Hamaker, Bregman & Sault (1995)
Mathematical model of an interferometer
The signal in a single arm
<math>\mathbf{e}_{out}(t)=\mathbf{J}\mathbf{e}_{in}(t)</math>
<math>\mathbf{e}=\begin{pmatrix} e_x \\ e_y \end{pmatrix}</math>
<math>\mathbf{v}=\mathbf{Q}\mathbf{e}</math>
The coherency vector
<math>\mathbf{e}=\left\langle\begin{pmatrix} e_{Ax}e_{Bx}^* \\ e_{Ax}e_{By}^* \\ e_{Ay}e_{Bx}^* \\ e_{Ay}e_{By}^* \end{pmatrix}\right\rangle = \langle\mathbf{e}_A\otimes\mathbf{e}_B^*\rangle</math>
<math>\otimes</math> represents outer product which is the missing link between the transformations of signals in the individual interferometer arms and those of the coherency vector in the interferometer as a whole.
Coordinate transformations
<math>\mathbf{x}_{new}=\mathbf{T}\mathbf{x}_{old}</math>
<math>\mathbf{J}_{new}=\mathbf{T}\mathbf{J}_{old}\mathbf{T}^{-1}</math>
In many cases a physical transformation is equivalent to a linear coordinate transformation (above equation).
Faraday rotation over an angle <math>\chi</math> is equivalent to rotating the feed over an angle <math>-\chi</math>.
Stokes representation of the coherency vector
The Stokes vector,
<math>\mathbf{e}^S=\begin{pmatrix} I \\ Q \\ U \\ V \end{pmatrix}=\mathbf{T}\mathbf{e}^+ \text{ ; } \mathbf{T}=\begin{pmatrix} 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & -1 \\ 0 & 1 & 1 & 0 \\ 0 & -i & i & 0 \\ \end{pmatrix}</math>
+ stands for cartesian xyz coordinate system, and S for Stokes coordinate system.
Stokes coordinate is an abstract one.
<math>\mathbf{S}=\mathbf{T}^{-1}=\frac{1}{2}\begin{pmatrix} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & i \\ 0 & 0 & 1 & -i \\ 1 & -1 & 0 & 0 \end{pmatrix}</math>
The signal path in an interferometer
The output from a correlator is literally,
<math>\mathbf{v}=\left\langle\begin{pmatrix} v_{Ax}v_{Bx}^* \\ v_{Ax}v_{By}^* \\ v_{Ay}v_{Bx}^* \\ v_{Ay}v_{By}^* \end{pmatrix}\right\rangle</math>
The interferometer equation
The incident signal is subject to the following transformations in a single arm,
- Faraday rotation in Earth's ionosphere, <math>\mathbf{F}</math>
- Parallactic angle rotation, <math>\mathbf{P}</math>. The angle is zero for an equatorial mount.
- Primary beam
- The feed <math>\mathbf{Q}</math> that can be modeled using two matrices,