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Subsections

• The feed-based instrumental Jones matrices
• The Jones matrix of a Tied Array feed
• Jones matrices for multiple beams

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,

$$\vec{V}_{{\sf i}{\sf j}} \left(\vphantom{\begin{array}{c} {\sf v}_{{\sf i}{\sf p}\,{\sf j... ...f j}{\sf p}}\\ {\sf v}_{{\sf i}{\sf q}\,{\sf j}{\sf q}} \end{array}}\right. \begin{array}{c} {\sf v}_{{\sf i}{\sf p}\,{\sf j}{\sf p}}\\ {... ... q}\,{\sf j}{\sf p}}\\ {\sf v}_{{\sf i}{\sf q}\,{\sf j}{\sf q}} \end{array} \left.\vphantom{\begin{array}{c} {\sf v}_{{\sf i}{\sf p}\,{\sf j... ...f j}{\sf p}}\\ {\sf v}_{{\sf i}{\sf q}\,{\sf j}{\sf q}} \end{array}}\right) \sf J_{{\sf i}}^{} \otimes \sf J_{{\sf j}}^{\ast} \sf S \left(\vphantom{\begin{array}{c} I\\ Q\\ U\\ V \end{array}}\right. \begin{array}{c} I\\ Q\\ U\\ V \end{array} \left.\vphantom{\begin{array}{c} I\\ Q\\ U\\ V \end{array}}\right)_{{\sf l},{\sf m}}^{}$$ (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 $\cal {M}$_{$\sf i$$\sf j$} [2] [3]. Using equation 1:

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

The feed-based instrumental Jones matrices

It will be assumed (for the moment) that all instrumental effects can be factored into feed-based contributions, i.e. any interferometer-based effects are assumed to be negligible (see section 3). The 4 x 4 interferometer response matrix $\sf J_{{\sf i}{\sf j}}^{}$ then consists of a `direct matrix product'<sup>2</sup> $\sf J_{{\sf i}}^{}$ $\otimes$ $\sf J_{{\sf j}}^{\ast}$ of two 2 x 2 feed-based response matrices, called `Jones matrices'. The reader will note that this factoring is the polarimetric generalisation of the familiar `Selfcal assumption', in which the (scalar) gains are assumed to be feed-based rather than interferometer-based.

The 2 x 2 Jones matrix $\sf J_{{\sf i}}^{}$ for feed $\sf i$ can be decomposed into a product of several 2 x 2 Jones matrices, each of which models a specific feed-based instrumental effect in the signal path:

$$\sf J_{{\sf i}}^{} \sf G_{{\sf i}}^{} \sf H_{{\sf i}}^{} \sf Y_{{\sf i}}^{} \sf B_{{\sf i}}^{} \sf K_{{\sf i}}^{} \sf T_{{\sf i}}^{} \sf F_{{\sf i}}^{} \sf G_{{\sf i}}^{} \sf H_{{\sf i}}^{} \sf Y_{{\sf i}}^{} \sf D_{{\sf i}}^{} \sf E_{{\sf i}}^{} \sf P_{{\sf i}}^{} \sf K_{{\sf i}}^{} \sf T_{{\sf i}}^{} \sf F_{{\sf i}}^{}$$ (3)

in which

  

 
$\sf F_{{\sf i}}^{}$($\vec{\rho}\,$,$\vec{r}_{{\sf i}}$) ionospheric Faraday rotation  

 
$\sf T_{{\sf i}}^{}$($\vec{\rho}\,$,$\vec{r}_{{\sf i}}$) atmospheric complex gain 

 
$\sf K_{{\sf i}}^{}$($\vec{\rho}\,$.$\vec{r}_{{\sf i}}$) factored Fourier Transform kernel  

 
$\sf P_{{\sf i}}^{}$
projected receptor orientation(s) w.r.t. the sky 

 
$\sf E_{{\sf i}}^{}$($\vec{\rho}\,$) voltage primary beam 

 
$\sf D_{{\sf i}}^{}$
position-independent receptor cross-leakage 

 
[$\sf Y_{{\sf i}}^{}$] commutation of IF-channels 

 
[$\sf H_{{\sf i}}^{}$] hybrid (conversion to circular polarisation                         coordinates) 

 
$\sf G_{{\sf i}}^{}$
electronic complex gain (feed-based                         contributions only)

Matrices between brackets ([ ]) are not present in all systems. $\sf B_{{\sf i}}^{}$ is the `Total Voltage Pattern' of an arbitrary feed, which is usually split up into three sub-matrices: $\sf D_{{\sf i}}^{}$ $\sf E_{{\sf i}}^{}$ $\sf P_{{\sf i}}^{}$. Jones matrices that model `image-plane' effects depend on the source position (direction) $\vec{\rho}\,$. Some also depend on the antenna position $\vec{r}_{{\sf i}}$. Of course most of them depend on time and frequency as well. The various Jones matrices are treated in some detail in section 5.

Since the Jones matrices do not always commute with each other, their order is important. In principle, they should be placed in the `physical' order, i.e. the order in which the signal is affected by them while traversing the instrument. In practice, this is not always possible or desirable. Section 6 discusses the implications of choosing a different order.

The Jones matrix of a Tied Array feed

The output signals from the two IF-channels of a `tied array' is the weighted sum of the IF-channel signals from n individual feeds. A tied array is itself a feed (see definition in appendix A), modelled by its own Jones matrix. For a single point source, we get:

$$\sf J_{{\sf i}}^{tied~array} \sf Q_{{\sf i}}^{} \sum_{n}^{} \sf i \sf J_{{\sf i}n}^{}$$ (4)

and for an interferometer between two tied arrays $\sf i$ and $\sf j$ with n and m constituent feeds respectively:

$$\sf J_{{\sf i}{\sf j}}^{} \sf J_{{\sf i}}^{} \otimes \sf J_{{\sf j}}^{\ast} \sf Q_{{\sf i}}^{} \otimes \sf Q_{{\sf j}}^{\ast} \sum_{n}^{} \sum_{m}^{} \sf i \sf j \sf J_{{\sf i}n}^{} \otimes \sf J_{{\sf j}m}^{\ast}$$ (5)

See also section 6.4. The matrix $\sf Q_{{\sf i}}^{}$ models electronic gain effects on the added signal of the tied array feed $\sf i$. The $\sf Q_{{\sf i}}^{}$ can be solved by the usual Selfcal methods, in contrast to instrumental errors in the constituent feeds before adding. The latter will often cause decorrellation, and thus closure errors in an interferometer.

Since a tied array feed can be modelled by a Jones matrix, it can be combined with any other type of feed to form an interferometer. Examples are the use of WSRT and VLA as tied arrays in VLBI arrays. Note that this is made possible by factoring the Fourier Transform kernel $\sf K_{{\sf i}{\sf j}}^{}$($\vec{u}_{{\sf i}{\sf j}}$.$\vec{\rho}\,$) into $\sf K_{{\sf i}}^{}$($\vec{r}_{{\sf i}}$.$\vec{\rho}\,$) and $\sf K_{{\sf j}}^{}$($\vec{r}_{{\sf j}}$.$\vec{\rho}\,$), and including the latter in the Jones matrices of the individual feeds (see equ 28).

Obviously, the primary beam of a tied array can be rather complicated, but it is fully modelled by equ 4. Moreover, the contributing feeds in a tied array are allowed to be quite dissimilar. It is nor even necessary for their receptors (dipoles) to be aligned with each other! Thus, equation 4 can also be used to model `difficult' telescopes like Ooty or MOST, or an element of the future Square Km Array (SKAI). This puts the crown on the remarkable power of the Measurement Equation.

Jones matrices for multiple beams

Using the definition in appendix A, each beam in a multiple beam system should be treated like a separate logical feed, modelled by its own Jones matrix. Any communality between them can be modelled in the form of shared parameters in the expressions for the various matrix elements.