Interferometry and Synthesis in Radio Astronomy pp 109151  Cite as
Geometrical Relationships, Polarimetry, and the Interferometer Measurement Equation
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Abstract
In this chapter, we start to examine some of the practical aspects of interferometry. These include baselines, antenna mounts and beam shapes, and the response to polarized radiation, all of which involve geometric considerations and coordinate systems. The discussion is concentrated on Earthbased arrays with tracking antennas, which illustrate the principles involved, although the same principles apply to other systems such as those that include one or more antennas in Earth orbit.
Keywords
Linear Polarization Circular Polarization Position Angle Stokes Parameter Radio AstronomyIn this chapter, we start to examine some of the practical aspects of interferometry. These include baselines, antenna mounts and beam shapes, and the response to polarized radiation, all of which involve geometric considerations and coordinate systems. The discussion is concentrated on Earthbased arrays with tracking antennas, which illustrate the principles involved, although the same principles apply to other systems such as those that include one or more antennas in Earth orbit.
4.1 Antenna Spacing Coordinates and (u, v) Loci
4.2 (u′, v′) Plane
4.3 Fringe Frequency
4.4 Visibility Frequencies
4.5 Calibration of the Baseline
Measurement of baseline parameters to an accuracy of order 1 part in 10^{7} (e.g., 3 mm in 30 km) implies timing accuracy of order 10^{−7}ω_{ e }^{−1} ≃ 1 ms. Timekeeping is discussed in Sects. 9.5.8 and 12.3.3
4.6 Antennas
4.6.1 Antenna Mounts
In most tracking arrays used in radio astronomy, the antennas are circularly symmetrical reflectors. A desirable feature is that the axis of symmetry of the reflecting surface intersect both the rotation axes of the mount. If this is not the case, pointing motions will cause the antenna to have a component of motion along the direction of the beam. It is then necessary to take account of phase changes associated with small pointing corrections, which may differ from one antenna to another. In most antenna mounts, however, whether of equatorial or altazimuth type, the reflector axis intersects the rotation axes with sufficient precision that phase errors of this type are negligible.
4.6.2 Beamwidth and BeamShape Effects
The interpretation of data taken with arrays containing antennas with nonidentical beamwidths is not always a straightforward matter. Each antenna pair responds to an effective intensity distribution that is the product of the actual intensity of the sky and the geometric mean of the normalized beam profiles. If different pairs of antennas respond to different effective distributions, then, in principle, the Fourier transform relationship between I(l, m) and \(\mathcal{V}(u,v)\) cannot be applied to the ensemble of observations. Mixed arrays are sometimes used in VLBI when it is necessary to make use of antennas that have different designs. However, in VLBI studies, the source structure under investigation is very small compared with the widths of the antenna beams, so the differences in the beams can usually be ignored. If cases arise in which different beams are used and the source is not small compared with beamwidths, it is possible to restrict the measurements to the field defined by the narrowest beam by convolution of the visibility data with an appropriate function in the (u, v) plane.
4.7 Polarimetry
Polarization measurements are very important in radio astronomy. Most synchrotron radiation shows a small degree of polarization that indicates the distribution of the magnetic fields within the source. As noted in Chap. 1, this polarization is generally linear (plane) and can vary in magnitude and position angle over the source. As frequency is increased, the percentage polarization often increases because the depolarizing action of Faraday rotation is reduced. Polarization of radio emission also results from the Zeeman effect in atoms and molecules, cyclotron radiation and plasma oscillations in the solar atmosphere, and Brewster angle effects at planetary surfaces. The measure of polarization that is almost universally used in astronomy is the set of four parameters introduced by Sir George Stokes in 1852. It is assumed here that readers have some familiarity with the concept of Stokes parameters or can refer to one of numerous texts that describe them [e.g., Born and Wolf (1999); Kraus and Carver (1973); Wilson et al. (2013)].
In considering the response of interferometers and arrays, up to this point we have ignored the question of polarization. This simplification can be justified by the assumption that we have been dealing with completely unpolarized radiation for which only the parameter I is nonzero. In that case, the response of an interferometer with identically polarized antennas is proportional to the total flux density of the radiation. As will be shown below, in the more general case, the response is proportional to a linear combination of two or more Stokes parameters, where the combination is determined by the polarizations of the antennas. By observing with different states of polarization of the antennas, it is possible to separate the responses to the four parameters and determine the corresponding components of the visibility. The variation of each parameter over the source can thus be imaged individually, and the polarization of the radiation emitted at any point can be determined. There are alternative methods of describing the polarization state of a wave, of which the coherency matrix is perhaps the most important (Ko 1967a,b). However, the classical treatment in terms of Stokes parameters remains widely used by astronomers, and we therefore follow it here.
4.7.1 Antenna Polarization Ellipse
In reception, an electric vector that rotates in a clockwise direction in Fig. 4.8 produces a voltage in the y′ dipole that leads the voltage in the x′ dipole by π/2 in phase, and the two signals therefore combine in phase at A. For counterclockwise rotation, the signals at A are in antiphase and cancel one another. Thus, the antenna in Fig. 4.8 receives righthanded waves incident from the positive z direction (that is, traveling toward negative z), and it transmits righthanded polarization in the direction toward positive z. To receive a righthanded wave propagating down from the sky (in the positive z direction), the polarity of one of the dipoles must be reversed, which requires that χ = −π∕4.
 1.
 2.
Indicate values of ψ, χ, and V ′ for the two antennas by subscripts m and n and calculate the correlator output, \(R_{mn} = G_{mn}\left \langle {{V{^\prime}}_{m}}{{V{^\prime}}_{n}}^{{\ast}}\right \rangle\), where G_{ mn } is an instrumental gain factor.
 3.
Substitute Stokes parameters for \(\mathcal{E}_{x},\mathcal{E}_{y},\delta _{x},\delta _{y}\) using Eq. (4.19) as follows:
4.7.2 Stokes Visibilities
Here, ψ_{ m } is the position angle of the antenna polarization measured from celestial north in the direction of east. The y polarization angle is equal to the x polarization angle plus π∕2. For ψ_{ m } equal to 0^{∘}, 45^{∘}, 90^{∘}, and 135^{∘}, the output R_{ xx } is proportional to (I_{ v } + Q_{ v }), (I_{ v } + U_{ v }), (I_{ v } − Q_{ v }), and (I_{ v } − U_{ v }), respectively. By using antennas with these polarization angles, I_{ v }, Q_{ v }, and U_{ v }, but not V_{ v }, can be measured. In many cases, circular polarization is negligibly small, and the inability to measure V_{ v } is not a serious problem. However, Q_{ v } and U_{ v } are often only a few percent of I_{ v }, and in attempting to measure them with identical feeds, one faces the usual problems of measuring a small difference in two much larger quantities. The same is true if one attempts to measure V_{ v } using identical circular feeds for which χ = ±π∕4 and the response is proportional to (I_{ v } ∓ V_{ v }). These problems are reduced by using oppositely polarized feeds to measure Q_{ v }, U_{ v }, or V_{ v }. For an example of measurement of V_{ v }, see Weiler and Raimond (1976).
Stokes visibilities vs. position angles
Position angles  

Stokes visibilities  
m  n  measured  
0^{∘}  0^{∘}  I_{ v } + Q_{ v }  Position angle I  
0^{∘}  90^{∘}  U_{ v } + jV_{ v }  ”  
90^{∘}  0^{∘}  U_{ v } − jV_{ v }  ”  
90^{∘}  90^{∘}  I_{ v } − Q_{ v }  ”  
45^{∘}  45^{∘}  I_{ v } + U_{ v }  Position angle II  
45^{∘}  135^{∘}  − Q_{ v } + jV_{ v }  ”  
135^{∘}  45^{∘}  − Q_{ v } − jV_{ v }  ”  
135^{∘}  135^{∘}  I_{ v } − U_{ v }  ” 
An example of a different arrangement of linearly polarized feeds, which has been used at the Westerbork Synthesis Radio Telescope, is described by Weiler (1973). The antennas are equatorially mounted and the parallactic angle of the polarization remains fixed as a source is tracked. The outputs of the antennas that are movable on rail track are correlated with those from the antennas in fixed locations. Table 4.2 shows the measurements when the position angles of the planes of polarization for the movable antennas are 45^{∘} and 135^{∘} and those of the fixed antennas 0^{∘} and 90^{∘}. Although the responses are reduced by a factor of \(\sqrt{ 2}\) relative to those in Table 4.1, there is no loss in sensitivity since each Stokes visibility appears at all four correlator outputs. Note that since only signals from antennas with different polarization configurations are crosscorrelated, this scheme does not make use of all possible polarization products.
Opposite circularly polarized feeds offer certain advantages for measurements of linear polarization. In determining the responses, an arbitrary position angle ψ_{ m } for antenna m is included to represent the effect of rotation caused, for example, by an altazimuth antenna mount. If the antennas provide simultaneous outputs for opposite senses of rotation (denoted by r and ℓ) and four correlation products are generated for each antenna pair, the outputs are proportional to the quantities in Table 4.3.
Stokes visibilities vs. position angles
Position angles  

m  n  Stokes visibilities measured  
0^{∘}  45^{∘}  \((I_{v} + Q_{v} + U_{v} + jV _{v})/\sqrt{2}\)  
0^{∘}  135^{∘}  \((I_{v}  Q_{v} + U_{v} + jV _{v})/\sqrt{2}\)  
90^{∘}  45^{∘}  \((I_{v}  Q_{v} + U_{v}  jV _{v})/\sqrt{2}\)  
90^{∘}  135^{∘}  \((I_{v}  Q_{v}  U_{v} + jV _{v})/\sqrt{2}\) 
Stokes visibilities vs. sense of rotation
Sense of rotation  

m  n  Stokes visibilities measured  
r  r  I_{ v } + V_{ v }  
r  ℓ  \((jQ_{v} + U_{v})e^{j2\psi _{m}}\)  
ℓ  r  \((jQ_{v}  U_{v})e^{\,j2\psi _{m}}\)  
ℓ  ℓ  I_{ v } − V_{ v } 
Here, we have made ψ_{ ℓ } = ψ_{ r } +π∕2, and χ = −π∕4 for right circular polarization and χ = π∕4 for left circular. The feeds need not be rotated during an observation, and the responses to Q_{ v } and U_{ v } are separated from those to I_{ v }. The expressions in Table 4.3 can be simplified by choosing values of ψ_{ r } such as π∕2, π∕4, or 0. For example, if ψ_{ r } = 0, the sum of the r ℓ and ℓ r responses is a measure of Stokes visibility U_{ v }. Again, the effects of the rotation of the position angle with altazimuth mounts must be taken into account. Conway and Kronberg (1969) appear to have been the first to use an interferometer with circularly polarized antennas to measure linear polarization in weakly polarized sources. Circularly polarized antennas have since been commonly used in radio astronomy.
4.7.3 Instrumental Polarization
4.7.4 Matrix Formulation
The description of polarimetry given above, using the ellipticity and orientation of the antenna response, is based on a physical model of the antenna and the electromagnetic wave, as in Eq. (4.29). Historically, studies of optical polarization have developed over a much longer period. A description of radio polarimetry following an approach originally developed in optics is given in Hamaker et al. (1996) and in more detail in four papers: Hamaker et al. (1996), Sault et al. (1996), Hamaker (2000), and Hamaker (2006). The mathematical analysis is largely in terms of matrix algebra, and in particular, it allows the responses of different elements of the signal path such as the atmosphere, the antennas, and the electronic system to be represented independently and then combined in the final solution. This approach is convenient for detailed analysis including effects of the atmosphere, ionosphere, etc.
4.7.5 Calibration of Instrumental Polarization
The fractional polarization of many astronomical sources is of magnitude comparable to that of the leakage and gain terms that are used above to define the instrumental polarization. Thus, to obtain an accurate measure of the polarization of a source, the leakage and gain terms must be accurately calibrated. It may be necessary to determine the calibration independently for each set of observations since the gain terms may be functions of the temperature and state of adjustment of the electronics and cannot be assumed to remain constant from one observing session to another. Making observations (i.e., measuring the coherency vector) of sources for which the polarization parameters are already known is clearly a way of determining the leakage and gain terms. The number of unknown parameters to be calibrated is proportional to the number of antennas, n_{ a }, but the number of measurements is proportional to the number of baselines, n_{ a }(n_{ a } − 1)∕2. The unknown parameters are therefore usually overdetermined, and a leastmeansquares solution may be the best procedure.
An observation of a single calibration source for which the four Stokes parameters are known enables four of the degrees of freedom to be determined. However, because of the relationships of the quantities involved, it takes at least three calibration observations to solve for all seven unknown parameters (Sault et al. 1996). In the calibration observations, it is useful to observe one unpolarized source, but observing a second unpolarized one would add no further solutions. At least one observation of a linearly polarized source is required to determine the relative phases of the two oppositely polarized channels, that is, the relative phases of the complex gain terms g_{ xm }g_{ yn }^{∗} and g_{ ym }g_{ xn }^{∗}, or g_{ rm }g_{ ℓ n }^{∗} and g_{ ℓ m }g_{ rn }^{∗}. Note that with antennas on altazimuth mounts, observations of a calibrator with linear polarization, taken at intervals between which large rotations of the parallactic angle occur, can essentially be regarded as observations of independent calibrators. Under these circumstances, three observations of the same calibrator will suffice for the full solution. Furthermore, the polarization of the calibrator need not be known in advance but can be determined from the observations.
In cases in which only an unpolarized calibrator can be observed, it may be possible to estimate two more degrees of freedom by introducing the constraint that the sum of the leakage factors over all antennas should be small. As shown by the expressions for the leakage terms in Appendix Appendix 4.2, this is a reasonable assumption for a homogeneous array, that is, one in which the antennas are of nominally identical design. However, the phase difference between the signal paths from the feeds to the correlator for the two orthogonal polarizations of each antenna remains unknown. This requires an observation of a calibrator with a component of linear polarization, or a scheme to measure the instrumental component of the phase. For example, on the compact array of the Australia Telescope (Frater and Brooks 1992), noise sources are provided at each antenna to inject a common signal into the two polarization channels (Sault et al. 1996). With such a system, it is necessary to provide an additional correlator for each antenna, or to be able to rearrange correlator inputs, to measure the relative phase of the injected signals in the two polarizations.
In the case of the approximations for weak polarization, Eqs. (4.38) and (4.43) show that if the gain terms are known, the leakage terms can be calibrated by observing an unpolarized source. For opposite circular polarizations, Eq. (4.43) shows that if V_{ v } is small, it is possible to obtain solutions for the gain terms from the outputs for the ℓ ℓ and rr combinations only, provided also that the number of baselines is several times larger than the number of antennas. The leakage terms can then be solved for separately. For crossed linear polarizations, Eq. (4.38) shows that this is possible only if the linear polarization (Q_{ v } and U_{ v } parameters) for the calibrator have been determined independently.
Optimum strategies for calibration of polarization observations is a subject that leads to highly detailed discussions involving the characteristics of particular synthesis arrays, the hour angle range of the observations, the availability of calibration sources (which can depend on the observing frequency), and other factors, especially if the solutions for strong polarization are used. Such discussions can be found, for example, in Conway and Kronberg (1969), Weiler (1973), Bignell (1982), Sault et al. (1991), Sault et al. (1996), and Smegal et al. (1997). Polarization measurements with VLBI involve some special considerations: see, for example, Roberts et al. (1991), Cotton (1993), Roberts et al. (1994), and Kemball et al. (1995).
For most large synthesis arrays, effective calibration techniques have been devised and the software to implement them has been developed. Thus, a prospective observer need not be discouraged if the necessary calibration procedures appear complicated. Some general considerations relevant to observations of polarization are given below.

Since the polarization of many sources varies on a timescale of months, it is usually advisable to regard the polarization of the calibration source as one of the variables to be solved for.

Two sources with relatively strong linear polarization at position angles that do not appear to vary are 3C286 and 3C138. These are useful for checking the phase difference for oppositely polarized channels.

For most sources, the circular polarization parameter V_{ v } is very small, ∼ 0. 2% or less, and can be neglected. Measurements with circularly polarized antennas of the same sense therefore generally give an accurate measure of I_{ v }. However, circular polarization is important in the measurement of magnetic fields by Zeeman splitting. As an example of positive detection at a very low level, Fiebig and Güsten (1989) describe measurements for which V∕I ≃ 5 × 10^{−5}. Zeeman splitting of several components of the OH line at 22.235 GHz was observed using a single antenna, the 100m paraboloid of the Max Planck Institute for Radio Astronomy, with a receiving system that switched between opposite circular polarizations at 10 Hz. Rotation of the feed and receiver unit was used to identify spurious instrumental responses to linearly polarized radiation, and calibration of the relative pointing of the two beams to 1^{ ′ ′ } accuracy was required.

Although the polarized emission from most sources is small compared with the total emission, it is possible for Stokes visibilities Q_{ v } and U_{ v } to be comparable to I_{ v } in cases in which there is a broad unpolarized component that is highly resolved and a narrower polarized component that is not resolved. In such cases, errors may occur if the approximations for weak polarization [Eqs. (4.38) and (4.43)] are used in the data analysis.

For most antennas, the instrumental polarization varies over the main beam and increases toward the beam edges. Sidelobes that are cross polarized relative to the main beam tend to peak near the beam edges. Thus, polarization measurements are usually made for cases in which the source is small compared with the width of the main beam, and for such measurements, the beam should be centered on the source.

Faraday rotation of the plane of polarization of incoming radiation occurs in the ionosphere and becomes important for frequencies below a few gigahertz; see Table 14.1 During polarization measurements, periodic observations of a strongly polarized source are useful for monitoring changes in the rotation, which varies with the total column density of electrons in the ionosphere. If not accounted for, Faraday rotation can cause errors in calibration; see, for example, Sakurai and Spangler (1994).

In some antennas, the feed is displaced from the axis of the main reflector, for example, when the Cassegrain focus is used and the feeds for different bands are located in a circle around the vertex. For circularly polarized feeds, this departure from circular symmetry results in pointing offsets of the beams for the two opposite hands. The pointing directions of the two beams are typically separated by ∼ 0. 1 beamwidths, which makes measurements of circular polarization difficult because V_{ v } is proportional to (R_{ rr } − R_{ ℓ ℓ }). For linearly polarized feeds, the corresponding effect is an increase in the crosspolarized sidelobes near the beam edges.

In VLBI, the large distances between antennas result in different parallactic angles at different sites, which must be taken into account.

The quantities m_{ ℓ } and m_{ t }, of Eqs. (4.20) and (4.22), have Rice distributions of the form of Eq. (6.63a), and the position angle has a distribution of the form of Eq. (6.63b). The percentage polarization can be overestimated, and a correction should be applied (Wardle and Kronberg 1974).
The following points concern choices in designing an array for polarization measurements.

The rotation of an antenna on an altazimuth mount, relative to the sky, can sometimes be used to advantage in polarimetry. However, the rotation could be a disadvantage in cases in which polarization imaging over a large part of the antenna beam is being attempted. Correction for the variation of instrumental polarization over the beam may be more complicated if the beam rotates on the sky.

With linearly polarized antennas, errors in calibration are likely to cause I_{ v } to corrupt the linear parameters Q_{ v } and U_{ v }, so for measurements of linear polarization, circularly polarized antennas offer an advantage. Similarly, with circularly polarized antennas, calibration errors are likely to cause I_{ v } to corrupt V_{ v }, so for measurements of circular polarization, linearly polarized antennas may be preferred.

Linearly polarized feeds for reflector antennas can be made with relative bandwidths of at least 2 : 1, whereas for circularly polarized feeds, the maximum relative bandwidth is commonly about 1. 4 : 1. In many designs of circularly polarized feeds, orthogonal linear components of the field are combined with ± 90^{∘} relative phase shifts, and the phaseshifting element limits the bandwidth. For this reason, linear polarization is sometimes the choice for synthesis arrays [see, e.g., James (1992)], and with careful calibration, good polarization performance is obtainable.

The stability of the instrumental polarization, which greatly facilitates accurate calibration over a wide range of hour angle, is perhaps the most important feature to be desired. Caution should therefore be used if feeds are rotated relative to the main reflector or if antennas are used near the high end of their frequency range.
4.8 The Interferometer Measurement Equation
The set of equations for the visibility values that would be measured for a given brightness distribution—taking account of all details of the locations and characteristics of the individual antennas, the path of the incoming radiation through the Earth’s atmosphere including the ionosphere, the atmospheric transmission, etc.—is commonly referred to as the measurement equation or the interferometer measurement equation. For any specified brightness distribution and any system of antennas, the measurement equation provides accurate values of the visibility that would be observed. The reverse operation, i.e., the calculation of the optimum estimate of the brightness distribution from the measured visibility values, is more complicated. Taking the Fourier transform of the observed visibility function usually produces a brightness function with physically distorted features such as negative brightness values in some places. However, starting with a physically realistic model for the brightness, the measurement equation can accurately provide the corresponding visibility values that would be observed. This provides a basis for derivation of realistic brightness distributions that represent the observed visibilities, using an iterative procedure.
The formulation of the interferometer measurement equation is based on the analysis of Hamaker et al. (1996) and further developed by Rau et al. (2009), Smirnov (2011a,b,c,d), and others. It traces the variations of the signals from a source to the output of the receiving system. Directiondependent effects include the direction of propagation of the signals, the primary beams of the antennas, polarization effects that vary with the alignment of the polarization of the source relative to that of the antennas, and also the effects of the ionosphere and troposphere. Directionindependent effects include the gains of the signal paths from the outputs of the antennas to the correlator. It is necessary to take account of all these various effects to calculate accurately the visibility values corresponding to the source model. Several of these effects are dependent upon the types of the interferometer antennas and the observing frequencies, so the details of the measurement equation are to some extent specific to each particular instrument to which it is applied.
4.8.1 Multibaseline Formulation
An example of the application of matrix formulation in radio astronomy is provided by the discussion of gain calibration by Boonstra and van der Veen (2003). Also, the eigenvectors of the matrix can be used to identify interfering signals that are strong enough to be distinguished in the presence of the noise. Such signals can then be removed from the data, as discussed, for example, by Leshem et al. (2000).
Footnotes
 1.
In VLBI observations, it is customary to set the X axis in the Greenwich meridian, in which case H is measured with respect to that meridian rather than a local one.
 2.
The first mention of elliptical loci appears to have been by Rowson (1963).
 3.
In comparing expressions for polarimetry by different authors, note that differences of signs or of the factor j can result from differences in the way the parallactic angle is defined with respect to the antenna, and similar arbitrary factors.
 4.
The diagonal terms are those that move downward from left to right, and the offdiagonal terms slope in the opposite direction.
 5.
Further explanation of Jones and Mueller matrices can be found in textbooks on optics [e.g., O’Neill (1963)].
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