Hamaker, Bregman & Sault (1995)

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Mathematical model of an interferometer

Hamaker et al. 1995 Fig1.png

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

Hamaker et al 1995 Fig2.png

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,