $M_{31}$ from 0.4% to 75% and $M_{32}$ from 1% to 50 %.

5.3. Frequency dependence

The pseudo-Stokes leakages due to the beam and offset analysed so far (Figure 7) were only for observations at 2 MHz. We know from Figure 6 that the beam pattern varies with frequency, and the delay due to the offset also varies with frequency (Figure 8). Thus, we can expect the Stokes leakage to vary in frequency. We calculated the Muller matrices similar to Figure 7 for frequencies 0.6 and 10 MHz, which approximately capture the low and high ends of the FARSIDE bandwidth. We then take the azimuthal averages of these absolute Muller matrices of each frequency and plot the result in Figure 9. The rows and columns correspond to respective Muller matrix positions (similar to Figure 7), with the rows corresponding to instrumental Stokes components( $\mathcal{V}$ ) and the columns to the sky Stokes components. Each subplot is the Stokes leakages (normalised to a maximum value of $M_{00}$ ) averaged along the azimuth axis, and the individual curves correspond to different frequencies: 0.6, 2, and 10 MHz. Considering just the co-located dipoles case, the plot shows that with an increase in frequency, the Muller matrix values also increase. And at 10 MHz, we notice higher variations with theta(viewing angle) as expected from Figure 6.When including a phase offset, similar to before, the offset effects increase in the leakages in the instrumental pseudo-Stokes $\mathcal{V}_U$ and $\mathcal{V}_V$ . But with an increase in frequency, the factor of increase in Stokes leakages is seen to decrease.

Figure 9. Azimuthally averaged absolute Muller matrix values for spatially co-located[dashed curves] and non-co-located [solid curves] orthogonal dipoles versus elevation angle. This is shown for three different frequencies within the operating band of the FARSIDE array. The effect of the offset is only on the last two rows of the Muller matrix. With increasing frequency, the absolute fractional mixing due to offset decreases. At the frequency of 10 MHz, the beam pattern of the antenna is no longer in the ideal dipole regime.

5.4. Intrinsic cross-polarisation metrics

We quantify the polarimetry performance of FARSIDE in terms of its intrinsic cross-polarisation ratio (IXR $_M$ ) for Stokes polarimetry in terms of Muller calculus defined by Carozzi & Woan (Reference Carozzi and Woan2011), Kohn et al. (Reference Kohn2019), Asad et al. (Reference Asad2016). This metric is independent of the choice of the coordinate systems, unlike the standard IEEE definition of cross-polarisation ratio. It is defined as:

(20) \begin{equation}IXR_M = \frac{1}{D} = \frac{|M_{00}|}{\sqrt{M_{10}^2+M_{20}^2+M_{30}^2}}\,.\end{equation}

This expression quantifies the instrumental polarisation (D), i.e. leakage of the unpolarised power into polarised power due to the instrument.

We adapt this metric and define two Figure of Merits that will appropriately capture the performance of FARSIDE for its two desired science cases:

1. 1.) Quantifying leakage into $\mathcal{V}_I$

   (21) \begin{equation}IXR_{M,I} = \frac{\sqrt{M_{01}^2 + M_{02}^2 + M_{03}^2}}{|M_{00}|}\,.\end{equation}

This will quantify how much of the sky Stokes Q, U, and V gets depolarised and leaks into instrumental Stokes I ( $\mathcal{V}_I$ ). This is important in the case of the Dark Ages science since the faint redshifted 21cm signal is expected to be unpolarised, and the brighter foreground will be partially polarised.

1. 2.) Quantifying leakage into $\mathcal{V}_V$

   (22) \begin{equation}IXR_{M,V}= \frac{\sqrt{M_{30}^2 + M_{31}^2 + M_{32}^2}}{|M_{33}|}\,.\end{equation}

This will quantify how much of the sky Stokes I, U and Q leaks into instrumental Stokes I ( $\mathcal{V}_I$ ). This is important in the case of the exoplanet magnetosphere emission science since the ECM mechanism is expected to produce 100% circular polarised radiation.

We plot these crosspolarisationn ratios in Figure 10 for three frequencies in the FARSIDE’s planned bandwidth $-0.6$ , 2 and 10 MHz. The IXM values increase with frequency. The IXM $_{M,I}$ is not affected by the phase offset between the dipoles. The offset increases IXM $_{M,V}$ by factors of 2 000, 100, and 10 at 0.6, 2, and 10 MHz, respectively. The Stokes leakage factors due to the offset are seen to decrease in frequency, as was noted with the azimuthally averaged plots too.

Figure 10. Intrinsic cross polarisation values for Stokes I and V for cases of no offset and with offset. Each row corresponds to a different frequency. The offset doesn’t add to the leakage into Stokes I, but it affects the leakages into Stokes V, which increases with frequency.

Figure 11. Block diagram of the complete flow of the interferometer pipeline developed for FARSIDE. It shows how the sky images are pre-processed before multiplying the beam. For the beam, the flowchart indicates the Muller matrices that take into account the beam and the spatial offset effects of the dipoles of the array.

6.1. Sky model

For the model sky, we use the GLEAM point source catalogueFootnote ^c surveyed by the Murchinson Widefield Array (Hurley-Walker et al. Reference Hurley-Walker2017). We note that this model does not include any extended/diffuse emission, and we will leave the analysis of that to future work. The GLEAM survey has flux values for radio sources between 72–230 MHz. We read in all the point sources and their associated spectral indices from the GLEAM catalogue using the pyradio sky python package.Footnote ^d Using pyradiosky, the spectral index, and the radio flux at 72 MHz (lowest frequency of data available with GLEAM), we estimate the radio flux at 2 MHz of each source and limit the number of sources by restricting the flux range between 1 Jy and 1.5 kJy. This sky that is clipped to 30 $^\circ$ from zenith is shown in Figure 12. We feed in only the Stokes I components of this output as the model sky into the pipeline described above. We obtain all four instrumental pseudo-Stokes dirty images for the FARSIDE array layout’s colocated and offset phase dipoles. The resulting images for both cases are shown as the first two columns of Figure 13.

Figure 12. The stokes I sky model using the GLEAM point sources between 1 Jy and 1.5 kJy within 30 deg of the zenith over the assumed FARSIDE phase center (ra = dec =0 $^\circ$ ). These sources are projected onto the same l and m grid as the beams of the FARSIDE array. This Stokes I sky model is processed following Section 6 to yield simulated observations (see Figure 13).

Figure 13. Simulated I, Q, U, and V images of the sources from the GLEAM catalogue through the developed pipeline that includes the beam, the snapshot PSF, and phase offset. The images are generated at the centre of the FARSIDE band, i.e. 2 MHz. The five image columns correspond to: co-located X and Y dipoles, with 50m offset between the X and Y dipoles, differences between co-located and non-colocated cases and offset corrected image.

As expected, the pseudo-Stokes I and Q images are identical for the colocated and spatially phase offset cases. The offset affects only the acquired stokes U and V images. The direction-dependent beam results in maximum $V/I$ to be $\approx 5\times 10^{-5}$ . Adding a spatial offset to the phase centres of the dipoles with this direction-dependent beam increases the maximum $V/I$ to $\approx 8\times10^{-3}$ , i.e. a two orders of magnitude increase in the leakage. The third column shows the differences between the colocated and non-colocated cases for each Stokes image. The differences are more prominent for viewing angles $>10^\circ$ from the zenith.

6.2. Correcting for the offset

Preliminary analysis of modelling a pure Stokes I sky through the FARSIDE beam and array shows that the offset between the phase centres of the dipoles within each node leads to considerable leakage into the instrumental pseudo-Stokes $\mathcal{V}_U$ and $\mathcal{V}_V$ dirty images. The increased leakage into Stokes $\mathcal{V}_V$ increases the confusion noise in searches for the circularly polarised emission from magnetospheres of exoplanets. Therefore, it is crucial to account for the offset’s effects. Here, we present a formulation for the correction.

The constructed pseudo-Stokes visibilities (ignoring the PSF of the array) from the model sky for co-located case (no spatial offset) are given by:

\begin{equation*}\begin{bmatrix}\mathcal{V}_I \\\mathcal{V}_Q \\\mathcal{V}_U \\\mathcal{V}_V\end{bmatrix} = M_{\mathrm{beam}} \begin{bmatrix} I \\Q \\U \\V\end{bmatrix}\end{equation*}

\begin{equation*} = S^{-1} (J_{\mathrm{beam}})\otimes(J_{\mathrm{beam}})^* S \begin{bmatrix} I \\Q \\U \\V\end{bmatrix}\,.\end{equation*}

Applying some math to enable the offset correction. This can be rewritten as:

\begin{equation*}\begin{split} \begin{bmatrix}\mathcal{V}_I \\\mathcal{V}_Q \\\mathcal{V}_U \\\mathcal{V}_V\end{bmatrix} = &S^{-1} (J_{\mathrm{offset}}^{-1}\times J_{\mathrm{offset}}\times J_{\mathrm{beam}})\\[-14pt]&\otimes(J_{\mathrm{offset}}^{-1}\times J_{\mathrm{offset}}\times J_{\mathrm{beam}})^* S \begin{bmatrix} I \\Q \\U \\V\end{bmatrix}\,.\end{split}\end{equation*}

Applying the Kronecker product rule again;

\begin{equation*}\begin{split} =& S^{-1} (J_{\mathrm{offset}}^{-1}\otimes (J_{\mathrm{offset}}^{-1})^*) \\&\times (J_{\mathrm{offset}}\times J_{\mathrm{beam}})\otimes( J_{\mathrm{offset}}\times J_{\mathrm{beam}})^* S \begin{bmatrix} I \\Q \\U \\V\end{bmatrix}\,.\end{split}\end{equation*}

Applying another set of math manipulations:

\begin{equation*}\begin{split} = &S^{-1} (J_{\mathrm{offset}}^{-1}\otimes (J_{\mathrm{offset}}^{-1})^*) S \\ &\{S^{-1}(J_{\mathrm{offset}}\times J_{\mathrm{beam}})\otimes( J_{\mathrm{offset}}\times J_{\mathrm{beam}})^* S\} \begin{bmatrix} I \\Q \\U \\V\end{bmatrix}\,.\end{split}\end{equation*}

The term within the curly braces corresponds to the Muller matrix for the beam and offset case. Thus,

\begin{equation*} \begin{bmatrix}\mathcal{V}_I \\\mathcal{V}_Q \\\mathcal{V}_U \\\mathcal{V}_V\end{bmatrix}_{\mathrm{beam}}= S^{-1}(J_{\mathrm{offset}}^{-1}\otimes (J_{\mathrm{offset}}^{-1})^*)S \begin{bmatrix}\mathcal{V}_I \\\mathcal{V}_Q \\\mathcal{V}_U \\\mathcal{V}_V\end{bmatrix}_{\mathrm{beam+offset}}\,.\end{equation*}

Since the offset Jones matrix is diagonal and unitary (Equation 8), it has the property of $J_{\mathrm{offset}}^{-1} = J_{\mathrm{offset}}^*$ . The correction factor in the above equation can be further simplified to:

\begin{equation*} \begin{bmatrix}\mathcal{V}_I \\\mathcal{V}_Q \\\mathcal{V}_U \\\mathcal{V}_V\end{bmatrix}_{\mathrm{beam}}= S^{-1}(J_{\mathrm{offset}}^{*}\otimes J_{\mathrm{offset}})S \begin{bmatrix}\mathcal{V}_I \\\mathcal{V}_Q \\\mathcal{V}_U \\\mathcal{V}_V\end{bmatrix}_{\mathrm{beam+offset}}\,.\end{equation*}

We apply this correction on the obtained offset images (second column of Figure 13) to remove the effect of dipole phase offsets and plot the resulting Stokes images in the last two columns of Figure 13). In the fourth column, we have applied the correction directly to the undeconvolved images. The correction does not affect the Stokes I and Q images as expected. The correction is seen to have improved both Stokes U and V images, with the levels of Stokes U leakage being similar to the co-located case. We didn’t exactly recover the polarisation levels of the non-offset cases because the correction was not applied to the deconvolved images, which will be the case when we have actual data from the array and use packages like CASA or WSClean to deconvolve the images. The residuals seen in corrected Stokes V images are due to the PSF and its sidelobes. To perform another test and show that the correction works as expected, we generate Stokes V images using the same pipeline developed, but assuming we have a perfect uv-coverage. This is shown in Figure 14 where each column corresponds to the Stokes V image from: No offset, with 50m offset, differences between the two cases, and corrected for offset. A perfect uv-coverage produces an ideal delta function for the PSF, implying the images generated don’t require a deconvolution step. Hence, when the correction for the offset is applied, it recovers the Stokes V levels to the case without offset.

Figure 14. Similar to Figure 13 but only for Stokes V and with an ideal PSF (generated with a uniform uv-coverage). The four image columns correspond to: co-located X and Y dipoles, with 50m offset between the X and Y dipoles, differences between co-located and non-colocated cases and offset corrected image.