File:  /aips++/nfra/185.latex Symbols File:  /aips++/nfra/megi-symbols.tex

Contents

1 INTRODUCTION

The matrix-based Measurement Equation (ME) of a Generic Radio Telescope was developed by Hamaker, Bregman and Sault [2] [3], based on earlier work by Bregman [1]. After discussion by Noordam [5] and Cornwell [6] [7] [8] [9] [10] [11], the M.E. has been adopted as the generic foundation of the uv-data calibration and imaging part of AIPS++. In the not too distant future, an ‘official’ AIPS++ description of the ME will be needed, with agreed conventions and nomenclature (see Appendix A). This note is a step towards that goal.

The heart of the M.E. is formed by the $2\times 2$ feed-based ‘Jones’ matrices, which describe the effects of various parts of the observing instrument on the signal. The main section of this document is devoted to describing the basic form of the Jones matrices in linear and circular polarisation coordinates. Another section discusses the conditions under which their order may be modified (matrices do not always commute).

It is expected that the details of the M.E. (and of this note) will be refined during the first few iterations of design and implementation of AIPS++. But the structure of the M.E. formalism as presented here appears to be rich enough to accomodate all existing and planned radio telescopes. This includes ‘exotic’ ones like cylindrical mirrors, phased arrays, and interferometer arrays with very dissimilar antennas. Further refinements should only require the addition of new Jones matrices, or devising new expressions for existing matrix elements.

In order to test this bold assertion, the various institutes might endeavour to model their own telescopes in terms of the precise and common language of the M.E., using this note as a reference. The following ‘rules’ are probably good ones:

• In modelling an instrument, stay as close to the actual physical situation as possible. Violations of this principle, for whatever reasons, will lead to problems sooner or later.
• It is counterproductive to try and simplify the M.E. to make it ‘look more tractable’. This practice introduces hidden assumptions, which tend to be forgotten by the programmer, and unknown to the user.
• Use the suggested nomenclature and conventions.

It is also good to realise that there are two basic forms of ME, which should not be confused: In the physical form, each instrumental effect is modelled separately by its own matrix. This is useful for simulation purposes. In the mathematical form, effects are ‘lumped together’ if they cannot be solved for separately. Example: the various contributions to the receiver gain, and tropospheric gain.

Acknowledgements: The author has greatly benefited from detailed discussions with Jayaram Chengalur, Jaap Bregman, Johan Hamaker, Tim Cornwell, Wim Brouw and Mark Wieringa.

2 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) $\overrightarrow{\rho }=\overrightarrow{\rho }\left(l,m\right)$ 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’ $\overrightarrow{V}ij$, which is related to the so-called ‘Stokes vector’ $\overrightarrow{I}\left(l,m\right)$ of the observed source by a matrix equation,

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

The ‘Stokes matrix’ $S$ is a constant $4\times 4$ coordinate transformation matrix. It is discussed in detail in section 4 below. The real heart of the M.E. is the ‘direct matrix product’ $Ji\otimes Jj\ast$ of two $2\times 2$ feed-based Jones matrices.

The ‘Stokes-to-Stokes’ transmission of a Stokes vector through an ‘optical’ element may be described by multiplication with a $4\times 4$ Mueller matrix $\mathcal{ℳ}ij$ [2] [3]. Using equation 1:

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

2.1 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 $4\times 4$ interferometer response matrix $Jij$ then consists of a ‘direct matrix product’² $Ji\otimes Jj\ast$ of two $2\times 2$ 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 $2\times 2$ Jones matrix $Ji$ for feed $i$ can be decomposed into a product of several $2\times 2$ Jones matrices, each of which models a specific feed-based instrumental effect in the signal path:

in which

Matrices between brackets ([ ]) are not present in all systems. $Bi$ is the ‘Total Voltage Pattern’ of an arbitrary feed, which is usually split up into three sub-matrices: $DiEiPi$. Jones matrices that model ‘image-plane’ effects depend on the source position (direction) $\overrightarrow{\rho }$. Some also depend on the antenna position $\overrightarrow{r}i$. 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.

2.2 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:

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

See also section 6.4. The matrix $Qi$ models electronic gain effects on the added signal of the tied array feed $i$. The $Qi$ 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 $Kij\left(\overrightarrow{u}ij.\overrightarrow{\rho }\right)$ into $Ki\left(\overrightarrow{r}i.\overrightarrow{\rho }\right)$ and $Kj\left(\overrightarrow{r}j.\overrightarrow{\rho }\right)$, 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.

2.3 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.

3 THE FULL MEASUREMENT EQUATION

3.1 Summing and averaging

For $k$ ‘real’ incoherent sources, observed with a ‘real’ telescope, equ 1 becomes:

The visibility vector $\overrightarrow{V}ij$ is integrated over the extent of the sources ($\int \; <!--nolimits-->dldm$), over the integration time ($\int \; <!--nolimits-->dt$) and over the channel bandwidth ($\int \; <!--nolimits-->df$). Integration over the aperture ($\int \; <!--nolimits-->dudv$) is taken care of by the primary beam properties.

There are only four integration coordinates, whose units are determined by the flux density units in which $\overrightarrow{I}$ is expressed: $energy∕sec∕Hz∕beam$. These coordinates define a 4-dimensional ‘integration cell’. If the variation of $\overrightarrow{V}\left(f,t,l,m\right)$ is linear over this cell, integration is not necessary:

in which ${\overrightarrow{V}}_{0k}$ is the value for source $k$ at the centre of the cell, for $\Delta f=1Hz$ and $\Delta t=1sec$. If the variation of $\overrightarrow{V}\left(f,t,l,m\right)$ over the cell can be approximated by a polynomial of order $\le 3$, then it is sufficient to calculate only the 2nd derivative(s) at the centre of the cell:

Here it is assumed that the 2nd derivatives are be constant over the cell, i.e. the cross-derivatives $\frac{\partial {\overrightarrow{V}}_0}{\partial p_1\partial p_2}$ are zero.

3.2 interferometer-based effects

Until now, we have assumed that all instrumental effects could be factored into feed-based contributions, i.e. we have ignored any interferometer-based effects. This is justified for a well-designed system, provided that the signal-to-noise ratio is large enough (thermal noise causes interferometer-based errors, albeit with a an average of zero). However, if systematic errors do occur, they can be modelled:

The $4\times 4$ diagonal matrix $X$, the ‘Correlator matrix’, represents interferometer-based corrections that are applied to the uv-data in software by the on-line system. Examples are the Van Vleck correction. In the newest correlators, it approaches a constant ($x$).

The $4\times 4$ diagonal matrix $M$ represents multiplicative interferometer-based effects.

The 4-element vector $\overrightarrow{A}ij$ represents additive interferometer-based effects. Examples are receiver noise, and correlator offsets.

In some cases, interferometer-based effects can be calibrated, e.g. when they appear to be constant in time. It will be interesting to see how many of them will disappear as a result of better modelling with the Measurement Equation. In any case, it is desirable that the cause of interferometer-based effects is properly understood (simulation!).

4 POLARISATION COORDINATES

In the $2\times 2$ signal domain, the electric field vector $\overrightarrow{E}$ of the incident plane wave can be represented either in a linear polarisation coordinate frame $\left(x,y\right)$ or a circular polarisation coordinate frame $\left(r,l\right)$. Jones matrices are linear operators in the chosen frame:

For linear polarisation coordinates, equation 1 becomes:

and there is a similar expression for circular polarisation coordinates. Thus, as emphasised in [2], the Stokes vector $\overrightarrow{I}\left(l,m\right)$ and the coherency vector $\overrightarrow{V}ij$ represent the same physical quantity, but in different abstract coordinate frames. A ‘Stokes matrix’ $S$ is a coordinate transformation matrix in the $4\times 4$ coherency domain: $S+$ transforms the representation from Stokes coordinates (I,Q,U,V) to linear polarisation coordinates ($xx,xy,yx,yy$). Similarly, $S\odot$ transforms to circular polarisation coordinates ($rr,rl,lr,ll$). Following the convention of [4], we write:³

$S$-matrices are almost unitary, i.e. except for a normalising constant: ${\left(S\right)}^{-1}=2{\left(S\right)}^{\ast T}$. $S$ cannot be factored into feed-based parts. The two Stokes matrices are related by:

with⁴

Most Jones matrices will have the same form in both polarisation coordinate frames. But if a Jones matrix is expressed in terms of parameters that are defined in one of the two frames, it will have two different but related forms. This is the case for Faraday rotation $Fi$, receptor orientation $Pi$, and receptor cross-leakage $Di$, in which the orientation w.r.t. the $x,ccY$ frame plays a role. The two forms of a Jones matrix $A$ can be converted into each other by the coordinate transformation matrix $\mathcal{\mathscr{H}}$ and its inverse:

The conversion may be done by hand, using (the elements $a,b,c,d$ may be complex):

Applying these general expressions to rotation $Rot\left(\alpha \right)$ and ellipticity $Ell\left(\alpha ,-\alpha \right)$ matrices (see Appendix for their definition), the conversions are:

Usually, all matrices in a ‘Jones chain’ will be defined in the same coordinate frame. An exception is the case where linear dipole receptors are used in conjunction with a ‘hybrid’ $Hi$ to create pseudo-circular receptors:

in which $Hi$ represents an electronic implementation of the coordinate transformation matrix $\mathcal{\mathscr{H}}$. All these expressions are equivalent in the sense that, in conjunction with the indicated Stokes matrix, they produce a coherency vector in circular polarisation coordinates. The choice of which expression to use depends on whether one wishes to model the feed explicitly in terms of its physical (dipole) properties, or whether one wishes to regard is as a ‘black box’ circular feed with unknown internal structure.

5 GENERIC FORM OF JONES MATRICES

In this section, the ‘generic’ form of various $2\times 2$ 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. $fixx$). 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: $yibp$ is an element of matrix $Y$ for feed $i$, which models the ‘coupling factor’ for the signal going from receptor $b$ to IF-channel $p$. Where possible, the expressions have been reduced to matrices like the diagonal matrix ($Diag$), rotation matrix ($Rot$) etc. These are defined in the Appendix.

5.1 Ionospheric Faraday rotation ($Fi\left(\overrightarrow{\rho },\overrightarrow{r}i\right)$)

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

In circular polarisation coordinates, the matrix $F\odot i$ 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 $\chi i$ (see above), we get:

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

5.2 Atmospheric gain ($Ti\left(\overrightarrow{\rho },\overrightarrow{r}i\right)$)

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

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 $ti={exp}^{i\psi i}$ in which the phase is assumed to be a low-order 2D polynomial as a function of the feed position $\overrightarrow{r}$: $\psi i=a_0\left(t\right)+a_1\left(t\right)\overrightarrow{r}i+a_2\left(t\right)\overrightarrow{r}{i}^2+\cdots$

5.3 Fourier Transform kernel ($Ki\left(\overrightarrow{r}i.\overrightarrow{\rho }\right)$)

The matrix $Ki$ 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 $Ki$ does not depend on the polarisation coordinate frame, there is only one form:

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

If $kiaa=kibb$, the interferometer matrix $Kij=\left(Ki\otimes Kj\ast \right)$ is a $4\times 4$ diagonal matrix with equal elements. This is equivalent to a multiplicative factor of the familiar form $kij=kik{j}^{\ast }=\frac{1}{n}{exp}^{i2\pi \left(\overrightarrow{r}i-\overrightarrow{r}j\right).\overrightarrow{\rho }∕\lambda }=\frac{1}{n}{exp}^{i\overrightarrow{u}ij.\overrightarrow{\rho }}$, i.e. the Fourier Transform kernel or ‘phase weight’ for the baseline $\overrightarrow{u}ij$. For small fields, $n\approx 1$, so $\overrightarrow{u}ij.\overrightarrow{\rho }=\left(ul+vm+w\left(n-1\right)\right)\approx \left(ul+vm\right)$ becomes a 2D FT.

The receptors of a feed are practically always co-located, i.e. they have the same phase-centre: $\overrightarrow{r}ia=\overrightarrow{r}ib=\overrightarrow{r}i$, so $kiaa=kibb=ki$. But note that it is possible to model a receptors that are not co-located, i.e. $\overrightarrow{r}ia\ne \overrightarrow{r}ib$. 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.

5.4 Projection matrix ($Pi$) if $\gamma xa=\gamma yb$

The ‘Projection matrix’ models the projected orientation of the receptors w.r.t. the electrical $x,y$ 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 $Pi$ (see section 4). For linear polarisation coordinates:

in which $\gamma xa$ is the projected angle between the positive $x$-axis and the orientation of receptor $a$ (see also Appendix ??). 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 ($\gamma xa\ne \gamma yb$).

For circular polarisation coordinates:

It is sometimes useful to introduce an intermediate coordinate frame, attached to the feed $i$. In that case: $\gamma xa=\gamma xi+\gamma ia=\beta +\gamma ia$. The ‘offset’ angle $\gamma ia$ between receptor $a$ and the frame of feed $i$ 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 $HA\left(t\right)$ for an alt-az feed:

5.5 Projection matrix ($Pi$) if $\gamma xa\ne \gamma yb$

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. $\gamma xa\ne \gamma yb$.

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

For linear polarisation coordinates $P+i$ becomes a ‘pseudo-rotation’ (compare with equ 29 above):

For circular polarisation coordinates:

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 $\gamma xa$ between a North-South (NS) dipole and the $x$-axis differs from the projected angle $\gamma yb$ between an East-West (EW) dipole and the $y$-axis (I hope this is correct now):

5.6 Voltage primary beam ($Ei\left(\overrightarrow{\rho }\right)$)

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 $Bi$, which fully describes the conversion of the incident electric field (V/m) into two voltages (V):

NB: Since the Jones matrix $Ji$ is feed-based, it deals with voltage beams. The power beam for interferometer $ij$ is modelled by $Bi\otimes Bj\ast$. 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 $Bi$ into a chain of sub-matrices:

• It is always possible to split off a projection matrix $Pi$:
  $Bi=\left(BiP{i}^{-1}\right)Pi=E{i}^{{}^{\prime }}Pi$
  See sections 5.4 and 5.5 above.
• It is always possible to split off a position-independent leakage matrix $Di$:
  $E{i}^{{}^{\prime }}Pi=Di\left(D{i}^{-1}E{i}^{{}^{\prime }}\right)Pi=DiEiPi$
  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 $Pi$ and $Di\left(\overrightarrow{\rho }\right)$ is an (approximately) real and diagonal matrix $Ei$ which decsribes the position-dependent primary beam attenuation and the position-dependent leakage (see also equation 38 below):

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

Note that the two receptor beams are each described in their own coordinate frame $l^{″}ia,m^{″}ia$ and $l^{″}ib,m^{″}ib$ projected on the sky (see Appendix A). The projection matrix $Pi$ 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 $\left(1+𝜖\right)$, 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 $Di$, they must be modelled as off-diagonal elements of $Ei$ itself.

In general, $Ei$ will be more complicated for antennas with less symmetry. In some exotic cases, it may not be very useful to split off $Di$ or even $Pi$, 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.

5.7 Position-independent receptor cross-leakage ($Di$)

The off-diagonal elements $eiba$ and $eiab$ of $E{i}^{{}^{\prime }}$ 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 $e{iba}^{{}^{\prime }}$ and $e{iab}^{{}^{\prime }}$ of this leakage into a separate matrix $Di$:

Usually, the position-dependent leakage coefficients $eiba$ and $eiab$ 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 $Di$. In that case, there is only one form:

But in many cases, position-independent leakage can be physically explained by deviations $\varphi$ from the nominal receptor position angles (see $Pi$), and by deviations $𝜃$ from nominal receptor ‘ellipticities’ $𝜃$. For linear polarisation coordinates:

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: $\varphi ia=\varphi ib$. One might also argue that ellipticity should be a reciprocal effect, so that $𝜃ib=-𝜃ia$. 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):

Again, the $\approx$ sign gives the approximation for $\varphi ia=\varphi ib$ and $𝜃ib=-𝜃ia$. See equation 34 for an expression for ($\mathcal{\mathscr{H}}Rot\left(\varphi ia,\varphi ib\right)\mathcal{\mathscr{H}}-1$) where $\varphi ia\ne \varphi ib$. The expression for ($\mathcal{\mathscr{H}}Ell\left(𝜃ia,𝜃ib\right)\mathcal{\mathscr{H}}-1$) with $𝜃ib\ne -𝜃ia$ is similar, but with real coefficients, as expected for circular polarisation coordinates.

5.8 Commutation ($Yi$)

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

5.9 Hybrid ($Hi$)

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 $\mathcal{\mathscr{H}}$ from linear to circular polarisation coordinates:

See equation 18 for the definition of $\mathcal{\mathscr{H}}$. If no hybrid is present, $Hi$ 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 $Gi$.

5.10 Electronic gain ($Gi$)

The matrix $Gi$ represents the product of all complex electronic gain effects per output IF-channel $p$ and $q$. 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).

The $\approx$ sign indicates that electronic cross-talk is assumed to be absent in well-designed systems, i.e. $gipq=giqp=0$. Since this kind of crosstalk is not necessarily reciprocal, $gipq\ne giqp$.

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

5.11 Do we need a configuration matrix ($Ci$)?

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’ $Ci$, 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 $Pi$ of the antenna w.r.t. the sky. $Ci$ also models the hybrid $Hi$ if present, but it ignores the primary beam $Ei$. Any deviations from this idealised behaviour are covered by the ‘leakage’ matrix $Di$.

However, the proposed $Ci$ 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, $Ci$ 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 $Bi$ does. This very general matrix can be split up if relevant into sub-matrices like $Pi$, $Ei$ and $Di$. The matrix $Hi$ has no part in this, since it represents a rearranging of electronic signals (V), just like $Yi$ (and will come after $Yi$ if present!). The projection matrix $Pi$ takes care of the entire orientation angle of the receptors w.r.t. the sky, which is the only thing that really counts.

6 THE ORDER OF JONES MATRICES

The Jones matrices in equation 3 generally do not commute, so their order is important. In principle, the matrices must be placed in the ‘physical’ order, i.e. the order of the signal propagation path. But in the equations that are enshrined in existing reduction packages, this is often not the case. This begs the question why these ‘wrong’ equations seem to produce so many good (even spectacular) results. The question is especially important since a different order often results in considerable gains in computational efficiency.

The answer is that, for existing (arrays of) circularly symmetric parabolic feeds, many Jones matrices can be approximated by matrices that do commute with at least some of the others.

6.1 Overview of commutation properties

We will analyse this in terms of those special matrices (see Appendix for their definition), whose commutation properties are:

• Unit matrices $\mathcal{𝒰}$ commute with all matrices.
• Multiplication matrices $Mult\left(a\right)$, i.e. diagonal matrices with equal elements $a$, are equivalent to a multiplicative factor. Therefore, they commute with all matrices.
• Diagonal matrices $Diag\left(a,b\right)$ with unequal elements $a,b$ commute with each other.
• Pure rotation matrices $Rot\left(\alpha \right)$ commute with each other.
• Pseudo rotation matrices $Rot\left(\alpha ,\beta \right)$ do not commute wit each other or with pure rotation matrices $Rot\left(\alpha \right)$. Moreover, there should only be one pseudo rotation matrix in the chain, and it should be to the left of (i.e. after) all other rotation matrices: $Rot\left(\alpha ,\beta \right)Rot\left(\gamma \right)=Rot\left(\alpha +\gamma ,\beta +\gamma \right)\ne Rot\left(\gamma \right)Rot\left(\alpha ,\beta \right)$.
• Ellipticity matrices $Ell\left(\alpha ,\beta \right)$ do not commute with each other , except when $\beta =-\alpha$. Moreover: $Ell\left(\alpha ,\beta \right)Ell\left(\gamma ,-\gamma \right)=Ell\left(\alpha +\gamma ,\beta -\gamma \right)\ne Ell\left(\gamma ,-\gamma \right)Ell\left(\alpha ,\beta \right)$.

In order to study the general implications of changing the order of multiplication, we take the two products $m.M$ and $M.m$ of two general matrices (whose elements may be complex):

The difference (i.e. commutation error) between the two matrix products can be expressed as a matrix $\Delta$:

Thus, by taking the wrong matrix order, one makes the following fractional errors of the following order in the result:  
- in the diagonal elements: of the order of $c∕a$, i.e. the ratio of non-diagonal and diagonal elements of the original matrices (which is often small).  
- in the off-diagonal elements: in the order of $\left(a-b\right)∕a$, i.e. they will be smaller as the diagonal elements of the original matrices are more equal.

If one of the two matrices is diagonal, e.g. $c=d=0$ then this reduces to:

The (not very surprising) conclusion is that the error caused by taking the wrong matrix order is smaller when one of the matrices is diagonal, and the values of its diagonal elements are almsot equal.

6.2 Overview of Jones matrix forms

It is sufficient to discuss the commutation properties of the feed-based Jones matrices because, if $Ai$ commutes with $Bi$ and $Aj$ with $Bj$, then $\left(Ai\otimes Aj\ast \right)$ commutes with $\left(Bi\otimes Bj\ast \right)$:

Inspecting the various Jones matrices separately: