# The Power of Low Frequencies: Faraday Tomography in the sub-GHz regime

Cameron L. Van Eck
Dunlap Institute for Astronomy and Astrophysics,
University of Toronto, 50 St. George Street, Toronto, ON M5S 3H4, Canada
Correspondence email<EMAIL_ADDRESS>
Department of Physics and Astronomy, University of Calgary, Calgary, Alberta,
T2N 1N4, Canada

###### Abstract

Faraday tomography, the study of the distribution of extended polarized
emission by strength of Faraday rotation, is a powerful tool for studying
magnetic fields in the interstellar medium of our Galaxy and nearby galaxies.
The strong frequency dependence of Faraday rotation results in very different
observational strengths and limitations for different frequency regimes. I
discuss the role these effects take in Faraday tomography below 1 GHz,
emphasizing the 100-200 MHz band observed by the Low Frequency Array and the
Murchison Widefield Array. With that theoretical context, I review recent
Faraday tomography results in this frequency regime, and discuss expectations
for future observations.

_K_ eywords polarization; ISM: magnetic fields; radio continuum: ISM;
techniques: polarimetric; Faraday tomography; Milky Way

##  1 Introduction

The study of magnetic fields in astrophysical systems has been a topic of
great interest for many years as, to paraphrase Woltjer (1967), it has often
been the case that aspects of observations that cannot be explained by models
are blamed on the effects of the (unmodelled) magnetic field. While many of
the basic methods of observing astrophysical magnetic fields have been used
for several decades, such as synchrotron polarization (Gardner & Whiteoak,
1962), Faraday rotation (Cooper & Price, 1962), starlight polarization
(Hiltner, 1949), polarized dust emission (Stein, 1966), and the Zeeman effect
(Hale, 1908), recent technological advances have drastically increased the
amount of information at our disposal.

Faraday rotation measurements in particular have become considerably more
flexible and powerful with the development of rotation measure (RM) synthesis
(Burn, 1966; Brentjens & de Bruyn, 2005). RM synthesis allows for measured
polarized signals to be separated by their strength of Faraday rotation,
called the Faraday depth, but requires large bandwidths with many frequency
channels in order to be effective and is still subject to a number of
limitations. The power of RM synthesis is now accessible through the
combination of broad-band receivers and high-capacity correlators/back-ends
that have been developed in recent years for new radio telescopes, such as the
Low Frequency Array (LOFAR) (van Haarlem et al., 2013) or the Australian
Square-Kilometer Array Pathfinder (ASKAP) (Johnston et al., 2007) and as
upgrades to existing telescopes, such as the Karl G. Jansky Very Large Array
(VLA). However, with the use of RM synthesis to analyze radio polarization
measurements also comes the need to understand the properties and limitations
of the RM synthesis method, as these affect how we can interpret the results
and what information we can extract on the underlying physics, particularly
the magnetic field in the region being studied. A detailed review the physical
motivations for studying radio polarization and Faraday rotation, particularly
in the Milky Way and other spiral galaxies, can be found in Beck (2015).

The Faraday depth, $\phi(x)$, describes the strength of Faraday rotation
experienced by polarized emission at a distance $x$ from the observer. It is
defined as

$\phi(x)=0.812\;{\rm
rad\,m^{-2}}\int_{x}^{0}\left(\frac{n_{\mathrm{e}}(l)}{{\rm
cm^{-3}}}\right)\left(\frac{\vec{B}(l)}{{\rm\upmu
G}}\right)\cdot\left(\frac{\vec{dl}}{{\rm pc}}\right)$ (1)

where $n_{e}(l)$ is the local thermal electron density, $\vec{B}(l)$ is the
magnetic field, and the product of these is integrated over the position $l$
along the line of sight, from $x$ to the observer. The Faraday depth,
multiplied by the observing wavelength, $\lambda$, squared gives the rotation
of the polarization angle for polarized emission at that wavelength and
distance. For lines of sight with only a single (dominant) source of polarized
emission at some distance, the measured polarization angle follows a linear
relationship with $\lambda^{2}$; prior to RM synthesis the most common method
of measuring Faraday depth was to determine the slope of this linear
relationship (e.g., Rand & Lyne, 1994; Brown et al., 2003), which was called
the rotation measure and equal to the Faraday depth only in this simple case.
More generally, the presence of multiple sources of polarized emission at
different distances, or polarized emission that is distributed over distance
with internal Faraday rotation along that distance, for a given line of sight
will result in the superposition of all these polarized signals being
observed. This superposition will generally not retain this linear
relationship, causing the RM to not accurately reflect the Faraday depth of
any of the polarized emission.

RM synthesis provides more flexibility, as it exploits the mathematical
properties of polarization and Faraday rotation to make a Fourier-like
transformation between the observed polarization as a function of wavelength
squared to the polarization as a function of Faraday depth. In this way,
multiple sources of emission with different Faraday depths can potentially be
distinguished from each other (subject to limitations in the effectiveness of
the transformation). The output of RM synthesis, for a given line of sight, is
a Faraday depth spectrum, giving the polarization for every possible Faraday
depth value. When this is applied to an entire field-of-view, for the study of
extended or diffuse polarized emission, it is often called Faraday tomography:
the use of Faraday depth cubes (two dimensions spanning the image plane, with
the third spanning Faraday depth) to analyze Faraday rotation along
potentially complex lines of sight. As a Fourier transform-based technique, RM
synthesis (and by extension Faraday tomography) has a number of properties and
limitations, including a point spread function and corresponding limited
resolution in Faraday depth, that are very similar to those found in radio
interferometry; these are a major subject of consideration in this article and
will be described in detail below. The strong frequency dependence of Faraday
rotation affects many of these properties, so even relatively moderate changes
in frequency can have very significant effects on the use of RM synthesis, and
in turn on the measurable physical properties of the astrophysical systems
being studied.

While early use of Faraday tomography was focused on interferometer
observations at frequencies below 1 GHz, such as de Bruyn & Brentjens (2005)
and Brentjens (2011) who used data from the Westerbork Synthesis Radio
Telescope (WRST) around 350 MHz, a lot of later work has been done at higher
frequencies, such as Heald et al. (2009); Harvey-Smith et al. (2010); Mao et
al. (2015) around 1.4 GHz; Mulcahy et al. (2017); Basu et al. (2017) at 2–4
GHz; and Irwin et al. (2017) around 6 GHz. Mao et al. (2012a) combined single-
dish and interferometer data at 1.4 GHz, and Sun et al. (2015) and Hill et al.
(2017) used single dish data at 1.4 GHz, to explore larger regions of the sky.
However, these different datasets must be interpreted very differently, as the
different frequency coverage drastically changes the volume of interstellar
medium being probed, as will be described in the next section.

In this proceedings, I review the current state of Faraday tomography at low
frequencies, which I define here as below 1 GHz, with a focus on the 100-200
MHz frequency range observed by LOFAR and the Murchison Widefield Array (MWA,
Tingay et al., 2013). I begin by examining the key parameters affecting the
limitations of RM synthesis, emphasizing their behaviour at low frequency and
describing in detail how this affects the physical interpretation of the
observed RM spectra, in Section 2. With that context, I review the published
work using low-frequency Faraday tomography in Section 3 and discuss
prospective future observations in Section 4. Section 5 provides a summary of
the article.

##  2 Theoretical Considerations

Since RM synthesis is based on the Fourier transform, most of its properties
follow directly, such as shape of the point spread function, aliasing,
convolution, and transform pairs. Many of these properties were laid out in a
general form in Brentjens & de Bruyn (2005), so below I focus on just a few of
these properties that show strongly frequency-dependent behaviour that may
have differences between the low-frequency (sub-GHz) regime and higher
frequencies.

###  2.1 Properties of RM synthesis at low frequencies

There are three key frequency-dependent parameters that determine the
limitations of RM synthesis: the Faraday depth resolution, the largest scale
in Faraday depth that can be measured, and the largest Faraday depth that can
be measured. For observations with a single continuous block of bandwidth,
these parameters are given by equations 61-63 of Brentjens & de Bruyn (2005).

The Faraday depth resolution, defined as the width of the Rotation Measure
Spread Function (RMSF), is inversely proportional to the range of
$\lambda^{2}$ values measured. As a result, low frequency observations with
even modest bandwidth will usually have much better Faraday depth resolution
than broadband higher frequency observations; an observation from 1–2 GHz,
possible with the VLA L-band receivers has a Faraday depth resolution of 50
rad m-2, while a typical observation in the LOFAR high-band from 110–170 MHz
has a resolution of 1 rad m-2, under the assumption no bandwidth is lost to
radio frequency interference. While it is possible to identify Faraday depth
structure on scales smaller than the resolution (Brown et al., 2017), it is
difficult to measure this structure unambiguously (Kumazaki et al., 2014) and
it requires high signal-to-noise. Low frequency observations are thus very
desirable for studies of objects with small Faraday depth variations, such as
giant radio galaxies or local interstellar medium (ISM) structure in the Milky
Way.

The largest measurable scale in Faraday depth is a critical parameter for
Faraday tomography, as it determines what types of polarization features will
be observable. Many early polarization studies relied on the assumption that
the polarized emission and Faraday rotation occurred in physically separated
regions, so that all the emission experienced the same Faraday rotation and
had a single, well-defined Faraday depth; this was called the Faraday-thin,
one-component regime (Vallee, 1980). However, in many regions the polarized
emission region is mixed with the Faraday rotation region, causing the
polarized emission to be distributed over a range of Faraday depths. As a
result, Faraday depth spectra can contain a mixture of very narrow (‘Faraday-
thin’) components and broader components. Due to the Fourier transform
relationship between the Faraday depth spectrum and the polarization as a
function of wavelength (squared), these broad components have more intensity
at shorter wavelengths and become depolarized at longer wavelengths (this can
be called wavelength-dependent depolarization or depth depolarization,
depending on whether the speaker is more oriented towards the wavelength or
Faraday depth domain).

Correspondingly, the largest measured scale in a Faraday depth spectrum is
inversely proportional to the shortest $\lambda^{2}$ value observed, or
equivalently, proportional to the highest frequency in an observation. More
specifically, the largest scale is approximately equal to (adapting Eq. 62 of
Brentjens & de Bruyn (2005)) 35 rad m-2
$(\nu_{\mathrm{max}}/\mathrm{GHz})^{2}$. Thus, a VLA L-band observation will
be sensitive to scales smaller than 140 rad m-2, while a LOFAR-high
observation is only sensitive to scales of 1.0 rad m-2 and smaller. The
largest scales present in a given astrophysical environment are expected to be
approximately the same as the largest Faraday depth contribution produced in
that environment, so environments like radio galaxies or galactic disks may
have Faraday depth features with widths of several hundred or even a few
thousand rad m-2 (Anderson et al., 2016), while local features or regions of
the Galactic halo may have features with scales of up to a few tens of rad m-2
(Mao et al., 2012b). As a result, low-frequency observations (without
supporting higher-frequency data) will be insensitive to much of the polarized
emission that we expect from these environments. However we do expect, and
observe, a population of Faraday-thin features; I will discuss this in Section
2.2.

Figure 1 demonstrates the effects of the RMSF and the largest measured scale.
The narrow feature in the left of the figure takes on the shape of the RMSF,
while the broad feature on the right is much broader than the largest measured
scale (1.1 rad m-2) is strongly depolarized except for the sharp edges, which
are partially recovered. This is due to the Fourier property that sharp edges
produce very broad features in the Fourier-conjugate domain ($\lambda^{2}$, in
RM synthesis) which equates to some polarized emission remaining at long
wavelengths; the result is that a Burn slab (Burn, 1966) that is significantly
broader than the largest scale will produce a ‘two-horned’ Faraday depth
feature (Beck et al., 2012; Van Eck et al., 2017).

Figure 1: A simulation of RM synthesis, where a simulated spectrum (dashed
line) containing a Faraday-thin (delta-function) component and a Burn slab
(top-hat) has been ‘observed’ with the frequency coverage (115–178 MHz) and
channel widths (24.4 kHz) of a sample LOFAR observation (Van Eck et al.,
2017). In the resulting RM spectrum (solid line), the Faraday component takes
on the shape of the RMSF, while the broad slab is mostly filtered out due to
being significantly broader than the largest measured scale (1.1 rad m-2).
Some emission is left at the edges of the slab, producing a characteristic
‘two-horn’ shape.

It is worth noting that the largest measured scale can be, and often is,
smaller than the Faraday depth resolution. As a result, it’s possible to have
observing configurations where no resolved features can be detected. Since the
resolution depends (inversely) on the range of wavelength squared observed,
$\Delta\lambda^{2}$, and the largest scale depends (inversely) on the shortest
wavelength squared, $\lambda^{2}_{\mathrm{min}}$, we can express this in terms
of the ratio of highest and lowest frequency. Equation 64 of Brentjens & de
Bruyn (2005) gives the condition to be able to resolve Faraday depth features
as $\lambda^{2}_{\textrm{min}}<\Delta\lambda^{2}$; this can be re-expressed in
terms of the ratio of highest and lowest frequencies as
$\nu_{\textrm{max}}/\nu_{\textrm{min}}>\sqrt{2}$. For many modern radio
telescopes, the large fractional bandwidth needed to meet this criterion is
now available. Typical LOFAR-high observations barely satisfy this criterion,
so only a very narrow range of scales could be resolved, and typical MWA
observations (e.g., Lenc et al., 2016) do not satisfy this criterion.
Correspondingly, polarization features observed with current low-frequency
instruments are almost always unresolved in Faraday depth, unless combined
with additional observations to extend the frequency coverage.

The third limiting parameter of RM synthesis is the largest Faraday depth that
can be measured. This limitation is caused by bandwidth depolarization, as
very large (positive or negative) Faraday depths will cause significant change
in polarization across the bandwidth of the channel. Therefore, the Faraday
depth at which the sensitivity has dropped to 50% is inversely proportional to
the channel width in $\lambda^{2}$. For low-frequency observations, where
small changes in frequency will produce larger changes in wavelength, it
becomes important to ensure that channel widths are kept as small as possible
to maximize sensitivity to larger RMs. This is reasonably straightforward for
dedicated polarimetric observations (but can introduce data volume challenges,
as the number of channels becomes large), but can be a problem with
observations that were not originally optimized for polarization analysis
(e.g. Van Eck et al., 2018). Typical values for this parameter are harder to
define, as most instruments allow the observer to select the channel width to
match their own preferences; observations for the LOFAR Two-meter Sky Survey
(LOTSS, Shimwell et al., 2017) have a largest Faraday depth of 170 rad m-2 (at
the bottom of the band; the higher frequencies within the observing bandwidth
will have lower depolarization and correspondingly higher largest Faraday
depths) due to channel-averaging, observations from the Galactic and
Extragalactic All-sky MWA survey ((GLEAM, Wayth et al., 2015)) have a largest
Faraday depth of 1937 rad m-2 (Riseley et al., 2018), and the fine-spectral
resolution data from the VLA Sky Survey will have a largest Faraday depth of
4800 rad m-2 (at the bottom of the band). It is possible to correct for the
reduced sensitivity at large Faraday depths (Schnitzeler & Lee, 2015), at the
cost of increased noise in those parts of the Faraday depth spectrum.

###  2.2 Producing Faraday-thin features

As low-frequency Faraday tomography is primarily sensitive to Faraday-thin
polarization features, the interpretation of such data requires an
understanding of the physical conditions that can lead to such features being
produced, as well as the conditions that will produce features too broad to be
observed. The criteria for defining such features were discussed by Van Eck et
al. (2017); a slightly more careful derivation of that result appears here.

The physical mapping between the polarized emission as a function of distance,
$\tilde{P}(x)$,111All quantities with a tilde denote complex quantities,
specifically polarization phasors. and the Faraday depth spectrum,
$\tilde{F}(\phi)$, can be written as (Bell et al., 2011):

$\tilde{F}(\phi)=\int_{0}^{\infty}\delta(\phi-\phi(x))\tilde{P}(x)dx,$ (2)

where $\phi(x)$ is the Faraday depth as a function of distance along the line
of sight.222It is important to note that while both $\phi$ and $\phi(x)$ are
called the Faraday depth, $\phi$ is the independent variable in the Faraday
depth spectrum and spans the range of possible amounts of Faraday rotation
($-\infty$ to $\infty$ rad m-2), while $\phi(x)$ is the specific Faraday depth
that would be observed for polarized emission produced at a given distance.
The Dirac delta function in this equation can be interpreted as selecting all
distances that produce the same Faraday depth, allowing them to be co-added to
the same Faraday depth in the resulting Faraday depth spectrum; this is
necessary to account for the possibility that multiple distances along a line
of sight can have the same Faraday depth. Exploiting a property of the Dirac
delta with a function as the argument, this can be rewritten as

$\tilde{F}(\phi)=\sum_{x_{\phi}}\frac{\tilde{P}(x_{\phi})}{\left|\frac{d}{dx}\phi(x)\right|_{x=x_{\phi}}}=\sum_{x_{\phi}}\frac{\tilde{P}(x_{\phi})}{n_{e}(x_{\phi})\left|B_{\parallel}(x_{\phi})\right|}$
(3)

where $x_{\phi}$ is the set of distances that satisfy $\phi-\phi(x_{\phi})=0$,
and we have replaced the derivative of Faraday depth in the denominator with
the free electron density and parallel magnetic field. The denominator in this
equation represents the physical effect where stronger Faraday rotation causes
the polarized emission to be more spread out in Faraday depth and
correspondingly lowers the amplitude at any given single Faraday depth.

A Faraday-thin feature implies a narrow peak in the Faraday depth spectrum,
which in turn requires that the right-hand side of Eq. 3 be much larger in a
small range of Faraday depths compared to the rest of the Faraday depth
spectrum. However, due to the high-pass filter effects of low-frequency
observations there are two additional scenarios in which an observed Faraday
spectrum mimicking a Faraday-thin feature may occur. First, a narrow region of
a Faraday depth spectrum with an anomalously low amplitude surrounded by broad
higher amplitude features will appear, after the high-pass filter removes the
broad features, as a peak with the opposite phase (perpendicular polarization
angle) in the observed spectrum. The second scenario occurs when a broad
feature has a sharp edge, which can occur in a Burn slab configuration (Burn,
1966), when the parallel magnetic field reverses sign (i.e., a Faraday
caustic, Bell et al., 2011), or when the Faraday depth converges to a fixed
value at large distances (e.g., some of the models investigated by Beck et
al., 2012). These features take the shape of a asymmetric peak, with a sharp
edge on one side and a gradual tail or other broad feature on the other.

In all of these cases, a sharp change in the amplitude of the Faraday depth
spectrum is required, so to understand the physical origins of such changes we
must consider the various physical parameters that go into the amplitude and
how they may have strong, localized variations. In the numerator of Eq. 3 is
the polarized emissivity, which we can decompose into three terms: the total-
intensity synchrotron emissivity ($\varepsilon$), the fractional polarization
($\Pi$), and a phasor defining the emitted polarization angle ($\theta_{0}$)
as perpendicular to the magnetic field projected on the sky-plane. The
synchrotron emissivity can be further decomposed, as it is proportional to the
density of cosmic ray electrons ($n_{cr}$) and the strength of the magnetic
field component perpendicular to the line of sight ($B_{\perp}$). The
fractional polarization is determined by the degree of order in the magnetic
field, on the scale of the angular resolution, making it affected by both
physical effects and observational parameters. The polarization angle does not
affect the amplitude (except through interference of overlapping Faraday depth
features) so I will not consider it further. The resulting equation looks like

$\tilde{F}(\phi)=\sum_{x_{\phi}}\frac{\Pi
e^{2i\theta_{0}(x_{\phi})}\varepsilon(n_{cr}(x_{\phi}),B_{\perp}(x_{\phi}))}{n_{e}(x_{\phi})\left|B_{\parallel}(x_{\phi})\right|}.$
(4)

Equation 4 has three important variables in the numerator: the fractional
polarization (which depends on the degree of order in the magnetic field), the
cosmic ray electron density, and the strength of the perpendicular component
of the magnetic field, and two parameters in the denominator: the density of
free (thermal) electrons and the strength of the parallel component of the
magnetic field. To produce a sharp amplitude change, one or more of these
parameters must change significantly in a localized region. More specifically,
the region over which these parameters are significantly different must have
internal Faraday rotation less than the largest measured scale in any
observation measuring the polarized feature. Following Van Eck et al. (2017),
I consider a few examples of physical conditions under which these conditions
could be satisfied and a Faraday-thin feature observed.

Shocks oriented perpendicular to the line of sight are a potential candidate
for producing Faraday-thin features. Such shocks could significantly enhance
the perpendicular component of the magnetic field, increasing synchrotron
emissivity, and may also increase the degree of order of the field. Shocks are
also thought to serve as sites for acceleration of cosmic rays (Bell, 2013,
and references therein), increasing the density of cosmic ray electrons.
However, shocks also have increased matter density over their surroundings,
which would increase the Faraday rotation. As a result, the effect of a shock
on the Faraday depth spectrum may be complicated and requires more careful
modelling to determine if a Faraday thin feature would be produced.

To produce high-amplitude, Faraday-thin Faraday depth features, conditions
under which the denominator becomes very small are an attractive possibility.
The strength of the parallel magnetic field is an obvious candidate, as it is
very easy to imagine magnetic field configurations with regions where the
magnetic field is nearly perpendicular to the line of sight. Regions where the
magnetic field reverses sign along the line of sight will produce Faraday
caustics (Bell et al., 2011) which have a high amplitude feature with a sharp
edge. Faraday caustics will generally not have an observational signature in
any other tracer, but can be identified in observations with sufficiently high
fractional bandwidth. Bell et al. (2011) also proposed that Faraday caustics
will occur in sheets, potentially producing diffuse polarized features with a
large angular size.

Regions of very low thermal electron density can also lead to high-amplitude
Faraday thin features. Such regions can occur in two forms: those occupied by
hot, low-density material (i.e., the hot ionized phase of the ISM, and the
intergalactic medium), and those occupied by higher density, very low
ionization fraction material (i.e., the warm neutral phase of the ISM). The
hot, thin phase of the ISM will occupy large parts of the ISM, particularly in
the Galactic halo, but these regions will often also have reduced synchrotron
emissivity and large path lengths which may prevent a sharp peak from forming
in some cases. Neutral regions, in the form of warm neutral clouds, may form
such peaks if they have a sufficiently low ionization fraction and/or a
sufficiently low density to prevent significant internal Faraday rotation
while still having enough path length to accumulate a significant amount of
synchrotron emission; Van Eck et al. (2017) used this to explain the the
polarization features seen in their field. The cold neutral phase,
specifically molecular clouds, are probably less likely to produce Faraday-
thin features, on account of their very high density offsetting the low
ionization fraction and leaving them with a significant free electron column,
which combined with the enhanced magnetic field found in those regions results
in significant internal Faraday rotation (Tahani et al., 2018).

###  2.3 Other wavelength-dependent depolarization effects

The sensitivity to only Faraday-thin features, or equivalently the
depolarization of Faraday-broad features, is the wavelength-dependent
depolarization effect most often considered in the context of RM synthesis.
However, there are several additional effects that may also be amplified at
low-frequency that should be considered.

One is the effects of beam depolarization, where the Faraday depth profiles
for all the lines of sight within the solid angle of the observing beam are
averaged together and can potentially cause partial cancellation. Some portion
of this can be caused by changes in the intrinsic (pre-Faraday rotation)
polarization angle of the emission, which will be wavelength-independent,
while the rest of the depolarization will occur by changes in the Faraday
rotation across the beam, and will correspondingly depend on wavelength. This
effect can be interpreted in two ways: in the Faraday depth domain, the beam
is averaging the Faraday depth profiles over some solid angle, which can cause
a spatially-varying Faraday-thin feature to become sufficiently Faraday-thick
on the scale of the beam as to depolarize significantly; alternatively, in the
wavelength domain a spatially-varying Faraday depth will cause the
polarization angle at a given frequency to vary significantly across the beam,
with longer wavelengths having stronger variations in polarization angle due
to experiencing more Faraday rotation.

Spatial variations in the Faraday depth of a polarized feature have the net
effect of driving polarized emission from large angular scales to smaller
scales (Schnitzeler et al., 2009). This has two main effects: the first is the
previously defined beam depolarization, where polarized emission is moved to
small angular scales which are not measured (in the case of interferometers,
due to the lack of sufficiently long baselines); the second is the reduction
in polarized emission on the largest angular scales. As a result, at very low
frequencies (i.e., the LOFAR and MWA regime) the effect of missing short-
spacing data may not be a major problem for large-area Faraday tomography with
interferometric observations, but this effect has not yet been quantified.

Wavelength dependent depolarization can also be strongly affected by the
specific shape of the Faraday depth profile. The simplest example of this is
comparing a Burn slab (Burn, 1966), which has a top-hat Faraday depth profile
and a corresponding $\sigma_{\phi}\lambda^{-2}$ depolarization law (where
$\sigma_{\phi}$ is the characteristic width of the profile), while a Gaussian
Faraday depth profile, which can be caused by a turbulent Faraday rotating
screen (Burn, 1966), has a $e^{-\sigma_{\phi}^{2}\lambda^{4}}$ depolarization
law which falls off much more rapidly with increasing wavelength. In the
Faraday depth domain, this can be interpreted in terms of the ‘high-pass
filter’ effect of RM synthesis, as a Gaussian is dominated by the broad
Fourier components while the top-hat function has significant emission in
narrow components to produce the sharp edges. As a result, different
distributions with the same $\sigma_{\phi}$ may experience very different
depolarization.

##  3 Recent Observations

The earliest low-frequency Faraday tomography was performed using observations
from the WRST at frequencies in the 115-175 MHz and 310-383 MHz ranges (e.g.,
de Bruyn & Brentjens, 2005; Schnitzeler et al., 2009; Brentjens, 2011; Pizzo
et al., 2011). These observations were primarily targeted on galaxy clusters,
but also identified diffuse polarization features which they attributed to the
Milky Way. de Bruyn & Brentjens (2005) and Brentjens (2011) used 300-MHz WRST
data to study the Perseus cluster, and identified several polarized features
on large (10′) angular scales, which were initially attributed to the cluster
(de Bruyn & Brentjens, 2005) but were later though to be within our Galaxy
(Brentjens, 2011). Pizzo et al. (2011) used WRST data in several frequency
ranges from 115-1795 MHz to observe the Abell 2255 cluster, and despite
deliberately filtering out large baselines to remove Galactic contributions
they identified polarized features that they attributed to our Galaxy.
Schnitzeler et al. (2009) used WSRT data in the 350 MHz band to observe
Galactic emission in a large field. Iacobelli et al. (2013) also used low-
frequency (139–155 MHz) WSRT data to observe the FAN region, an area of sky
previously known to have unusually high polarized intensity; they identified
several distinct polarized features with small separations in Faraday depth,
and attempted to associate them with known structures in the ISM. Many of
these studies had limited fields of view, opening questions about the full
extent the observed structures and how much of the sky is occupied by such
features, and also had limited short-baseline coverage, opening question about
whether features on larger angular scales were being missed.

With the construction of LOFAR and MWA, a new capability was opened up to
explore polarized emission on larger angular scales and larger regions of the
sky. Jelić et al. (2014), Jelić et al. (2015), and Van Eck et al. (2017)
applied Faraday tomography to individual LOFAR observations and found many
diffuse polarized features on large scales. The 3C196 field of Jelić et al.
(2015) yielded a number of interesting features: bright diffuse Galactic
polarization was observed, but with one long narrow filament clearly offset in
Faraday depth from its surroundings, indicating a filamentary foreground
screen; an alignment between low-frequency polarized features and dust
polarization (Zaroubi et al., 2015); and a series of long and very straight
depolarization canals (Jelić et al., 2018). Van Eck et al. (2017) found two
distinct polarized features in their field, overlapping on the sky but
separated in Faraday depth by less than 10 rad m-2, and and associated these
polarized features with neutral clouds within 1 kpc of the Sun.

Lenc et al. (2016) applied Faraday tomography to MWA observations over a very
large region (625 deg2) near the South Galactic Pole at low resolution (50′).
They found diffuse polarized emission in their field at a high intensity level
(4–11 K polarized brightness temperature at 154 MHz), indicating that this
region had either strong polarized emission or that beam depolarization was
not extreme in that region. They found that this emission was also very local,
within 220 pc, although this assumed a uniform Faraday rotating medium and did
not account for the presence of the Local Bubble, which is expected to have
very low Faraday rotation.

Van Eck (2017) combined Faraday depth cubes from 60 LOFAR observations from
LOTSS to create a 568 deg2 mosaic covering a region in the northern Galactic
halo. A collapsed version of this cube, showing the Faraday depth and
polarized intensity of the brightest peak in the Faraday depth spectrum of
each pixel, is shown in Figures 2 and 3. In this mosaic there are several
polarized features that extend beyond ten degrees in length, much larger than
the field of view of a single LOFAR observation. Many of these features could
be clearly seen as continuous sheets of (Faraday-thin) polarized emission,
covering tens of square degrees, with gradients in the Faraday depth of the
peak. Van Eck (2017) proposed that the origin of the Faraday-thin emission may
be Faraday caustics, and that the gradients in Faraday depth (especially where
the sign of Faraday depth changes) could be caused by loops in the magnetic
field, but they also acknowledged that other interpretations are possible.

For Faraday tomography of extragalactic objects, such as nearby galaxies, low-
frequency data has not been as successful. Mulcahy et al. (2014) and Farnes et
al. (2013) used observations from LOFAR at 150 MHz and the Giant Meterwave
Radio Telescope at 610 MHz respectively to search for polarized emission from
M51, but did not detect significant polarization. Faraday tomography has been
successfully applied to nearby galaxies at higher frequencies (e.g. Mao et
al., 2012b), so this is a sign of strong wavelength dependent depolarization.
Based on the results from Milky Way Faraday tomography, it seems quite
reasonable to think that Faraday thin features exist in other galaxies, but
gradients in Faraday depth will cause them to appear as broad features when
averaged over the beam, causing them to be appear depolarized at low
frequencies.

Figure 2: A map of Faraday depth of the strongest peak (encoded as hue) and
peak polarized intensity (encoded as brightness, capped at 3 mJy PSF-1
RMSF-1), calculated from the western region of the Faraday depth cube of Van
Eck (2017). Continuous gradients in Faraday depth can be observed in several
regions. Instrumental leakage of bright point sources shows up as small
bright-blue features throughout the image. Figure 3: As Figure 2, but for the
eastern region of the mosaic. The instrumental leakage from 3C295 dominates
the center, but patches of diffuse polarized emission can be seen on the right
and a long, straight feature can be seen below the center, which traces the
outer edge of a neutral hydrogen filament (not shown).

##  4 Future Observations

The observations to date have shown low-frequency Faraday tomography can be
highly successful in finding polarization features in the ISM of the Milky
Way. However, the parameter space of sky area, Faraday depth resolution, and
sensitivity to Faraday-broad features is still very sparsely explored. Figure
4 shows the regions of the sky that have been mapped to date, which represents
a small fraction of the total sky.

Figure 4: A map of the sky, in a Hammer projection of Galactic coordinates,
showing the regions covered by current low-frequency Faraday tomography.
Observations below 250 MHz are shown in blue, and observations between 250 and
1000 MHz are shown in red (this considers only the lowest frequency included
in the RM synthesis, so some fields will have included data above 1000 MHz).
Combined, these observations cover less than 4% of the total sky.

Both LOFAR and MWA are conducting all-sky total intensity surveys (LOTSS and
GLEAM/GLEAM-X, respectively), and the data for each has been shown to be
suitable for polarization analysis (Van Eck, 2017, and Riseley et al,
submitted, respectively). As a result, there is the opportunity to conduct
commensal polarization and Faraday tomography surveys with these data.
Together these would give full sky coverage at 150 MHz with arcminute angular
resolution and baselines down to a few tens of meters or less.

Another avenue that is being explored is single-dish polarization surveys,
which are sensitive to large-scale Galactic features. The Global Magneto-Ionic
Medium Survey (GMIMS, Wolleben et al., 2009) is an ongoing project to observe
the entire sky from approximately 300 to 1800 MHz, through 6 sub-surveys
dividing the frequency range into three segments and the sky into the northern
and southern hemispheres. The first two sub-surveys, covering the northern
hemisphere from $\sim$1.3–1.8 GHz and the southern hemisphere from 300–480
MHz, are expected to be released soon (T. Landecker, private communication).
While GMIMS will suffer from limited angular resolution due to the limits of
single-dish and compact interferometer telescopes at low frequencies, the very
wide frequency coverage will offer unprecedented opportunity to explore
Faraday complexity, by giving both the Faraday depth resolution of the lowest
frequencies and the sensitivity to broad features of the higher frequencies.

Low-frequency Faraday tomography of nearby galaxies will remain challenging,
as very high angular resolution will be required to overcome beam
depolarization of Faraday thin features. The LOTSS data may be the most
promising possibility in the near future, as it can achieve 5′′ resolution,
but further progress may have to wait for the Square Kilometre Array to
provide higher resolution and sensitivity at low frequencies.

##  5 Summary

Faraday tomography is a very powerful tool for studying cosmic magnetic fields
because it provides us with information on the distribution of Faraday
rotation along a line of sight. Using Faraday tomography with low-frequency
radio polarization data enhances this advantage, by providing improved Faraday
depth resolution, but this comes with the cost of not detecting features that
are broadened in Faraday depth. However, this limitation can be exploited as
it provides constraints on the thermal electron and magnetic field
distributions that can lead to a detectable low-frequency polarization
feature. In particular, low-frequency Faraday tomography is sensitive to
polarized emission from comparatively small regions (required to produce
Faraday thin features), making it particularly effective in probing structure
in the magnetic field on smaller spatial scales when compared with higher
frequency data where the entire line of sight is blended together.

Recent results have demonstrated our capability to detect and analyze low-
frequency polarization with new instruments such as LOFAR and MWA. These
observations have found a number of interesting, previously known polarization
features of predominantly Galactic origin, and have shown that some of these
features have very interesting relationships with other aspects of the ISM.

However, we have only scratched the surface of exploring the low-frequency
Faraday depth sky, as most of the sky remains unexplored. Future polarization
surveys with LOFAR and MWA have the potential to cover the whole sky at 150
MHz and arcminute to arcsecond resolution, while GMIMS will cover the whole
sky from 300–1800 MHz at degree resolution. These surveys, once complete, will
dramatically advance our knowledge of polarized emission, Faraday rotation,
and the associated magnetic fields in our Galaxy and in extragalactic radio
sources.

## Acknowledgements

The author wishes to thank Marijke Haverkorn and the three reviewers for their
comments on the manuscript.

The preparation of this manuscript made use of Astropy, a community-developed
core Python package for Astronomy (Astropy Collaboration et al., 2013); NumPy
(van der Walt et al., 2011); IPython (Pérez & Granger, 2007); and matplotlib
(Hunter, 2007).

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