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Figure 1: Simple schematic overview of two antennas (p and q) each with two dipoles (x and y) that receive a signal, where one is delayed w.r.t. the other.

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The table below gives an overview of the various types of gain matrices

G_{p} = \begin{pmatrix}
A_{xx}e^{\phi_{xx}} & A_{xy}e^{\phi_{xy}}\\
A_{yx}e^{\phi_{yx}} & A_{yy}e^{\phi_{yy}}
\end{pmatrix}

G_{p} = \begin{pmatrix}
A_{xx}e^{\phi_{xx}} & 0\\
0 & A_{yy}e^{\phi_{yy}}
\end{pmatrix}

G_{p} = \begin{pmatrix}
e^{\phi_{xx}} & 0\\
0 & e^{\phi_{yy}}
\end{pmatrix}

G_{p} = \begin{pmatrix}
e^{\phi} & 0\\
0 & e^{\phi}
\end{pmatrix}

LINC

The LINC (LOFAR INitial Calibration pipeline) produces direction-independent calibrated visibilities, a wide-field image of the target field, calibration solutions and diagnostic plots. LINC corrects for various instrumental and ionospheric effects for HBA (and soon to come LBA) observations. The benefit of using LINC over other processing methods (e.g. CASA, AIPS) is that it is largely automated, so it requires little input from the users as well as preparing the data for direction-dependent calibration using dedicated pipelines.

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The (Faraday) rotation matrix describes the relationship between the X and Y components of the voltages before and after the rotation. These components change, hence the Faraday rotation affects the amplitudes of the components as well as the phase.

A_{x} = A \cos \phi \\
A_{y} = A \sin \phi \\
A^{'}_{x} = A \cos \psi = A \cos (\phi - \alpha) \\
A^{'}_{y} = A \sin \psi = A \sin (\phi - \alpha) \\
A^{'}_{x} = A \cos \phi \cos \alpha - A\sin \phi \sin \alpha \\
A^{'}_{y} = A \sin \phi \cos \alpha + A \cos \phi \sin \alpha \\
A^{'}_{x} = A_{x} \cos \alpha - A_{y} \sin \alpha \\
A^{'}_{y} = A_{y} \cos \alpha + A_{x} \sin \alpha \\
\left(  \begin{matrix}A^{'}_{x} \\
A^{'}_{y} \end{matrix} \right) = \left(  \begin{matrix} \cos \alpha & -\sin \alpha \\
\sin \alpha & \cos \alpha  \end{matrix} \right) \left(  \begin{matrix} A_{x} \\
A_{y}  \end{matrix} \right)

Figure 5: Schematic of the mathematical principles of polarisation misalignment being a rotation effect.

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