A&A 441, 12171228 (2005)
DOI: 10.1051/00046361:20052990
M. A. Brentjens^{1,2}  A. G. de Bruyn^{2,1}
1  Kapteyn Astronomical Institute, University of Groningen,
PO Box 800, 9700 AV Groningen, The Netherlands
2 
ASTRON, PO Box 2, 7990 AA Dwingeloo, The Netherlands
Received 4 March 2005 / Accepted 20 June 2005
Abstract
We extend the rotation measure work of Burn (1966, MNRAS, 133, 67) to
the cases of limited sampling of
space and nonconstant
emission spectra. We introduce the rotation measure transfer function
(RMTF), which is an excellent predictor of
ambiguity problems
with the
coverage. Rotation measure synthesis can be
implemented very efficiently on modern computers. Because the analysis
is easily applied to wide fields, one can conduct very fast RM surveys
of weak spatially extended sources. Difficult situations, for example
multiple sources along the line of sight, are easily detected and
transparently handled. Under certain conditions, it is even possible
to recover the emission as a function of Faraday depth within a single
cloud of ionized gas. Rotation measure synthesis has already been
successful in discovering widespread, weak, polarized emission
associated with the Perseus cluster (de Bruyn & Brentjens 2005, A&A, 441, 931). In
simple, high signal to noise situations it is as good as traditional
linear fits to
versus
plots. However, when the
situation is more complex or very weak polarized emission at high
rotation measures is expected, it is the only viable option.
Key words: methods: data analysis  techniques: polarimetric  magnetic fields  polarization  ISM: magnetic fields  Cosmology: largescale structure of Universe
Polarization observations at radio frequencies are an important diagnostic tool in the study of galactic and extragalactic magnetic fields (e.g. Widrow 2002; Vallee 1997; Kronberg 1994). Due to birefringence of the magnetoionic medium, the polarization angle of linearly polarized radiation that propagates through the plasma is rotated as a function of frequency. This effect is called Faraday rotation. There exist many papers describing aspects of Faraday rotation work. The most relevant ones for this work are Burn (1966), Gardner & Whiteoak (1966), Sokoloff et al. (1998), Sokoloff et al. (1999), and Vallee (1980).
Assuming that the directions of the velocity vectors of the
electrons gyrating in a magnetized plasma are isotropically distributed,
Le Roux (1961) showed that the intrinsic degree of polarization of
synchrotron radiation from plasma in a uniform magnetic field
is given by
(2) 
From observations of the Crab nebula by Woltjer (1958), Westfold (1959) determined that . This would imply a polarization fraction of approximately 67%, independent of frequency. In many radio sources, the observed polarization fractions are much lower. Usually the polarization fraction decreases steeply with increasing wavelength (Strom & Conway 1985; Conway & Strom 1985).
Burn (1966) discusses this depolarization effect extensively. One of the mechanisms he discusses is Faraday dispersion: emission at different Faraday depths along the same line of sight.
Following Burn (1966), we make a clear distinction between
Faraday depth ()
and rotation measure (RM). We define the
Faraday depth of a source as
The rotation measure is commonly defined as the slope of a
polarization angle
versus
plot:
(4) 
(5) 
P can be written as
If there is only one source along the line of sight, which
in addition has no internal Faraday rotation, and does not suffer from
beam depolarization, then the Faraday depth of that source is equal to
its rotation measure at all wavelengths:
A simple example illustrates this. Imagine a classical double
radio galaxy, of which the lobe closest to us is at a Faraday depth of
.
The lobe itself is
Faraday thin and has an intrinsic polarized flux density of
0.25 Jy beam^{1} (positive Stokes Q). At low frequencies, there
is usually some polarized Galactic foreground emission between us and
the radio galaxy. The Galactic foreground is modelled as a uniform slab with a constant,
uniform magnetic field. The total integrated polarized surface
brightness of the Galactic foreground at
is 1 Jy beam^{1}(positive Stokes Q). The Faraday dispersion function
is a
top hat function:
(10) 
Figure 1 plots , , and for Eq. (13). is the real part of . We have taken and . At low , the foreground dominates over the lobe, forcing Stokes Q of the sum of the polarizations to be positive, while U can be both positive and negative. In this regime, oscillates around zero. However, when the foreground is significantly depolarized, the lobe starts to dominate the total (Q, U) vector. This point is reached somewhere near m^{2}. From there on the total (Q, U) vector runs through all four quadrants. As the polarized flux of the foreground vanishes, the total polarization angle approaches more and more a straight line corresponding to a RM of +10 rad m^{2}.
Figure 2 shows an example of a fairly complex line of sight. There are three areas with polarized emission (A, B, and C), of which two (A and B) also have internal Faraday rotation. The middle panel shows the nonmonotonic relation between Faraday depth and physical depth. Although area B is larger in physical depth, area A is larger in Faraday depth due to the high absolute value of .
A physical interpretation of this example would be that region A and its adjacent rotationonly areas reside in our Galaxy, area B and its neighboring rotationonly areas are a galaxy cluster, and area C represents a collection of distant polarized sources without any internal Faraday rotation of their own. Line of sight 1 goes through the cluster, while line of sight 2 just misses it. This causes C to be at different Faraday depth in the two lines of sight.
Because of the Fourier nature of both Eq. (6) and radio synthesis imaging, there exist many analogies between the two. Examples are uv plane sampling versus sampling and synthesized beam versus RMTF. Therefore we prefer to call the process of inversion "Rotation Measure synthesis'' ("RMsynthesis'' for short).
Similar methods have recently been applied to pulsar observations (Mitra et al. 2003; Weisberg et al. 2004). de Bruyn (1996) applied the method for the first time to an entire field of view. He also introduced the concept of a Rotation Measure Transfer Function (RMTF, see also Sect. 2 of this work). When applied to a complete field of view instead of just one line of sight, the output of a RMsynthesis is a socalled "RMcube''. The RMcube has axes , , and . It is the Faraday rotation equivalent of a 21 cm line cube. The application to wide fields allowed the discovery of widespread, very faint polarized emission associated with the Perseus cluster (de Bruyn & Brentjens 2005).
Modern correlator backends, like the ones installed at the WSRT, the GMRT, and the ATCA and the one to be installed at the EVLA deliver the visibilities in many (32 to 1024) narrow channels across a wide band (typically 16 to 160 MHz). The narrow channels move the bandwidth depolarization limit to much higher rotation measures. The wider bands yield very high sensitivities if the full bandwidth can be used. Thanks to these backends RMsynthesis has finally become a practical, even necessary observing method.
Section 2 discusses the generally incomplete sampling of . We formally derive the RMTF. Section 3 treats modifications to the assumption that is frequency independent. In Sect. 4 we treat the relation between the RMTF and ambiguities in traditional RM fitting. Section 5 describes RMsynthesis with Stokes Q or U only. Section 6 gives advice on designing Faraday rotation experiments, taking the findings of this work into account. Section 7 concludes this work. Appendix A expands on error estimation in RM work and Appendix B treats an example simulation illustrating a few important concepts presented in this work.
Symbol  Description 
Polarization angle (N through E)  
Polarization angle at  
Frequency  
Channel width in frequency  
Central frequency of a channel  
Wavelength  
Wavelength to which all polarization vectors are derotated  
Central wavelength squared of a channel  
Channel width in wavelength squared  
Total bandwidth in wavelength squared.  
Faraday depth  
FWHM of the main peak of the RMTF  
Rotation measure  
Weight function  
w_{i}  Weight of the ith data point 
K  One over the integral of W or one over the sum of weights 
Faraday dispersion function without spectral dependence  
Reconstructed approximation to  
General form of the Faraday dispersion function  
Spectral dependence in I, normalized to unity at  
Frequency spectral index  
Complex polarized surface brightness  
Observed P:  
Complex polarization fraction  
Rotation Measure Transfer Function (RMTF)  
Magnetic induction  
Position vector  
Thermal electron density  
Spectral index of the relativistic electron energy distribution  
Real part of z  
Imaginary part of z  
Merit function for traditional linear least squares fitting of rotation measures. Defined in Eq. (49) 

rms noise in a single channel map  
,  rms noise in single Q or U channel maps 
,  Standard error of and in individual channel maps 
,  Standard error in Faraday depth and position angle at 
Standard deviation of the distribution of
values that are sampled. This is a measure of the effective width of the sampling 

Dirac delta function 
The goal of this section is to approximate
by Fourier inverting a generalized version of Eq. (6).
Table 1 summarizes the symbols that
are used throughout this paper. We generalize Eq. (6) by introducing the weight
function
.
is also called the sampling
function. It is nonzero at all
points where measurements
are taken. It is zero elsewhere. Obviously,
for
because of the lack of measurements there. The
observed polarized flux density, or surface brightness in the case of
spatially extended sources, is
(19) 
The above set of equations is not yet our final result. Figure 3 displays the rotation measure transfer function corresponding to the sampling of our Perseus data set (de Bruyn & Brentjens 2005). It only shows a small part of the RMTF close to the peak. The response function displays a rapid rotation of the (real, imaginary) vector. Because one usually samples space at finite intervals, this rotation makes it very difficult to correctly estimate the polarization angle at or near the maximum of . If the Faraday depth of a frame is only a tenth of the width of the RMTF away from the actual Faraday depth of the source, the (real, imaginary) vector may already be rotated by several tens of degrees.
Equations (22) and (23) correspond to derotating all polarization vectors back to their position at . At first this appears sensible, because the polarization vector at is directly related to the electric field vector in the plane of the sky without any Faraday rotation. Nevertheless no information is lost by derotating to some other common .
The more general versions of Eqs. (22) and (23) are
The simplest way to see this is to consider the case when
.
This changes the
convolution in Eq. (25) into a
multiplication. Hence the result of the righthand side of Eq. (25) can be written as
(27) 
Ideally, the response in the entire main peak of the RMTF
should be parallel to the actual polarization vector at
.
The best way of achieving that is keeping the orthogonal
response as close to zero as possible. We set the derivative of the
imaginary part at
to zero:
A drawback of having
is that the polarization
angle that one derives still needs to be transformed to a polarization
angle at
,
if one wants information on the orientation
of the electric field direction in the source. In case of a
high S/N ratio, this is very easy:
(33) 
Figure 4: RMTF of the same dataset as described in Fig. 3. This time, however, all vectors have been derotated to the average . It is seen that the imaginary part remains almost constant within the central peak of the RMTF. 
Figure 4 shows the same RMTF as Fig. 3, except that is set to the weighted average . The improvement with respect to the orthogonal response is evident. The response function is almost completely real between the first minima. The only drawback is that one cannot convert the observed polarization angle at to a vector in a straightforward way. In order to accomplish reliable derotation to , one needs a sufficiently high S/N ratio to determine the Faraday depth with an accuracy well within the full width at half maximum (FWHM) of the RMTF. This is not a problem for bright sources that are already detected in individual channels, but for faint emission that is only detectable after RMsynthesis, one cannot usually do this. These signal to noise statements are quantified in Sect. 4 and Appendix A.
In most correlators, all channels have equal bandwidth ,
centred
around
,
the central frequency of the channel. Our
prime coordinate is ,
not .
If we assume a top hat
channel bandpass, we have for every channel:
If
for all channels, we may approximate
the integrals in Eqs. (25) and (26) by sums:
In this section we investigate the effect of the emission spectrum of
a source on the method. We start with the most general case of an
arbitrary spectrum at each Faraday depth. We substitute
(39) 
(40) 
The second case is a specialization of the first case. Equation (44) reduces to Eq. (15) in case of a flat
spectrum. The approximate Faraday dispersion function compensated for
a nonflat spectrum is given by:
(45) 
Equation (42) applies only in some very specific scenarios. It holds for example in optically thin synchrotronemitting and Faradayrotating clouds that have the same relativistic electron energy distribution throughout the cloud. It also holds if multiple optically thin clouds along the line of sight happen to have the same spectral dependence. Optically thin synchrotron radiation has a spectrum that is proportional to over a large range of frequencies (Conway et al. 1963). For most sources, is in the range . In extreme cases the spectral index of optically thin emission can approach 0 (e.g. the Crab nebula) or 3 (for halo or relic sources in galaxy clusters).
In general, spectral indices vary across a map. One can of course easily correct for the spectra of sources that are reliably detected in individual channel maps. This is impossible for sources that are much fainter and only show up after averaging the full band. For those objects it makes sense to estimate some "average'' spectral index and apply that to the entire map.
What is the effect of using the wrong spectral index in correcting for the spectrum of a single source along the line of sight? The contributions of multiple sources along the line of sight is simply the sum of their individual responses. Because the spectrum is an amplitude only effect, it has no influence on the location of the maximum of the Faraday dispersion function of the source. Therefore its derived Faraday depth is unaffected. It does distort the RMTF associated with the source at points away from the main peak. This complicates deconvolution algorithms slightly.
Figure 5 gives the Faraday dispersion functions of Faraday thin model sources with spectral indices 3 to 0. It is seen that the largest effect occurs close to the nulls of the RMTF. The difference between and is small over the 17% total frequency bandwidth in the simulation. It will not be noticeable if the emission has such low S/N that it is invisible in individual channels. For comparison, the normalized of a Faraday thick uniform slab model is included. The slab emits at . It is seen that the effect of even a tiny amount of structure in the source is much larger than the effect of changing the spectral index by .
The general case of an arbitrary spectral dependence at multiple Faraday depths is not invertible. One can only recover the Faraday dispersion function if the spectral dependence is the same at all Faraday depths along the line of sight. One should then divide the observed polarization by the spectral dependence in I. Figure 5 shows that if the spectral index is estimated with an absolute uncertainty less than 1, the maximum absolute error of the estimated flux density at a certain Faraday depth is less than 25% of the brightest emission along the line of sight. This accuracy is easily exceeded for sources that are visible in total intensity. Sources that have not been detected in total intensity should generally be assigned a spectral index of 1. This worked very well in our observations of the Perseus cluster, where we see large, faint polarized features that have no detectable counterpart in total intensity (de Bruyn & Brentjens 2005).
The traditional way to compute the rotation measure of a source is to measure its polarization angle at several wavelengths and determine the slope of a straight line through the polarization angle as a function of . This method suffers from socalled ambiguity problems. If only a few data points are available, there may exist multiple RM solutions that are equally good, but have the polarization angle of the data points wrapped around one or more turns. Complicated methods have been devised to attempt to circumvent these problems, some of which are quite successful. An example is the "Pacerman'' method (Vogt et al. 2004; Dolag et al. 2004), which operates on images, and does a good job in finding and correcting ambiguities using spatial continuity arguments.
In this section we show that the RMTF is an excellent indicator of possible ambiguity problems. By analyzing the RMTF, one can take measures to minimize or even eradicate any potential problems in the experiment in advance. We also show that using only RMsynthesis to determine Faraday depths is as accurate as traditional fitting, but has the added value of straightforward ambiguity problem detection.
We first consider traditional
fitting. This is done by
linear least squares minimization of a merit function .
If the
estimated errors of all points are equal, then the merit function
looks like Eq. (49)
Figure 7: Both plots show, from top to bottom, merit function and . The lefthand panel uses eight sample points, equally spaced in frequency. The frequency sampling of the righthand panel is identical to the sampling used in Fig. 6. The sampling in the lefthand panel has the same extent in space as the sampling of the righthand panel. 
Figure 8: This plot shows the effect of tweaking the exact frequencies of eight sampling points. The lefthand panel shows the same RMTF as the lefthand panel of Fig. 7. In the righthand plot, however, we stretched the frequency intervals such that low frequency intervals are wider than high frequency intervals. This eliminates the resonances from the lefthand plot. 
If the model RM is sufficiently different from the actual RM, one
expects the errors l_{i} to be approximately uniformly distributed in
the range
.
Because the square is taken, this is
equivalent to a uniform distribution in the range
.
The average value of l_{i}^{2} is then
given by
(51) 
An interesting aspect seen in Fig. 6 is that the envelope of looks like when . Deep minima of are associated with high peaks in the RMTF. In fact, they appear to be approximately proportional to . These deep minima are closely related to socalled ambiguities in traditional RM measurements. They are points that fit the data (almost) equally well as the "true'' solution.
The similarity between the envelope of and is better demonstrated in Fig. 7. It shows both and over a large range in . The lefthand panel displays and for 8 points, equally spaced in frequency. To facilitate comparison, the total width of the pattern, , has been scaled to match the width of the sampling of Fig. 6. The righthand panel of Fig. 7 shows and based on the same input data as Fig. 6. It is obvious that the RMTF of a 126 point sampling has much lower side lobes than an 8point sampling. ambiguities are completely eliminated.
The lefthand panel of Fig. 8 shows the same RMTF as the lefthand panel of Fig. 7. The two resonances to the left and right are due to the nearregularity of the sampling points in space. If the frequency intervals at the lower frequencies are stretched more than at the intervals at higher frequencies, for example by making them decrease linearly with increasing frequency, one can make the pattern in space less regular. The result is shown in the righthand panel of Fig. 8. The resonances are now lower and wider. If one requires the highest side lobes to be at least 5lower than unity, then a total S/N of 20 (7 per channel) is sufficient to prevent ambiguities outside the main peak of the RMTF. Using the same requirement, in case of the 126 points, a S/N of 6 in total (0.6 per channel) is enough to prevent ambiguities outside the main peak of the RMTF.
ambiguities are conceptually closely related to the grating response of a regularly spaced interferometer like the WSRT. When an interferometer only has baselines that are a multiple of some fixed distance, then its instantaneous synthesized beam is a collection of parallel fan beams. Each fan beam has the same peak amplitude. Therefore, without any further constraints it is impossible to determine in which fan beam the source is actually located. The same holds true for sampling. If one only samples space at regular intervals, there exist multiple solutions that fit the data equally well. These solutions correspond to peaks of unit amplitude in the RMTF: grating responses.
Figure 6 also shows that there are multiple minima of within the main lobe of the RMTF. These indicate uncertainties in the RM smaller than the width of the main peak. We shall now investigate the uncertainty in RM within the main peak of .
The standard error in the RM when obtained by fitting a straight line
to a plot of
versus
is given by
(54) 
(55) 
In RMsynthesis, one determines the RM of a single source along the line of sight by fitting, for example, a parabola to the main peak of . The detailed procedure is to first find the brightest point in a critically sampled Faraday dispersion function (23 points per ), covering a wide range in . This is followed by oversampling the region around the peak by a large factor. A parabolic fit to the 1020 points directly surrounding the peak then yields the RM of the source.
We have simulated this procedure in order to get a quantitative idea of the typical error in RM that one obtains, given a certain noise level in the Stokes Q and U images, and a certain set of sample points . The results are shown in Fig. 9. The total signaltonoise ratio is equal to . The solid line is Eq. (52). The points are standard deviations in RM computed from 1000 iterations per S/N ratio point. We have assumed the noise in Stokes Q and U to be equal and Gaussian. We see excellent agreement with the error expected for traditional fitting (the straight line). At a S/N ratio less than 4, the points deviate strongly from the line. This is due to the fact that the nonGaussianity of the noise in P is only noticeable close to the origin of the complex plane. It is stressed that a total S/N of 4 when having 126 channels implies a S/N per channel of slightly less than 0.4. It is impossible to determine a polarization angle with such a low S/N in the case of standard fitting.
It is also possible to perform a RMsynthesis with Stokes Q or Uonly. There exist many radio observations that have produced only Stokes I and Q, for example spectral line work with arrays equipped with linearly polarized feeds, or data from backends with limited correlator capacity. However, by using only one of the two Stokes parameters, one loses information about the sign of the Faraday depth.
Figure 9: Comparison between the standard error in RM obtained by traditional line fitting (line) to simulated RMsynthesis experiments where a parabola was fit to the main peak of the Faraday dispersion function (dots). The 126 points used for this figure are the same as the ones used in Fig. 3. 
The derivation is started with Eq. (25). The identities
(59)  
(60) 
Figure 10 compares results of a complete RMsynthesis of data of the Perseus cluster, taken with the WSRT (de Bruyn & Brentjens 2005), to results of a Qonly RMsynthesis of the same dataset. It compares both the Galactic foreground emission at low Faraday depth, and the emission at higher Faraday depth that we attribute to the Perseus cluster. It is clearly seen that the noise in the Qonly images is increased with respect to the complete RMsynthesis. The barlike feature at , is already visible in the Qonly images. This demonstrates that one actually can detect faint emission at high Faraday depths using only Stokes Q or U images. Unless the situation is simple, meaning only one discrete source along the line of sight, these images are unfortunately not useful in a quantitative sense. However, it is an efficient way to discover weak, Faraday rotated, polarized emission in existing datasets, which can then be followed up with full polarization observations.
Three main parameters are involved when planning a rotationmeasure
experiment, namely the channel width
,
the width of
the
distribution
,
and the shortest
wavelength squared
.
They are summarized in
Fig. 11. These parameters determine
respectively the maximum observable Faraday depth, the resolution in
space, and the largest scale in
space to which one is
sensitive. Estimates for the FWHM of the main peak of the RMTF, the
scale in
space to which sensitivity has dropped to 50% and
the maximum Faraday depth to which one has more than 50% sensitivity
are approximately
It is interesting to compare Eqs. (61) and (62). This is where the analogy between RMsynthesis and regular synthesis imaging breaks down. In synthesis imaging, the width of the synthesized beam is inversely proportional to the maximum absolute uv vector. That is, the distance between the origin and the uvpoint most distant from it. The maximum scale that one can measure depends on the shortest baseline. Therefore one is always maximally sensitive to structures smaller than the width of the synthesized beam.
Figure 11: The three instrumental parameters that determine the output of a Faraday rotation experiment. 
This is quite different in RMsynthesis. In RMsynthesis it
is possible that a source is unresolved in the sense that its extent
in
is less than the width of the RMTF, yet "resolved'' out
because one has not sampled the typical scale of the source due
to lack of small
points. Equation (61) shows that the width of the RMTF
depends on the width of the
distribution, not on the
largest
measured. Nevertheless the largest scale in that one is sensitive to is set by the smallest
as is
shown in Eq. (62). In order to truly
resolve Faraday thick clouds in
space in the sense that one
could see internal structure, the main peak of the RMTF should be
narrower than the maximum scale to which one is sensitive. Because
,
the requirement for resolving Faraday thick
structures is
For deconvolution the RMTF should be known as accurately as possible for all sources within the field of view and along the line of sight. The main problems are:
After primary beam correction, one should align the channel maps spectrally. Our preferred method is to determine the average total intensity of a large sample of sources, and scale the images until the average of the ensemble in a particular channel map matches the value at . Using this approach, the spectra of as yet undetected emission should be approximately flat to within a spectral index range of . Of course one could flatten source spectra on an individual basis. This is only useful if one is interested in bright sources that are easily detected in individual channels.
A convenient property of RMsynthesis is that moreorless frequency independent instrumental problems end up at , convolved with the RMTF. This means that instrumental problems are highly reduced at higher absolute Faraday depths. In other words: at high Faraday depth, we "windup'' the instrumental polarization problems, while "unwinding'' the Faraday rotated cosmic polarization signals.
We have extended the work of Burn (1966) to the cases of limited sampling of space and some spectral dependencies. We have introduced the RMTF, which is an excellent predictor of ambiguity problems in the frequency setup. RMsynthesis can be implemented very efficiently on modern computers. For example, a RM synthesis of 126 input maps of 1024^{2} pixels, yielding output maps of 1024^{2} pixels (P, Q, and U) takes less than 5 min on a laptop equipped with a 2 GHz Intel Pentium processor and 512 MB of RAM.
Because the analysis is easily applied to wide fields, one can conduct very fast RM surveys of weak sources. Difficult situations, for example multiple sources along the line of sight, are easily detected. Under certain conditions, it is even possible to recover the emission as a function of Faraday depth within a single cloud of ionized gas.
Instrumental problems that are weakly frequency independent, or have a very characteristic frequency dependence, are easily separated from cosmic signals that are only subject to Faraday rotation.
Rotation measure synthesis has already been successful in discovering widespread, weak, polarized emission associated with the Perseus cluster (de Bruyn & Brentjens 2005). In simple, high signal to noise situations it is as good as traditional linear fits to versus plots. However, when the situation is more complex, or very weak polarized emission at high rotation measures is expected, it is the only viable option.
Acknowledgements
We acknowledge Robert Braun, Torsten Enßlin and Peter Katgert for useful and vivid discussions on the subject. The Westerbork Synthesis Radio Telescope is operated by ASTRON (Netherlands Foundation for Research in Astronomy) with support from the Netherlands Foundation for Scientific Research (NWO).
The expected standard errors in RM/Faraday depth and are useful quantities when planning a rotation measure experiment. In this appendix we present a formal derivation.
(A.1)  
(A.2) 
The derivation is done in two steps. First we derive the standard error in the polarization angle and total polarization, and of measurements in individual channels. Then we apply standard results for the least squares fit of a straight line to obtain and , the standard errors in rotation measure / Faraday depth and the polarization angle at .
Error propagation (Squires 2001) gives us
(A.9)  
(A.10) 
(A.11) 
(A.13) 
Equation (A.16) may now be simplified by
substituting Eqs. (A.17) and (A.12). The final result is:
(A.18) 
(A.19) 
(A.20) 
In this appendix we show, as an illustration, three model runs of an RMsynthesis of an artificial Faraday dispersion function, measured with a realistic frequency sampling. We hope that these figures aid in understanding the most important aspects of RMsynthesis specifically and rotation measure work in general.
Sources that are extended in the plane of the sky have their surface brightness measured in Jy per steradian. For point sources the flux in Jy is sufficient to characterize it. The respective brightness units for sources that are both extended in the plane of the sky and in Faraday depth are Jy steradian^{1} (rad m^{2})^{1} or Jy m^{2}rad^{3}. Sources that are extended in the plane of the sky and pointlike in space have their brightness in space measured in Jy steradian^{1}. The brightness of the measured Faraday dispersion function has units of Jy (beam on the sky)^{1}(rmtf)^{1}. Sources that are pointlike in the plane of the sky have the steradian^{1} or (beam on the sky)^{1} removed.
In order to keep the units simple, we made all simulated sources pointlike in the sky plane. Hence the units used in the figures in this appendix are:
The RMTF in all three figures is the same because the pattern and width of the coverage is exactly the same for all of them. The only difference is the absolute position of the pattern. Figure B.1 has , Fig. B.2 has , and Fig. B.3 has .
The three sources in this simulation have different properties to illustrate different cases.
Source B represents the other extreme. Being several RMTFs wide, one requires in order to recover the full flux of the source. Only Fig. B.1 meets this requirement. In Fig. B.2, only two bumps at the edges of the source remain. Because in Fig. B.2 we only sample smaller scales in due to the larger , the only parts of source B that remain are the parts where these smaller scales are important: the edges. Source B has practically disappeared in Fig. B.3.
Source C is of an intermediate type. Because its typical scale is narrower than source B, there is a larger fraction of the total flux recovered in Figs. B.2 and B.3.
In analogy to radio interferometric observations, one could state that the sampling in Fig. B.1 corresponds to a connected element array, where one samples all scales up to approximately equally well. Figure B.3 corresponds to a VLBI observation, where one misses the short spacings and therefore is insensitive to extended emission. A fundamental difference with radio interferometry is that the resolution in space is determined by the width of the distribution, , and not by the largest sampled. Hence one could encounter situations where a source is not resolved in the sense that the thickness of the source in is much less than the width of the RMTF, while at the same time it is resolved out in the sense that one has not sampled sufficiently short points to detect the source.