One of the key techniques used to obtain information about the cluster magnetic fields strength and geometry is the Faraday rotation analysis of radio sources in the background of clusters or in the galaxy clusters themselves.
RM calculated from extragalactic radio sources can be considered as the sum of three integrals which represent the contribution of three different regions, namely internal to the source itself, due to our own Galaxy, and occurring in the ICM. The latter is the RM in which we are interested here.
Typical values of the RM of Galactic origin are of the order of 10 rad m-2 for most sources, and up to 300 rad m-2 for sources at low Galactic latitudes . Once the contribution of our Galaxy is subtracted, however, the RM of radio galaxies located inside or behind clusters should be dominated by the contribution of the ICM.
High resolution RM studies of Cygnus A  were the first to demonstrate that the high RM values with large gradients on arcsec scales cannot be either of Galactic origin or due to a thermal gas mixed with the radio plasma, but must arise in an external screen of magnetized, ionized plasma. Similarly, the asymmetric depolarization found in double radio lobes embedded in galaxy clusters can be understood as resulting from a difference in the Faraday depth of the two lobes [130, 131, 132] (Laing-Garrington effect). Indeed, the radio source lobe pointing towards the observer is less depolarized than the lobe pointing away.
The observing strategy to get information on the cluster magnetic field intensity and structure is to obtain high resolution RM maps of sources located at different impact parameters of a cluster, then derive the average value of the rotation measure <RM> and the value of its dispersion RM. As described in Sec. 3.5.2, the RM values are combined with measurements of the thermal gas density ne to estimate the cluster magnetic field along the line of sight. Such studies have been carried out on both statistical samples and on individual objects.
The first successful statistical demonstrations of Faraday rotation from radio sources seen through a cluster atmosphere were presented by Lawler and Dennison  for a dozen of radio galaxies and by Vallée et al.  for A2319. In both studies, a broadening of the values of RM was found in the cluster sources, with respect to the sources in a control sample.
Kim et al.  investigated the magnetic field in the Coma cluster using 18 radio sources, and found a significant enhancement of the RM in the inner parts of the clusters. They deduced a field strength of ~ 2 µG. For the magnetic field structure, they assumed the simple model with a single typical length for field reversal, i.e. a cluster field consisting of cells of uniform size, with the same electron density and magnetic field strength, but with a random field orientation. They obtained a cell size in the range 10 - 30 kpc. In the following year, Kim et al.  improved the statistics by analyzing a much larger sample of 106 radio sources, and deduced that magnetic fields strengths in the cluster gas are of the order of 1 µG. In a more recent statistical study, Clarke et al.  analyzed the RMs for a representative sample of 27 cluster sources, plus a control sample, and found a statistically significant broadening of the RM distribution in the cluster sample, and a clear increase in the width of the RM distribution toward smaller impact parameters. Their estimates give a magnetic field of 4 - 8 µG, assuming a cell size of ~ 15 kpc.
The first detailed studies of RM on individual clusters have been performed in cooling core clusters, owing to the presence of powerful radio galaxies at their centers. Extreme values of RMs are found to be associated with these radiogalaxies, with the magnitude of the RMs roughly proportional to the cooling rate. . Magnetic fields, from ~ 5 µG up to the values of ~ 30 µG are deduced in the innermost regions of these clusters, e.g. Hydra A  and 3C295 [138, 139].
Polarization data from sources at different cluster locations have been obtained in clusters without cooling cores, i.e. Coma , A119 , A514 , 3C129 , A400 , A2634 . In the Coma cluster, Feretti et al.  derived a magnetic field of 7 µG tangled on scales of ~ 1 kpc, in addition to a weaker field component of ~ 0.2 µG, ordered on a scale of about one cluster core radius. Generally, a decreasing |<RM>| and RM with an increasing projected distance from the cluster center is found. RM gradients are detected across the sources, indicating the presence of structure in the intracluster magnetic field. The data lead to magnetic field estimates of ~ 2 - 8 µG, with patchy structures of ~ 5 - 15 kpc.
Overall, the data are consistent with cluster atmospheres containing µG fields, with perhaps an order of magnitude scatter in field strength between clusters, or within a given cluster, and with extreme field values in cluster cooling cores. These estimates of the magnetic field strength from RM data crucially depend on the magnetic field structure and geometry. The RM distribution is generally patchy, indicating that large-scale magnetic fields are not regularly ordered on cluster scales, but have structures on scales as low as 10 kpc or less. In Fig. 6 we show the RM images obtained for the two central radio galaxies in A119 .
Figure 6. VLA contour plot at 21 cm and RM images (insets) of the two tailed radio galaxies 0053-015 (left) and 0053-016 (right) in A119 . The contour plot is overlaid onto the ROSAT X-ray image (gray-scale) of the cluster. The two radio galaxies are located at a projected distance from the cluster center of ~ 0.45rc and 1.2rc respectively. Both the RM images show fluctuations on small scales (~ 10 kpc). The RM values in 0053-015 are between -350 rad m-2 and +450 rad m-2 with <RM> = 28 rad m-2, and RM = 152 rad m-2. The RM values in 0053-016 are between -300 rad m-2 and +200 rad m-2, with <RM>= -79 rad m-2 and RM = 91 rad m-2.
In many cases, high resolution RM images show a nearly Gaussian RM distribution, suggesting an isotropic distribution of the field component along the line-of-sight. However, many RM distributions show clear evidence for a non-zero mean <RM> if averaged over areas comparable with the radio source size, even after the Galactic contribution is subtracted. These <RM> offsets are likely due to fluctuations of the cluster magnetic fields on scales greater than the typical source size, i.e. considerably larger than those responsible for the RM dispersion. The random magnetic field must therefore both be tangled on sufficiently small scales, in order to produce the smallest structures observed in the RM images and also fluctuate on scales one, or even two, orders of magnitude larger, to account for the non-zero RM average. For this reason, it is necessary to consider cluster magnetic field models where both small and large scale structures coexist.
So far very little attention has been given in the literature to the determination of the power spectrum of the intracluster magnetic field fluctuations. Very recently Enßlin and Vogt  and Vogt and Enßlin  pointed out that the single scale cell model is not realistic because it does not satisfy the condition div = 0. By using a semi-analytic technique, they showed that the magnetic field power spectrum can be estimated by Fourier transforming RM maps if very detailed RM images are available. Moreover, they derived that the autocorrelation length of the RM fluctuations is in general larger than the magnetic field autocorrelation length.
An alternative numerical approach to investigate the strength and structure of cluster magnetic fields through Monte Carlo simulations is presented in Murgia et al. . A brief description of the capability of such a numerical approach is presented in Sec. 10.
It is worth mentioning here that some authors have suggested the
possibility that the RM observed in radio galaxies is not
associated with the foreground ICM, but may arise locally to the radio
either in a thin skin of dense warm gas mixed along the edge of
the radio emitting plasma, or in its immediate
surroundings. There are, however, several
arguments against this interpretation:
i) the trends of RM versus the cluster impact parameter in both statistical studies and individual cluster investigations,
ii) the relation between the RM and the cooling flow rate in relaxed clusters ,
iii) the Laing-Garrington effect [130, 131, 132],
iv) statistical tests on the scatter plot of RM versus polarization angle, for the radio galaxy PKS1246-410 ,
v) the very consistent scenario drawn by all the results presented in this section.
Thus, we conclude that local effects might give some contribution to the RM, however the major factor responsible for the Faraday Rotation should be the ICM. Future high resolution RM studies with the next generation radio telescopes (e.g. EVLA, LOFAR, SKA) should help in distinguishing the local effects, as well as possible effects arising internally to the radio sources.