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• Outline
• Glossary
• 7. Observing Systems
  • Previous: 7.1. Jones notation
  • Next: 7.3. Analogue Electronics

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7.2. The Radio Interferometer Measurement Equation (RIME)

The RIME naturally emerges as soon as we apply Jones calculus to an interferometric measurement. The purpose of an aperture synthesis interferometer is to measure complex visibilities, corresponding to the coherency between the E-fields at two different positions, $\langle e_1 e_2^* \rangle$. Let's take a look at how this can be measured.

7.2.1. A simplified model of a two-element interferometer and correlator

The technical details of interferometric measurements will be covered later on in $\S$ 7.3 ➞; for now let us consider a simplified mathematical model. Consider a two-element interferometer. Let us designate the two antennas by the indices $p$ and $q$. Let us designate the E-field vectors at the antenna's nominal positions as $\mathbf{e}_p$ and $\mathbf{e}_q$.

Each antenna has a dual-receptor feed (because we want to capture full polarization information, i.e. both the $e_x$ and $e_y$ components of the incipient EM field, we must use two receptors, e.g. two orthogonal dipoles. See $\S$ 7.6 ➞); the incoming wavefront induces electrical currents in the receptors, which is what our electronics then sample. For now, think of the measured quantities at each antenna as two sets of complex voltages $v_{p1},v_{p2}$ and $v_{q1},v_{q2}$, or two voltage vectors:

$$ \mathbf{v}_p = \left[ \begin{array}{c}v_{p1} \\ v_{p2} \end{array} \right],~~~ \mathbf{v}_q = \left[ \begin{array}{c}v_{q1} \\ v_{q2} \end{array} \right]. $$

These voltages are then fed into a correlator, which computes four complex cross-products called correlations:

$$ V_{11}=\langle v_{p1}v_{q1}^*\rangle,~ V_{12}=\langle v_{p1}v_{q2}^*\rangle,~V_{21}=\langle v_{p2}v_{q1}^*\rangle,~V_{22}=\langle v_{p2}v_{q2}^*\rangle. $$

Note that at this point we're implicitly assuming that:

• the averaging operation $\langle\cdot\rangle$ is done over a time bin of finite (small) extent $\Delta t$

• the radiation is confined to a narrow frequency bin of $[\nu-\Delta\nu/2,\nu+\Delta\nu/2]$. (In actual fact a correlator will explicitly channelize the incoming signal into a set of narrow frequency channels, so the correlations above are computed per time bin, per frequency channel.)

7.2.1.1. The correlation matrix

It proves very convenient to treat the four correlations as a 2x2 matrix, because we can then represent it as the matrix product of $\mathbf{v}_p$ and $\mathbf{v}_q^H$, where $\cdot^H$ is the Hermitian (or conjugate) transpose operator:

$$ \mathbf{V}_{pq} = 2 \left[ \begin{array}{cc}V_{11} & V_{12} \\ V_{21} & V_{22} \end{array} \right ] = 2\left\langle \left[ \begin{array}{c}v_{p1} \\ v_{p2} \end{array} \right] \left[ \begin{array}{cc}v_{q1}^* & v_{q2}^* \end{array} \right ] \right\rangle = 2\langle \mathbf{v}_p \mathbf{v}_q^H \rangle . $$

(NB: the factor of 2 is used as a matter of convention, see Revisiting the radio interferometer measurement equation. I. A full-sky Jones formalism ⤴ for details).

$\mathbf{V}_{pq}$ is called the correlation matrix, or often the visibility matrix. The two terms are used somewhat interchangeably -- as we'll see below, the measured correlations correspond to the "true" underlying visibilities, "corrupted" by the instrument response.

7.2.1.2. Enter Jones

Recall from the previous section ($\S$7.1)➞ that the voltage vectors at each antenna can be expressed as a product of the "source" EM field vector $\mathbf{e}$ and a 2x2 Jones matrix describing the entire signal propagation path from source to receiver. Since the signal comes to the two antennas via two different paths, each antenna will have its own Jones matrix:

$$ \mathbf{v}_p = \mathbf{J}_p \mathbf{e},~~~ \mathbf{v}_q = \mathbf{J}_q \mathbf{e}. $$

Substituting this into the correlation matrix above, we have:

$$ \mathbf{V}_{pq} = 2\langle (\mathbf{J}_p \mathbf{e}) ( \mathbf{J}_p \mathbf{e})^H \rangle = 2\langle \mathbf{J}_p \mathbf{e}\mathbf{e}^H \mathbf{J}_q^H \rangle = \mathbf{J}_p \, 2\langle \mathbf{e}\mathbf{e}^H \rangle \, \mathbf{J}_q^H~~~~~~~~~(\mathrm{using} (\mathbf{AB})^H = \mathbf{B}^H\mathbf{A}^H). $$

Note that this implicitly assumes that ***Jones matrices are constant over the averging interval***.

GSF:use a different script style for the brightness matrix, prehaps mathsf{} or mathcal{}? have been asked by students from clarification with respect to the bandpass Jones term.

Let us now assume a single source of radiation in the sky. The quantity in the angle brackets, $2\langle \mathbf{e}\mathbf{e}^H\rangle$, is then called the source brightness matrix, designated $\mathbf{B}$ (Note: though the notation is the same, this is different from the the B-Jones term which represents the bandpass). Using the definition of the Stokes parameters in the previous section ($\S$7.1)➞, we get a particularly elegant expression for $\mathbf{B}$ in terms of $IQUV$:

$$ \mathbf{B} = 2\langle \mathbf{e}\mathbf{e}^H\rangle = 2 \left[ \begin{array}{cc}e_x e_x^* & e_x e_y^* \\ e_y e_x^* & e_y e_y^* \end{array} \right ] = \left[ \begin{array}{cc}I+Q & U+iV \\ U-iV & I-Q \end{array} \right ] $$

7.2.2. The basic RIME of a point source

This gives us the most basic form of the RIME:

$$ \mathbf{V}_{pq} = \mathbf{J}_{p} \mathbf{B} \mathbf{J}_{q}^H, $$

that is, the measured correlation matrix is a product of three 2x2 matrices -- a Jones matrix representing the signal path to antenna $p$, a brightness matrix capturing the intrinsic properties (flux, polarization) of the source, and a Jones matrix representing the effects of the signal path to antenna $q$.

Recalling the concept of Jones Chains ($\S$7.1.5 ➞ ), with mutiple effects in the signal path, we have:

$$ \mathbf{J}_p = \mathbf{J}_{p,n} \mathbf{J}_{p,n-1} \dots \mathbf{J}_{p,1}, $$

which gives us the "onion form" of the RIME:

$$ \mathbf{V}_{pq} = \mathbf{J}_{p,n}(\dots( \mathbf{J}_{p,1} \mathbf{B} \mathbf{J}_{q,1}^H)\dots)\mathbf{J}_{q,n}^H. $$

The onion form explicitly shows how successive "layers" of propagation effects are applied to the signal as it makes its way from the source to the telescope.

7.2.2.1. Specific RIMEs vs the general RIME

The general formalism above is commontly referred to as The Radio Interferometry Measurement Equation, or simply The Measurement Equation. When we want to describe a specific instrument and/or observation, we string together specific Jones terms, and thus derive a RIME or a measurement equation. This is a very important distinction to make!!

For example, here's a RIME describing a very simple observation that takes into account only complex receiver gains and geometric delay:

$$ \mathbf{V}_{pq} = \mathbf{G}_{p} K_p \mathbf{B} K_q^H \mathbf{G}_{q}^H. $$

Let us now assume a perfect instrument (i.e. gains of unity) and concentrate on the geometric delay term. Our RIME then simplifies to

$$ \mathbf{V}_{pq} = K_p \mathbf{B} K_q^H. $$

Note: the K-Jones matrix is scalar, hence the use of Roman rather than boldface font to designate it.

7.2.2.2. Geometric delays

In a sense, the K-Jones matrix is the most important Jones term of all, since it is exactly what makes interferometry possible. Consider the geometry of our instrument, as we had done in $\S$4.2 ➞. Let us pick a coordinate system $xyz$ with the $z$ axis pointing towards some reference direction on the celestial sphere $S_0$ (i.e. the direction we want to observe in). By convention, $x$ then is chosen to point East, and $y$ North on the sky. Note that this coordinate system is fixed to the sky, and the Earth rotates relative to it. The coordinate origin $O$ can be chosen arbitrarily. Let us designate the antenna positions in this coordinate system by $\mathbf{u}_p=(u_p,v_p,w_p)$ and $\mathbf{u}_q=(u_q,v_q,w_q)$. Let's assume a source of radiation at direction $S$; the unit length vector towards this direction is designated as $\mathbf{\sigma}$.

Figure 7.2.1: UVW write up description, or refer to relevant section 4 the baseline

The pathlength difference of a wavefront arriving from direction $\mathbf{\sigma}$ between the origin $O$ and antenna $p$ is the simply the scalar product $\mathbf{u}_p\cdot\mathbf{\sigma}.$ The K-Jones matrix corresponding to this pathlength difference is a scalar matrix:

$$ K_p = \mathrm{e}^{-2\pi i (\mathbf{u}_p\cdot\mathbf{\sigma})/\lambda} $$

where $\lambda$ is the wavelength of the monochromatic signal. Substituting this into the RIME above, and remembering that scalar matrices commute, we have

$$ \mathbf{V}_{pq} = K_p \mathbf{B} K_q^H = \mathbf{B} K_p K_q^H = \mathbf{B}\,\mathrm{e}^{-2\pi i ((\mathbf{u}_p - \mathbf{u_q})\cdot\mathbf{\sigma})/\lambda} = \mathbf{B}\,\mathrm{e}^{-2\pi i (u_{pq}l+v_{pq}m+w_{pq}n)/\lambda}, $$

where $u_{pq},v_{pq},w_{pq}$ are the components of the baseline vector $\mathbf{u}_{pq}=\mathbf{u}_{p}-\mathbf{u}_{q}$, and $l,m,n$ are the components of the direction vector $\mathbf{\sigma}$ (also called the direction cosines, since they are equal to the cosines of the angle formed by $\mathbf{\sigma}$ w.r.t. the coordinate axes). Note that $n=\sqrt{1-l^2-m^2}$, since $\mathbf{\sigma}$ is a unit length vector.

The complex exponent given by the product of the two antennas' K-terms above corresponds to the Fourier transform. In that sense, the K-Jones term of the RIME is at the heart of all interferometry.

7.2.2.3. Effects of averaging & fringe stopping

In deriving the RIME, we assumed that Jones matrices are constant over the chosen time - and frequency - averaging intervals. For K-Jones, this is clearly not necessarily true -- the $uvw$ coordinate vector changes over time due to the Earth's rotation, and the frequency dependence is explicit in the $1/\lambda$ scaling of the exponent. This results in the phase term $K_{pq}=K_p K_q^H$ being a complex quantity with a rotating complex phase, and averaging a rotating complex phase (i.e. a vector average!) always reduces amplitude. In interferometry, this effect is known as time/frequency smearing. In exteme cases, if the phase rotates through a cycle or more over the averaging interval, the vector average will be close to 0!

This is easy to illustrate with a simple plot:

The figure above plots three series of rotating complex vectors (shown by a segment of the arc that the vector rotates through), as well as their vector average (shown by the arrow). The blue arc and arrow corresponds to a vector rotating through $10^\circ$, the red to $90^\circ$, and the green to $180^\circ$. The amplitude loss due to smearing is quite apparent in the last two cases.

Fortunately, we can mimimize the effect of smearing in a particular direction of interest. Consider again the coordinates figure above -- for a wavefront arriving from direction $S_0$ (our directon of interest), the direction vector is $\mathbf{\sigma}_0 = (0,0,1)$, and the corresponding phase term is

$$ \mathrm{e}^{-2\pi i (\mathbf{u}_{pq}\cdot\mathbf{\sigma_0})/\lambda} = \mathrm{e}^{-2\pi i w_{pq}/\lambda} $$

This phase term corresponds to the excess pathlength $\tau_0$ (see the figure above) associated with baseline $pq$ and direction $S_0$. Since $w_{pq}$ can change as a function of time, averaging over this complex exponent will lead to smearing (i.e. amplitude loss). This can be avoided by having the correlator insert an artificial delay of $-\tau_0$ into its signal chain. The effective K-Jones towards direction $\sigma$ is then given by

$$ K_{pq} = \mathrm{e}^{-2\pi i (u_{pq}l+v_{pq}m+w_{pq}n)/\lambda} \mathrm{e}^{-2\pi i w_{pq}n/\lambda} = \mathrm{e}^{-2\pi i (u_{pq}l+v_{pq}m+w_{pq}(n-1))/\lambda}, $$

and for our direction of interest ($\sigma=\sigma_0$) it is simply the constant 1. This technique is known as fringe stopping, since we are effectively "stopping" the complex fringe in the direction $\sigma_0$ from rotating as a function of frequency and time. The effect of fringe stopping is to minimize both the phase and the resulting phase variations of $K_{pq}$ around the reference direction $S_0$. This is the case here, since $l,m,n-1$ are all small numbers for $S$ close to $S_0$.

In a fringe stopping correlator, the K-Jones term becomes

$$ K_p = \mathrm{e}^{-2\pi i (u_{p}l+v_{p}m+w_{p}(n-1))/\lambda}. $$

Fringe stopping is critically important, since it minimizes amplitude loss due to smearing. It is (almost) universally employed by modern radio correlators.

7.2.3. The all-sky RIME

Consider again the basic RIME of a point source ⤵. The real sky does not contain a single point source, but must rather be described with a brightness distribution. We can express this by making $\mathbf{B}$ a function of direction: $\mathbf{B}(\mathbf{\sigma})$. The signal path from each direction is also different in principle: the Jones matrix should thus also be a function of direction, $\mathbf{J}_p(\mathbf{\sigma})$. Again assuming a linear system, the aggregate correlation $\mathbf{V}_{pq}$ recorded by our interferometer can be obtained by integrating the RIME over the entire celestial spehere:

$$ \mathbf{V}_{pq} = \iint\limits_{4\pi} \mathbf{J}_p(\mathbf{\sigma}) \mathbf{B}(\mathbf{\sigma}) \mathbf{J}_q^H(\mathbf{\sigma}) \mathrm{d}\mathbf{\sigma} $$

To make this integral a little more tractable, let us change variables to $l,m$. This is equivalent to projecting (half of) the celestial spehere onto a plane tangential at $S_0$. We can safely assume that the other half of the sphere is not visible to our instrument: the Earth is in the way...there are a few exotic exceptions to this e.g. space interferometry, but it is generally true. We have:

$$ \mathbf{V}_{pq} = \iint\limits_{lm} \mathbf{J}_p(l,m) \mathbf{B}(l,m) \mathbf{J}_q^H(l,m) \frac{\mathrm{d}\mathbf{\sigma}}{\mathrm{d}l\mathrm{d}m} \mathrm{d}l \mathrm{d}m = \iint\limits_{lm} \mathbf{J}_p(l,m) \mathbf{B}(l,m) \mathbf{J}_q^H(l,m) \frac{1}{n} \mathrm{d}l \mathrm{d}m, $$

where the $1/n$ term arises due to a change in the integration volume from $\mathrm{d}\sigma$ to $\mathrm{d}l\mathrm{d}m$.

Let us now split $\mathbf{J}_p$ into direction-dependent and direction-independent parts. Recall that $\mathbf{J}_p$ is a product of a Jones chain ($\S$7.1 ➞ ) of Jones matrices representing successive propagation effects:

$$ \mathbf{J}_p = \mathbf{J}_{p,n} \mathbf{J}_{p,n-1} ... \mathbf{J}_{p,1}, $$

Some of the terms in the chain are functions of $l,m$ and are thus called direction-dependent effects (DDEs), while others do not depend on $l,m$, and are therefore direction-independent effects (DIEs). As a general trend (but by no means a rule), effects on the left side of the chain ("in the instrument"), such us electronic gain, will be direction-indpendent effects, while effects on the right of the chain ("in the sky") will be direction-dependent effects. Note that there's always at least one DDE present, and that is the geometric delay term $K_p$. Being scalar, we can commute it to any position in the chain, in particular the right most one. Now, let us designate the left most remaining DDE in the chain by $k$ (it is possible that $k=n$, i.e. there are no DIEs). We have:

$$ \mathbf{J}_p(l,m) = (\mathbf{J}_{p,n} ... \mathbf{J}_{p,k+1}) (\mathbf{J}_{p,k} ... \mathbf{J}_{p,2}) K_p = \mathbf{G}_p \mathbf{E}_p(l,m) K_p(l,m) $$

Here, the direction-independent part is designated by $\mathbf{G}$ and the direction-dependent part by $\mathbf{E}$. Substituting this into the integral, and expanding $K$, we have:

$$ \mathbf{V}_{pq} = \mathbf{G}_p \left( \iint\limits_{lm} \frac{1}{n} \mathbf{E}_p(l,m) \mathbf{B}(l,m) \mathbf{E}_q^H(l,m) \mathrm{e}^{-2\pi i(u_{pq}l+v_{pq}m+w_{pq}(n-1))/\lambda} \mathrm{d}l \mathrm{d}m \right ) \mathbf{G}_q^H, $$

which is starting to look like a 2D Fourier transform. One final wrinkle remains: the so-called w-term. The $w_{pq}(n-1)$ term in the exponent can be split into two per-antenna parts, since $w_{pq}=w_p-w_q$. We can define a notional W-Jones as:

$$ W_p = \frac{1}{\sqrt{n}} \mathrm{e}^{w_p(n-1)}, $$

and treat it as a DDE that can be absorbed in the $\mathbf{E}_p$ part of the Jones chain. Without loss of generality, we can then write:

$$ \mathbf{V}_{pq} = \mathbf{G}_p \left( \iint\limits_{lm} \mathbf{B}_{pq}(l,m) \mathrm{e}^{-2\pi i(u_{pq}l+v_{pq}m)/\lambda} \mathrm{d}l \mathrm{d}m \right ) \mathbf{G}_q^H,~~~\mathbf{B}_{pq} = \mathbf{E}_p \mathbf{B} \mathbf{E}_q^H $$

If we express the $uv$ coordinates in units of $\lambda$, the $1/\lambda$ term disappears, and the exponent under the integral is simply a 2D Fourier transform kernel. This explicitly shows that the measured correlations are related to a 2D Fourier transform of the sky.

GSF: start fold in of the old 7.3 section

Figure 7.2.2: This diagram illustrates the terms of the Radio Interferometer Measurement Equation (RIME). The emissions of a radio source are characterised by a brightness matrix $\mathbf{B_{t\lambda s}}$. These emission intensities are converted into the frequency domain by a Fourier transform $\mathbf{K_{tp\lambda s}}$, whose frequencies are subject to direction-dependent and -independent effects, labelled $\mathbf{E_{tp\lambda s}}$ and $\mathbf{G_{tp\lambda}}$ respectively. Direction-dependent terms frequently model antenna beam, ionospheric and tropospheric effects in a radio source direction. By contrast, direction-independent terms are not associated with a particular direction, and can incorporate effects such as temperature fluctuations and polarisation leakages on the physical antenna. The RIME outputs model visibilites $\mathbf{V_{tpq\lambda}}$.

Another example of a Jones chain is

$$ \mathbf{J} = \mathbf{BG_\mathrm{instr}DE_\mathrm{PB}PT} $$

where

$\mathbf{B}$ is the Jones matrix corresponding to the bandpass gain, a direction-idependent effect,

$\mathbf{G}_\mathrm{instr}$ is the Jones matrix corresponding to the receiver/instrumental gain, a direction-independent effect,

$\mathbf{D}$ is the Jones matrix corrsponding to polarization leakage, a direction-independent effect,

$\mathbf{E}_\mathrm{PB}$ is the Jones matrix corresponding to the antenna primary beam, a direction-dependent effect,

$\mathbf{P}$ is the Jones matrix corresponding to parallactic angle rotation, a direction-independent effect,

$\mathbf{T}$ is the Jones matrix corresponding to ionospheric and tropospheric effects, which are direction-dependent effects.

GSF: stop fold in of the old 7.3 section

7.2.3.1. When is it a Fourier transform?

In $\S$ 4 ➞ we implicitly assumed that an interferometer directly measures visibilities that correspond to a 2D Fourier transform of the sky. This is the underlying assumption of classical interferometry. In the literature you will often see mention of DDEs and "wide field" and the "$w$-term" in the context of breaking this assumption, and thus introducing distortions. The RIME formulation above makes it easy to understand the Fourier transform assumption and what happens when it's broken.

The matrix $\mathbf{B}_{pq}$ in the equation above is known as the apparent sky seen by baseline $pq$. If we assume for a minute that every baseline sees the same apparent sky $\mathbf{B}_\mathrm{app}$ throughout the duration of the measurement (and remember that it may take several hours to fill the $uv$-plane), then the RIME becomes a true Fourier transform:

$$ \mathbf{V}_{pq} = \mathbf{G}_p \left( \iint\limits_{lm} \mathbf{B}_\mathrm{app}(l,m) \mathrm{e}^{-2\pi i(u_{pq}l+v_{pq}m)/\lambda} dl dm \right ) \mathbf{G}_q^H $$

this is often written in shorthand as

$$ \mathbf{V}_{pq} = \mathbf{G}_p \mathbf{X}_{pq} \mathbf{G}_q^H $$

and interpreted as follows: each baseline, at each point in time, observes a coherency that corresponds to a 2D Fourier transform of the sky brightness distribution, multiplied by per-antenna G-Jones matrices corresponding to direction-independent instrumental effects. Basic direction-independent calibration ($\S$8.3 ➞) (also known as second-generation calibration, or 2GC) consists of fitting the $\mathbf{G}$ terms in some sort of least-squares way, and the inverse of the solutions $\mathbf{\tilde G}$ to the observed data, to obtain corrected data

$$ \mathbf{V}_{pq}^\mathrm{(corr)} = \mathbf{\tilde G}_p^{-1} \mathbf{V}_{pq} \mathbf{\tilde G}_q^{-H} \approx \mathbf{X}_{pq} $$

which is then imaged to obtain an image of the sky.

7.2.3.2. Apparent skies and classical interferometry

The question is, when can we assume that every baseline sees the same apparent sky? Since $\mathbf{B}_{pq} = \mathbf{E}_p \mathbf{B} \mathbf{E}_q^H$, the abstract answer is simple: when the DDE term $\mathbf{E}_p$ is the same across all antennas, and constant in time. We call such DDEs trivial DDEs:

$$ \mathrm{if}~~\mathbf{E}_p(t) \equiv \mathbf{E}~~\mathrm{for~all}~~t,p,~~\mathrm{then}~~\mathbf{B}_\mathrm{app}=\mathbf{EBE}^H $$

To understand whether a DDE is trivial or not, we need to consider what effects go into $\mathbf{E}_p$ more closely. For starters, we introduced a W-Jones above and absorbed it into $\mathbf{E}_p$:

$$ W_p = \frac{1}{\sqrt{n}} \mathrm{e}^{w_p(n-1)}, $$

This is generally not a trivial DDE: the $w_p$ coordinate can be different for all antennas, and varies as a function of time (since the Earth rotates). However, if one of two conditions is met:

• the field-of-view is small, therefore $n=\sqrt{1-l^2-m^2}\approx1$

• the array is co-planar and remains so for the duration of the observation (i.e. all antennas lie in a plane), in which case we can choose a coordinate system where all $w_bp=0$

...then the W-Jones term becomes unity and can be neglected. This is called the narrow-field assumption and the coplanarity assumption. Narrow fields-of-view are the usual case with larger dishes, and/or at higher frequencies. Coplanarity is only possible if:

• we do a short ("snapshot") observation with a small array, or

• our antennas are laid out along an East-West line

New instruments (LOFAR, MeerKAT, SKA1) tend to be wide-field and thus need to deal with $W$. This is called the wide-field imaging problem, and is extensively discussed in $\S$ 5.5.

7.2.4. Mueller calculus

Mueller calculus is an alternative to (or rather, extension of) Jones calculus. The basic rationale of Mueller calculus is as follows. From the Jones formalism, it is obvious that the interferometer response is linear with respect to the four Stokes parameters $I,Q,U,V$. Instead of defining a 2x2 brightness matrix $\mathbf{B}$, let us define the Stokes vector $\mathbf{b}$ as:

$$ \mathbf{b} = \left[ \begin{array}{c} I \\ Q \\ U \\ V \end{array} \right ] $$

We can also define the measured Stokes vector associated with baseline $pq$. In the case of linear feeds, this is defined as

$$ \mathbf{b}_{pq} = \left[ \begin{array}{c} I' \\ Q' \\ U' \\ V' \end{array} \right ] = \frac{1}{2} \left[ \begin{array}{c} V_{11}+V_{22} \\ V_{11}-V_{22} \\ V_{12}+V_{21} \\ -\imath(V_{12}-V_{21}) \end{array} \right ] $$

For circular feeds, the definition is slightly different:

$$ \mathbf{b}_{pq} = \left[ \begin{array}{c} I' \\ Q' \\ U' \\ V' \end{array} \right ] = \frac{1}{2} \left[ \begin{array}{c} V_{11}+V_{22} \\ V_{12}+V_{21} \\ -\imath(V_{12}-V_{21}) \\ V_{11}-V_{22} \end{array} \right ] $$

In the case of a point source at phase centre, and a perfect telescope with no propagation effects (i.e. Jones matrices of unity), we would have $\mathbf{b}'_{pq} \equiv \mathbf{b}$. If we take propagation effects into account, this equality no longer holds. However: the basic assumption of Jones calculus is that the response of the system is linear. This means that for each baseline, the response may be represented by a 4x4 matrix $\mathbf{M}_{pq}$:

$$ \mathbf{b}_{pq} = \mathbf{M}_{pq} \mathbf{b} $$

$\mathbf{M}_{pq}$ is called the Mueller matrix of baseline $pq$.

7.2.4.1. Mueller vs. Jones matrices

There is a simple relationship between Mueller and Jones matrices. Following Understanding radio polarimetry. I ⤴, we can stack the 4 complex visibilities into a visibility vector :

$$ \mathbf{v}_{pq} = 2 \left[ \begin{array}{c} V_{11} \\ V_{12} \\ V_{21} \\ V_{22} \end{array} \right ] = 2 \mathbf{v}_p \otimes \mathbf{v}^*_q, $$

where $\otimes$ denotes the Kronecker or outer product, and $\mathbf{v}^*_q$ is the element-by-element complex conjugate. This leads to a natural defition of a corresponding coherency vector $\mathbf{x}$. Since $\mathbf{v}_p = \mathbf{J}_p \mathbf{e}$, we have (in a linear polarization basis)

$$ \mathbf{v}_{pq} = (\mathbf{J}_p \otimes \mathbf{J}^*_q)(2 \mathbf{e} \otimes \mathbf{e}^*) = (\mathbf{J}_p \otimes \mathbf{J}^*_q) \left[ \begin{array}{c} I+Q \\ U+\imath V \\ U-\imath V \\ I-Q \end{array} \right ] = (\mathbf{J}_p \otimes \mathbf{J}^*_q) \mathbf{x} $$

The coherency vector $\mathbf{x}$ relates to the Stokes vector $\mathbf{b}$ via a simple conversion matrix $\mathbf{S}$. In the linear basis:

$$ \mathbf{x} = \left[ \begin{array}{c} I+Q \\ U+\imath V \\ U-\imath V \\ I-Q \end{array} \right ] = \left[ \begin{array}{cccc} 1&1&0&0 \\ 0&0&1&\imath \\ 0&0&1&-\imath \\ 1&-1&0&0 \end{array} \right ] \left[ \begin{array}{c} I \\ Q \\ U \\ V \end{array} \right ] = \mathbf{S} \,\mathbf{b} $$

and the opposite relation is given by the inverse of the conversion matrix:

$$ \mathbf{b} = \left[ \begin{array}{c} I \\ Q \\ U \\ V \end{array} \right ] = \frac{1}{2} \left[ \begin{array}{cccc} 1&0&0&1 \\ 1&0&0&-1\\ 0&1&1&0 \\ 0&-\imath&\imath&0 \end{array} \right ] \left[ \begin{array}{c} I+Q \\ U+\imath V \\ U-\imath V \\ I-Q \end{array} \right ] = \mathbf{S}^{-1}\mathbf{x} $$

We can now write the measured Stokes vector as $\mathbf{b}_{pq}=\mathbf{S}^{-1} \mathbf{v}_{pq}$. In the Mueller formalism, the RIME then becomes

$$ \mathbf{b}_{pq} = \mathbf{S}^{-1} (\mathbf{J}_p \otimes \mathbf{J}^*_q) \mathbf{S} \, \mathbf{b} $$

The interferometer Mueller matrix is then given by $\mathbf{M}_{pq}=\mathbf{S}^{-1} (\mathbf{J}_p \otimes \mathbf{J}^*_q) \mathbf{S}$.

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7.2.4.2. Mueller chains

By analogy with $\S$7.1.5. Jones chains ➞, when the signal chain contains multiple propagation effects, each one can be described by its own Mueller matrix. The Mueller form of the RIME then becomes

$$ \mathbf{b}_{pq} = \mathbf{M}_{pq,n} \mathbf{M}_{pq,n-1} \dots \mathbf{M}_{pq,1} \mathbf{b}, $$

Each Mueller matrix in the chain takes the form of $\mathbf{M}_{pq,k} = \mathbf{S}^{-1} (\mathbf{J}_{p,k} \otimes \mathbf{J}^*_{q,k}) \mathbf{S}$, and the whole chain can be written as

$$ \mathbf{b}_{pq} = \mathbf{S}^{-1} (\mathbf{J}_{p,n} \otimes \mathbf{J}^*_{q,n})(\mathbf{J}_{p,n-1} \otimes \mathbf{J}^*_{q,n-1}) \dots (\mathbf{J}_{p,1} \otimes \mathbf{J}^*_{q,1}) \mathbf{S} \, \mathbf{b} $$

This is the Mueller equivalent of the onion form of the 2x2 RIME ⤵.

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• Outline
• Glossary
• 7. Observing Systems
  • Previous: 7.1. Jones notation
  • Next: 7.3. Analogue Electronics

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