THE M.E. FOR A SINGLE POINT SOURCE

For the moment, it will be assumed that there is a single point source at an arbitrary position (direction) $\vec{{\rho}}\,$ = $\vec{{\rho}}\,$ ( $\sf l$ , $\sf m$ ) w.r.t. the fringe-tracking centre, and that observing bandwidth and integration time are negligible. Multiple and extended sources, and the effects of non-zero bandwidth and integration time will be treated for the Full Measurement Equation in section 3.

For a given interferometer, the measured visibilities can be written as a 4-element `coherency vector' $\vec{{V}}_{{{{\sf i}{\sf j}}}}^{{}}$ , which is related to the so-called `Stokes vector' $\vec{{I}}\,$ ( $\sf l$ , $\sf m$ ) of the observed source by a matrix equation,

$\displaystyle \vec{{V}}_{{{{\sf i}{\sf j}}}}^{{}}$ = $\displaystyle \left(\vphantom{\begin{array}{c} {\sf v}_{{\sf i}{\sf p}\,{\sf j... ...f j}{\sf p}}\\ {\sf v}_{{\sf i}{\sf q}\,{\sf j}{\sf q}} \end{array}}\right.$ $\displaystyle \begin{array}{c} {\sf v}_{{\sf i}{\sf p}\,{\sf j}{\sf p}}\\ {... ... q}\,{\sf j}{\sf p}}\\ {\sf v}_{{\sf i}{\sf q}\,{\sf j}{\sf q}} \end{array}$ $\displaystyle \left.\vphantom{\begin{array}{c} {\sf v}_{{\sf i}{\sf p}\,{\sf j... ...f j}{\sf p}}\\ {\sf v}_{{\sf i}{\sf q}\,{\sf j}{\sf q}} \end{array}}\right)$ = ( $\displaystyle \sf J_{{{\sf i}}}^{}$ $\displaystyle \otimes$ $\displaystyle \sf J_{{{\sf j}}}^{{\ast}}_{{}}$ ) $\displaystyle \sf S$ $\displaystyle \left(\vphantom{\begin{array}{c} I\\ Q\\ U\\ V \end{array}}\right.$ $\displaystyle \begin{array}{c} I\\ Q\\ U\\ V \end{array}$ $\displaystyle \left.\vphantom{\begin{array}{c} I\\ Q\\ U\\ V \end{array}}\right)_{{{\sf l},{\sf m}}}^{}$

(1)

The subscripts $\sf i$ and $\sf j$ are the labels of the two feeds that make up the interferometer. The subscripts $\sf p$ and $\sf q$ are the labels of the two output IF-channels from each feed.¹

The `Stokes matrix' $\sf S$ is a constant 4 x 4 coordinate transformation matrix. It is discussed in detail in section 4 below. The real heart of the M.E. is the `direct matrix product' $\sf J_{{{\sf i}}}^{}$ $\otimes$ $\sf J_{{{\sf j}}}^{{\ast}}_{{}}$ of two 2 x 2 feed-based Jones matrices.

The `Stokes-to-Stokes' transmission of a Stokes vector through an `optical' element may be described by multiplication with a 4 x 4 Mueller matrix $\cal {M}$ [2] [3]. Using equation 1:

$\displaystyle \vec{{I}}^{{{out}}}_{{}}$ ( $\displaystyle \sf l$ , $\displaystyle \sf m$ ) = $\displaystyle \sf S^{{-1}}_{}$ $\displaystyle \vec{{V}}_{{{{\sf i}{\sf j}}}}^{{}}$ = $\displaystyle \sf S^{{-1}}_{}$ ( $\displaystyle \sf J_{{{\sf i}}}^{}$ $\displaystyle \otimes$ $\displaystyle \sf J_{{{\sf j}}}^{{\ast}}_{{}}$ ) $\displaystyle \sf S$ $\displaystyle \vec{{I}}^{{{in}}}_{{}}$ ( $\displaystyle \sf l$ , $\displaystyle \sf m$ ) = $\displaystyle \cal {M}$ ( $\displaystyle \sf l$ , $\displaystyle \sf m$ ) $\displaystyle \vec{{I}}^{{{in}}}_{{}}$ ( $\displaystyle \sf l$ , $\displaystyle \sf m$ )

(2)

Mueller matrices are useful in simulation, when studying the effect of instrumental effects on a test source $\vec{{I}}\,$ ( $\sf l$ , $\sf m$ ). They can be easily generalised to the full M.E. (see section 3).