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Subsections

• Ionospheric Faraday rotation ( $\sf F_{{\sf i}}^{}$($\vec{\rho}\,$,$\vec{r}_{{\sf i}}$))
• Atmospheric gain ( $\sf T_{{\sf i}}^{}$($\vec{\rho}\,$,$\vec{r}_{{\sf i}}$))
• Fourier Transform kernel ( $\sf K_{{\sf i}}^{}$($\vec{r}_{{\sf i}}$.$\vec{\rho}\,$))
• Projection matrix ( $\sf P_{{\sf i}}^{}$) if $\gamma_{{\sf x}{\sf a}}$ = $\gamma_{{\sf y}{\sf b}}$
• Projection matrix ( $\sf P_{{\sf i}}^{}$) if $\gamma_{{\sf x}{\sf a}}$ $\neq$ $\gamma_{{\sf y}{\sf b}}$
• Voltage primary beam ( $\sf E_{{\sf i}}^{}$($\vec{\rho}\,$))
• Position-independent receptor cross-leakage ( $\sf D_{{\sf i}}^{}$)
• Commutation ( $\sf Y_{{\sf i}}^{}$)
• Hybrid ( $\sf H_{{\sf i}}^{}$)
• Electronic gain ( $\sf G_{{\sf i}}^{}$)
• Do we need a configuration matrix ( $\sf C_{{\sf i}}^{}$)?

GENERIC FORM OF JONES MATRICES

In this section, the `generic' form of various 2 x 2 feed-based instrumental Jones matrices (operators) will be treated in some detail.

It will be noted that for each matrix, the 4 elements have been given an `official' name (e.g. $\sf f_{{\sf i}{{\sf x}}{{\sf x}}}^{}$). The (possibly naive) idea is that, if the structure of the Measurement Equation is more or less complete, these `standard' matrix elements could be referred to explicitly by their official names in other AIPS++ documents (and code), for instance to replace them with specific expressions for particular telescopes or purposes.

The subscript convention is as follows: $\sf y_{{\sf i}{\sf b}{\sf p}}^{}$ is an element of matrix $\sf Y$ for feed $\sf i$, which models the `coupling factor' for the signal going from receptor $\sf b$ to IF-channel $\sf p$. Where possible, the expressions have been reduced to matrices like the diagonal matrix ( $\sf Diag$), rotation matrix ($\sf Rot$) etc. These are defined in the Appendix.

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

The matrix $\sf F^{+}_{{\sf i}}$ represents (ionospheric) Faraday rotation of the electric vector over an angle $\chi_{{\sf i}}$ w.r.t. the celestial $\sf x$,$\sf y$-frame. Since $\chi_{{\sf i}}$ is defined in one of the polarisation coordinate frames, there will be two different forms for $\sf F_{{\sf i}}^{}$ (see also section 4). For linear polarisation coordinates:

$$\sf F^{+}_{{\sf i}} \vec{\rho}\, \vec{r}_{{\sf i}} \left(\vphantom{\begin{array}{cc}{\sf f}_{{\sf i}{{\sf x}}{{\sf x... ...i}{{\sf x}}{{\sf y}}} & {\sf f}_{{\sf i}{{\sf y}}{{\sf y}}} \end{array}}\right. \begin{array}{cc}{\sf f}_{{\sf i}{{\sf x}}{{\sf x}}} & {\sf f}_{{... ...}_{{\sf i}{{\sf x}}{{\sf y}}} & {\sf f}_{{\sf i}{{\sf y}}{{\sf y}}} \end{array} \left.\vphantom{\begin{array}{cc}{\sf f}_{{\sf i}{{\sf x}}{{\sf x... ...i}{{\sf x}}{{\sf y}}} & {\sf f}_{{\sf i}{{\sf y}}{{\sf y}}} \end{array}}\right) \sf Rot \chi_{{\sf i}}$$ (25)

In circular polarisation coordinates, the matrix $\sf F^{\odot}_{{\sf i}}$ is a diagonal matrix which introduces a phase difference, or rather a delay difference. It expresses the fact that ionospheric Faraday rotation is caused by a (strongly frequency-dependent) difference in propagation velocity between right-hand and left-hand circularly polarised signals when travelling through a charged medium like the ionosphere. In terms of the Faraday rotation angle $\chi_{{\sf i}}$ (see above), we get:

$$\sf F^{\odot}_{{\sf i}} \vec{\rho}\, \vec{r}_{{\sf i}} \left(\vphantom{\begin{array}{cc}{\sf f}_{{\sf i}{\sf r}{\sf r}} ... ...}_{{\sf i}{\sf r}{\sf l}} & {\sf f}_{{\sf i}{\sf l}{\sf l}} \end{array}}\right. \begin{array}{cc}{\sf f}_{{\sf i}{\sf r}{\sf r}} & {\sf f}_{{\sf ... ... {\sf f}_{{\sf i}{\sf r}{\sf l}} & {\sf f}_{{\sf i}{\sf l}{\sf l}} \end{array} \left.\vphantom{\begin{array}{cc}{\sf f}_{{\sf i}{\sf r}{\sf r}} ... ...}_{{\sf i}{\sf r}{\sf l}} & {\sf f}_{{\sf i}{\sf l}{\sf l}} \end{array}}\right) \cal {H} \sf F^{+}_{{\sf i}} \cal {H} \sf Diag \chi_{{\sf i}} \chi_{{\sf i}}$$ (26)

In principle, the Faraday rotation angle is a function of source direction and feed position: $\chi_{{\sf i}}$ = $\chi_{{\sf i}}$($\vec{\rho}\,$,$\vec{r}_{{\sf i}}$). However, Faraday rotation is a large-scale effect, so it will usually have the same value for all sources in the primary beam: $\chi_{{\sf i}}$ = $\chi$($\vec{r}_{{\sf i}}$). For arrays smaller than a few km, the rotation angle will usually also be the same for all feeds: $\chi_{{\sf i}}$ = $\chi$(t). These assumptions reduce the number of independent parameters considerably.

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

The matrix $\sf T^{+}_{{\sf i}}$ represents complex atmospheric gain: refraction, extinction and perhaps non-isoplanaticity. Since $\sf T^{+}_{{\sf i}}$ does not depend on a polarisation coordinate frame, there is only one form:

$$\sf T^{+}_{{\sf i}} \sf T^{\odot}_{{\sf i}} \sf T_{{\sf i}}^{} \vec{\rho}\, \vec{r}_{{\sf i}} \approx \left(\vphantom{\begin{array}{cc}{\sf t}_{{\sf i}} & 0\\ 0 & {\sf t}_{{\sf i}} \end{array}}\right. \begin{array}{cc}{\sf t}_{{\sf i}} & 0\\ 0 & {\sf t}_{{\sf i}} \end{array} \left.\vphantom{\begin{array}{cc}{\sf t}_{{\sf i}} & 0\\ 0 & {\sf t}_{{\sf i}} \end{array}}\right) \sf Mult \sf t_{{\sf i}}^{}$$ (27)

The matrix is diagonal because the atmosphere does is not supposed to cause cross-talk. The diagonal elements are assumed to be equal, because the atmosphere is not supposed to affect polarisation.

Atmospheric effects in the `pupil-plane' (i.e. originating directly above the feeds) can be modelled with a complex gain. It is less clear how to deal with effects that originate higher up in the atmosphere, i.e. between pupil plane and image plane.

A phase screen over the array can be modelled as $\sf t_{{\sf i}}^{}$ =  exp^{i$\psi_{{\sf i}}$} in which the phase is assumed to be a low-order 2D polynomial as a function of the feed position $\vec{r}\,$: $\psi_{{\sf i}}$  =  a₀(t)  +  a₁(t) $\vec{r}_{{\sf i}}$  +  a₂(t) $\vec{r}_{{\sf i}}^{2}$  +   ^...

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

The matrix $\sf K_{{\sf i}}^{}$ represents the Fourier Transform kernel, which can also be seen as a phase weight factor). It is factored into feed-based parts in order to be able to model a tied array (see section 2.2). Since $\sf K_{{\sf i}}^{}$ does not depend on the polarisation coordinate frame, there is only one form:

$$\sf K^{+}_{{\sf i}} \sf K^{\odot}_{{\sf i}} \sf K_{{\sf i}}^{} \vec{r}_{{\sf i}} \vec{\rho}\, \left(\vphantom{\begin{array}{cc}{\sf k}_{{\sf i}{\sf a}{\sf a}} & 0\\ 0 & {\sf k}_{{\sf i}{\sf b}{\sf b}} \end{array}}\right. \begin{array}{cc}{\sf k}_{{\sf i}{\sf a}{\sf a}} & 0\\ 0 & {\sf k}_{{\sf i}{\sf b}{\sf b}} \end{array} \left.\vphantom{\begin{array}{cc}{\sf k}_{{\sf i}{\sf a}{\sf a}} & 0\\ 0 & {\sf k}_{{\sf i}{\sf b}{\sf b}} \end{array}}\right) \left(\vphantom{\begin{array}{cc}{\sf k}_{{\sf i}} & 0\\ 0 & {\sf k}_{{\sf i}} \end{array}}\right. \begin{array}{cc}{\sf k}_{{\sf i}} & 0\\ 0 & {\sf k}_{{\sf i}} \end{array} \left.\vphantom{\begin{array}{cc}{\sf k}_{{\sf i}} & 0\\ 0 & {\sf k}_{{\sf i}} \end{array}}\right) \sf Mult \sf k_{{\sf i}}^{}$$ (28)

in which $\sf k_{{\sf i}}^{}$ = ${\frac{1}{\sqrt{{\sf n}}}}$exp^{i2$\pi$$\vec{r}_{{\sf i}}$.$\vec{\rho}\,$/$\lambda$}, which depends on the projected feed position $\vec{r}_{{\sf i}}$ and the source direction $\vec{\rho}\,$ = $\vec{\rho}\,$($\sf l$,$\sf m$) w.r.t. the fringe tracking centre $\vec{\rho}_{ftc}$, and $\sf n$ = $\sqrt{1-{\sf l}^2-{\sf m}^2}$ $\approx$ 1 - 0.5($\sf l^{2}_{}$ + $\sf m^{2}_{}$).

If $\sf k_{{\sf i}{\sf a}{\sf a}}^{}$ = $\sf k_{{\sf i}{\sf b}{\sf b}}^{}$, the interferometer matrix $\sf K_{{\sf i}{\sf j}}^{}$ = ($\sf K_{{\sf i}}^{}$ $\otimes$ $\sf K_{{\sf j}}^{\ast}$) is a 4 x 4 diagonal matrix with equal elements. This is equivalent to a multiplicative factor of the familiar form $\sf k_{{\sf i}{\sf j}}^{}$ = $\sf k_{{\sf i}}^{}$$\sf k_{{\sf j}}^{\ast}$ = ${\frac{1}{{\sf n}}}$exp^{i2$\pi$($\vec{r}_{{\sf i}}$ - $\vec{r}_{{\sf j}}$).$\vec{\rho}\,$/$\lambda$} = ${\frac{1}{{\sf n}}}$exp^{i$\vec{u}_{{\sf i}{\sf j}}$.$\vec{\rho}\,$}, i.e. the Fourier Transform kernel or `phase weight' for the baseline $\vec{u}_{{\sf i}{\sf j}}$. For small fields, $\sf n$ $\approx$ 1, so $\vec{u}_{{\sf i}{\sf j}}$.$\vec{\rho}\,$ = ($\sf u$$\sf l$ + $\sf v$$\sf m$ + $\sf w$($\sf n$ - 1)) $\approx$ ($\sf u$$\sf l$ + $\sf v$$\sf m$) becomes a 2D FT.

The receptors of a feed are practically always co-located, i.e. they have the same phase-centre: $\vec{r}_{{\sf i}{\sf a}}$ = $\vec{r}_{{\sf i}{\sf b}}$ = $\vec{r}_{{\sf i}}$, so $\sf k_{{\sf i}{\sf a}{\sf a}}^{}$ = $\sf k_{{\sf i}{\sf b}{\sf b}}^{}$ = $\sf k_{{\sf i}}^{}$. But note that it is possible to model a receptors that are not co-located, i.e. $\vec{r}_{{\sf i}{\sf a}}$ $\neq$ $\vec{r}_{{\sf i}{\sf b}}$. It is not immediately obvious why one would want to do such a thing, but it is good to know that the formalism allows it.

Projection matrix ( $\sf P_{{\sf i}}^{}$) if $\gamma_{{\sf x}{\sf a}}$ = $\gamma_{{\sf y}{\sf b}}$

The `Projection matrix' models the projected orientation of the receptors w.r.t. the electrical $\sf x$,$\sf y$ frame on the sky, as seen from the direction of the source (see also section 5.6 below). Since the orientations are defined in one of the polarisation coordinate frames, there will be two different forms for $\sf P_{{\sf i}}^{}$ (see section 4). For linear polarisation coordinates:

$$\sf P^{+}_{{\sf i}} \left(\vphantom{\begin{array}{cc}{\sf p}_{{\sf i}{{\sf x}}{\sf a}... ...\sf i}{{\sf x}}{\sf b}} & {\sf p}_{{\sf i}{{\sf y}}{\sf b}} \end{array}}\right. \begin{array}{cc}{\sf p}_{{\sf i}{{\sf x}}{\sf a}} & {\sf p}_{{\s... ...sf p}_{{\sf i}{{\sf x}}{\sf b}} & {\sf p}_{{\sf i}{{\sf y}}{\sf b}} \end{array} \left.\vphantom{\begin{array}{cc}{\sf p}_{{\sf i}{{\sf x}}{\sf a}... ...\sf i}{{\sf x}}{\sf b}} & {\sf p}_{{\sf i}{{\sf y}}{\sf b}} \end{array}}\right) \equiv \left(\vphantom{\begin{array}{cc}\cos{{\gamma}_{{\sf x}{\sf a}}} ... ...\gamma}_{{\sf x}{\sf a}}} & \cos{{\gamma}_{{\sf x}{\sf a}}} \end{array}}\right. \begin{array}{cc}\cos{{\gamma}_{{\sf x}{\sf a}}} & -\sin{{\gamma}... ... \sin{{\gamma}_{{\sf x}{\sf a}}} & \cos{{\gamma}_{{\sf x}{\sf a}}} \end{array} \left.\vphantom{\begin{array}{cc}\cos{{\gamma}_{{\sf x}{\sf a}}} ... ...\gamma}_{{\sf x}{\sf a}}} & \cos{{\gamma}_{{\sf x}{\sf a}}} \end{array}}\right) \sf Rot \gamma_{{\sf x}{\sf a}}$$ (29)

in which $\gamma_{{\sf x}{\sf a}}$ is the projected angle between the positive $\sf x$-axis and the orientation of receptor $\sf a$ (see also Appendix A.3). There is an implicit assumption here that the feed has perpendicular receptors and is fully steerable, which is the case for the majority of existing telescopes. See the next section for the case where the projected orientations are not perpendicular ( $\gamma_{{\sf x}{\sf a}}$ $\neq$ $\gamma_{{\sf y}{\sf b}}$).

For circular polarisation coordinates:

$$\sf P^{\odot}_{{\sf i}} \left(\vphantom{\begin{array}{cc}{\sf p}_{{\sf i}{\sf r}{\sf a}} ... ...}_{{\sf i}{\sf r}{\sf b}} & {\sf p}_{{\sf i}{\sf l}{\sf b}} \end{array}}\right. \begin{array}{cc}{\sf p}_{{\sf i}{\sf r}{\sf a}} & {\sf p}_{{\sf ... ... {\sf p}_{{\sf i}{\sf r}{\sf b}} & {\sf p}_{{\sf i}{\sf l}{\sf b}} \end{array} \left.\vphantom{\begin{array}{cc}{\sf p}_{{\sf i}{\sf r}{\sf a}} ... ...}_{{\sf i}{\sf r}{\sf b}} & {\sf p}_{{\sf i}{\sf l}{\sf b}} \end{array}}\right) \cal {H} \sf P^{+}_{{\sf i}} \cal {H} \sf Diag \gamma_{{\sf x}{\sf a}} \gamma_{{\sf x}{\sf a}}$$ (30)

It is sometimes useful to introduce an intermediate coordinate frame, attached to the feed $\sf i$. In that case: $\gamma_{{\sf x}{\sf a}}$ = $\gamma_{{\sf x}{\sf i}}$ + $\gamma_{{\sf i}{\sf a}}$ = $\beta$ + $\gamma_{{\sf i}{\sf a}}$. The `offset' angle $\gamma_{{\sf i}{\sf a}}$ between receptor $\sf a$ and the frame of feed $\sf i$ will be zero in most cases. The angle $\beta$ is the parallactic angle, i.e. the angle between two great circles through the source, and through the celestial North Pole and the local zenith respectively. This parallactic angle is zero for an equatorial feed, and varies smoothly with HA($\sf t$) for an alt-az feed:

$$\beta$$ = cosLAT sinHA

$$\beta$$ = cosDEC sinLAT  -  sinDEC cosLAT cosHA (31)

Projection matrix ( $\sf P_{{\sf i}}^{}$) if $\gamma_{{\sf x}{\sf a}}$ $\neq$ $\gamma_{{\sf y}{\sf b}}$

The M.E. formalism must also be able to deal with more `exotic' antennas like parabolic cylinders (Arecibo, MOST) or horizontal dipole arrays (SKAI). In those cases, the projected angles of the two receptors will generally not be equal, i.e. $\gamma_{{\sf x}{\sf a}}$ $\neq$ $\gamma_{{\sf y}{\sf b}}$.

NB: The angle $\gamma_{{\sf y}{\sf b}}$ of receptor $\sf b$ is defined w.r.t. the $\sf y$-axis rather than the $\sf x$-axis. This ensures that $\gamma_{{\sf y}{\sf b}}$ = $\gamma_{{\sf x}{\sf a}}$, so that matrix $\sf P^{+}_{{\sf i}}$ reduces to a simple rotation $\sf Rot$($\gamma_{{\sf x}{\sf a}}$), in the common case described in section 5.4 above.

For linear polarisation coordinates $\sf P^{+}_{{\sf i}}$ becomes a `pseudo-rotation' (compare with equ 29 above):

$$\sf P^{+}_{{\sf i}} \left(\vphantom{\begin{array}{cc}{\sf p}_{{\sf i}{{\sf x}}{\sf a}... ...\sf i}{{\sf x}}{\sf b}} & {\sf p}_{{\sf i}{{\sf y}}{\sf b}} \end{array}}\right. \begin{array}{cc}{\sf p}_{{\sf i}{{\sf x}}{\sf a}} & {\sf p}_{{\s... ...sf p}_{{\sf i}{{\sf x}}{\sf b}} & {\sf p}_{{\sf i}{{\sf y}}{\sf b}} \end{array} \left.\vphantom{\begin{array}{cc}{\sf p}_{{\sf i}{{\sf x}}{\sf a}... ...\sf i}{{\sf x}}{\sf b}} & {\sf p}_{{\sf i}{{\sf y}}{\sf b}} \end{array}}\right) \equiv \left(\vphantom{\begin{array}{cc}\cos{{\gamma}_{{\sf x}{\sf a}}} ... ...\gamma}_{{\sf y}{\sf b}}} & \cos{{\gamma}_{{\sf y}{\sf b}}} \end{array}}\right. \begin{array}{cc}\cos{{\gamma}_{{\sf x}{\sf a}}} & -\sin{{\gamma}... ... \sin{{\gamma}_{{\sf y}{\sf b}}} & \cos{{\gamma}_{{\sf y}{\sf b}}} \end{array} \left.\vphantom{\begin{array}{cc}\cos{{\gamma}_{{\sf x}{\sf a}}} ... ...\gamma}_{{\sf y}{\sf b}}} & \cos{{\gamma}_{{\sf y}{\sf b}}} \end{array}}\right) \sf Rot \gamma_{{\sf x}{\sf a}} \gamma_{{\sf y}{\sf b}}$$ (32)

For circular polarisation coordinates:

$$\sf P^{\odot}_{{\sf i}} \left(\vphantom{\begin{array}{cc}{\sf p}_{{\sf i}{\sf r}{\sf a}} ... ...}_{{\sf i}{\sf r}{\sf b}} & {\sf p}_{{\sf i}{\sf l}{\sf b}} \end{array}}\right. \begin{array}{cc}{\sf p}_{{\sf i}{\sf r}{\sf a}} & {\sf p}_{{\sf ... ...\ {\sf p}_{{\sf i}{\sf r}{\sf b}} & {\sf p}_{{\sf i}{\sf l}{\sf b}} \end{array} \left.\vphantom{\begin{array}{cc}{\sf p}_{{\sf i}{\sf r}{\sf a}} ... ...}_{{\sf i}{\sf r}{\sf b}} & {\sf p}_{{\sf i}{\sf l}{\sf b}} \end{array}}\right) \cal {H} \sf P^{+}_{{\sf i}} \cal {H}$$ = (33)

$$\left(\vphantom{\begin{array}{cc}\cos{{\gamma}_{{\sf x}{\sf a}}}+... ...\gamma}_{{\sf x}{\sf a}}}+ \sin{{\gamma}_{{\sf y}{\sf b}}}) \end{array}}\right. \begin{array}{cc}\cos{{\gamma}_{{\sf x}{\sf a}}}+ \cos{{\gamma}_{... ...i(\sin{{\gamma}_{{\sf x}{\sf a}}}+ \sin{{\gamma}_{{\sf y}{\sf b}}}) \end{array} \left.\vphantom{\begin{array}{cc}\cos{{\gamma}_{{\sf x}{\sf a}}}+... ...\gamma}_{{\sf x}{\sf a}}}+ \sin{{\gamma}_{{\sf y}{\sf b}}}) \end{array}}\right)$$ =

The future large radio telescopes may have feeds in the form of dipole arrays, possibly tilted over an angle $\alpha$ towards the South w.r.t. the local horizontal plane. In that case, the projected angle $\gamma_{{\sf x}{\sf a}}$ between a North-South (NS) dipole and the $\sf x$-axis differs from the projected angle $\gamma_{{\sf y}{\sf b}}$ between an East-West (EW) dipole and the $\sf y$-axis (I hope this is correct now):

$$\gamma_{{\sf x}{\sf a}} \alpha \alpha$$ =

$$\gamma_{{\sf x}{\sf a}} \alpha$$ =

$$\gamma_{{\sf y}{\sf b}}$$ = cosHA

$$\gamma_{{\sf y}{\sf b}}$$ = -sinHA sinDEC (34)

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

The effects of the primary beam are ignored by [2], which deals implicitly with on-axis sources observed by feeds with fully steerable parabolic mirrors. The AIPS++ M.E. must of course deal with the general case, including `exotic' telescopes like Arecibo, MOST and SKAI. To this end, we define a total voltage pattern matrix $\sf B_{{\sf i}}^{}$, which fully describes the conversion of the incident electric field (V/m) into two voltages (V):

$$\sf B^{+}_{{\sf i}} \vec{\rho}\, \left(\vphantom{\begin{array}{cc}{\sf b}_{{\sf i}{{\sf x}}{\sf a}... ...\sf i}{{\sf x}}{\sf b}} & {\sf b}_{{\sf i}{{\sf y}}{\sf b}} \end{array}}\right. \begin{array}{cc}{\sf b}_{{\sf i}{{\sf x}}{\sf a}} & {\sf b}_{{\s... ...sf b}_{{\sf i}{{\sf x}}{\sf b}} & {\sf b}_{{\sf i}{{\sf y}}{\sf b}} \end{array} \left.\vphantom{\begin{array}{cc}{\sf b}_{{\sf i}{{\sf x}}{\sf a}... ...\sf i}{{\sf x}}{\sf b}} & {\sf b}_{{\sf i}{{\sf y}}{\sf b}} \end{array}}\right) \sf B^{\odot}_{{\sf i}} \vec{\rho}\, \left(\vphantom{\begin{array}{cc}{\sf b}_{{\sf i}{\sf r}{\sf a}} ... ...}_{{\sf i}{\sf r}{\sf b}} & {\sf b}_{{\sf i}{\sf l}{\sf b}} \end{array}}\right. \begin{array}{cc}{\sf b}_{{\sf i}{\sf r}{\sf a}} & {\sf b}_{{\sf ... ... {\sf b}_{{\sf i}{\sf r}{\sf b}} & {\sf b}_{{\sf i}{\sf l}{\sf b}} \end{array} \left.\vphantom{\begin{array}{cc}{\sf b}_{{\sf i}{\sf r}{\sf a}} ... ...}_{{\sf i}{\sf r}{\sf b}} & {\sf b}_{{\sf i}{\sf l}{\sf b}} \end{array}}\right)$$ (35)

NB: Since the Jones matrix $\sf J_{{\sf i}}^{}$ is feed-based, it deals with voltage beams. The power beam for interferometer $\sf i$$\sf j$ is modelled by $\sf B_{{\sf i}}^{}$ $\otimes$ $\sf B_{{\sf j}}^{\ast}$. Note that the formalism deals implicitly with interferometers between feeds with quite dissimilar primary beams.

In practice, it is often convenient to split the matrix $\sf B_{{\sf i}}^{}$ into a chain of sub-matrices:

• It is always possible to split off a projection matrix $\sf P_{{\sf i}}^{}$:
  $\sf B_{{\sf i}}^{}$  =  ($\sf B_{{\sf i}}^{}$ $\sf P_{{\sf i}}^{-1}$) $\sf P_{{\sf i}}^{}$  =  $\sf E_{{\sf i}}^{'}$ $\sf P_{{\sf i}}^{}$
  See sections 5.4 and 5.5 above.

• It is always possible to split off a position-independent leakage matrix $\sf D_{{\sf i}}^{}$:
  $\sf E_{{\sf i}}^{'}$ $\sf P_{{\sf i}}^{}$  =  $\sf D_{{\sf i}}^{}$ ($\sf D_{{\sf i}}^{-1}$ $\sf E_{{\sf i}}^{'}$) $\sf P_{{\sf i}}^{}$  =  $\sf D_{{\sf i}}^{}$ $\sf E_{{\sf i}}^{}$ $\sf P_{{\sf i}}^{}$
  See section 5.7 below.

This is most useful in the common case of a fully steerable parabolic antenna. The voltage patterns of its feed(s) have a fixed shape, which are rotated and translated w.r.t. the sky when pointing the antenna in different directions. What remains after splitting off $\sf P_{{\sf i}}^{}$ and $\sf D_{{\sf i}}^{}$($\vec{\rho}\,$) is an (approximately) real and diagonal matrix $\sf E_{{\sf i}}^{}$ which decsribes the position-dependent primary beam attenuation and the position-dependent leakage (see also equation 38 below):

$$\sf E^{+}_{{\sf i}} \vec{\rho}\, \sf E^{\odot}_{{\sf i}} \vec{\rho}\, \sf E_{{\sf i}}^{} \vec{\rho}\, \left(\vphantom{\begin{array}{cc}{\sf e}_{{\sf i}{\sf a}{\sf a}} ... ...}_{{\sf i}{\sf a}{\sf b}} & {\sf e}_{{\sf i}{\sf b}{\sf b}} \end{array}}\right. \begin{array}{cc}{\sf e}_{{\sf i}{\sf a}{\sf a}} & {\sf e}_{{\sf ... ... {\sf e}_{{\sf i}{\sf a}{\sf b}} & {\sf e}_{{\sf i}{\sf b}{\sf b}} \end{array} \left.\vphantom{\begin{array}{cc}{\sf e}_{{\sf i}{\sf a}{\sf a}} ... ...}_{{\sf i}{\sf a}{\sf b}} & {\sf e}_{{\sf i}{\sf b}{\sf b}} \end{array}}\right) \approx \sf Diag \sf e_{{\sf i}{\sf a}{\sf a}}^{} \sf e_{{\sf i}{\sf b}{\sf b}}^{}$$ (36)

As an example, the diagonal elements of $\sf E^{+}_{{\sf i}}$ for an idealised axially symmetric gaussian beam and dipole receptorswould look like:

$$\sf e_{{\sf i}{\sf a}{\sf a}}^{} {\frac{{{\sf l}^{''}_{{{\sf i}{\sf a}}}}}{{\sigma}_{\sf a}(1+{\epsilon}_{\sf a})}} {\frac{{{\sf m}^{''}_{{{\sf i}{\sf a}}}}}{{\sigma}_{\sf a}(1-{\epsilon}_{\sf a})}}$$   =  

$$\sf e_{{\sf i}{\sf b}{\sf b}}^{} {\frac{{{\sf l}^{''}_{{{\sf i}{\sf b}}}}}{{\sigma}_{\sf b}(1+{\epsilon}_{\sf b})}} {\frac{{{\sf m}^{''}_{{{\sf i}{\sf b}}}}}{{\sigma}_{\sf b}(1-{\epsilon}_{\sf b})}}$$   =  

Note that the two receptor beams are each described in their own coordinate frame $\sf l^{''}_{{{\sf i}{\sf a}}}$,$\sf m^{''}_{{{\sf i}{\sf a}}}$ and $\sf l^{''}_{{{\sf i}{\sf b}}}$,$\sf m^{''}_{{{\sf i}{\sf b}}}$ projected on the sky (see Appendix A). The projection matrix $\sf P_{{\sf i}}^{}$ only takes care of electrical rotation, but not of the rotation of the voltage beam on the sky!.

Equation 37 illustrates that the voltage beam of a dipole receptor will be slightly elongated in the direction of the dipole by a factor (1 + $\epsilon$), even if the mirror is perfectly circular and symmetrical. Obviously, the two asymmetric voltage beams of a feed will not coincide, because they are oriented differently. The resulting position-dependent difference is one cause of off-axis instrumental polarisation.

In reality, things will be more complicated, especially for off-axis sources. For instance, standing waves between the primary mirror and the frontend box, or scattering off support legs, may cause position-dependent leakage terms. Since these cannot be part of $\sf D_{{\sf i}}^{}$, they must be modelled as off-diagonal elements of $\sf E_{{\sf i}}^{}$ itself.

In general, $\sf E_{{\sf i}}^{}$ will be more complicated for antennas with less symmetry. In some exotic cases, it may not be very useful to split off $\sf D_{{\sf i}}^{}$ or even $\sf P_{{\sf i}}^{}$, although it is always allowed. In any case, the M.E. formalism offers a framework for the ful description of the primary beam of any radio telescope that can be conceived.

Position-independent receptor cross-leakage ( $\sf D_{{\sf i}}^{}$)

The off-diagonal elements $\sf e_{{\sf i}{\sf b}{\sf a}}^{}$ and $\sf e_{{\sf i}{\sf a}{\sf b}}^{}$ of $\sf E_{{\sf i}}^{'}$ describe `leakage' between receptors, i.e. the extent to which each receptor is sensitive to the radiation that is supposed to be picked up by the other one.

It is customary to split off the position-independent part $\sf e_{{\sf i}{\sf b}{\sf a}}^{'}$ and $\sf e_{{\sf i}{\sf a}{\sf b}}^{'}$ of this leakage into a separate matrix $\sf D_{{\sf i}}^{}$:

$$\sf E_{{\sf i}}^{'} \vec{\rho}\, \left(\vphantom{\begin{array}{cc}{\sf e}_{{\sf i}{\sf a}{\sf a}} ... ...}_{{\sf i}{\sf a}{\sf b}} & {\sf e}_{{\sf i}{\sf b}{\sf b}} \end{array}}\right. \begin{array}{cc}{\sf e}_{{\sf i}{\sf a}{\sf a}} & {\sf e}_{{\sf ... ...sf e}^{'}_{{\sf i}{\sf a}{\sf b}} & {\sf e}_{{\sf i}{\sf b}{\sf b}} \end{array} \left.\vphantom{\begin{array}{cc}{\sf e}_{{\sf i}{\sf a}{\sf a}} ... ...}_{{\sf i}{\sf a}{\sf b}} & {\sf e}_{{\sf i}{\sf b}{\sf b}} \end{array}}\right)$$ =

$$\approx \left(\vphantom{\begin{array}{cc}1 & {\sf e}_{{\sf i}{\sf b}{\sf ... ...f i}{\sf a}{\sf b}}^{'}/{\sf e}_{{\sf i}{\sf a}{\sf a}} & 1 \end{array}}\right. \begin{array}{cc}1 & {\sf e}_{{\sf i}{\sf b}{\sf a}}^{'}/{\sf e}_... ... e}_{{\sf i}{\sf a}{\sf b}}^{'}/{\sf e}_{{\sf i}{\sf a}{\sf a}} & 1 \end{array} \left.\vphantom{\begin{array}{cc}1 & {\sf e}_{{\sf i}{\sf b}{\sf ... ...f i}{\sf a}{\sf b}}^{'}/{\sf e}_{{\sf i}{\sf a}{\sf a}} & 1 \end{array}}\right) \left(\vphantom{\begin{array}{cc}{\sf e}_{{\sf i}{\sf a}{\sf a}} ... ...}_{{\sf i}{\sf a}{\sf b}} & {\sf e}_{{\sf i}{\sf b}{\sf b}} \end{array}}\right. \begin{array}{cc}{\sf e}_{{\sf i}{\sf a}{\sf a}} & {\sf e}_{{\sf ... ...sf e}^{'}_{{\sf i}{\sf a}{\sf b}} & {\sf e}_{{\sf i}{\sf b}{\sf b}} \end{array} \left.\vphantom{\begin{array}{cc}{\sf e}_{{\sf i}{\sf a}{\sf a}} ... ...}_{{\sf i}{\sf a}{\sf b}} & {\sf e}_{{\sf i}{\sf b}{\sf b}} \end{array}}\right)$$

$$\left(\vphantom{\begin{array}{cc}{\sf d}_{{\sf i}{\sf a}{\sf a}} ... ...}_{{\sf i}{\sf a}{\sf b}} & {\sf d}_{{\sf i}{\sf b}{\sf b}} \end{array}}\right. \begin{array}{cc}{\sf d}_{{\sf i}{\sf a}{\sf a}} & {\sf d}_{{\sf ... ...\ {\sf d}_{{\sf i}{\sf a}{\sf b}} & {\sf d}_{{\sf i}{\sf b}{\sf b}} \end{array} \left.\vphantom{\begin{array}{cc}{\sf d}_{{\sf i}{\sf a}{\sf a}} ... ...}_{{\sf i}{\sf a}{\sf b}} & {\sf d}_{{\sf i}{\sf b}{\sf b}} \end{array}}\right) \left(\vphantom{\begin{array}{cc}{\sf e}_{{\sf i}{\sf a}{\sf a}} ... ...}_{{\sf i}{\sf a}{\sf b}} & {\sf e}_{{\sf i}{\sf b}{\sf b}} \end{array}}\right. \begin{array}{cc}{\sf e}_{{\sf i}{\sf a}{\sf a}} & {\sf e}_{{\sf ... ...sf e}^{'}_{{\sf i}{\sf a}{\sf b}} & {\sf e}_{{\sf i}{\sf b}{\sf b}} \end{array} \left.\vphantom{\begin{array}{cc}{\sf e}_{{\sf i}{\sf a}{\sf a}} ... ...}_{{\sf i}{\sf a}{\sf b}} & {\sf e}_{{\sf i}{\sf b}{\sf b}} \end{array}}\right) \sf D_{{\sf i}}^{} \sf E_{{\sf i}}^{} \vec{\rho}\,$$ = (37)

Usually, the position-dependent leakage coefficients $\sf e_{{\sf i}{\sf b}{\sf a}}^{}$ and $\sf e_{{\sf i}{\sf a}{\sf b}}^{}$ are assumed to be zero, but that is not always justified.

If the leakage coefficients are determined empirically by calibration, it is not necessary to know the details of the leakage mechanism. It is sufficient to solve for the elements of $\sf D_{{\sf i}}^{}$. In that case, there is only one form:

$$\sf D^{+}_{{\sf i}} \sf D^{\odot}_{{\sf i}} \sf D_{{\sf i}}^{} \left(\vphantom{\begin{array}{cc}{\sf d}_{{\sf i}{\sf a}{\sf a}} ... ...}_{{\sf i}{\sf a}{\sf b}} & {\sf d}_{{\sf i}{\sf b}{\sf b}} \end{array}}\right. \begin{array}{cc}{\sf d}_{{\sf i}{\sf a}{\sf a}} & {\sf d}_{{\sf ... ... {\sf d}_{{\sf i}{\sf a}{\sf b}} & {\sf d}_{{\sf i}{\sf b}{\sf b}} \end{array} \left.\vphantom{\begin{array}{cc}{\sf d}_{{\sf i}{\sf a}{\sf a}} ... ...}_{{\sf i}{\sf a}{\sf b}} & {\sf d}_{{\sf i}{\sf b}{\sf b}} \end{array}}\right)$$ (38)

But in many cases, position-independent leakage can be physically explained by deviations $\phi$ from the nominal receptor position angles (see $\sf P_{{\sf i}}^{}$), and by deviations $\theta$ from nominal receptor `ellipticities' $\theta$. For linear polarisation coordinates:

$$\sf D^{+}_{{\sf i}} \left(\vphantom{\begin{array}{cc}{\sf d}_{{\sf i}{\sf a}{\sf a}} ... ...}_{{\sf i}{\sf a}{\sf b}} & {\sf d}_{{\sf i}{\sf b}{\sf b}} \end{array}}\right. \begin{array}{cc}{\sf d}_{{\sf i}{\sf a}{\sf a}} & {\sf d}_{{\sf ... ...\ {\sf d}_{{\sf i}{\sf a}{\sf b}} & {\sf d}_{{\sf i}{\sf b}{\sf b}} \end{array} \left.\vphantom{\begin{array}{cc}{\sf d}_{{\sf i}{\sf a}{\sf a}} ... ...}_{{\sf i}{\sf a}{\sf b}} & {\sf d}_{{\sf i}{\sf b}{\sf b}} \end{array}}\right) \sf Ell \theta_{{\sf i}{\sf a}} \theta_{{\sf i}{\sf b}} \sf Rot \phi_{{\sf i}{\sf a}} \phi_{{\sf i}{\sf b}}$$ =

$$\approx \sf Ell \theta_{{\sf i}{\sf a}} \theta_{{\sf i}{\sf a}} \sf Rot \phi_{{\sf i}{\sf a}}$$ (39)

The $\approx$ sign gives the approximation for a well-designed system. Often the two receptors are mounted in a single unit, so position angle deviations caused by mechanical bending of the feed structure are the same for both: $\phi_{{\sf i}{\sf a}}$ = $\phi_{{\sf i}{\sf b}}$. One might also argue that ellipticity should be a reciprocal effect, so that $\theta_{{\sf i}{\sf b}}$ = - $\theta_{{\sf i}{\sf a}}$. This is roughly consistent with WSRT experience, and these two assumptions are implicit in equ 27 of [3]. However, for high accuracy polarisation measurements, the parameters for each receptor should be at least partly independent.

For circular polarisation coordinates (see equ 22):

$$\sf D^{\odot}_{{\sf i}} \cal {H} \sf D^{+}_{{\sf i}} \cal {H} \cal {H} \sf Ell \theta_{{\sf i}{\sf a}} \theta_{{\sf i}{\sf b}} \cal {H} \cal {H} \sf Rot \phi_{{\sf i}{\sf a}} \phi_{{\sf i}{\sf b}} \cal {H}$$ =

$$\approx \sf Rot \theta_{{\sf i}{\sf a}} \sf Diag \phi_{{\sf i}{\sf a}} \phi_{{\sf i}{\sf a}}$$ (40)

Again, the $\approx$ sign gives the approximation for $\phi_{{\sf i}{\sf a}}$ = $\phi_{{\sf i}{\sf b}}$ and $\theta_{{\sf i}{\sf b}}$ = - $\theta_{{\sf i}{\sf a}}$. See equation 34 for an expression for ( $\cal {H}$ $\sf Rot$($\phi_{{\sf i}{\sf a}}$,$\phi_{{\sf i}{\sf b}}$) $\cal {H}$^-1) where $\phi_{{\sf i}{\sf a}}$ $\neq$ $\phi_{{\sf i}{\sf b}}$. The expression for ( $\cal {H}$ $\sf Ell$($\theta_{{\sf i}{\sf a}}$,$\theta_{{\sf i}{\sf b}}$) $\cal {H}$^-1) with $\theta_{{\sf i}{\sf b}}$ $\neq$ - $\theta_{{\sf i}{\sf a}}$ is similar, but with real coefficients, as expected for circular polarisation coordinates.

Commutation ( $\sf Y_{{\sf i}}^{}$)

In some systems, the receptor signals can be switched (commuted) between IF-channels for calibration.

$$\sf Y_{{\sf i}}^{} \left(\vphantom{\begin{array}{cc}1 & 0\\ 0 & 1 \end{array}}\right. \begin{array}{cc}1 & 0\\ 0 & 1 \end{array} \left.\vphantom{\begin{array}{cc}1 & 0\\ 0 & 1 \end{array}}\right) \sf Y_{{\sf i}}^{} \left(\vphantom{\begin{array}{cc}0 & 1\\ 1 & 0 \end{array}}\right. \begin{array}{cc}0 & 1\\ 1 & 0 \end{array} \left.\vphantom{\begin{array}{cc}0 & 1\\ 1 & 0 \end{array}}\right)$$ (41)

Hybrid ( $\sf H_{{\sf i}}^{}$)

In some cases, circularly polarised receptors consist of linearly polarised dipoles, followed by a `hybrid'. The latter is an electronic implementation of the coordinate transformation matrix $\cal {H}$ from linear to circular polarisation coordinates:

$$\sf H_{{\sf i}}^{} \approx \cal {H}$$ (42)

See equation 18 for the definition of $\cal {H}$. If no hybrid is present, $\sf H_{{\sf i}}^{}$ is the unit matrix. Any gain effects in these electronic components are ignored, or rather they are assumed to be `absorbed' by the gain matrix $\sf G_{{\sf i}}^{}$.

Electronic gain ( $\sf G_{{\sf i}}^{}$)

The matrix $\sf G_{{\sf i}}^{}$ represents the product of all complex electronic gain effects per output IF-channel $\sf p$ and $\sf q$. It models the effects of all feed-based electronics (amplifiers, mixers, LO, cables etc). (The correlator causes interferometer-based effects, which are discussed in section 3).

$$\sf G^{+}_{{\sf i}} \sf G^{\odot}_{{\sf i}} \left(\vphantom{\begin{array}{cc}{\sf g}_{{\sf i}{\sf p}{\sf p}} ... ...}_{{\sf i}{\sf p}{\sf q}} & {\sf g}_{{\sf i}{\sf q}{\sf q}} \end{array}}\right. \begin{array}{cc}{\sf g}_{{\sf i}{\sf p}{\sf p}} & {\sf g}_{{\sf ... ... {\sf g}_{{\sf i}{\sf p}{\sf q}} & {\sf g}_{{\sf i}{\sf q}{\sf q}} \end{array} \left.\vphantom{\begin{array}{cc}{\sf g}_{{\sf i}{\sf p}{\sf p}} ... ...}_{{\sf i}{\sf p}{\sf q}} & {\sf g}_{{\sf i}{\sf q}{\sf q}} \end{array}}\right) \approx \left(\vphantom{\begin{array}{cc}{\sf g}_{{\sf i}{\sf p}} & 0\\ 0 & {\sf g}_{{\sf i}{\sf q}} \end{array}}\right. \begin{array}{cc}{\sf g}_{{\sf i}{\sf p}} & 0\\ 0 & {\sf g}_{{\sf i}{\sf q}} \end{array} \left.\vphantom{\begin{array}{cc}{\sf g}_{{\sf i}{\sf p}} & 0\\ 0 & {\sf g}_{{\sf i}{\sf q}} \end{array}}\right) \sf Diag \sf g_{{\sf i}{\sf p}}^{} \sf g_{{\sf i}{\sf q}}^{}$$ (43)

The $\approx$ sign indicates that electronic cross-talk is assumed to be absent in well-designed systems, i.e. $\sf g_{{\sf i}{\sf p}{\sf q}}^{}$ = $\sf g_{{\sf i}{\sf q}{\sf p}}^{}$ = 0. Since this kind of crosstalk is not necessarily reciprocal, $\sf g_{{\sf i}{\sf p}{\sf q}}^{}$ $\neq$ $\sf g_{{\sf i}{\sf q}{\sf p}}^{}$.

In reality, $\sf G_{{\sf i}}^{}$ will be a product of many electronic gain matrices, one for each linear electronic component in the system: $\sf G_{{\sf i}}^{}$  =  $\sf G_{{\sf i}}^{LNA}$ $\sf G_{{\sf i}}^{mixers}$ $\sf G_{{\sf i}}^{cables}$ $\sf G_{{\sf i}}^{IF-system}$  ^... Although a solver will not be able to distinguish these different effects from each other, but it is useful for simulation of instrumental effects.

Do we need a configuration matrix ( $\sf C_{{\sf i}}^{}$)?

NB: This section is a little polemical, and should disappear when things are more settled.

There has been some debate about the concept of a `configuration matrix' $\sf C_{{\sf i}}^{}$, as proposed by [2], which models the nominal feed configuration. It represents an idealised coordinate transformation `from the frame of the rotating antenna mount to the electronic voltage frame'. It models any rotation of the receptors w.r.t. `the antenna mount', which must be added to the `parallactic' rotation $\sf P_{{\sf i}}^{}$ of the antenna w.r.t. the sky. $\sf C_{{\sf i}}^{}$ also models the hybrid $\sf H_{{\sf i}}^{}$ if present, but it ignores the primary beam $\sf E_{{\sf i}}^{}$. Any deviations from this idealised behaviour are covered by the `leakage' matrix $\sf D_{{\sf i}}^{}$.

However, the proposed $\sf C_{{\sf i}}^{}$ is most suitable for the special case of fully steerable parabolic antennas. The introduction of an intermediate antenna coordinate frame seems an unnecessary complication in those cases where the mirror is not steerable, or is absent entirely (like in a dipole array). Moreover, $\sf C_{{\sf i}}^{}$ violates the rules of modelling by lumping together two effects that have nothing to do with each other, and do not even occur at the same point in the signal path.

In principle it is a good idea to have one matrix that models the transition from electric fields (V/m) to electric voltages (V), and this is precisely what $\sf B_{{\sf i}}^{}$ does. This very general matrix can be split up if relevant into sub-matrices like $\sf P_{{\sf i}}^{}$, $\sf E_{{\sf i}}^{}$ and $\sf D_{{\sf i}}^{}$. The matrix $\sf H_{{\sf i}}^{}$ has no part in this, since it represents a rearranging of electronic signals (V), just like $\sf Y_{{\sf i}}^{}$ (and will come after $\sf Y_{{\sf i}}^{}$ if present!). The projection matrix $\sf P_{{\sf i}}^{}$ takes care of the entire orientation angle of the receptors w.r.t. the sky, which is the only thing that really counts.