###### Abstract

Here, I overview one of the available techniques for the analysis of broad-
band spectropolarimetric data, the Stokes QU-fitting. Since broad-band
receivers have been installed at most radio facilities, the collection of
radio data, both the total intensity and the linear polarization, is revealing
interesting features in their spectra. The polarized light, and therefore its
properties, i.e. the fractional polarization p and the polarized angle $\chi$,
are now finally well sampled in wide wavelength ranges. The new complex
behaviors revealed by the data can be studied using the Stokes QU-fitting,
which consists of modeling the Stokes parameters Q and U using wavelength-
dependent analytical models, available in the literature. This technique
provides a very good diagnostic of the nature and structure of the magnetized
plasma, with the possibility to identify complex structures, internal or
external, of the source of study. A summary of the available and most used
models describing the polarization behavior, is presented. Moreover, some of
the most significant observational works which use this technique are also
summarized.

###### keywords:

spectropolarimetry; polarization; magneto-ionic medium

1 1 0 Academic Editor: Margo Aller, Jose L. Gómez and Eric Perlman 13 April
2021 1 August 2021 https://doi.org/ Message in a Bottle: Unveiling the
Magneto-Ionic Complexity of AGNs through the Stokes QU-Fitting Technique
Message in a Bottle: Unveiling the Magneto-Ionic Complexity of AGNs through
the Stokes QU-Fitting Technique Alice Pasetto 1,2Alice Pasetto Pasetto, A.

##  1 Introduction

An important astrophysical topic that is still a matter of investigation is
the understanding of the physical properties of magnetised plasma. It is still
unclear under which conditions, and how, the magnetic field influences the
formation and evolution of astrophysical objects, what its role is in
fundamental processes such as galaxy formation, particle acceleration or
confinement of the Active Galactic Nuclei (AGN) jets. It is indisputable that
the magnetic field (B-field) is present everywhere in the Universe and at
different physical scales. It is possible to investigate the B-field with
different observational methods, at very different wavelengths, directly
through polarized observations in the electromagnetic spectrum, or indirectly
through, for example, magnetic Kelvin–Helmholtz (KH) instabilities (e.g.,
Vikhlinin et al. (2001)). In the radio regime, an important radiative process,
which provides information about the magnetic field, is the synchrotron
radiation. Measuring the wavelength dependence of the polarized light from
synchrotron sources can reveal valuable and interesting constraints on the
nature of the objects observed, such as insights into the magneto–ionic medium
internal to the source of investigation or about their surroundings. The
quantities extracted directly from the observations are the Stokes parameters,
I, Q, U and V, where I represents the total intensity, Q and U represent the
linear polarized signal and V the circular polarization. In this review, only
the properties of the linear polarization will be discussed; therefore, the
Stokes V is not taken into consideration. From the Stokes parameters I, Q and
U, one can estimate the fractional polarization (p) and the polarized angle
($\chi$). The first gives an idea of the ratio of the turbulent over ordered
magnetic field strength projected in the plane of the sky, while the latter
provides information about the orientation of the magnetic field projected on
the sky. Moreover, the synchrotron polarized light can experience two effects:
Faraday rotation and Faraday depolarization. Both these effects can alter the
polarization properties (p and $\chi$) of the radiation. The study of both
phenomena allows us to obtain information about the strength, the direction
and the complexity (ordered or turbulent) of the magnetic field, identified by
the parameters of Faraday Depth (FD - $\phi$) or the Rotation Measure (RM)
dispersion ($\sigma_{RM}$). Measurements and analysis of these quantities can
help to define a 3-dimensional (3D) view of the magneto–ionic medium in
astrophysical sources.

In general radio polarimetric studies have been considered very challenging.
Since the polarized light is only a small fraction of the total emission from
a source, it is necessary to observe with high sensitivity in order to perform
a proper polarization analysis. Moreover, the collection of sparse and narrow-
band polarized data makes difficult to fully describe the polarization
behavior of the subject of study. Nevertheless, important works focused on
identifying the magneto–ionic structure of radio sources were performed in the
last decades using different radio telescopes, different frequencies and
different angular resolutions (e.g., Cotton et al. (1984); Dreher et al.
(1987); Attridge et al. (1999); Berkhuijsen et al. (2003); Zavala and Taylor
(2003)). Many spectral studies of source-integrated, radio band linear
polarization from the Michigan program, starting in the mid 1960s, and from
the Brandeis group using the Very Long Baseline Array (VLBA) data, gave
important results (see Wardle, 2004, and references therein). Simultaneously,
important theoretical works have also been developed in order, for example, to
explain the observed brightness and polarization asymmetries between the jets
and counter-jets for some Fanaroff–Riley Class I (FR I) radio galaxies. The
asymmetries are explained with a model in which decelerating backflows,
containing a toroidal magnetic field, are covering the central jets’ spine
Laing and Bridle (2012, 2014). Moreover, besides using the traditional
determination and interpretation of the RM (i.e., as the linear fit of the
polarized angle in a range of sparse wavelength squared) to extract
information about the magnetic structure of astrophysical objects, an
alternative technique was developed. It combines the study of the spectral
index (called spectral tomography Katz-Stone and Rudnick (1997)) with the
polarized emission information. This technique also produced important
results, such as the detection of the jet spine and sheath components in wide-
angle tails’ radio sources (see Katz-Stone et al. (1999); Hardcastle (1999)).

Thanks to recent upgrades performed at some radio facilities, for example, at
the Jansky Very Large Array (Jansky VLA) and at the Australia Telescope
Compact Array (ATCA), and the new incoming generation of ultrasensitive radio
telescopes, such as the Low-Frequency Array (LOFAR van Haarlem et al. (2013)),
the Murchison Widefield Array (MWA Tingay et al. (2013)), the Australian
Square Kilometer Array Pathfinder (ASKAP McConnell et al. (2016)) or the
future Square Kilometer Array (SKA), which are (or will be) equipped with
broad-band receivers, radio polarization observations have become a much more
captivating and fashionable research area. Thanks to this new capability, our
ability to interpret polarised sources has dramatically improved. Now, we are
able to collect well sampled polarized data and therefore to follow the
behavior of the polarization properties in a broader frequency domain. It is
possible to use the Faraday effect to create a tomographic reconstruction of
magnetized structures along the line of sight (LOS) and, therefore, to
determine the distribution, and analyze the properties, of the magneto–ionic
material along the LOS internal or external of an astronomical object. This
process is called Faraday tomography (see the Galaxies MDPI special issue of
the workshop—The Power of Faraday Tomography—for more information
https://www.mdpi.com/journal/galaxies/special_issues/FaradayTomography,
accessed on November 2018), which allows to add information on the origin and
structure of magnetic fields. As a consequence, new computational techniques
for the analysis of the wide-band spectropolarimetric data have been
developed. The two most commonly used techniques are the Faraday RM synthesis
Brentjens and de Bruyn (2005) and the Stokes QU-fitting Farnsworth et al.
(2011); O’Sullivan et al. (2012). The first is a Fourier transform operation
on the polarized radiation in the wavelength domain; it assumes the target as
a sum of emitters at different Faraday depths. Because of its nature, the RM
synthesis can be considered a non-parametric technique. There are several
works in the literature using this technique (e.g., see recent works by Loi et
al. (2019); Krause et al. (2020); Stuardi et al. (2020)), a detailed
explanation of which is out of the scope of this work. In contrast, the QU-
fitting technique, which is the focus of this review, is a model fitting
method (usually computationally more expensive but more powerful than RM
Synthesis Sun et al. (2015)), which consists of fitting both the Stokes Q and
U parameters, wavelength ($\lambda$) dependent, using analytical models
available in the literature. This technique provides an excellent diagnostic
of the nature and structure of magnetized plasma. It has the advantage that
physically motivated models can be fit directly to the polarization data,
therefore yielding to physical constraints on the magneto–ionic structure of
the sources. These models assume the presence of a regular and/or a turbulent
magnetic field (e.g., Burn (1966); Tribble (1991); Sokoloff et al. (1998)).
For this reason, the Stokes QU-fitting needs some assumptions on the nature of
the medium. The main parameters in the models are the intrinsic fractional
polarization (${p_{0}}$), the intrinsic polarization angle ($\chi_{0}$), the
Faraday Depth ($\phi$) and the RM dispersion ($\sigma_{RM}$). Generally, the
fit model attempts to describe the complex physics that the radio object is
experiencing. For that reason, sometimes it is necessary to use multiple
Faraday components (i.e., a combination of models) to better describe the
polarized behavior.

This work is structured as follows: in Section 2 I summarize the most commonly
used models for the Stokes QU-fitting technique, in Section 3, I review some
of the recent and most significant observational works on spectropolarimetry
with a special focus on AGN jets, and in Section 4, a summary and conclusions
are given.

##  2 Polarization Models

Analytical models that attempt to describe polarization behavior have been
available since the 1960s. They explore different scenarios in which uniform
and/or turbulent magnetic fields are involved, according to the results of
radio observations (for details see the works by Burn (1966); Tribble (1991);
Sokoloff et al. (1998); Rossetti et al. (2008)). For an exhaustive review on
the polarization of extragalactic radio sources, please see Saikia and Salter
(1988).

To interpret the information of the polarized light of a source which passes
through a medium, it is necessary to define the observables. The synchrotron
radiation from a radio source is intrinsically partially polarized and its
polarization signal can be represented by its Stokes parameters, I, Q, U and
V. These parameters have the advantage that they are directly measured from
the radio telescope and describe the polarization characteristics of a source.
Following Sokoloff et al. (1998), these observables can be combined to
describe the linear polarized signal as a complex number:

$P=Q+iU=pIe^{2i\chi},$ (1)

where $\chi$ is the observed electric E-vector polarization angle. Using the
notation in Farnsworth et al. (2011) we define the fractional Stokes
parameters, q = Q/I and u = U/I, so that the fractional (linear) polarization
p is defined as:

$p=\sqrt{q^{2}+u^{2}},$ (2)

and the polarized angle is obtained from:

$\chi[rad]=\frac{1}{2}{\rm arctan}(\frac{u}{q}).$ (3)

Faraday rotation and, consequently, the Faraday depolarization, are two
phenomena that can complicate the behavior of the linearly polarized optically
thin synchrotron emission. In fact, when one, or both, of these events take
place, the polarized angle and the fractional polarization can be altered. The
polarized wave can be decomposed into two opposite-handed circularly polarized
components. When the radiation passes through a magneto–ionic medium, these
opposite-handed waves experience different phase velocities within the
material. This results in a rotation of the polarized plane of the linearly
polarized wave, called the Faraday rotation. The general expression of the
Faraday depth is given by the equation:

$\phi[rad/m^{2}]=8.1\times 10^{5}\int_{L}n_{e}B_{\parallel}dL,$ (4)

where $n_{e}$ [cm-3] is the electron density in the plasma, $B_{\parallel}$
[Gauss] is the magnetic field along the line of sight and L is the path length
[pc]. Narrow-band multi-wavelength polarization observations have shown that
the polarization angle changes linearly as a function of $\lambda^{2}$ so
that:

$\Delta\chi[rad]=\chi(\lambda)-\chi(0)=\phi\lambda^{2}.$ (5)

From these equations, it becomes clear that a detailed study of the
polarization properties across the spectrum can provide physical information
about the source and its surroundings. In the presence of a uniform magnetic
field in the medium, the Faraday depth is identical to the polarized emission-
weighted mean value, called the Rotation Measure (RM), which is therefore
determined by fitting the linear equation
$\chi(\lambda^{2})=\chi_{0}+RM\lambda^{2}$. However, this equation is true
only when one restricts the RM fitting to regions of the $\lambda^{2}$ space,
where the fractional polarization, p is constant (e.g., Simard-Normandin et
al. (1981)). In this scenario, the complex representation of the polarized
wave in the presence of a single Faraday RM screen, which only rotates the
polarized plane of the light, leaving the fractional polarization constant,
becomes:

$P=p_{0}e^{2i(\chi_{0}+RM\lambda^{2})},$ (6)

where $p_{0}$ and $\chi_{0}$ are the intrinsic fractional polarization and the
intrinsic polarization angle, respectively. This is the only situation where
d$\chi$/d$\lambda^{2}$ and p($\lambda^{2}$) are constant, that is, only a
single uniform foreground Faraday screen is present. Real scenarios are much
more complicated and more sophisticated modeling is required. Since the
spectropolarimetric data obtained from broad-band receivers have become
available, the behavior of the main polarization parameters, p and $\chi$,
have been observed to behave in a complex way. The general behavior of $\chi$
is very far from a simple linear variation with $\lambda^{2}$, and the p is
usually reduced at longer wavelengths producing a Faraday depolarization
(e.g., Law et al. (2011); O’Sullivan et al. (2012); Anderson et al. (2015,
2016); Pasetto et al. (2016, 2018)). Thus, it is necessary to study these
properties across the spectrum in order to properly obtain the physical
parameters of the sources Farnsworth et al. (2011). The first exhaustive work
that examines the polarized emission for a subsample of sources taken from the
NRAO VLA Sky Survey (NVSS) catalogue was done by Farnes et al. (2014).
Although they collected the available narrow-band polarized data, they could
already fit the depolarization behaviors, using simple descriptions of the
more complex depolarization models.

Depolarization can be caused internally within the source, when the
synchrotron emitting and the Faraday rotating regions are spatially
coincident, or externally to the source, when the Faraday rotation region is
located between the source and the observer. Moreover, both scenarios can be
described, considering the presence of either a uniform magnetic field, a
turbulent magnetic field or a combination of both. For an exhaustive
description of the different scenarios, see Sokoloff et al. (1998).
Depolarization effects should therefore be added to the complex linear
polarization Equation (6). When the synchrotron radiation passes through an
external magneto–ionic medium, that is, a region devoid of relativistic
electrons, containing a turbulent magnetic field, the plane of polarization
undergoes a random walk which leads to depolarization as long as many
turbulent cells are within the beam area. For a regular magnetic field,
depolarization can occur when variations in the strength or direction of the
field within the observing beam, exist. This depolarization is called External
Faraday Dispersion/Beam depolarization (EFD/BD) and it is represented by the
equation

$P=p_{0}e^{-2\sigma^{2}_{RM}\lambda^{4}}e^{2i(\chi_{0}+\phi\lambda^{2})},$ (7)

where $\sigma_{RM}$ is the Faraday dispersion of the random field within the
volume traced by the telescope beam Farnsworth et al. (2011) and $\phi$ is the
Faraday depth across the source on the sky and it assumes the form of RM when
considering the presence of a regular magnetic field.

When the synchrotron-emitting and the Faraday rotating regions are embedded
together, the depolarization is called internal. The main equations which
describe this depolarization scenario are: (1) internal Faraday dispersion
(IFD) and (2) the differential Faraday rotation (DFR) Burn (1966); Sokoloff et
al. (1998). When the synchrotron-emitting and the Faraday-rotating regions
contain a mixture of a turbulent and regular magnetic fields, the equation
which takes into account both scenarios is the following:

$P=p_{0}e^{2i\chi_{0}}\left(\frac{1-e^{-S}}{S}\right),$ (8)

where S = 2$\sigma^{2}_{RM}\lambda^{4}$–2i$\phi\lambda^{2}$. This equation
includes the DFR as the imaginary term and the IFD as the real term. Here, a
random walk of the plane of polarization through the region, occurs. In this
equation, $\sigma_{RM}$ is the Faraday dispersion of the internal random field
and again $\phi$ is the Faraday depth through the region. When the turbulent
magnetic field component is neglected ($\sigma_{RM}$ = 0), the emitting and
rotating regions are only co-spatial in the presence of a regular magnetic
field. In this situation, the DFR takes place and the complex degree of
polarization is given by:

$P=p_{0}e^{2i(\chi_{0}+\frac{1}{2}\phi\lambda^{2})}\left(\frac{{\rm
sin}(\phi\lambda^{2})}{(\phi\lambda^{2})}\right).$ (9)

In this case, the radiation coming from the most distant part of the region
with respect to the observer suffers a different amount of Faraday rotation
(in this case $\phi$ is a parameter whose value is to be estimated) with
respect to the radiation coming from the nearest part of that region.
Therefore, this equation also embraces the scenario in which a smooth change
in the Faraday depth occurs within a source. Here, the Faraday depth $\phi$
assumes the form of a simple RM, which produces a gradient of RM ($\Delta$RM)
across the telescope beam (Berkhuijsen and Beck, Berkhuijsen; Sokoloff et al.,
1998). Equation (9) becomes:

$P=p_{0}e^{2i(\chi_{0}+RM\lambda^{2})}\left(\frac{{\rm sin}(\Delta
RM\lambda^{2})}{(\Delta RM\lambda^{2})}\right).$ (10)

However, the parameter $\Delta$RM can describe both internal and external
Faraday depolarization effects; for example, a linear gradient in RM across
the emission region, or internal Faraday rotation in a uniform field (Sokoloff
et al., 1998; Schnitzeler et al., 2015). The internal case is of great
interest since it should represent the simplest scenario occurring within a
radio jet where a smooth change of the magnetic field configuration across the
jet is expected (Laing, 1981; Blandford, 1993).

More generally, the polarization dependencies given by Equations (7)–(10)
could not be uniquely associated with the given distributions of emitting and
thermal plasma. For example, the oscillatory behavior of the fractional
polarization is not uniquely associated with the presence of the mixing of
thermal and non-thermal plasma. In fact, it can be produced by a combination
of several narrow Faraday screens of the Faraday dispersion function (FDF,
equation (6)). This scenario describes a regular magnetic field but it does
not specify between the foreground or the mixed model. The same occurs when
talking about the external Faraday rotation. It can be represented by the FDF,
located in the foreground of a mixed plasma, but represented with a Gaussian-
like shape. Equations (7)–(10), correspond to FDF that can be produced by
specific and more complex configurations of thermal and non-thermal plasma.
Therefore, it is necessary to support the analysis of the polarized data with
additional information, for example, the angular resolution, or from the
analysis of the continuum synchrotron emission to draw a more complete picture
of the magneto–ionic medium of the object of study.

To investigate the complex behavior of the polarized signal, it is therefore
necessary to fit simultaneously the wide band Stokes Q and U spectra as
proposed by Farnsworth et al. (2011); O’Sullivan et al. (2012). To capture a
broad range of possible Faraday rotation behaviors, we use a general equation
which attempts to describe all the above mentioned depolarization models
(describing the effects of both random and uniform magnetic fields), which is
the following complex polarization equation:

$P=p_{0}e^{2i(\chi_{0}+\frac{1}{2}\phi\lambda^{2})}sinc(\phi\lambda^{2})e^{-2\sigma^{2}_{RM}\lambda^{4}},$
(11)

where p0 is the intrinsic fractional polarization, $\chi_{0}$ is the intrinsic
polarization angle, and $\phi$ and $\sigma_{RMj}$ describe the variation of
the Faraday depth in a regular and turbulent magnetic field, respectively.
This equation uses the simplest possible parameterization of the Faraday
depolarization from uniform and random fields. It is also implicitly assumed
that the polarized emission is coming from optically thin regions with similar
spectral indices. However, the above equation is too simple for explaining the
complex scenarios revealed by real astrophysical data. Instead, multiple
emitting and/or rotating Faraday components unresolved within the telescope
beam seem to better represent the reality in most cases (see, e.g., O’Sullivan
et al. (2012); Anderson et al. (2016); Pasetto et al. (2018)). Therefore, to
model more complicated behaviors, extra RM components need to be used. A
multiple Faraday components model is simply constructed as P = P1 \+ P2 +…+ Pn
O’Sullivan et al. (2012). A representation of the models mentioned above is
shown in Figure 1.

The operational procedure to perform the fitting is explained below. Firstly
we need to collect wide-band polarized data, which, after a standard cleaning
process, result in a sample of Stokes I, Q and U images at each available
frequency (within the wide-band frequency range). The subsequent step depends
on the scientific case to address. The polarized Stokes parameters can be
extracted at each frequency considering either the object of study in its
totality (in the case of both resolved or unresolved radio sources) or by
selecting specific areas. Lastly, the parameters are then used for the model
fit, applying Equation (11). The results of the fitting are the intrinsic
polarized properties of the magnetized layer/s ($p_{0j}$, $\chi_{0j}$,
$\phi_{j}$ and $\sigma_{RMj}$ ) which describe a scenario where the Faraday
screen/s is/are either internal or external to the source of study, depending
on the best resulting fit model, or a combination of models. The common
statistic used to determine the goodness of the fitting is the standard
${\chi^{2}}$ test. In general, the ${\chi^{2}}$ value decreases with an
increasing number of initial parameters. However, choosing a model with too
many parameters can result in an overfitting. Therefore, two other statistical
methods are usually used in combination with the ${\chi^{2}}$ test: the Akaike
Information Criterion (AIC) and the Bayesian Information Criterion (BIC). AIC
estimates the quality of each model, relative to each of the other models
considered in the study, while BIC is a criterion for model selection among a
sample of models.

Figure 1: Representation of de/polarization models: (a) Equation (6), Faraday
rotation (only one Faraday component does not depolarize the emission); (b)
Equation (7), External Faraday depolarization (considering turbulent magnetic
field); (c) Equation (9), Internal Faraday depolarization (considering regular
magnetic field, only); (d) Equation (11), Mixture of Internal and External
Faraday depolarization. Comparison between the behaviors of low and large RM
values and low and large $\sigma_{RM}$ and $\Delta_{RM}$ are represented by
dashed and straight lines, respectively. The used values of RM, $\sigma_{RM}$
and $\Delta_{RM}$, are reported in the labels on the right side of the plots.

2

##  3 Observational Broad-Band Spectropolarimetry Works

###  3.1 Unresolved Sources

Although early analyses performing model fit on sparse narrow-band fractional
polarization and polarization angle (using the traditional linear fit of
$\chi$ versus $\lambda^{2}$) radio data, are present in the literature, (e.g.,
Laing and Bridle (1987); Laing et al. (2008); Rossetti et al. (2008); Guidetti
et al. (2010)), the first call of attention about the importance of fitting
the Stokes Q and U in broad $\lambda^{2}$ domain was made by Farnsworth et al.
(2011). They presented a polarization analysis of 585 compact radio sources
observed with the WSRT at 350 MHz. On some of the sources, they performed
different analyses: the traditional $\chi(\lambda^{2})$ linear fit, the Stokes
QU-fitting and the RM Synthesis techniques. They fit the sources using two
simple depolarization scenarios: a foreground screen and two interfering RM
components. They found that, although a single representative RM may be found
using all the model fitting techniques, the Faraday structure of the sources
is not always adequately represented. They stressed the importance of
considering both the degree and the angle of polarization (or equivalently, q
and u) over a range of $\lambda^{2}$ as wide as possible, in order to obtain a
more global picture of the polarization behavior. They favour QU-fitting as
the technique which allows a more accurate description of the Faraday
structure/s.

A subsequent work opened the door to the broad-band spectropolarimetry study
on AGN jets: O’Sullivan et al. O’Sullivan et al. (2012) presented a detailed
study of the Faraday depth structure of four bright (>1 Jy), strongly
polarized, point-like radio-loud quasars. Using the ATCA telescope at
frequencies from 1 to 3 GHz (2 GHz of instantaneous bandwidth) they fit
various Faraday rotation models to the data. From the results of the broad-
band fitting it was claimed that the polarized structures arose from the
compact inner regions of the radio sources themselves (either from their
intrinsic emissions or from their local environments) and not from polarized
emissions from galactic or intergalactic foreground regions. This result has
important implications for using background extragalactic radio sources to
probe the Galactic and intergalactic magneto–ionic media. In fact, narrow-band
observations can produce erroneous results whenever a source shows multiple
interfering Faraday components. Moreover, this result suggests that RM time
variability in extragalactic point sources may be due to the evolution of
polarized components, which reveal different parts of a magneto–ionic medium
surrounding the radio jet.

The first VLA broad-band (1–12 GHz) spectropolarimetric study of a sample of
point-like AGN was performed by Pasetto et al. (2018). The sample (14 sources)
consisted of very compact sources (with linear sizes smaller than $\sim$5 kpc)
that are unpolarized at 1.4 GHz in the NVSS survey. The sources are highly
polarized at high frequency and therefore characterized by large Faraday RM
values. Highly reliable complexity (with very high signal to noise -S/N-
detection of $\sim$1000 for the continuum data and $\sim$100 for the polarized
signal), has been detected. The sources show several total intensity
synchrotron and Faraday components (up to three Faraday screens) in both their
continuum and polarized spectra. It is worth mentioning that more components
(synchrotron components or Faraday screens) are more likely to be fitted with
large S/N data. Depolarization modeling was performed on the polarized data
and a better knowledge on the AGN jet environment for that sample, was
achieved. The work concluded that for the majority of the targets, the very
large RMs and strong depolarization were due to turbulent magnetic fields
local to the sources. The synchrotron radiation from the jet passes through
several clumpy or highly turbulent regions which cover the central jet.
Similar conclusions were obtained by Anderson et al. (2016), by analyzing a
sample of unresolved sources. They also detected complex behaviors in almost
every source in the sample mostly due to Faraday dispersive medium intrinsic
to the source themselves. Examples of the depolarization model fitting on
point-like sources are shown in Figure 2.

O’Sullivan et al. O’Sullivan et al. (2017) presented the results of a broad-
band (1 to 3 GHz), spectropolarimetry study from 100 extragalactic point like
radio sources with the ATCA, selected to be highly linearly polarized at 1.4
GHz. The science case behind the project was to investigate whether some
correlation between the broad-band polarized behavior of the radio source and
their host galaxy accretion states existed. They divided the sample into:
“radiative-mode” (objects with strong, high-ionization, optical emission lines
powered by accretion onto the central super-massive black holes at rates in
excess of $\sim$1 per cent of the Eddington limit) and “jet-mode” (objects
with weak or non-existent optical emission lines, possibly due to radiatively
inefficient accretion) AGN. They fit multiple Faraday components, up to three
distinct polarized emission regions, to the majority of their sources and they
found no significant difference between the Faraday rotation or Faraday
depolarization properties of jet-mode and radiative-mode AGN. However, a
statistically significant difference in the intrinsic degree of polarization
between the two types has been found: the jet-mode sources showed more
intrinsically ordered magnetic field structures than the radiative-mode
sources. A preferred perpendicular orientation of the intrinsic magnetic field
structure to the axes of FRI jets, is considered a possible explanation for
the high intrinsic degree of polarization in the “jet-mode” sources. The
authors also found that almost all FRs are hosted by low-excitation/jet-mode
AGN. No clear magnetic field orientation preference was found for the
radiative-mode sources. This may be related to the inner jet regions of FRI
radio galaxies where the magnetic field is expected to have a high degree of
order.

Figure 2: Depolarization Stokes QU-fitting on the sources: (a) PKSB1039-47 and
(b) 0958+3224. The four panels represent: the Stokes parameters q, u (top
left), the fractional polarization p (top right), the polarized angle $\chi$
[deg] (bottom left) all versus wavelength squared $\lambda^{2}$[m2] and the
Stokes u versus Stokes q (bottom right). Straight lines are the resulting
model fit. Both sources have been fitted with three internal Faraday screens
(Equation (10)). Images adapted from O’Sullivan et al. (2012); Pasetto et al.
(2018).

2

Finally, it is also worth mentioning the work by Anderson et al. (2019), in
which they could detect the evolution of polarized structures on a sample of
blazars through the analysis of broad-band spectropolarimetic polarization
emission variability. The sources were selected from the catalogues in
O’Sullivan et al. (2012); Anderson et al. (2016), which are dominated by sub-
kpc-scale emission structures. They analyzed temporal changes between two
epochs separated by $\sim$5 years using ATCA data. Polarized complexity with
the presence of several Faraday components and variability in their
polarization emission, were detected. They concluded that the observed
spectral changes most likely originate because of the evolution in the parsec
to decaparsec scale structure of the blazar jets themselves. Figure 3 shows an
example of the polarized spectral changes between the two epochs.

Figure 3: Stokes I data (black points, top row of panels), Stokes q and u data
(red and blue points, middle row) and fractional polarisation data (p, black
points, last row) for PKS B0454-810 in the 2012 and 2017 epochs (left and
right column of panels, respectively). To facilitate comparison between
epochs, the best-fit model for both the 2012 and 2017 epochs are mirrored on
the opposite epoch axes as dashed orange lines for each of the Stokes I,
(q,u), and p plots. Straight lines (black, red and blue) are the best model
fit. The sub-axes on each panel are the standardised residuals (measured in
standard deviations) for the difference between the data and the best-fit
model. Image adapted from Anderson et al. (2019).

2

All the above works show how the QU-fitting technique is essential to
investigating the true Faraday depth structure; this technique is necessary
for probing the magnetized radio source environment and to finally spectrally
resolve the polarized components of unresolved radio sources.

###  3.2 Resolved Sources

Recently, the community is putting lot of efforts in studying the
magneto–ionic media surrounding AGN jets of nearby, well resolved radio
objects. Pioneer works on understanding the magneto–ionic media around
resolved radio jets are those, for example, in Owen et al. (1990); Laing et
al. (2008), in which the authors observed M87 and 3C31 with the VLA at four
and six discrete narrow-band frequencies, respectively. The objective was to
explore the spatial scale and the origin of the RM in these sources. They
concluded that, in both cases, the main reason for Faraday rotation and
depolarization behaviors are due to foreground material. Now, with the
available broad-band data, the community can perform more sophisticated QU-
fitting modeling, unveiling much more complex structures within or around the
sources of study.

Banfield et al. Banfield et al. (2019) showed broad-band radio polarization
observations of NGC 612 taken with the ATCA from 1 to 3 GHz. They examined the
linear polarization, Faraday rotation, and depolarization distributions
resulting from the interaction between the lobes of the radio galaxy NGC 612
and the circumgalactic gas. They found that the polarized properties are
tracing a behavior, likely due to an external Faraday rotation characterized
by a turbulent magnetic field in the intra-group medium. Evidence of a
correlation between the Faraday structure and the radio morphology was found.
In particular, the increase in the RM and the depolarization at the hotspot is
most likely explained by the compression of the ambient gas due to the bow-
shock region on to the lobe.

The radio lobes of the radio galaxy Fornax A have been analyzed by Anderson et
al. (2018). A detailed study of both the continuum and the polarized
radiations, were performed using the ATCA at frequencies from 1 to 3 GHz. They
could map with great detail the magnetic field of the radio lobes. The Stokes
QU-fitting revealed an oscillatory depolarization behavior, therefore leaving
open two scenarios where the inhomogeneities in the magneto–ionic structure of
the thermal material, or a co-located suffusion of synchrotron-emitting
plasma, are responsible for the large Faraday dispersions and/or differential
Faraday depths which are generating the associated depolarization. The toy
model proposed suggests the presence of advection instabilities in the lobes
where material from the interstellar medium (ISM) are mixing with the lobes’
synchrotron radiation, producing filamentary structures where the complex
polarized behaviors are detected.

Zooming into a nearby relativistic radio jet and to study its
spectropolarimetric properties, is the objective of the work by Pasetto et al.
(2021). Broad-band VLA data, from 4 to 18 GHz, of the kpc jet of M87 is
presented and the Stokes QU-fitting analysis is performed to investigate the
nature of its magnetic field. M87 has a powerful jet which emits synchrotron
radiation in a broad frequency domain (i.e., up to optical wavelengths) and
therefore has been extensively studied in the past. There have been
observations in the radio and optical regimes, and its magneto–ionic medium
analyzed (e.g., Owen et al. (1980, 1989, 1990); Capetti et al. (1997); Heinz
and Begelman (1997); Perlman et al. (1999); Zavala and Taylor (2003); Avachat
et al. (2016)). Recently, important studies on the Faraday rotation have been
performed at different radio frequencies and different radio linear scales
(e.g., Chen et al. (2011); Algaba et al. (2016); Park et al. (2019);
Kravchenko et al. (2020)). These works claim the detected Faraday rotation to
be associated with an external Faraday screen, located in the vicinity of the
jet. In particular, the claimed external layer is attributed either to an un-
collimated wind launched from hot accretion flows Park et al. (2019);
Kravchenko et al. (2020) or to the sheath of the jet with a non-negligible
additional contribution from the lobe (part of which should be located in
front of knot C Algaba et al. (2016)). However, the resolution and the
sensitivity of these sparse narrow-band radio data did not allow the authors
to resolve the polarized emission across the M87 jet nor to properly study the
polarization properties across the frequency domain. Nevertheless, the Stokes
QU-fitting has been performed on sparse narrow-band VLBA data to test a
possible internal Faraday rotation Park et al. (2019), reaching no significant
conclusion on this matter.

With the new broad-band, well sampled, VLA data, Reference Pasetto et al.
(2021) wanted to test the specific model of the internal Faraday rotation
which would suggest a gradual change in the direction of the magnetic field,
hence the presence of a helical magnetic field. For a helical configuration,
it is expected to observe gradients of the fractional polarization and the
Faraday depth perpendicular to the jet flow as the line-of-sight magnetic
field changes its direction (see for details: Laing (1981); Blandford (1993);
Lyutikov et al. (2005)). Thanks to the high sensitivity data, depolarization
modeling (by using the Stokes QU-fitting) at each pixel has been performed.
Maps of the polarization properties, that is, intrinsic fractional
polarization and intrinsic polarized angle, have been obtained from the
fitting. For the first time, a highly detailed map, that is, with a high
significance polarization signal, of the magnetic field configuration (with
spatial resolution of tens of pc, see Figure 4), has been obtained. At the
position of the core, the magnetic field configuration shows a change in its
direction recalling a convex angle (see zoom of the VLA core in Figure 4).
Along the jet, the B-field is well ordered at least up to knot A. However, it
also shows regions, mainly between the knots, where the B-field lines behave
like a spiral (between knot D and E and between F and I) or they open like a
funnel (between know E and F). Moreover, changes by 90 degrees at the position
of knot A (see zoom of knot A in Figure 4), where a shock front occurs, are
well visible. Drastic changes at knot C mark a more chaotic configuration at
the termination of the jet (see zoom of knot C in Figure 4). An RM map of the
entire kpc M87 jet showing variations of RM along the jet (see Figure 5) has
been obtained. Internal depolarization model fits have been performed
integrating the polarized signal over some specific areas, especially at the
position of knots A, B and C. The result of the depolarization fit are shown
in Figure 6 and the resulting parameters are listed in Table 6. The Stokes QU-
fitting highlights that at least three internal Faraday screens are needed at
each of these positions to explain the polarization behavior. The internal
depolarization behavior and the presence of variations of the RM values, are
features which suggest the presence of a helical magnetic field configuration
in a jet Laing (1981); Blandford (1993); Lyutikov et al. (2005). The Stokes
QU-fitting allows us to better constrain the configuration and the nature of
the B-field. An exhaustive analysis on this matter will soon be published in a
forthcoming paper.

Polarized emission coming from nearby spiral galaxies with the objective to
understand the nature and the evolution of their large-scale ordered magnetic
fields has also been extensively studied. To pursue this objective, several
fitting techniques have been performed in the past. The three-dimensional
model of the magnetic field represented in terms of a Fourier series has been
applied to study the spiral galaxy M51. This method, using multi-frequency,
narrow-band polarization observations, allows the identification of the
existence of a magneto–ionic halo and to show that the magnetic fields in the
disk and the halo have different configurations Berkhuijsen et al. (1997).
With broad-band data, it is now possible to also use the Stokes QU-fitting for
the analysis of the polarization emission, to enrich the knowledge of and to
map the galaxies’ magneto–ionic media. For example, the system of the Antennae
galaxies has been studied using both the techniques of Rotation Measure
Synthesis and Faraday depolarization analysis. Detailed analysis of the
intrinsic polarization properties (i.e., the intrinsic fractional polarization
and the intrinsic polarization angle, both results of the depolarization
modeling) revealed the presence of turbulent magnetic field within the merging
bodies but also a large-scale regular field structure in the outskirts, most
likely generated by the stretching of the galactic disc during the interaction
Basu et al. (2017). Finally, the Stokes QU-fitting has been used to unveil and
analyze the Faraday structures of a lensing disk galaxy. Its coherent $\mu$G
magnetic field has been detected with similar strength and geometry to local
volume galaxies. This has strong implications for the formation and evolution
of the magnetic fields, in particular the observed coherent magnetic field
property is compatible with a mean-field dynamo origin Mao et al. (2017).

The Stokes QU-fitting technique has been used for several other topics, for
example to study the ambient medium in galaxy clusters and to study the
magnetic field in cosmic web filaments (e.g., the recent work by Ozawa et al.
(2015); Mizener and Williams (2020); Di Gennaro et al. (2021)).

Figure 4: Line integral convolution (LIC) map of the sky-projected magnetic
field orientation of M87 and zooms of the core, knot A and knot C. Figure 5:
RM map of the kpc jet of M87. Color scale: from $-$500 to 500 rad/m2.
Variations on the RM values are detected. Figure 6: Depolarization modelling
of the knots A, B and C. Polarized Stokes parameters q and u of the three
knots have been fitted using three internal Faraday screens. Top panels: the
magenta points are the Stokes q, the green points are the Stokes u and the
black points are the fractional polarization p. Bottom panels: the black
points are the polarized angle $\chi$ [deg]. Gray dot-dashed lines and red
straight lines are the resulted model fit.

[H] Depolarization parameters resulted from the model fit. Integration has
been performed over the knots Knot p01 p02 p03 $\chi_{01}$ $\chi_{02}$
$\chi_{03}$ RM01 RM02 RM03 $\Delta$RM01 $\Delta$RM02 $\Delta$RM03 Std [deg]
[rad/m2] A 0.20 0.11 0.11 $-$45 49 34 155 $-$1420 $-$573 $-$240 10,420 17,160
0.005 B 0.17 0.03 0.09 32 21 $-$133 160 355 $-$850 $-$300 2270 $-$17,460 0.007
C 0.01 0.08 0.06 120 113 $-$0.1 1945 730 $-$520 $-$300 $-$210 12,200 0.01 2

###  3.3 Future Spectropolarimetric Surveys

It is worth mentioning three, already ongoing, important surveys which aim to
inspect the polarized sky with the ultimate goal of investigating cosmic
magnetism and the magnetic properties of individual sources: the Polarisation
Sky Survey of the Universe’s Magnetism (POSSUM) (Gaensler et al., 2010), the Q
and U Observations at Cm wavelengths and Km baselines with the ATCA (QUOCKA)
surveys and the VLA Sky Survey (VLASS).

The POSSUM survey is one of the eight major surveys to be undertaken by ASKAP.
POSSUM will be a continuum polarization survey of the entire sky south of +30
degrees, covering the frequency range 1130–1430 MHz to an RMS sensitivity of
10 $\upmu$y/beam at 10-arcsec resolution. The main result will be a catalogue
of Faraday RMs for $\sim$a million extragalactic radio sources. This dataset
will help to determine the 3D geometry of the Milky Way’s magnetic field, to
test dynamo and other models for magnetic field generation, and to carry out a
comprehensive census of magnetic fields as a function of redshift in galaxies,
in clusters and in the overall intergalactic medium (for more information see:
https://askap.org/possum/. The first results using commissioning data from
POSSUM have been published by Anderson et al. (2021), in which a first Faraday
RM grid study of the Fornax cluster, which is presently undergoing a series of
mergers, is presented. An RM grid is a powerful tool for probing foreground
magnetic fields (i.e., cases where the emitting and Faraday rotating regions
are well separated), and where the study of these external Faraday screens can
be used to extract the global magneto-ionic properties of the intervening
material (Johnston-Hollitt et al., 2015).

The QUOCKA project (https://research.csiro.au/quocka/publications/, accessed
on August 2018), studies radio galaxies across the 1–8.5 GHz range, which is
essential to uncovering their detailed polarized structure. The primary aims
of the QUOCKA survey are to solve outstanding issues like understanding the
complexity exhibited by polarised radio galaxies, and how the observed
polarization properties are related to the internal structure and the
evolution of the host objects. QUOCKA is building an extensive library of
broadband polarization spectra, unveiling the internal structure of the
sources, and revealing the connection to other source properties (e.g.,
environment, degree of activity, redshift, projected proximity to foreground
sources). QUOCKA project progress is shared on the website:
https://research.csiro.au/quocka.

In the northern sky, the most important ongoing survey is the VLA Sky Survey
(VLASS), one of the largest all-sky radio observations in 40 years. It will
map 80 percent of the sky in three phases over 7 years and is expected to
catalog approximately 10 million radio sources in the frequency range of 2 to
4 GHz (S band). Its polarization data will be extremely useful for
spectropolarimetric analysis of all the radio sources the survey will detect.
For more information see https://science.nrao.edu/vlass/.

Future surveys with Square Kilometre Array (SKA) are also planned (for a
complete summary on the projects see: Gaensler et al. (2015)). For example,
Beck et al. Beck and Gaensler (2004) proposed a 1.4-GHz polarisation survey
with the SKA covering 10,000 deg2, resulting in a closely spaced Faraday RM
grid of AGN.

##  4 Summary and Conclusions

I reviewed a technique for the analysis of broad-band spectropolarimetric
data: Stokes QU-fitting. This technique consists of the modeling of both the
Stokes parameters Q and U over a large range of wavelengths using analytical
models available in literature. An overview of the polarization models has
been presented. The principal characteristic is whether the change of the
polarized plane (the Faraday depth) and the decreased polarized signal (the
depolarization effects), is considered internal or external to the source of
study. The presented models consider the presence of a regular or turbulent
magnetic field configuration or a combination of both. The main characteristic
of all the observational studies available over the last 10 years is the
presence of complex behaviors of the polarized light; therefore, with an
abrupt decrease of the fractional polarization with wavelength and a non
linear behavior of the polarized angle with the wavelength. This is mainly due
to multiple Faraday screens close to the sources of study (depending on the
angular resolution of the observations) in which the magnetic field could be
either regular or with a mixture of regular and turbulent magnetic fields.
Radio sources are revealing themselves in their complexity also under the
polarized light. The Stokes QU-fitting is an exquisite tool to analyze those
complex substructures. A new messenger is available to decipher and to
indirectly map the magneto–ionic medium of AGN.

This research received no external funding.

Not applicable

Not applicable

Not applicable

###### Acknowledgements.

A.P. acknowledges the program Cátedras CONACyT. The author declares no
conflict of interest. References yes

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