# The correct sense of Faraday rotation

K. Ferrière,1 J. L. West,2 T. R. Jaffe3
1Institut de Recherche en Astrophysique et Planétologie (IRAP), Université de
Toulouse, CNRS, 9 avenue du Colonel Roche, BP 44346, 31028 Toulouse Cedex 4,
France
2Dunlap Institute for Astronomy and Astrophysics University of Toronto,
Toronto, ON M5S 3H4, Canada
3NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA E-mail:
<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The phenomenon of Faraday rotation of linearly polarized synchrotron emission
in a magneto-ionized medium has been understood and studied for decades. But
since the sense of the rotation itself is irrelevant in most contexts, some
uncertainty and inconsistencies have arisen in the literature about this
detail. Here, we start from basic plasma theory to describe the propagation of
polarized emission from a background radio source through a magnetized,
ionized medium in order to rederive the correct sense of Faraday rotation. We
present simple graphics to illustrate the decomposition of a linearly
polarized wave into right and left circularly polarized modes, the temporal
and spatial propagation of the phases of those modes, and the resulting
physical rotation of the polarization orientation. We then re-examine the case
of a medium that both Faraday-rotates and emits polarized radiation and show
how a helical magnetic field can construct or destruct the Faraday rotation.
This paper aims to resolve a source of confusion that has arisen between the
plasma physics and radio astronomy communities and to help avoid common
pitfalls when working with this unintuitive phenomenon.

###### keywords:

ISM – magnetic fields – polarization

††pubyear: 2015††pagerange: The correct sense of Faraday rotation–B

##  1 Introduction

Faraday rotation of linearly polarized synchrotron radiation at radio
wavelengths is one of the primary tools used to study galactic and
extragalactic magnetic fields. The Faraday rotation measure (RM) is commonly
derived from observational data by taking measurements at a variety of
wavelengths, $\lambda$, and examining how the polarization angle (PA) changes.
In the simplest case of a background polarized source, a plot of PA vs
$\lambda^{2}$ shows a linear relation and the slope of the line gives the RM.
The conventions now widely known and adopted in observational radio astronomy
were first put forth by Manchester (1972): when ${\rm RM}>0$, the magnetic
field, $\vec{\mbox{\boldmath{$B$}}}$, is on average directed toward the
observer, and when ${\rm RM}<0$, $\vec{\mbox{\boldmath{$B$}}}$ is on average
directed away from the observer. However, the physical sense of rotation of
the electric field vector of the wave is not so well known, as this
information is generally not relevant in astrophysical problems. Many
astrophysics textbooks contain a comparatively short section on this topic and
tend to gloss over some of the finer points that are critical to a complete
understanding. Even the astrophysics textbooks that do provide a detailed
derivation (Spitzer, 1978; Rybicki & Lightman, 1979; Bowers & Deeming, 1984;
Shu, 1991; Elitzur, 1992) tend to lay the emphasis on the physical mechanism
rather than the actual sense of rotation.

A complete mathematical derivation of Faraday rotation can be found in several
textbooks of plasma physics (e.g., Nicholson, 1983; Chen, 2016) or optics and
radiation theory (e.g., Stone, 1963; Papas, 1965; Stutzman, 1993; Goldstein,
2011; Collett & Schaefer, 2012). However, few discuss the actual sense of
rotation explicitly, and those who do could easily leave radio astronomers
confused, either because they measure the sense of rotation from a different
point of view (Stone, 1963; Papas, 1965; Collett & Schaefer, 2012, who use the
same conventions as optical astronomers; see Sect. 3.1) or because their
reasoning is flawed (Chen, 2016, see Sect. 3.3).

There are cases where knowing the sense of Faraday rotation is of critical
importance. For instance, several studies have looked at a possible link
between Faraday rotation and magnetic helicity (Volegova & Stepanov, 2010;
Brandenburg & Stepanov, 2014). Our own recent work (West et al., 2020)
confirmed the existence of such a link, but the correlation that we measured
systematically had the opposite sign to that obtained in certain previous
studies. In fact, we were able to recover the results of these previous
studies provided we flipped the sense of Faraday rotation. This naturally led
us to question our adopted sense of Faraday rotation. We consulted several
reference textbooks (including those cited above) and we asked several leading
experts, both in plasma physics and in radio astronomy. We were quite
surprised by the variety, and sometimes the uncertainty, in the answers we
found or received. Thus, in the course of our "investigation", we realized
that what was supposed to be a trivial question did not have a simple and
unanimous answer in the community. The purpose of this paper is to clarify the
situation and, whenever possible, either reconcile the different approaches or
explain why and where they disagree.

In Sect. 2, we rederive the correct sense of Faraday rotation as well as the
exact expression of the Faraday RM from basic plasma theory. Along the way, we
provide detailed physical interpretations and simple graphical illustrations
of our equations, while the more mathematical aspects of the derivation are
relegated to Appendices A and B. In Sect. 3, we discuss four possible sources
of confusion and explain how they can lead to erroneous reasoning and/or
faulty conclusions. In Sect. 4, we return to the problem that motivated the
present paper and re-examine the link between Faraday rotation and magnetic
helicity. In Sect. 5, we conclude our study.

Throughout the paper, we use the IAU reference frame for polarization,111 IAU
General Assembly Meeting, 1973, Commission 40 (Radio Astronomy), 8.
POLARIZATION DEFINITIONS.  which is shown in Figure 1. Accordingly, we take
the position angle of linear polarization, simply referred to as the
polarization angle, $\psi$, to increase counterclockwise in the sky, starting
from North. As a note of caution for CMB radio astronomers, let us mention
that the IAU convention for the sign of $\psi$ is opposite to that chosen for
cosmological analysis in much of the CMB community.

Figure 1:  IAU coordinate and polarization conventions used in this paper. $X$
and $Y$ are the Cartesian coordinates in the plane of the sky (increasing
toward North and East, respectively), $Z$ is the line-of-sight coordinate
(increasing toward the observer), the red double-headed arrow indicates the
orientation of linear polarization, and $\psi$ denotes the associated position
angle, also referred to as the polarization angle (measured counterclockwise
from North).

##  2 Mathematical derivation

###  2.1 The two circularly polarized modes of parallel propagation

Consider a cold, magnetized plasma with electron density $n_{\rm e}$ and
magnetic field $\vec{\mbox{\boldmath{$B$}}}$. Two characteristic frequencies
of this plasma are the plasma frequency, $\omega_{\rm
e}=\sqrt{\frac{\displaystyle 4\pi n_{\rm e}e^{2}}{\displaystyle m_{\rm e}}}$,
and the electron gyro-frequency, $\Omega_{\rm e}=\frac{\displaystyle q_{\rm
e}B}{\displaystyle m_{\rm e}c}$, where $m_{\rm e}$ is the mass of the
electron, $q_{\rm e}=-e$ the electric charge of the electron ($q_{\rm e}<0$),
$B$ the magnetic field strength ($B>0$), and $c$ the speed of light. Note that
$\Omega_{\rm e}$, as defined by plasma physicists, is negative. However, for
convenience, $\Omega_{\rm e}$ is often redefined as a positive quantity
($\Omega_{\rm e}=\frac{\displaystyle eB}{\displaystyle m_{\rm e}c}$) outside
the plasma community. Here, to avoid any possible confusion, we will work with
the absolute value of $\Omega_{\rm e}$: $|\Omega_{\rm e}|=\frac{\displaystyle
eB}{\displaystyle m_{\rm e}c}$, which is unambiguously positive. In the
interstellar medium (ISM), most of the free electrons reside in the warm
ionized medium (WIM), where $n_{\rm e}\sim 0.2~{}{\rm cm}^{-3}$ and $B\sim
5~{}\mu{\rm G}$, leading to $\omega_{\rm e}\sim 25~{}{\rm kHz}$ and
$|\Omega_{\rm e}|\sim 90~{}{\rm Hz}$ (e.g., Ferrière, 2020).

Let us now study the propagation of a radio electromagnetic wave with given
angular frequency $\omega>0$ (imposed at the source), corresponding to
frequency $\nu=\frac{\displaystyle\omega}{\displaystyle 2\pi}$. Assume that
this wave has wave vector $\vec{\mbox{\boldmath{$k$}}}$ (directed from the
source to the observer), corresponding to wavenumber
$k=|\vec{\mbox{\boldmath{$k$}}}|>0$, propagation direction
$\hat{\mbox{\boldmath{$e$}}}_{k}=\frac{\displaystyle\vec{\mbox{\boldmath{$k$}}}}{\displaystyle
k}$, and wavelength $\lambda=\frac{\displaystyle 2\pi}{\displaystyle k}$. The
wavenumber in any propagation direction is not imposed at the source, but it
is given by the dispersion relation, which depends on the properties of the
traversed medium.

Let us first focus on the case of parallel propagation, when
$\vec{\mbox{\boldmath{$k$}}}\parallel\vec{\mbox{\boldmath{$B$}}}$. The
dispersion relation can then be written as (see Appendix A)

$\omega^{2}\ =\ c^{2}\,k^{2}\ +\ \frac{\omega_{\rm
e}^{2}}{1\mp\frac{\displaystyle|\Omega_{\rm e}|}{\displaystyle\omega}}\ ,$ (1)

where the $\mp$ sign in the denominator arises from the existence of two
solutions: a right circularly polarized mode, for which the electric field
vector of the wave rotates in a right-handed sense about
$\vec{\mbox{\boldmath{$B$}}}$ (upper sign), and a left circularly polarized
mode, for which the electric field vector of the wave rotates in a left-handed
sense about $\vec{\mbox{\boldmath{$B$}}}$ (lower sign). Potential confusion
regarding the concepts and definitions of right and left circularly polarized
modes will be discussed in Sect. 3.1.

The waves of interest here typically have $\nu\gtrsim 100~{}{\rm MHz}$, so
that $\omega_{\rm e},\,|\Omega_{\rm e}|\lll\omega$. Under these conditions,
Eq. (1) can be approximated by

$\omega^{2}\ =\ c^{2}\,k^{2}+\omega_{\rm e}^{2}\pm\frac{\omega_{\rm
e}^{2}\,|\Omega_{\rm e}|}{\omega}\ \cdot$ (2)

The first term in the right-hand side represents electromagnetic wave
propagation in free space. The next two terms describe the diamagnetic effect
of the plasma, with a dominant contribution to the electric current arising
from the electric force (second term) and a much weaker contribution arising
from the magnetic force (third term). The latter adds to [subtracts from] the
electric force when the electric field vector rotates in the same sense as
[the opposite sense to] the natural gyration of electrons about
$\vec{\mbox{\boldmath{$B$}}}$.

The expression of the phase velocity, $V_{\phi}$, directly follows from Eq.
(2):

$V_{\phi}\ \equiv\ \frac{\omega}{k}\ =\ c\ \left(1+\frac{\omega_{\rm
e}^{2}}{2\,\omega^{2}}\pm\frac{\omega_{\rm e}^{2}\,|\Omega_{\rm
e}|}{2\,\omega^{3}}\right)\ ,$ (3)

where again the upper [lower] sign in the third term pertains to the right (R)
[left (L)] mode. Clearly, although both modes have the same angular frequency
$\omega$ (imposed at the source), they have slightly different phase
velocities, with $V_{\phi,{\rm R}}>V_{\phi,{\rm L}}$. As a result, if they
start off with the same phase at the source, a phase difference will arise and
grow between them as they propagate away from the source. The sign of this
phase difference, which is critical to the sense of Faraday rotation, depends
on how exactly the phase is defined.

In a homogeneous plasma, the phase can be defined as either
$\phi=(\omega\,t-k\,s+\varphi_{0})$ or $\phi=(k\,s-\omega\,t+\varphi_{0})$,
where $t$ denotes the time since a certain initial time, $s$ denotes the
distance from the radio source along the propagation direction,
$\hat{\mbox{\boldmath{$e$}}}_{k}$, and $\varphi_{0}$ is an arbitrary phase
offset. In an inhomogeneous plasma, the above expressions should be replaced
by $\phi=\left(\omega\,t-\int_{0}^{s}k\,ds^{\prime}+\varphi_{0}\right)$ and
$\phi=\left(\int_{0}^{s}k\,ds^{\prime}-\omega\,t+\varphi_{0}\right)$,
respectively. For convenience, we set the phase offset to $\varphi_{0}=0$,
such that $\phi=0$ at the source ($s=0$) and initial time ($t=0$).

Definition of the phase | Behavior | Behavior
---|---|---
|  | with increasing $t$ | with increasing $s$
|  | at constant $s$ | at constant $t$
$\displaystyle\phi=\omega\,t-\int_{0}^{s}k\,ds^{\prime}$ | (Eq. 4) |  $\phi$ increases |  $\phi$ decreases
$\displaystyle\phi=\int_{0}^{s}k\,ds^{\prime}-\omega\,t$ | (Eq. 13) |  $\phi$ decreases |  $\phi$ increases
Table 1:  Two alternative definitions of the phase, $\phi$, of a radio wave
with angular frequency $\omega$ and wavenumber $k$, at time $t$ and distance
$s$ from the source (first column), and corresponding behaviors with
increasing $t$ at constant $s$ (second column) and with increasing $s$ at
constant $t$ (third column).

With the first definition of the phase, $\phi$ at a given distance $s$
increases with increasing time $t$, while $\phi$ at a given time $t$ decreases
with increasing distance $s$ (see Table 1); this means that $\phi$ can be
taken equal to the angle through which the electric field vector has rotated
(in a right-handed [left-handed] sense about $\vec{\mbox{\boldmath{$B$}}}$ for
the right [left] mode) from its initial direction at the source. With the
second definition, $\phi$ at a given distance $s$ decreases with increasing
time $t$, while $\phi$ at a given time $t$ increases with increasing distance
$s$; this means that $\phi$ can be taken equal to minus the angle through
which the electric field vector has rotated from its initial direction at the
source. An alternative interpretation of $\phi$ in the latter case will be
provided in Sect. 3.2.

Here, and for the rest of the paper (except in Sect. 3.2), we adopt the first
definition, which directly gives the angle of the electric field vector and
which conforms to the IEEE standard:222 IEEE Standard Definitions of Terms for
Radio Wave Propagation (IEEE Std 211-1997).

$\phi\ =\ \omega\,t-\int_{0}^{s}k\,ds^{\prime}\ \cdot$ (4)

However, we emphasize that both definitions are valid and both lead to the
same sense of Faraday rotation. This will become clearer in Sect. 3.2, where
we discuss the implications of adopting the opposite definition of the phase.

###  2.2 Propagation of a linearly polarized wave

Consider a source of linearly polarized synchrotron radiation, e.g., a
Galactic pulsar, at a given angular frequency $\omega$. In reality,
synchrotron radiation is not fully polarized, but we will only retain its
linearly polarized component, which in practice is measured through the Stokes
parameters $Q$ and $U$.333 In general, the polarization state of
electromagnetic radiation can be described by the four Stokes parameters: the
total intensity, $I$; the linear polarization parameters, $Q$ and $U$, whose
quadratic sum gives the linearly polarized intensity, $|P|$ (see Eq. 23
below), and whose ratio yields the polarization angle, $\psi$ (see Eq. 24);
and the circular polarization parameter, $V$, which gives the circularly
polarized intensity. Synchrotron radiation is partially linearly polarized,
with $|P|<I$ and $V=0$.  Accordingly, we consider that the electric field
vector, $\vec{\mbox{\boldmath{$E$}}}_{\ell}$, oscillates at angular frequency
$\omega$ along an axis with given orientation, which defines the polarization
orientation. For future reference, we select one of the two unit vectors in
the polarization orientation at the source (subscript $\star$) and denote it
by $\hat{\mbox{\boldmath{$e$}}}_{\star}$. If we choose the initial time
appropriately, we can then write the electric field vector at the source as

$\vec{\mbox{\boldmath{$E$}}}_{\ell\star}=E_{0}\ \cos(\omega\,t)\
\hat{\mbox{\boldmath{$e$}}}_{\star}\ \cdot$ (5)

A linearly polarized wave with angular frequency $\omega$ propagating parallel
to the magnetic field, $\vec{\mbox{\boldmath{$B$}}}$, can be seen as the
superposition of a right (R) and a left (L) circularly polarized mode with the
same $\omega$. The electric field vectors of the right and left modes,
$\vec{\mbox{\boldmath{$E$}}}_{\rm R}$ and $\vec{\mbox{\boldmath{$E$}}}_{\rm
L}$, rotate in opposite senses, symmetrically with respect to the polarization
orientation, in such a way that their vector sum equals the electric field
vector of the linearly polarized wave, $\vec{\mbox{\boldmath{$E$}}}_{\ell}$:

$\vec{\mbox{\boldmath{$E$}}}_{\ell}=\vec{\mbox{\boldmath{$E$}}}_{\rm
R}+\vec{\mbox{\boldmath{$E$}}}_{\rm L}\ \cdot$ (6)

At the source and initial time, $\vec{\mbox{\boldmath{$E$}}}_{\ell}$ is at its
maximum extent in the $+\hat{\mbox{\boldmath{$e$}}}_{\star}$ direction (see
Eq. 5), which implies that $\vec{\mbox{\boldmath{$E$}}}_{\rm R}$ and
$\vec{\mbox{\boldmath{$E$}}}_{\rm L}$ are both pointing in the
$+\hat{\mbox{\boldmath{$e$}}}_{\star}$ direction. Therefore,
$\hat{\mbox{\boldmath{$e$}}}_{\star}$ represents the initial direction of
$\vec{\mbox{\boldmath{$E$}}}_{\rm R}$ [$\vec{\mbox{\boldmath{$E$}}}_{\rm L}$]
at the source, which, as explained above Eq. (4), defines the reference
direction from which the current angle of $\vec{\mbox{\boldmath{$E$}}}_{\rm
R}$ [$\vec{\mbox{\boldmath{$E$}}}_{\rm L}$] (measured in a right-handed [left-
handed] sense about $\vec{\mbox{\boldmath{$B$}}}$) can be equated to the
current phase of the right [left] mode, $\phi_{\rm R}$ [$\phi_{\rm L}$], as
defined by Eq. (4).

At the source ($s=0$), the right and left modes have the same phase,
$\phi_{\star}(t)=\omega\,t$. At a distance $s$ from the source, the two modes
have a phase difference, $\Delta\phi\equiv\phi_{\rm R}-\phi_{\rm L}$, given by

$\displaystyle\Delta\phi$ $\displaystyle=$ $\displaystyle-\int_{0}^{s}\Delta
k\ ds^{\prime}$ (7) $\displaystyle=$ $\displaystyle\int_{0}^{s}\frac{\Delta
V_{\phi}}{V_{\phi}}\ k\ ds^{\prime}$ $\displaystyle=$
$\displaystyle\int_{0}^{s}\frac{\omega_{\rm e}^{2}\,|\Omega_{\rm
e}|}{\omega^{3}}\ k\ ds^{\prime}\ ,$

where we have successively used Eq. (4) with $\Delta\omega\equiv\omega_{\rm
R}-\omega_{\rm L}=0$ (remember that both modes have the same $\omega$) and
$\Delta k\equiv k_{\rm R}-k_{\rm L}$, the first equality in Eq. (3) with
$\Delta V_{\phi}\equiv V_{\phi,{\rm R}}-V_{\phi,{\rm L}}$, and the second
equality in Eq. (3) with the upper [lower] sign in the last term corresponding
to $V_{\phi,{\rm R}}$ [$V_{\phi,{\rm L}}$]. The phase difference between the
right and left modes is positive, which means that, at any distance $s$ from
the source, the right mode is more advanced in phase than the left mode.

Figure 2:  Cartoon illustrating the propagation of the phases $\phi_{\rm L}$
and $\phi_{\rm R}$ of the left (blue) and right (red) circularly polarized
components of a linearly polarized radio wave with period
$T=\frac{\displaystyle 2\pi}{\displaystyle\omega}$. Time, $t$, increases
rightward, by steps of $\frac{T}{12}$, from the initial time ($t=0$) to half a
period ($\frac{T}{2}$). Distance from the source, $s$, increases continuously
downward, from the source ($s=0$) to a third of the right mode’s wavelength
($s=\frac{\lambda_{\rm R}}{3}$). With the phase of a mode defined through Eq.
(4),a $\phi_{\rm L}$ and $\phi_{\rm R}$ can be identified with the angles
through which the electric field vectors of the left and right modes,
$\vec{\mbox{\boldmath{$E$}}}_{\rm L}$ and $\vec{\mbox{\boldmath{$E$}}}_{\rm
R}$, have respectively rotated (in a left-handed and right-handed sense about
the magnetic field, $\vec{\mbox{\boldmath{$B$}}}$) from their common initial
direction at the source (see inset in the lower-left corner). The planes in
which $\vec{\mbox{\boldmath{$E$}}}_{\rm L}$ and
$\vec{\mbox{\boldmath{$E$}}}_{\rm R}$ rotate are actually perpendicular to the
propagation direction (the $s$-axis) and hence perpendicular to the plane of
the page. However, for the reader to see them face-on, we inclined them by
$90^{\circ}$, as if $\vec{\mbox{\boldmath{$B$}}}$ were pointing toward the
reader. The inclined views show $\vec{\mbox{\boldmath{$E$}}}_{\rm L}$ (thick,
blue arrows) and $\vec{\mbox{\boldmath{$E$}}}_{\rm R}$ (thick, red arrows),
together with their respective sense of rotation (thin, blue and red curved
arrows) and the circles traced out by their rotation (dotted, blue and red
circles). At the source (top row), $\vec{\mbox{\boldmath{$E$}}}_{\rm L}$ and
$\vec{\mbox{\boldmath{$E$}}}_{\rm R}$ rotate symmetrically with respect to
their common initial direction (far-left column), consistent with $\phi_{\rm
L}=\phi_{\rm R}$. Each $(\vec{\mbox{\boldmath{$E$}}}_{\rm
L},\vec{\mbox{\boldmath{$E$}}}_{\rm R})$ pair plotted at the source can be
followed step by step as $\vec{\mbox{\boldmath{$E$}}}_{\rm L}$ and
$\vec{\mbox{\boldmath{$E$}}}_{\rm R}$ propagate away from the source at the
phase velocities of the left and right modes, $V_{\phi,{\rm L}}$ and
$V_{\phi,{\rm R}}$, respectively. For instance, the propagation of the
$(\vec{\mbox{\boldmath{$E$}}}_{\rm L},\vec{\mbox{\boldmath{$E$}}}_{\rm R})$
pair starting at the source at time $\frac{T}{6}$, with $\phi_{\rm
L}=\phi_{\rm R}=\frac{\pi}{3}$, can be followed through the highlighted
sequence. Since $V_{\phi,{\rm R}}>V_{\phi,{\rm L}}$,b
$\vec{\mbox{\boldmath{$E$}}}_{\rm R}$ propagates faster than
$\vec{\mbox{\boldmath{$E$}}}_{\rm L}$, and the right mode is ahead in phase
relative to the left mode. $\delta s$ and $\delta t$ represent the spatial and
temporal phase lags of the left mode with respect to the right mode at
$t=\frac{T}{2}$ and $s=\frac{\lambda_{\rm R}}{3}$. Figure produced with the
help of Theophilus Britt Griswold (NASA Goddard).

${}^{\rm a}\ $ If the phase of a mode was defined through Eq. (13), the figure
would look exactly the same, with this difference that the angles of the
electric field vectors (which themselves would remain unchanged) would be
labeled $-\phi_{\rm L}$ and $-\phi_{\rm R}$ instead of $\phi_{\rm L}$ and
$\phi_{\rm R}$.
${}^{\rm b}\ $ Here, for illustrative purposes, we took $V_{\phi,{\rm
R}}=1.25\ V_{\phi,{\rm L}}$. In reality, the phase velocity difference between
the right and left modes is tremendously smaller, with
$\frac{\displaystyle\Delta V_{\phi}}{\displaystyle
V_{\phi}}=\frac{\displaystyle\omega_{\rm e}^{2}\,|\Omega_{\rm
e}|}{\displaystyle\omega^{3}}\lll 1$ (see Eq. 3). For instance, with
$\omega=1~{}{\rm GHz}$, $\omega_{\rm e}=25~{}{\rm kHz}$, and $|\Omega_{\rm
e}|=90~{}{\rm Hz}$ (see beginning of Sect. 2.1),
$\frac{\displaystyle\omega_{\rm e}^{2}\,|\Omega_{\rm
e}|}{\displaystyle\omega^{3}}\simeq 2\times 10^{-19}$.

This can be understood physically with the help of Figure 2, which shows how
the right and left modes propagate their phases, $\phi_{\rm R}$ and $\phi_{\rm
L}$ (equal to the angles through which the electric field vectors
$\vec{\mbox{\boldmath{$E$}}}_{\rm R}$ and $\vec{\mbox{\boldmath{$E$}}}_{\rm
L}$ have respectively rotated from their common initial direction at the
source, $\hat{\mbox{\boldmath{$e$}}}_{\star}$), away from the source. At the
source, both modes at any time $t_{\star}$ have the same phase,
$\phi_{\star}(t_{\star})=\omega\,t_{\star}$, which is an increasing function
of $t_{\star}$. Both modes propagate this phase $\phi_{\star}(t_{\star})$ away
from the source at their own phase velocities. Since the right mode has a
slightly larger phase velocity, it is faster to propagate
$\phi_{\star}(t_{\star})$ out to a given distance $s$ from the source. In
other words, a given $\phi_{\star}(t_{\star})$ reaches $s$ earlier for the
right mode than for the left mode. By the time $t$ that
$\phi_{\star}(t_{\star})$ reaches $s$ for the right mode,
$\phi_{\star}(t_{\star})$ is behind, say, at $s-\delta s$, for the left mode
and the phase of the left mode at $s$ is behind $\phi_{\star}(t_{\star})$,
say, equal to $\phi_{\star}(t_{\star}-\delta t)$. Mathematically,
$\phi_{\star}(t_{\star})=\phi_{\rm R}(t,s)=\phi_{\rm L}(t,s-\delta s)$ and
$\phi_{\rm L}(t,s)=\phi_{\star}(t_{\star}-\delta t)=\phi_{\rm R}(t-\delta
t,s)$. In Figure 2, the propagation of $\phi_{\star}(t_{\star})$ is
illustrated for $t_{\star}=\frac{T}{6}$ by the highlighted sequence of
electric field vectors at phase $\phi_{\star}(t_{\star})=\frac{\pi}{3}$. The
spatial lag, $\delta s$, and time lag, $\delta t$, are shown for
$t=\frac{T}{2}$ and $s=\frac{\lambda_{\rm R}}{3}$, with $\lambda_{\rm R}$ the
wavelength of the right mode.

At a distance $s$ from the source, the superposition of the right and left
modes still forms a linearly polarized wave. However, because the right and
left modes now have different phases ($\phi_{\rm R}\not=\phi_{\rm L}$), their
electric field vectors, $\vec{\mbox{\boldmath{$E$}}}_{\rm R}$ and
$\vec{\mbox{\boldmath{$E$}}}_{\rm L}$, have rotated through different angles;
as a result, the electric field vector of the linearly polarized wave,
$\vec{\mbox{\boldmath{$E$}}}_{\ell}$, has rotated with respect to its initial
direction at the source, $\hat{\mbox{\boldmath{$e$}}}_{\star}$ (see Figure 3).
The associated rotation of the polarization orientation is known as Faraday
rotation. This rotation occurs in the sense that the phase of the leading mode
increases, i.e., the sense that $\vec{\mbox{\boldmath{$E$}}}_{\rm R}$ rotates,
which is the right-handed sense about $\vec{\mbox{\boldmath{$B$}}}$. The angle
through which the polarization orientation rotates is half the phase
difference between the right and left modes (see Appendix B for the exact
derivation):

$\displaystyle\Delta\psi^{\scriptscriptstyle[B]}$ $\displaystyle=$
$\displaystyle{\textstyle\frac{1}{2}}\ \Delta\phi$ (8) $\displaystyle=$
$\displaystyle\frac{1}{2}\ \int_{0}^{s}\frac{\omega_{\rm e}^{2}\,|\Omega_{\rm
e}|}{\omega^{3}}\ k\ ds^{\prime}$ $\displaystyle=$
$\displaystyle\left(\frac{e^{3}}{2\pi\,m_{\rm e}^{2}\,c^{4}}\
\int_{0}^{s}n_{\rm e}\ B\ ds^{\prime}\right)\ \lambda^{2}\ ,$

where we have used Eq. (7) for the second equality and $\omega_{\rm
e}^{2}=\frac{\displaystyle 4\pi n_{\rm e}e^{2}}{\displaystyle m_{\rm e}}$,
$|\Omega_{\rm e}|=\frac{\displaystyle eB}{\displaystyle m_{\rm e}c}$,
$\omega=c\,k$, and $k=\frac{\displaystyle 2\pi}{\displaystyle\lambda}$ for the
third equality. Superscript ${\scriptstyle[B]}$ indicates that
$\psi^{\scriptscriptstyle[B]}$ is an angle about the magnetic field,
$\vec{\mbox{\boldmath{$B$}}}$ (measured in a right-handed sense) – as opposed
to $\psi$, which is an angle in the plane of the sky (measured
counterclockwise; see Figure 1).

From the perspective of the observer, located at a distance $d$ from the
source, the Faraday rotation angle over the entire path from the source
($s=0$) to the observer ($s=d$) is

$\displaystyle\Delta\psi$ $\displaystyle=$
$\displaystyle\pm\,\Delta\psi^{\scriptscriptstyle[B]}(s=d)$ (9)
$\displaystyle=$ $\displaystyle\left(\frac{e^{3}}{2\pi\,m_{\rm e}^{2}\,c^{4}}\
\int_{0}^{d}n_{\rm e}\ (\pm B)\ ds\right)\ \lambda^{2}\ ,$

with the $+$ [$-$] sign applying if $\vec{\mbox{\boldmath{$B$}}}$ points
toward [away from] the observer.

Figure 3:  Cartoon showing the electric field vectors of a linearly polarized
radio wave, $\vec{\mbox{\boldmath{$E$}}}_{\ell}$ (far right) and of its left
(blue) and right (red) circularly polarized components,
$\vec{\mbox{\boldmath{$E$}}}_{\rm L}$ and $\vec{\mbox{\boldmath{$E$}}}_{\rm
R}$, as defined with respect to the magnetic field,
$\vec{\mbox{\boldmath{$B$}}}$ (plasma physics convention), and as seen when
$\vec{\mbox{\boldmath{$B$}}}$ points toward the reader (as in Figure 2). The
wave vector, $\vec{\mbox{\boldmath{$k$}}}$, is parallel to
$\vec{\mbox{\boldmath{$B$}}}$, but its actual direction
($+\vec{\mbox{\boldmath{$B$}}}$ or $-\vec{\mbox{\boldmath{$B$}}}$) is
irrelevant. Top row: at the source ($s=0$) and initial time ($t=0$). Bottom
row: at distance $s$ from the source and time $t$ greater than the phase
travel time to $s$. The top and bottom rows correspond to the
$(\vec{\mbox{\boldmath{$E$}}}_{\rm L},\vec{\mbox{\boldmath{$E$}}}_{\rm R})$
pairs plotted in the upper-left and lower-right corners of Figure 2. The
dashed lines show the (linear or circular) figures traced out by the electric
field vectors. The curved arrows in the top row indicate the sense of rotation
of $\vec{\mbox{\boldmath{$E$}}}_{\rm L}$ and $\vec{\mbox{\boldmath{$E$}}}_{\rm
R}$. The curved arrows in the bottom row show the angles through which
$\vec{\mbox{\boldmath{$E$}}}_{\rm L}$ and $\vec{\mbox{\boldmath{$E$}}}_{\rm
R}$ have rotated from their common initial direction at the source, which is
along the orientation of linear polarization at the source. These angles are
exactly equal to the phases $\phi_{\rm L}$ and $\phi_{\rm R}$ of the left and
right modes, when the phase of a mode is defined through Eq. (4). Clearly,
$\vec{\mbox{\boldmath{$E$}}}_{\rm R}$ has rotated more than
$\vec{\mbox{\boldmath{$E$}}}_{\rm L}$, so the orientation of linear
polarization has rotated in the sense of the right mode. Hence, Faraday
rotation is right-handed about $\vec{\mbox{\boldmath{$B$}}}$. After the
corrected Figure 8.1 of Rybicki & Lightman (1979).

Eq. (9) was derived in the case of wave propagation parallel to the magnetic
field, $\vec{\mbox{\boldmath{$B$}}}$. It can be shown that this equation
remains valid for other propagation directions provided the factor $(\pm B)$
in the integral be replaced by the magnetic field component in the (positive)
propagation direction,
$\vec{\mbox{\boldmath{$B$}}}\\!\cdot\\!\hat{\mbox{\boldmath{$e$}}}_{k}$. This
field component is equivalent to the field component along the line of sight,
$B_{\parallel}$, as defined in the Faraday rotation community, which considers
that $B_{\parallel}$ is positive [negative] if $\vec{\mbox{\boldmath{$B$}}}$
points toward [away from] the observer. Eq. (9) can then be recast in the form

$\Delta\psi\ =\ {\rm RM}\ \lambda^{2}\ ,$ (10)

where

${\rm RM}\ \equiv\ \frac{e^{3}}{2\pi\,m_{\rm e}^{2}\,c^{4}}\
\int_{0}^{d}n_{\rm e}\ B_{\parallel}\ ds$ (11)

is the so-called rotation measure. Note that the convention used here for the
sign of $B_{\parallel}$ was precisely chosen to match the sign of ${\rm RM}$
(Manchester, 1972).

Finally, the observed polarization angle of the incoming radiation, $\psi_{\rm
obs}$, is related to the intrinsic polarization angle at the source,
$\psi_{\rm src}$, and to the Faraday rotation angle, $\Delta\psi$, through

$\psi_{\rm obs}\ =\ \psi_{\rm src}+\Delta\psi\ \cdot$ (12)

To summarize, what we have learnt in this section is that Faraday rotation is
right-handed about the magnetic field, $\vec{\mbox{\boldmath{$B$}}}$. The
implication for an observer looking toward a source of linearly polarized
synchrotron radiation is that Faraday rotation appears counterclockwise in the
plane of the sky if $\vec{\mbox{\boldmath{$B$}}}$ points toward the observer
and clockwise if $\vec{\mbox{\boldmath{$B$}}}$ points away from the observer.
Those are physical results, independent of any particular convention
regarding, e.g., the definition of the right and left circularly polarized
modes (Sect. 3.1), the definition of the phase (Sect. 3.2), or the sign of
$B_{\parallel}$ (Sect. 3.4).

##  3 Possible sources of confusion

###  3.1 Right and left modes

Circularly polarized waves are divided into right and left modes, based on the
sense of rotation of the electric field vector, $\vec{\mbox{\boldmath{$E$}}}$.
However, the reference vector about which the sense of rotation is measured
can be chosen in different ways (see summary in Table 2).

For plasma physicists, the right (R) and left (L) circularly polarized modes
are defined with respect to the magnetic field, $\vec{\mbox{\boldmath{$B$}}}$,
such that the electric field vector of the R [L] mode,
$\vec{\mbox{\boldmath{$E$}}}_{\rm R}$ [$\vec{\mbox{\boldmath{$E$}}}_{\rm L}$],
rotates in a right-handed [left-handed] sense about
$\vec{\mbox{\boldmath{$B$}}}$ (see, e.g., Nicholson, 1983; Chen, 2016).

For astronomers, the right circularly polarized (RCP) and left circularly
polarized (LCP) modes are defined with respect to the wave vector,
$\vec{\mbox{\boldmath{$k$}}}$, i.e., from the perspective of an observer
looking at the sky and watching the wave approach. Radio astronomers use the
IEEE convention, according to which the electric field vector of the RCP [LCP]
mode, $\vec{\mbox{\boldmath{$E$}}}_{\rm\scriptscriptstyle RCP}$
[$\vec{\mbox{\boldmath{$E$}}}_{\rm\scriptscriptstyle LCP}$], rotates in a
right-handed [left-handed] sense about $\vec{\mbox{\boldmath{$k$}}}$, i.e.,
counterclockwise [clockwise] as viewed by the observer. Optical astronomers
use the opposite convention (see, e.g., Robishaw & Heiles, 2018).

Thus, in radio astronomy, the R [L] wave is seen as an RCP [LCP] wave if
$\vec{\mbox{\boldmath{$B$}}}$ points toward the observer (see Figure 4a) and
as an LCP [RCP] wave if $\vec{\mbox{\boldmath{$B$}}}$ points away from the
observer (see Figure 4b). Let us now examine the properties of the RCP and LCP
modes in more detail.

Figure 4:  3D patterns formed by the tips of the electric field vectors of a
linearly polarized radio wave (right panels) and of its left and right
circularly polarized components, defined either with respect to the magnetic
field, $\vec{\mbox{\boldmath{$B$}}}$, as the L and R modes (plasma physics
convention), or with respect to the wave vector,
$\vec{\mbox{\boldmath{$k$}}}$, as the LCP and RCP modes (radio astronomy
convention). The electric field vector of the linearly polarized wave,
$\vec{\mbox{\boldmath{$E$}}}_{\ell}$, defines the orientation of linear
polarization (right panels). The latter undergoes Faraday rotationa between
the source (far side) and the observer (near side), and this rotation has the
same handedness as the propagation helix of the L mode (left panels). (a)
$\vec{\mbox{\boldmath{$B$}}}$ points toward the observer, so the L [R] mode
corresponds to the LCP [RCP] mode and Faraday rotation is right-handed about
$\vec{\mbox{\boldmath{$k$}}}$, i.e., counterclockwise for the observer. (b)
$\vec{\mbox{\boldmath{$B$}}}$ points away from the observer, so the L [R] mode
now corresponds to the RCP [LCP] mode and Faraday rotation is left-handed
about $\vec{\mbox{\boldmath{$k$}}}$, i.e., clockwise for the observer. An
animated version of this plot, which more explicitly demonstrates the
propagation in time, is available as online supplementary material on the
journal website. The video can also be found at https://youtu.be/rB_giDghgr4.

${}^{\rm a}\ $ Here, for illustrative purposes, the Faraday rotation of the
polarization orientation has been hugely exaggerated with respect to the
rotation of the electric field vectors $\vec{\mbox{\boldmath{$E$}}}_{\rm L}$
and $\vec{\mbox{\boldmath{$E$}}}_{\rm R}$.

Regardless of the direction of $\vec{\mbox{\boldmath{$B$}}}$ with respect to
the observer, the temporal behavior (with increasing $t$) of
$\vec{\mbox{\boldmath{$E$}}}_{\rm\scriptscriptstyle RCP}$
[$\vec{\mbox{\boldmath{$E$}}}_{\rm\scriptscriptstyle LCP}$] at the source is a
right-handed [left-handed] rotation about $\vec{\mbox{\boldmath{$k$}}}$. This
temporal behavior at the source propagates toward the observer, such that the
state at a distance $s$ from the source is delayed with respect to the state
at the source. Hence, moving from the source toward the observer at a given
time $t$ is equivalent to going backward in time at the source. As a result,
the spatial behavior (with increasing $s$) of
$\vec{\mbox{\boldmath{$E$}}}_{\rm\scriptscriptstyle RCP}$
[$\vec{\mbox{\boldmath{$E$}}}_{\rm\scriptscriptstyle LCP}$] at a given time
$t$ is a left-handed [right-handed] rotation about
$\vec{\mbox{\boldmath{$k$}}}$. Stated differently, the tip of
$\vec{\mbox{\boldmath{$E$}}}_{\rm\scriptscriptstyle RCP}$
[$\vec{\mbox{\boldmath{$E$}}}_{\rm\scriptscriptstyle LCP}$] at a given time
$t$ traces out a left-handed [right-handed] helix along the propagation
direction.444 This explains the convention adopted by optical astronomers.
While radio astronomers base their definition of right and left circularly
polarized modes on the sense of rotation (in time) of
$\vec{\mbox{\boldmath{$E$}}}_{\rm\scriptscriptstyle RCP}$ and
$\vec{\mbox{\boldmath{$E$}}}_{\rm\scriptscriptstyle LCP}$ at a given position,
optical astronomers rely on the handedness of the propagation helix (in space)
of $\vec{\mbox{\boldmath{$E$}}}_{\rm\scriptscriptstyle RCP}$ and
$\vec{\mbox{\boldmath{$E$}}}_{\rm\scriptscriptstyle LCP}$ at a given time.
This is illustrated in the middle panel of Figure 4a and left panel of Figure
4b [left panel of Figure 4a and middle panel of Figure 4b]. Note, in
preparation for Sect. 4, that the magnetic field vector of each mode describes
the same propagation helix as the electric field vector, with the implication
that the associated current helicity (defined in Sect. 4) is positive
[negative].

The above considerations offer an alternative way of obtaining the sense of
Faraday rotation. As we saw in Sect. 2.1, the L mode has a slightly slower
phase velocity, and hence a slightly shorter wavelength
$\left(\lambda=2\pi\frac{\displaystyle V_{\phi}}{\displaystyle\omega}\right)$,
than the R mode at the same angular frequency $\omega$ (see Eq. 3).
Accordingly, the propagation helix of the L mode (left panels in Figure 4)
goes through slightly more cycles between the source and the observer than the
propagation helix of the R mode (middle panels). The effect is weak and can
only be noticed on close examination of the figure. The resulting Faraday
rotation (right panels) occurs in the sense that the helix with more cycles,
i.e., the helix of the L mode, winds up while approaching the observer, which
is the right-handed sense about $\vec{\mbox{\boldmath{$k$}}}$ –
counterclockwise for the observer – if $\vec{\mbox{\boldmath{$B$}}}$ points
toward the observer (Figure 4a) and the left-handed sense about
$\vec{\mbox{\boldmath{$k$}}}$ – clockwise for the observer – if
$\vec{\mbox{\boldmath{$B$}}}$ points away from the observer (Figure 4b).
Clearly, this conclusion is identical to that reached in Sect. 2.2, based on
the phase difference, $\Delta\phi=\phi_{\rm R}-\phi_{\rm L}$.

An obvious pitfall here is to mistake the sense of rotation of a mode with
increasing time $t$, at a given distance $s$ from the source, for the sense of
rotation with increasing distance $s$, at a given time $t$. The possible
confusion is discussed in some detail by Kliger et al. (1990); Shurcliff
(1963); Goldstein (2011). It is also illustrated in the animated version of
Figure 4 (at https://youtu.be/xXPEJjlcZ2w), which shows the propagation in
time much more clearly than the static figure. The above mistake leads to the
erroneous conclusion that the RCP [LCP] mode describes a right-handed [left-
handed] helix along the propagation direction, which in turn yields the wrong
sense of Faraday rotation.

Concerned | Electric | Sense of rotation | Sense of rotation | Reference
---|---|---|---|---
scientific | field | with increasing $t$ | with increasing $s$ | vector
community | vectors | at constant $s$ | at constant $t$ | or plane
Plasma | $\vec{\mbox{\boldmath{$E$}}}_{\rm R}$ |  right-handed |  left-handed |  about $\vec{\mbox{\boldmath{$B$}}}$
physics |  |  |  |
| $\vec{\mbox{\boldmath{$E$}}}_{\rm L}$ |  left-handed |  right-handed |  about $\vec{\mbox{\boldmath{$B$}}}$
Radio | $\vec{\mbox{\boldmath{$E$}}}_{\rm\scriptscriptstyle RCP}$ |  right-handed |  left-handed |  about $\vec{\mbox{\boldmath{$k$}}}$
astronomy |  |  or CCW |  or CW |  in the sky
| $\vec{\mbox{\boldmath{$E$}}}_{\rm\scriptscriptstyle LCP}$ |  left-handed |  right-handed |  about $\vec{\mbox{\boldmath{$k$}}}$
|  |  or CW |  or CCW |  in the sky
Optical | $\vec{\mbox{\boldmath{$E$}}}_{\rm\scriptscriptstyle RCP}$ |  left-handed |  right-handed |  about $\vec{\mbox{\boldmath{$k$}}}$
astronomy |  |  or CW |  or CCW |  in the sky
| $\vec{\mbox{\boldmath{$E$}}}_{\rm\scriptscriptstyle LCP}$ |  right-handed |  left-handed |  about $\vec{\mbox{\boldmath{$k$}}}$
|  |  or CCW |  or CW |  in the sky
Table 2:  Definitions of the right (R or RCP) and left (L or LCP) circularly
polarized modes in different scientific communities (first column), based on
the sense of rotation of the associated electric field vectors (second
column), both with increasing time $t$ at constant distance $s$ from the
source (third column) and with increasing $s$ at constant $t$ (fourth column).
The reference vector about which the sense of rotation is measured (fifth
column) is either the magnetic field, $\vec{\mbox{\boldmath{$B$}}}$ (in plasma
physics), or the wave vector, $\vec{\mbox{\boldmath{$k$}}}$ (in radio
astronomy and in optical astronomy). Right-handed [left-handed] rotation about
$\vec{\mbox{\boldmath{$k$}}}$ is equivalent to counterclockwise [clockwise]
rotation in the plane of the sky. The sense of rotation about
$\vec{\mbox{\boldmath{$k$}}}$ in the fourth column corresponds to the
handedness of the propagation helix in Figure 4.

###  3.2 Phase of a wave

The phase of a wave can also be defined through the opposite of Eq. (4),
namely,

$\phi\ =\ \int_{0}^{s}k\,ds^{\prime}-\omega\,t\ \cdot$ (13)

Let us see how proceeding from Eq. (13) instead of Eq. (4) affects the derived
sense of Faraday rotation. Consider again a linearly polarized radio wave with
angular frequency $\omega$ propagating parallel to the magnetic field,
$\vec{\mbox{\boldmath{$B$}}}$. As before, this wave can be decomposed into
right (R) and left (L) circularly polarized modes with the same $\omega$.

At the source ($s=0$), the right and left modes have again the same phase,
$\phi_{\star}(t)$, but now $\phi_{\star}(t)=-\omega\,t$. At a distance $s$
from the source, the two modes have a phase difference,
$\Delta\phi\equiv\phi_{\rm R}-\phi_{\rm L}$, which is now given by

$\displaystyle\Delta\phi$ $\displaystyle=$ $\displaystyle\int_{0}^{s}\Delta k\
ds^{\prime}$ (14) $\displaystyle=$ $\displaystyle-\int_{0}^{s}\frac{\Delta
V_{\phi}}{V_{\phi}}\ k\ ds^{\prime}$ $\displaystyle=$
$\displaystyle-\int_{0}^{s}\frac{\omega_{\rm e}^{2}\,|\Omega_{\rm
e}|}{\omega^{3}}\ k\ ds^{\prime}\ ,$

instead of Eq. (7). This phase difference is now negative, which means that,
at any distance $s$ from the source, the right mode is less advanced in phase
than the left mode.

There are two ways to interpret this result physically, both based on the
right mode having the larger phase velocity ($V_{\phi,{\rm R}}>V_{\phi,{\rm
L}}$). The first interpretation is analogous to that proposed in Sect. 2.2,
below Eq. (7). At the source, the right and left modes at any time $t_{\star}$
have the same phase, $\phi_{\star}(t_{\star})=-\omega\,t_{\star}$, which is
now a decreasing function of $t_{\star}$. The right mode is faster to
propagate this decreasing $\phi_{\star}(t_{\star})$ out to a given distance
$s$ from the source, so that the phase of the right mode at distance $s$ is
smaller (or more negative) than the phase of the left mode.

Alternatively, we can identify the phase of a mode, which is now an increasing
function of $s$, with the angle through which the electric field vector turns
along the propagation helix (see middle [left] panels in Figure 4 for the
right [left] mode), starting from the point corresponding to the initial
direction at the source and moving in the propagation direction (i.e., in a
left-handed [right-handed] sense about $\vec{\mbox{\boldmath{$B$}}}$ for the
right [left] mode). Since the helix of the left mode winds up at a faster rate
than the helix of the right mode (see Sect. 3.1), the phase of the left mode
at a given distance $s$ is larger than the phase of the right mode.

In both cases, the left mode is ahead in phase relative to the right mode. The
first interpretation is more directly relevant for times greater than the
phase travel times to a given distance $s$ ($t>\frac{\displaystyle
s}{\displaystyle V_{\phi,{\rm L}}}>\frac{\displaystyle s}{\displaystyle
V_{\phi,{\rm R}}}$), whereas the second interpretation is easier to picture
for distances greater than the phase travel distances on a given time $t$
($s>V_{\phi,{\rm R}}\,t>V_{\phi,{\rm L}}\,t$). However, both interpretations
can be generalized to all times and distances.

The consequence for the linearly polarized wave is again that the polarization
orientation rotates in the sense that the phase of the leading mode increases,
which is now the sense that the propagation helix of the left mode winds up
while approaching the observer. Regardless of whether
$\vec{\mbox{\boldmath{$B$}}}$ points toward the observer (Figure 4a) or away
from the observer (Figure 4b), this is again the right-handed sense about
$\vec{\mbox{\boldmath{$B$}}}$. Thus, Faraday rotation remains right-handed
about the magnetic field.

###  3.3 Two examples of incorrect derivation

To illustrate the subtleties and pitfalls discussed in Sects. 3.1 and 3.2, we
now critically examine two derivations presented in widely read textbooks,
where a subtle error in the sense of rotation of the electric field vectors of
the right and left modes, $\vec{\mbox{\boldmath{$E$}}}_{\rm R}$ and
$\vec{\mbox{\boldmath{$E$}}}_{\rm L}$, has led to the wrong sense of Faraday
rotation.

Let us first look at Chapter 4 of the plasma textbook by Chen (2016). There,
the phase of a mode is defined through an equation (Eq. 4.7) similar to our
Eq. (13), stating that the phase increases with increasing distance from the
source. In his Sect. 4.17.2, Chen argues that the left mode undergoes more
cycles over a given distance than the right mode, from which he correctly
concludes that, at a distance $d$ from the source, the left mode is more
advanced in phase than the right mode. Besides, he implicitly identifies the
phase of the right [left] mode with the angle through which
$\vec{\mbox{\boldmath{$E$}}}_{\rm R}$ [$\vec{\mbox{\boldmath{$E$}}}_{\rm L}$]
turns upon traversing the plasma, i.e., in our terminology, the angle through
which $\vec{\mbox{\boldmath{$E$}}}_{\rm R}$ [$\vec{\mbox{\boldmath{$E$}}}_{\rm
L}$] turns along the propagation helix between $s=0$ and $s=d$.

The problem is that Chen takes $\vec{\mbox{\boldmath{$E$}}}_{\rm R}$
[$\vec{\mbox{\boldmath{$E$}}}_{\rm L}$] to turn (in space) along the
propagation helix in the same sense as $\vec{\mbox{\boldmath{$E$}}}_{\rm R}$
[$\vec{\mbox{\boldmath{$E$}}}_{\rm L}$] rotates (in time) at a given position,
i.e., in a right-handed [left-handed] sense about
$\vec{\mbox{\boldmath{$B$}}}$ (see his Figure 4.44), while in reality
$\vec{\mbox{\boldmath{$E$}}}_{\rm R}$ [$\vec{\mbox{\boldmath{$E$}}}_{\rm L}$]
turns in the opposite sense (see middle [left] panels in our Figure 4). This
error leads him to incorrectly conclude that, at a distance $d$ from the
source, $\vec{\mbox{\boldmath{$E$}}}_{\rm L}$ has turned more in a left-handed
sense than $\vec{\mbox{\boldmath{$E$}}}_{\rm R}$ in a right-handed sense, and,
by implication, that Faraday rotation is left-handed about the magnetic field.

A similar mistake was probably made in Sect. 8.2 of the astrophysics textbook
by Rybicki & Lightman (1979). Their exact line of thought is more difficult to
follow, because the derivation is very compact, some key information is
missing, and unfortunately a critical sign error is present. To start with,
Rybicki & Lightman adopt the optical convention for the right and left modes,
according to which the electric field vector of the right [left] mode,
$\vec{\mbox{\boldmath{$E$}}}_{\rm R^{\prime}}$
[$\vec{\mbox{\boldmath{$E$}}}_{\rm L^{\prime}}$],555 We added a prime to the
indices ${\rm R}$ and ${\rm L}$ of Rybicki & Lightman (1979) to distinguish
their definition of right and left modes from ours.  rotates in a left-handed
[right-handed] sense about the wave vector, $\vec{\mbox{\boldmath{$k$}}}$.
They define the phase of a mode through the same equation as Chen (2016),
similar to our Eq. (13). Based on these two premises, the time variation of
$\vec{\mbox{\boldmath{$E$}}}_{\rm R^{\prime}}$
[$\vec{\mbox{\boldmath{$E$}}}_{\rm L^{\prime}}$] is described by their Eq.
(8.24) with the upper [lower] sign.666 In reality, the $\pm$ sign in Eq.
(8.24) should be replaced by a $\mp$ sign.  Since the ambient magnetic field,
$\vec{\mbox{\boldmath{$B$}}}_{0}$ (their Eq. 8.25), is assumed to point in the
same direction as $\vec{\mbox{\boldmath{$k$}}}$ (toward the reader in their
Figure 8.1), $\vec{\mbox{\boldmath{$E$}}}_{\rm R^{\prime}}$
[$\vec{\mbox{\boldmath{$E$}}}_{\rm L^{\prime}}$] corresponds to
$\vec{\mbox{\boldmath{$E$}}}_{\rm L}$ [$\vec{\mbox{\boldmath{$E$}}}_{\rm R}$]
in our notation (see Table 2).

The dispersion relation (their Eq. 8.27) is given in the form of an equation
for the dielectric constant, $\epsilon=\frac{\displaystyle
c^{2}k^{2}}{\displaystyle\omega^{2}}$ (their Eq. 8.9), with the plasma
frequency, $\omega_{p}$ (their Eq. 8.11), equivalent to our $\omega_{\rm e}$
and the cyclotron frequency, $\omega_{B}$ (their Eq. 8.21), equivalent to our
$|\Omega_{\rm e}|$. It is easily verified that their Eq. (8.27) is equivalent
to our Eq. (1), with the right and left modes swapped. Their Eq. (8.27) then
leads to an expression for the wavenumber, $k$ (their Eq. 8.29), which implies
$k_{\rm R^{\prime}}>k_{\rm L^{\prime}}$, such that
$\vec{\mbox{\boldmath{$E$}}}_{\rm R^{\prime}}$ turns (in space) at a faster
rate than $\vec{\mbox{\boldmath{$E$}}}_{\rm L^{\prime}}$.

For illustration, Figure 8.1 of Rybicki & Lightman (1979) shows the electric
field vectors at two different positions (say, $s=0$ in Fig. 8.1a and $s=d$ in
Fig. 8.1b), with $\vec{\mbox{\boldmath{$E$}}}_{\rm R^{\prime}}$ (left panel)
[$\vec{\mbox{\boldmath{$E$}}}_{\rm L^{\prime}}$ (middle panel)] rotating (in
time) in a left-handed [right-handed] sense about
$\vec{\mbox{\boldmath{$B$}}}_{0}$. The sense that
$\vec{\mbox{\boldmath{$E$}}}_{\rm R^{\prime}}$
[$\vec{\mbox{\boldmath{$E$}}}_{\rm L^{\prime}}$] turns from $s=0$ to $s=d$ is
not clearly indicated in the figure, but since
$\vec{\mbox{\boldmath{$E$}}}_{\rm R^{\prime}}$ must turn through a larger
angle than $\vec{\mbox{\boldmath{$E$}}}_{\rm L^{\prime}}$ (as implied by
$k_{\rm R^{\prime}}>k_{\rm L^{\prime}}$), we gather that
$\vec{\mbox{\boldmath{$E$}}}_{\rm R^{\prime}}$
[$\vec{\mbox{\boldmath{$E$}}}_{\rm L^{\prime}}$] turns in a left-handed
[right-handed] sense about $\vec{\mbox{\boldmath{$B$}}}_{0}$, consistent with
Figure 8.1b (right panel) showing left-handed Faraday rotation.

This again is incorrect, and again the error comes from mistaking the sense
that $\vec{\mbox{\boldmath{$E$}}}_{\rm R^{\prime}}$
[$\vec{\mbox{\boldmath{$E$}}}_{\rm L^{\prime}}$] rotates (in time) at a given
position for the sense that $\vec{\mbox{\boldmath{$E$}}}_{\rm R^{\prime}}$
[$\vec{\mbox{\boldmath{$E$}}}_{\rm L^{\prime}}$] turns (in space) in the
propagation direction at a given time.

###  3.4 Sign of $B_{\parallel}$

Two opposite conventions for the sign of the magnetic field component along
the line of sight, $B_{\parallel}$, are being used by radio astronomers (see,
e.g., Robishaw & Heiles, 2018).

In the Zeeman splitting community, the sign convention for $B_{\parallel}$ is
analogous to the sign convention adopted for the line-of-sight velocity,
$v_{\parallel}$, measured through the Doppler effect. In the same way as
$v_{\parallel}$ is taken to be positive for motions away from the observer,
$B_{\parallel}$ is taken to be positive when the magnetic field,
$\vec{\mbox{\boldmath{$B$}}}$, points away from the observer (Verschuur,
1969).

In the Faraday rotation community, the opposite sign convention is used,
namely, $B_{\parallel}$ is taken to be positive when
$\vec{\mbox{\boldmath{$B$}}}$ points toward the observer. This convention
dates back to Manchester (1972), who chose to have the sign of $B_{\parallel}$
match the sign of ${\rm RM}$ (see Sect. 2.2).

The only impact of the chosen convention for the sign of $B_{\parallel}$ is in
the expression of ${\rm RM}$. With the sign convention from the Zeeman
splitting community, $B_{\parallel}$ would have the opposite sign to ${\rm
RM}$, which would then be given by

${\rm RM}\ =\ -\frac{e^{3}}{2\pi\,m_{\rm e}^{2}\,c^{4}}\ \int_{0}^{d}n_{\rm
e}\ B_{\parallel}\ ds\ \cdot$

But again, the derived sense of Faraday rotation would remain unchanged.

The existence of opposite conventions for the sign of $B_{\parallel}$ could
potentially be a source of error. Green et al. (2012) compared the magnetic
field directions inferred from Zeeman splitting of 18-cm OH masers in Galactic
star-forming regions with those inferred from Faraday rotation of Galactic
pulsars and extragalactic radio sources along nearby lines of sight. Both
kinds of measurements indicate coherent magnetic field directions across the
observed region, but the field directions inferred from Zeeman splitting and
from Faraday rotation are in opposition. Since the two methods probe different
media, the discrepencay could very well have a physical origin. However, it
could also be that an error was made either in the Zeeman splitting convention
(as discussed by the authors) or in the Faraday rotation convention.

###  3.5 Limits of integration

There has been some confusion in the literature regarding the order of the
integration limits in the expression of the rotation measure. Our derivation
in Sect. 2.2 led us to write ${\rm RM}$ as an integral from the source ($s=0$)
to the observer ($s=d$):

${\rm RM}\ =\ {\cal C}\ \int_{s=0}^{s=d}n_{\rm e}\ B_{\parallel}\ ds\ ,$ (15)

where ${\cal C}\equiv\frac{\displaystyle e^{3}}{\displaystyle 2\pi\,m_{\rm
e}^{2}\,c^{4}}$ is the prefactor appearing in the right-hand side of Eq. (11),
and $B_{\parallel}$ is the line-of-sight component of the magnetic field,
$\vec{\mbox{\boldmath{$B$}}}$, taken to be positive when
$\vec{\mbox{\boldmath{$B$}}}$ points from the source to the observer. In view
of this convention for the sign of $B_{\parallel}$, Eq. (15) can be recast in
vectorial form:

${\rm RM}\ =\ {\cal C}\ \int_{\rm src}^{\rm obs}n_{\rm e}\
\vec{\mbox{\boldmath{$B$}}}\cdot d\vec{\mbox{\boldmath{$s$}}}\ \cdot$ (16)

Figure 5:  The two complementary coordinates used in this paper to locate an
arbitrary point between a radio source and an observer: $s$ is the distance
from the source along the propagation direction and $r$ is the distance from
the observer along the line of sight. Also shown is the convention for the
sign of $B_{\parallel}$ used in the Faraday rotation community:
$B_{\parallel}>0$ when the magnetic field points from the source to the
observer, i.e., in the opposite direction to increasing $r$.

In practice, it is often more convenient to place the origin of the line-of-
sight coordinate at the observer, rather than at the source. If we denote by
$r$ the line-of-sight distance from the observer (see Figure 5), such that
$r=d-s$, we can rewrite Eq. (15) as an integral over $r$. Noting that $s=0$
corresponds to $r=d$ (at the source), $s=d$ corresponds to $r=0$ (at the
observer), and $ds=-dr$, we obtain

${\rm RM}\ =\ {\cal C}\ \int_{r=d}^{r=0}n_{\rm e}\ B_{\parallel}\ (-dr)\ ,$

which is equivalent to

${\rm RM}\ =\ {\cal C}\ \int_{r=0}^{r=d}n_{\rm e}\ B_{\parallel}\ dr\ \cdot$
(17)

Thus, the expression of ${\rm RM}$ can be written as an integral either from
the source ($s=0$) to the observer ($s=d$) or from the observer ($r=0$) to the
source ($r=d$), with exactly the same integrand, $n_{\rm e}\,B_{\parallel}$.
The only requirement is that one sticks to the convention that $B_{\parallel}$
is positive [negative] when $\vec{\mbox{\boldmath{$B$}}}$ points toward [away
from] the observer. In particular, if $n_{\rm e}$ and $B_{\parallel}$ are both
constant along the line of sight, Eqs. (15) and (17) reduce to the same
expression, ${\rm RM}={\cal C}\,n_{\rm e}\,B_{\parallel}\,d$, with the same
sign.

Several authors have mistakenly rewritten Eq. (16) in the form

${\rm RM}\ =\ {\cal C}\ \int_{d}^{0}n_{\rm e}\
\vec{\mbox{\boldmath{$B$}}}\cdot d\vec{\mbox{\boldmath{$s$}}}\hfill({\rm a})$

(e.g., Van Eck et al., 2017; Van Eck, 2018) or

${\rm RM}\ =\ {\cal C}\ \int_{d}^{0}n_{\rm e}\ B_{\parallel}\ dr\hfill({\rm
b})$

(e.g., Sun et al., 2015; Dickey et al., 2019; Thomson et al., 2019; Stein et
al., 2020). Clearly, the idea was to keep the order of the integration limits
from the source to the observer. However, Eq. (a) is ill-defined (the
integration limits should be position vectors, not scalars) and Eq. (b) misses
a minus sign, which arises from the transformation
$\vec{\mbox{\boldmath{$B$}}}\cdot
d\vec{\mbox{\boldmath{$s$}}}=B_{\parallel}\,ds=-B_{\parallel}\,dr$. Without
this minus sign, Eq. (b) is equivalent to

${\rm RM}\ =\ -{\cal C}\ \int_{0}^{d}n_{\rm e}\ B_{\parallel}\ dr\ ,$

which, together with ${\cal C}>0$, $n_{\rm e}\geq 0$, and $dr\geq 0$ ($r$
increasing from $0$ to $d$), implies that ${\rm RM}$ has the opposite sign to
(the line-of-sight averaged) $B_{\parallel}$ – in contradiction with our
adopted convention. Note, however, that the mistake made here is purely
formal, with no impact on the derived sign of $B_{\parallel}$ or on the
inferred magnetic field direction.

##  4 Faraday rotation, synchrotron emission, and helicity

In Sects. 2 and 3, we considered the Faraday rotation of a background source
of linearly polarized synchrotron radiation by a foreground magnetized plasma.
Here, we drop the notion of a background source and assume instead that the
magnetized plasma itself, in addition to causing Faraday rotation, is also the
site of synchrotron emission. Such diffuse synchrotron emission pervades the
ISM and is commonly used to study magnetic fields in the Milky Way Galaxy
(e.g., Haverkorn, 2015). Thus, we switch from a situation where synchrotron
emission and Faraday rotation are well separated in space to a situation where
they are spatially mixed.

Synchrotron emission is linearly polarized perpendicularly to the local
magnetic field projected onto the plane of the sky,
$\vec{\mbox{\boldmath{$B$}}}_{\perp}$. If we denote by $\psi_{B_{\perp}}$ the
angle of $\vec{\mbox{\boldmath{$B$}}}_{\perp}$ (measured counterclockwise from
North; see Figure 1), the intrinsic polarization angle of synchrotron emission
is

$\psi_{\rm src}\ =\ \psi_{B_{\perp}}\pm\frac{\pi}{2}\ \cdot$ (18)

As explained in Sect. 2.2, the polarization angle of the emission produced at
distance $r$ from the observer undergoes Faraday rotation, changing from
$\psi_{\rm src}(r)$ at the source to $\psi_{\rm obs}(r)$ at the observer,
where, according to Eqs. (12), (10), and (17),

$\displaystyle\psi_{\rm obs}(r)$ $\displaystyle=$ $\displaystyle\psi_{\rm
src}(r)+\Delta\psi(r)$ (19) $\displaystyle=$ $\displaystyle\psi_{\rm
src}(r)+\Phi(r)\ \lambda^{2}$

and

$\Phi(r)\ \equiv\ {\cal C}\ \int_{0}^{r}n_{\rm e}\ B_{\parallel}\ dr^{\prime}\
\cdot$ (20)

$\Phi(r)$ is the so-called Faraday depth777 Faraday depth is generally denoted
by $\phi$ (Burn, 1966; Gardner & Whiteoak, 1966; Brentjens & de Bruyn, 2005;
Van Eck, 2018). Here, we denote it by $\Phi$ to distinguish it from the phase
of a mode defined earlier (Eqs. 4 and 13).  at distance $r$ from the observer.
It has basically the same formal expression as ${\rm RM}$ in Eq. (17), but is
conceptually different: whereas ${\rm RM}$ is a purely observational quantity,
which can be meaningfully defined only for a background synchrotron source,
$\Phi(r)$ is a truly physical quantity, which can be defined at any point of
the ISM, independent of any background source. The notion of Faraday depth is
particularly useful in the present context, where synchrotron emission and
Faraday rotation are spatially mixed.

If the relativistic electrons responsible for synchrotron emission have a
power-law energy spectrum described by $f(E)=K_{e}\,E^{-\gamma}$, the
synchrotron emissivity at frequency $\nu$ is given by

${\cal E}\ =\ {\rm fc}(\gamma)\ K_{e}\ B_{\perp}^{\frac{\gamma+1}{2}}\
\nu^{-\frac{\gamma-1}{2}}\ ,$ (21)

where ${\rm fc}(\gamma)$ is a known function of the electron spectral index,
and the intrinsic degree of linear polarization is $p_{\rm
src}=\frac{\gamma+1}{\gamma+\frac{7}{3}}$ (e.g., Ginzburg & Syrovatskii,
1965). Then, for any given line of sight, the synchrotron total intensity is

$I\ =\ \int_{0}^{\infty}{\cal E}\ dr$ (22)

and the (complex) synchrotron polarized intensity is

$P\ \equiv\ Q+i\,U\ =\ \int_{0}^{\infty}p_{\rm src}\ {\cal E}\ e^{2i\psi_{\rm
obs}}\ dr\ \cdot$ (23)

The observed polarization angle is then given by

$\psi=\frac{1}{2}\ \arctan\frac{U}{Q}\ ,$ (24)

with $\arctan$ the two-argument arctangent function defined from
$-180^{\circ}$ to $+180^{\circ}$, and the observed degree of linear
polarization is given by

$p=\frac{|P|}{I}=\frac{\sqrt{Q^{2}+U^{2}}}{I}\ \cdot$ (25)

Because of the factor $e^{2i\psi_{\rm obs}}$ in Eq. (23), depolarization
occurs when the synchrotron emission produced at different distances along the
line of sight reaches the observer with different polarization angles.
According to Eq. (19), this happens either when the intrinsic polarization
angle (Eq. 18) varies along the line of sight or when the Faraday depth (Eq.
20) varies along the line of sight, i.e., when Faraday rotation is present.
Eq. (19) also shows that Faraday rotation by a magnetic field pointing toward
the observer ($B_{\parallel}>0$, such that $\Phi(r)$ increases positively with
increasing $r$) acts in the same sense as counterclockwise rotation of
$\vec{\mbox{\boldmath{$B$}}}_{\perp}$ away from the observer (such that
$\psi_{\rm src}(r)$ increases positively with increasing $r$).

A magnetic field with $\vec{\mbox{\boldmath{$B$}}}_{\perp}$ rotating
counterclockwise away from the observer forms a "spiral-slide" helical
magnetic field, for which the tip of $\vec{\mbox{\boldmath{$B$}}}_{\perp}$
traces out a left-handed helix (see Figure 8 in Brandenburg & Stepanov (2014)
and top panels in Figure 6). Such a helical magnetic field can be written in
IAU coordinates (see Figure 1) as

$\vec{\mbox{\boldmath{$B$}}}=B_{\perp}\left(\cos\psi_{B_{\perp}}\
\hat{\mbox{\boldmath{$e$}}}_{X}+\sin\psi_{B_{\perp}}\
\hat{\mbox{\boldmath{$e$}}}_{Y}\right)+B_{\parallel}\
\hat{\mbox{\boldmath{$e$}}}_{Z}\ ,$ (26)

with $\frac{\displaystyle\partial\psi_{B_{\perp}}}{\displaystyle\partial
Z}=-\frac{\displaystyle\partial\psi_{B_{\perp}}}{\displaystyle\partial r}<0$.
If $\vec{\mbox{\boldmath{$B$}}}$ varies only along the line of sight, the
associated current helicity,

$H_{j}\ =\
\vec{\mbox{\boldmath{$B$}}}\cdot\nabla\times\vec{\mbox{\boldmath{$B$}}}\ =\
-B_{\perp}^{2}\ \frac{\partial\psi_{B_{\perp}}}{\partial Z}\ ,$ (27)

is positive, independent of the sign of $B_{\parallel}$. Noting that
$\frac{\displaystyle\partial\psi_{B_{\perp}}}{\displaystyle\partial
Z}=-\frac{\displaystyle\partial\psi_{\rm src}}{\displaystyle\partial r}$, Eq.
(27) can be integrated to give

$\psi_{\rm src}(r)-\psi_{\rm src}(0)\ =\
\int_{0}^{r}\frac{H_{j}}{B_{\perp}^{2}}\ dr^{\prime}\ \cdot$ (28)

The similarity between Eqs. (28) and (20) reflects the similarity between the
two depolarization mechanisms mentioned above. On the one hand, the current
helicity, $H_{j}$, of a simple spiral-slide helical magnetic field causes a
line-of-sight variation of the intrinsic polarization angle, $\psi_{\rm
src}(r)$. On the other hand, the line-of-sight magnetic field,
$B_{\parallel}$, in a magnetized plasma causes a line-of-sight variation of
the Faraday depth, $\Phi(r)$, which indicates that Faraday rotation is
occurring.

Figure 6:  Cartoon illustrating how Faraday rotation can either reinforce or
counteract the depolarization effect of a helical magnetic field. The
perspective is the same as in Figure 4, with the observer on the near side.
However, here, the background source is replaced by extended synchrotron
emission throughout the Faraday-rotating plasma. The magnetic field,
$\vec{\mbox{\boldmath{$B$}}}$, is described by Eq. (26), with
$\frac{\displaystyle\partial\psi_{B_{\perp}}}{\displaystyle\partial Z}<0$, so
the current helicity, given by Eq. (27), is positive ($H_{j}>0$). The left
panels show the intrinsic polarization orientation of the synchrotron emission
produced at various distances $r$ from the observer (first term in the right-
hand side of Eq. 19), which is simply the orientation of the local
$\vec{\mbox{\boldmath{$B$}}}_{\perp}$ rotated by $90^{\circ}$ (see Eq. 18).
The middle panels, directly taken from the far-right column in Figure 4, show
how the polarization orientation is Faraday-rotated as the signal propagates
along the line of sight (second term in the right-hand side of Eq. 19), (a)
when $\vec{\mbox{\boldmath{$B$}}}_{\parallel}$ points toward the observer
($B_{\parallel}>0$), and (b) when $\vec{\mbox{\boldmath{$B$}}}_{\parallel}$
points away from the observer ($B_{\parallel}<0$). The right panels show the
polarization orientation of the synchrotron emission produced at various
distances $r$, after Faraday rotation, i.e., as it reaches the observer (sum
of both terms in the right-hand side of Eq. 19). In scenario (a), the twisted
polarization orientation from the helical field is further wound by the
Faraday rotation, with the greatest amount of rotation at the back of the
slab; this yields a less coherent polarization signal, equivalent to the
signal from a more strongly helical field. In scenario (b), the twisted
polarization orientation from the helical field is unwound by the Faraday
rotation, with again the greatest amount of rotation at the back of the slab;
this yields a strongly coherent polarization signal, equivalent to the signal
from a non-helical field.

The two depolarization mechanisms can either reinforce or counteract each
other, as illustrated in Figures 6a and 6b, respectively. If $H_{j}$ and
$B_{\parallel}$ have the same sign, Eqs. (28) and (20) together imply that
$\psi_{\rm src}(r)-\psi_{\rm src}(0)$ and $\Phi(r)$ also have the same sign;
in that case, Faraday rotation amplifies the line-of-sight variation of the
intrinsic polarization angle, thereby increasing depolarization. In contrast,
if $H_{j}$ and $B_{\parallel}$ have opposite signs, $\psi_{\rm
src}(r)-\psi_{\rm src}(0)$ and $\Phi(r)$ also have opposite signs, in which
case Faraday rotation reduces the line-of-sight variation of the intrinsic
polarization angle, thereby decreasing depolarization. In consequence, if the
ambient magnetic field has positive [negative] current helicity, synchrotron
emission from regions at positive [negative] Faraday depths is strongly
depolarized, while synchrotron emission from regions at negative [positive]
Faraday depths is more weakly depolarized.

Nevertheless, there is an important difference between the effect of the
current helicity, $H_{j}$, and the effect of the line-of-sight magnetic field,
$B_{\parallel}$: $H_{j}$ is intrinsic to the considered helical magnetic
field, and so is the handedness of the associated helix, whereas
$B_{\parallel}$ depends on the position of the observer, and so does the sense
– or the effective handedness – of the associated Faraday rotation.

To illustrate this difference, let us revisit, and hopefully correct, the
thought experiment described in the Appendix of Brandenburg & Stepanov (2014).
Consider two hypothetical observers O1 and O2 located on opposite sides of a
uniform plasma in which both synchrotron emission and Faraday rotation take
place. Assume that the plasma is embedded in a simple spiral-slide helical
magnetic field, with a line-of sight component pointing away from observer O2
and toward observer O1. Both observers see a helical magnetic field with the
same $H_{j}$, and hence the same handedness, but they see opposite
$B_{\parallel}$, and hence opposite Faraday rotation. For observer O1,
$B_{\parallel}>0$, Faraday rotation is counterclockwise (see right panel in
Figure 4a), the synchrotron-emitting region lies at positive Faraday depths
($\Phi>0$), and the measured synchrotron emission is strongly [weakly]
depolarized if $H_{j}>0$ [$H_{j}<0$]. For observer O2, $B_{\parallel}<0$,
Faraday rotation is clockwise (see right panel in Figure 4b), the synchrotron-
emitting region lies at negative Faraday depths ($\Phi<0$), and the measured
synchrotron emission is weakly [strongly] depolarized if $H_{j}>0$
[$H_{j}<0$].

The idea that a helical magnetic field can compensate the depolarization
effect of Faraday rotation was first proposed by Sokoloff et al. (1998) and
later studied through numerical simulations by Volegova & Stepanov (2010) and
Brandenburg & Stepanov (2014).888 Both Volegova & Stepanov (2010) and
Brandenburg & Stepanov (2014) consider magnetic helicity, as opposed to
current helicity, but we believe that the conclusions they obtain with
magnetic helicity would remain qualitatively the same with current helicity.
Furthermore, Volegova & Stepanov (2010) and (in some places) Brandenburg &
Stepanov (2014) use the expression ”rotation measure” to refer to what we call
”Faraday depth”.  Volegova & Stepanov (2010) find that positive [negative]
magnetic helicity leads to a positive [negative] correlation between rotation
measure and polarization degree. Similarly, Brandenburg & Stepanov (2014)
conclude that "positive [negative] magnetic helicity could be detected by
observing positive [negative] RM in highly polarized regions and negative
[positive] RM in weakly polarized regions. Both conclusions are opposite to
our own conclusion (West et al., 2020). The reason is that the authors
implicitly take Faraday rotation to be left-handed about the magnetic field.
Although not directly stated in their papers, this can be inferred from two
premises of their models: (1) The $Z$-axis points from the observer to the
source (opposite to the IAU convention; see Figure 1). Accordingly the
polarization angle, $\psi$, and the Faraday rotation angle,
$\Phi\,\lambda^{2}$, increase clockwise for the observer. In other words, the
Faraday depth, $\Phi$, is positive when Faraday rotation is clockwise for the
observer. (2) The Faraday depth is positive when the mean magnetic field
points toward the observer (standard convention). Together, these two premises
incorrectly imply that Faraday rotation is left-handed about the magnetic
field.999 Brandenburg & Stepanov (2014) take $B_{\parallel}=B_{Z}$, which
implies that $B_{\parallel}$ is positive when the magnetic field points away
from the observer (opposite to the standard convention). This is consistent
with their Eq. (3), which indicates that $\Phi$ has the opposite sign to
$B_{\parallel}$.

##  5 Conclusions

Our paper aims to present a clear and unambiguous description of the true,
physical, sense of Faraday rotation, which we found lacking in existing texts.
To that end, we rederived the equations describing the propagation of a
linearly polarized radio electromagnetic wave undergoing Faraday rotation in a
magnetized plasma with magnetic field $\vec{\mbox{\boldmath{$B$}}}$. By
decomposing the electric field vector of the linearly polarized wave,
$\vec{\mbox{\boldmath{$E$}}}_{\ell}$, into the contributions from a right
circularly polarized mode (whose electric field vector,
$\vec{\mbox{\boldmath{$E$}}}_{\rm R}$, rotates in a right-handed sense about
$\vec{\mbox{\boldmath{$B$}}}$) and a left circularly polarized mode (whose
electric field vector, $\vec{\mbox{\boldmath{$E$}}}_{\rm L}$, rotates in a
left-handed sense about $\vec{\mbox{\boldmath{$B$}}}$) and noting that the
right mode has a slightly larger phase velocity than the left mode, we found
that Faraday rotation is right-handed about $\vec{\mbox{\boldmath{$B$}}}$.

This sense of Faraday rotation is easily understood physically. In brief,
Faraday rotation results from the action of $\vec{\mbox{\boldmath{$B$}}}$ on
the motion of free electrons accelerated by
$\vec{\mbox{\boldmath{$E$}}}_{\ell}$. This action of
$\vec{\mbox{\boldmath{$B$}}}$, which occurs through the Lorentz force (second
term in the right-hand side of the momentum equation, Eq. (31)), is always in
the sense of a right-handed rotation about $\vec{\mbox{\boldmath{$B$}}}$ –
exactly as for the simple electron gyro-motion about magnetic field lines.
Since the motion of electrons produces an electric current, which in turn acts
back on the electric field (second term in the right-hand side of Maxwell-
Ampère’s equation, Eq. (29)), the right-handed rotation that
$\vec{\mbox{\boldmath{$B$}}}$ tends to impart to electrons is automatically
passed on to $\vec{\mbox{\boldmath{$E$}}}_{\ell}$.

Our derived sense of Faraday rotation is by no means a new result. It can be
found in several textbooks (e.g., Stone, 1963; Papas, 1965; Collett &
Schaefer, 2012). But the opposite result – that Faraday rotation is left-
handed about $\vec{\mbox{\boldmath{$B$}}}$ – is also found in the literature
(e.g., Rybicki & Lightman, 1979; Chen, 2016). This has been the source of much
confusion and uncertainty in the astrophysics community. Our recent work on
the link between Faraday rotation and magnetic helicity (West et al., 2020)
prompted us to produce a clear, systematic, and complete derivation, which is
both rigorous from the point of view of plasma physicists and physically
transparent for radio astronomers. In that sense, our paper will help bridge
any convention-related gap between plasma physics and radio astronomy.

We also discussed the possible pitfalls that came to light in the course of
our investigation. We often realized that an apparently correct and convincing
reasoning could lead to the wrong conclusion for a very subtle reason. This is
why we took great pains to explain all the steps of our derivation in detail
and we carefully examined alternative approaches.

The most frequent errors and misconceptions that we encountered arise from a
confusion between the sense that the electric field vector of a mode rotates
with increasing time at a given distance from the source and the sense that
the electric field vector of this mode turns with increasing distance at a
given time. As a reminder (see Table 2), at a given distance,
$\vec{\mbox{\boldmath{$E$}}}_{\rm R}$ [$\vec{\mbox{\boldmath{$E$}}}_{\rm L}$]
rotates with increasing time in a right-handed [left-handed] sense about
$\vec{\mbox{\boldmath{$B$}}}$, whereas at a given time,
$\vec{\mbox{\boldmath{$E$}}}_{\rm R}$ [$\vec{\mbox{\boldmath{$E$}}}_{\rm L}$]
turns with increasing distance, i.e., in the propagation direction, in a left-
handed [right-handed] sense about $\vec{\mbox{\boldmath{$B$}}}$. As explained
in Sect. 3.1, the reason why the two senses are opposite is because moving
away from the source at a given time is equivalent to going backward in time
at a given distance.

Similarly for radio astronomers, who define the right circularly polarized
(RCP) and left circularly polarized (LCP) modes with respect to the wave
vector, $\vec{\mbox{\boldmath{$k$}}}$: At a given position, the electric field
vector of the RCP [LCP] mode,
$\vec{\mbox{\boldmath{$E$}}}_{\rm\scriptscriptstyle RCP}$
[$\vec{\mbox{\boldmath{$E$}}}_{\rm\scriptscriptstyle LCP}$], rotates in a
right-handed [left-handed] sense about $\vec{\mbox{\boldmath{$k$}}}$, whereas
at a given time, the propagation helix of the RCP [LCP] mode (plotted in the
middle panel of Figure 4a and left panel of Figure 4b [left panel of Figure 4a
and middle panel of Figure 4b]) is left-handed [right-handed].

To conclude our paper, we present in Figure 7 a summary plot which clearly
illustrates the physical process of Faraday rotation and unambiguously shows
the sense of rotation both with respect to the magnetic field as well as from
the perspective of the observer. Since radio astronomers commonly use the
Stokes parameters $Q$ and $U$ (defined through Eq. 23) to measure the sky, we
include an inset in the upper-left corner, which shows the IAU coordinate and
polarization conventions, including the $Q$ and $U$ lines. In this manner,
Figure 7 provides a complete "takeaway" figure, which connects our theoretical
discussion to observable quantities.

Figure 7:  Summary plot showing qualitatively how the electric field vector of
a linearly polarized radio wave oscillates along the propagation direction,
between the source (far side) and the observer (near side), and how the
polarization orientation (red double-headed arrow) undergoes Faraday rotation
as the wave passes through a magneto-ionized region (blue shaded region).a
Faraday rotation is always right-handed about the magnetic field,
$\vec{\mbox{\boldmath{$B$}}}$. When $\vec{\mbox{\boldmath{$B$}}}$ points
toward [away from] the observer, Faraday rotation is counterclockwise
[clockwise] in the plane of the sky; this corresponds to a positive [negative]
rotation angle, $\Delta\psi$ (Eq. 9), in the IAU definition of the
polarization angle (measured counterclockwise from North; see inset in the
upper-left corner). For completeness, the inset also shows the axes of the
Stokes parameters $Q$ and $U$ (defined through Eq. 23), i.e., the (faded
green) lines where $Q=\pm|P|$ (or $U=0$) and the (faded blue) lines where
$U=\pm|P|$ (or $Q=0$). Image credit: NASA Goddard/Theophilus Britt Griswold.

${}^{\rm a}\ $ Quantitatively, this plot is not realistic at all for the ISM,
where the Faraday rotation rate is tremendously smaller than the oscillation
rate of the electric field vector. Typically, Faraday rotation occurs over
parsec-scale distances, while the wavelength of a radio wave is
$\lambda=(30~{}{\rm cm})\ \Big{(}\\!\frac{\displaystyle\nu}{\displaystyle
1~{}{\rm GHz}}\\!\Big{)}^{-1}$.

## Acknowledgements

Our paper was triggered by discussions about helicity (in particular, with
Anvar Shukurov) at the IMAGINE workshop of April, 2019. We express our
gratitude to the colleagues we contacted in the course of our investigation
(Axel Brandenburg, Jo-Anne Brown, Gabriel Fruit, JinLin Han, George Heald,
Anna Ordog, Rodion Stepanov) for their replies to our initial query "What is
the correct sense of Faraday rotation?", for their willingness to engage in a
longer discussion with us, and for our constructive exchanges. We also thank
Theophilus Britt Griswold for his great help in producing Figures 2 and 7 and
Bryan Gaensler for his contributions to improving Figure 7. J.L.W.
acknowledges support of the Dunlap Institute, which is funded through an
endowment established by the David Dunlap family and the University of
Toronto. This research has made use of the NASA Astrophysics Data System
(ADS). Finally, we extend our sincere thanks to the referee for their
extremely constructive and detailed report, which helped us considerably
improve the clarity and readability of our paper.

## Data Availability

No new data were generated or analysed in support of this research.

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##  Appendix A Dispersion relation of parallel electromagnetic waves

The equations governing the propagation of an electromagnetic wave in a cold,
magnetized plasma with electron density $n_{\rm e}$ and magnetic field
$\vec{\mbox{\boldmath{$B$}}}_{0}=B_{0}\,\hat{\mbox{\boldmath{$e$}}}_{z}$101010
In the appendices, the ambient magnetic field is denoted by
$\vec{\mbox{\boldmath{$B$}}}_{0}$, to distinguish it from the wave magnetic
field vector, which is called $\vec{\mbox{\boldmath{$B$}}}$ by analogy with
the wave electric field vector, $\vec{\mbox{\boldmath{$E$}}}$. In the rest of
the paper, where the wave magnetic field vector is not discussed, the ambient
magnetic field is denoted by $\vec{\mbox{\boldmath{$B$}}}$. There is also a
distinction between the unit vector $\hat{\mbox{\boldmath{$e$}}}_{z}$ in the
direction of $\vec{\mbox{\boldmath{$B$}}}_{0}$ (used in the appendices) and
the unit vector $\hat{\mbox{\boldmath{$e$}}}_{Z}$ in the direction of the wave
vector, $\vec{\mbox{\boldmath{$k$}}}$ (shown in Figure 1 and used in the rest
of the paper).  are the linearized evolution equations for the wave electric
field vector, $\vec{\mbox{\boldmath{$E$}}}$, the wave magnetic field vector,
$\vec{\mbox{\boldmath{$B$}}}$, and the electron velocity,
$\vec{\mbox{\boldmath{$V$}}}_{\rm e}$, i.e., Maxwell-Ampère’s equation,

$\frac{\partial\vec{\mbox{\boldmath{$E$}}}}{\partial t}\ =\ c\
\nabla\times\vec{\mbox{\boldmath{$B$}}}-4\pi\ \vec{\mbox{\boldmath{$j$}}}\ ,$
(29)

with the electric current density $\vec{\mbox{\boldmath{$j$}}}=q_{\rm
e}\,n_{\rm e}\,\vec{\mbox{\boldmath{$V$}}}_{\rm e}$, Maxwell-Faraday’s
equation,

$\frac{\partial\vec{\mbox{\boldmath{$B$}}}}{\partial t}\ =\ -c\
\nabla\times\vec{\mbox{\boldmath{$E$}}}\ ,$ (30)

and the electron momentum equation,

$m_{\rm e}\ \frac{\partial\vec{\mbox{\boldmath{$V$}}}_{\rm e}}{\partial t}\ =\
q_{\rm e}\ \left(\vec{\mbox{\boldmath{$E$}}}+\frac{1}{c}\
\vec{\mbox{\boldmath{$V$}}}_{\rm
e}\times\vec{\mbox{\boldmath{$B$}}}_{0}\right)\ \cdot$ (31)

In the above equations, $c$ is the speed of light, $q_{\rm e}=-e$ the charge
of the electron, and $m_{\rm e}$ the mass of the electron. In principle, the
electric current should include contributions from both the electrons and the
ions. However, at the high frequencies considered here ($\omega\ggg\omega_{\rm
e},\,|\Omega_{\rm e}|$; see Sect. 2.1), only the electrons are sufficiently
mobile to respond to the wave and contribute to the current – the much more
massive ions remain virtually motionless.

Following the standard procedure, we look for complex solutions,
$\vec{\mbox{\boldmath{$\sf E$}}}$, $\vec{\mbox{\boldmath{$\sf B$}}}$, and
$\vec{\mbox{\boldmath{$\sf V$}}}_{\rm e}$, that vary as $\displaystyle
e^{i(\omega\,t-\vec{\mbox{\boldmath{$k$}}}\cdot\vec{\mbox{\boldmath{$r$}}})}$,
where $\omega$ is the angular frequency ($\omega>0$),
$\vec{\mbox{\boldmath{$k$}}}$ the wave vector, and
$\vec{\mbox{\boldmath{$r$}}}$ the vector position of the considered point.We
may then replace $\frac{\displaystyle\partial}{\partial t}$ by $i\omega$ and
$\nabla$ by $-i\vec{\mbox{\boldmath{$k$}}}$ in Eqs. (29)–(31). In the case of
parallel propagation,
$\vec{\mbox{\boldmath{$k$}}}\parallel\vec{\mbox{\boldmath{$B$}}}_{0}\parallel\hat{\mbox{\boldmath{$e$}}}_{z}$,
and these equations become

$\omega\ \vec{\mbox{\boldmath{$\sf E$}}}\ =\ -c\,k_{z}\
\hat{\mbox{\boldmath{$e$}}}_{z}\times\vec{\mbox{\boldmath{$\sf B$}}}\ +\
4\pi\,i\,q_{\rm e}\,n_{\rm e}\,\vec{\mbox{\boldmath{$\sf V$}}}_{\rm e}\ ,$
(32) $\omega\ \vec{\mbox{\boldmath{$\sf B$}}}\ =\ c\,k_{z}\
\hat{\mbox{\boldmath{$e$}}}_{z}\times\vec{\mbox{\boldmath{$\sf E$}}}\ ,$ (33)
$\omega\ \vec{\mbox{\boldmath{$\sf V$}}}_{\rm e}\ =\ -i\,\frac{q_{\rm
e}}{m_{\rm e}}\ \vec{\mbox{\boldmath{$\sf E$}}}\ -\ i\,\Omega_{\rm e}\
\vec{\mbox{\boldmath{$\sf V$}}}_{\rm e}\times\hat{\mbox{\boldmath{$e$}}}_{z}\
,$ (34)

where $\Omega_{\rm e}=\frac{\displaystyle q_{\rm e}B_{0}}{\displaystyle m_{\rm
e}c}$ is the electron gyro-frequency ($\Omega_{\rm e}<0$).

The set of equations (32)–(34) admits two types of solutions: electrostatic
solutions, for which $\vec{\mbox{\boldmath{$\sf
E$}}},\,\vec{\mbox{\boldmath{$\sf V$}}}_{\rm
e}\parallel\hat{\mbox{\boldmath{$e$}}}_{z}$ and $\vec{\mbox{\boldmath{$\sf
B$}}}=0$, and electromagnetic solutions, for which $\vec{\mbox{\boldmath{$\sf
E$}}},\,\vec{\mbox{\boldmath{$\sf B$}}},\,\vec{\mbox{\boldmath{$\sf V$}}}_{\rm
e}\perp\hat{\mbox{\boldmath{$e$}}}_{z}$. Here, we only consider the latter.
Taking the cross-product of Eq. (34) with $\hat{\mbox{\boldmath{$e$}}}_{z}$,
we then obtain an equation for $\vec{\mbox{\boldmath{$\sf V$}}}_{\rm
e}\times\hat{\mbox{\boldmath{$e$}}}_{z}$, which can be re-injected into Eq.
(34) to give

$\left(\omega^{2}-\Omega_{\rm e}^{2}\right)\,\vec{\mbox{\boldmath{$\sf
V$}}}_{\rm e}\ =\ -i\,\omega\ \frac{q_{\rm e}}{m_{\rm e}}\
\vec{\mbox{\boldmath{$\sf E$}}}\ -\ \Omega_{\rm e}\ \frac{q_{\rm e}}{m_{\rm
e}}\ \vec{\mbox{\boldmath{$\sf E$}}}\times\hat{\mbox{\boldmath{$e$}}}_{z}\
\cdot$ (35)

We now use Eqs. (33) and (35) to eliminate $\vec{\mbox{\boldmath{$\sf B$}}}$
and $\vec{\mbox{\boldmath{$\sf V$}}}_{\rm e}$ from Eq. (32), whereupon we
obtain an equation for $\vec{\mbox{\boldmath{$\sf E$}}}$ alone:

$\omega^{2}\ \vec{\mbox{\boldmath{$\sf E$}}}\ =\ c^{2}\,k^{2}\
\vec{\mbox{\boldmath{$\sf E$}}}\ +\ \frac{\omega_{\rm e}^{2}\
\omega^{2}}{\omega^{2}-\Omega_{\rm e}^{2}}\ \left(\vec{\mbox{\boldmath{$\sf
E$}}}-i\,\frac{\Omega_{\rm e}}{\omega}\ \vec{\mbox{\boldmath{$\sf
E$}}}\times\hat{\mbox{\boldmath{$e$}}}_{z}\right)\ ,$ (36)

where $k=|k_{z}|$ is the wavenumber and $\omega_{\rm
e}^{2}=\frac{\displaystyle 4\pi n_{\rm e}q_{\rm e}^{2}}{\displaystyle m_{\rm
e}}$ the plasma frequency squared. Finally, in the same way as we proceeded
with Eq. (34), we take the cross-product of Eq. (36) with
$\hat{\mbox{\boldmath{$e$}}}_{z}$ and re-inject the resulting equation for
$\vec{\mbox{\boldmath{$\sf E$}}}\times\hat{\mbox{\boldmath{$e$}}}_{z}$ into
Eq. (36):

$\left(\omega^{2}-c^{2}\,k^{2}-\frac{\omega_{\rm e}^{2}\
\omega^{2}}{\omega^{2}-\Omega_{\rm
e}^{2}}\right)^{2}\,\vec{\mbox{\boldmath{$\sf E$}}}\ =\
\left(\frac{\omega_{\rm e}^{2}\ \Omega_{\rm e}\ \omega}{\omega^{2}-\Omega_{\rm
e}^{2}}\right)^{2}\,\vec{\mbox{\boldmath{$\sf E$}}}\ \cdot$ (37)

The dispersion relation directly follows from Eq. (37):

$\omega^{2}-c^{2}\,k^{2}-\frac{\omega_{\rm e}^{2}\
\omega^{2}}{\omega^{2}-\Omega_{\rm e}^{2}}\ =\ \pm\ \frac{\omega_{\rm e}^{2}\
|\Omega_{\rm e}|\ \omega}{\omega^{2}-\Omega_{\rm e}^{2}}$ (38)

or, equivalently,

$\omega^{2}\ =\ c^{2}\,k^{2}\ +\ \frac{\omega_{\rm e}^{2}\
\omega}{\omega\mp|\Omega_{\rm e}|}\ ,$ (39)

which is identical to Eq. (1). The $\mp$ sign in Eq. (39) indicates the
existence of two distinct modes, the nature of which can be better understood
by inserting Eq. (38) back into Eq. (36). The result is simply

$\vec{\mbox{\boldmath{$\sf E$}}}=\pm\ i\,\vec{\mbox{\boldmath{$\sf
E$}}}\times\hat{\mbox{\boldmath{$e$}}}_{z}\ ,$ (40)

which can be successively rewritten as

$\displaystyle\vec{\mbox{\boldmath{$\sf E$}}}$ $\displaystyle=$
$\displaystyle{\sf E}_{x}\,\left(\hat{\mbox{\boldmath{$e$}}}_{x}\mp
i\,\hat{\mbox{\boldmath{$e$}}}_{y}\right)$ (41) $\displaystyle=$
$\displaystyle{\sf E}_{x0}\
e^{i(\omega\,t-k_{z}\,z)}\,\left(\hat{\mbox{\boldmath{$e$}}}_{x}\mp
i\,\hat{\mbox{\boldmath{$e$}}}_{y}\right)$ $\displaystyle=$ $\displaystyle{\sf
E}_{x0}\ \left[e^{i\omega\,t}\
\hat{\mbox{\boldmath{$e$}}}_{x}+e^{i(\omega\,t\mp\frac{\pi}{2})}\
\hat{\mbox{\boldmath{$e$}}}_{y}\right]\ e^{-ik_{z}\,z}\ ,$

with ${\sf E}_{x0}$ generally complex. Eq. (41) clearly shows that $|{\sf
E}_{x}|=|{\sf E}_{y}|$, while ${\sf E}_{y}$ is behind [ahead of] ${\sf E}_{x}$
by a quarter period for the mode corresponding to the upper [lower] sign.

Going back to physical space, the electric field vector,
$\vec{\mbox{\boldmath{$E$}}}$, is simply the real part of the complex electric
field vector, $\vec{\mbox{\boldmath{$\sf E$}}}$. It then follows that $E_{x}$
and $E_{y}$ have the same amplitude, while $E_{y}$ is behind [ahead of]
$E_{x}$ by a quarter period. This means that $\vec{\mbox{\boldmath{$E$}}}$
rotates circularly in a right-handed [left-handed] sense about
$\hat{\mbox{\boldmath{$e$}}}_{z}$ – which, we recall, was taken to lie in the
$+\vec{\mbox{\boldmath{$B$}}}_{0}$ direction. In other words, the wave is
circularly polarized in a right-handed [left-handed] sense about
$\vec{\mbox{\boldmath{$B$}}}_{0}$. Therefore, the mode corresponding to the
upper [lower] sign is called the right [left] circularly polarized mode.

The reason for the existence of two circularly polarized modes can be found in
the electron momentum equation, Eq. (35), where the two terms in the right-
hand side represent the electric force and the magnetic force, respectively.
Using Eq. (40), one can easily show that the latter is equal to
$\pm\frac{\displaystyle|\Omega_{\rm e}|}{\displaystyle\omega}$ times the
former. Hence, the magnetic force is much smaller than the electric force, and
it is directed in the same sense as [the opposite sense to] the electric force
for the right [left] circularly polarized mode. Physically, this is because
the (dominant) electric force accelerates the electrons in a right-handed
[left-handed] sense about $\vec{\mbox{\boldmath{$B$}}}_{0}$ for the right
[left] mode, whereas the magnetic force always acts on the electrons in a
right-handed sense about $\vec{\mbox{\boldmath{$B$}}}_{0}$.

Had we taken the complex fields, $\vec{\mbox{\boldmath{$\sf E$}}}$,
$\vec{\mbox{\boldmath{$\sf B$}}}$, and $\vec{\mbox{\boldmath{$\sf V$}}}_{\rm
e}$, to vary as $\displaystyle
e^{i(\vec{\mbox{\boldmath{$k$}}}\cdot\vec{\mbox{\boldmath{$r$}}}-\omega\,t)}$,
instead of $\displaystyle
e^{i(\omega\,t-\vec{\mbox{\boldmath{$k$}}}\cdot\vec{\mbox{\boldmath{$r$}}})}$,
we would have obtained the exact same dispersion relation, Eq. (39), but we
would have found the opposite sign in the right-hand side of Eq. (40), so that
the last equality in Eq. (41) would have been

$\vec{\mbox{\boldmath{$\sf E$}}}\ =\ {\sf E}_{x0}\ \left[e^{-i\omega\,t}\
\hat{\mbox{\boldmath{$e$}}}_{x}+e^{-i(\omega\,t\mp\frac{\pi}{2})}\
\hat{\mbox{\boldmath{$e$}}}_{y}\right]\ e^{ik_{z}\,z}\ \cdot$

In that case, it would have remained true that $E_{y}$ is behind [ahead of]
$E_{x}$ by a quarter period, and therefore that $\vec{\mbox{\boldmath{$E$}}}$
rotates circularly in a right-handed [left-handed] sense about
$\vec{\mbox{\boldmath{$B$}}}_{0}$, for the mode corresponding to the upper
[lower] sign.

##  Appendix B Electric field vector of a linearly polarized electromagnetic
wave

Consider again a cold, magnetized plasma with magnetic field
$\vec{\mbox{\boldmath{$B$}}}_{0}=B_{0}\,\hat{\mbox{\boldmath{$e$}}}_{z}$, and
now examine the parallel propagation
($\vec{\mbox{\boldmath{$k$}}}\parallel\vec{\mbox{\boldmath{$B$}}}_{0}$) of a
linearly polarized electromagnetic wave with angular frequency $\omega$.
$\hat{\mbox{\boldmath{$e$}}}_{\ell}\perp\hat{\mbox{\boldmath{$e$}}}_{k}$. To
fix ideas, assume that the electric field vector,
$\vec{\mbox{\boldmath{$E$}}}_{\ell}$, at the source (subscript $\star$)
oscillates along the $x$-axis with an amplitude $E_{0}$ and reaches a crest at
the initial time, $t=0$:

$\vec{\mbox{\boldmath{$E$}}}_{\ell\star}=E_{0}\ \cos(\omega\,t)\
\hat{\mbox{\boldmath{$e$}}}_{x}\ \cdot$ (42)

Let us now decompose the linearly polarized wave into a right (R) and a left
(L) circularly polarized mode:

$\vec{\mbox{\boldmath{$E$}}}_{\ell\star}=\vec{\mbox{\boldmath{$E$}}}_{\rm
R\star}+\vec{\mbox{\boldmath{$E$}}}_{\rm L\star}\ \cdot$ (43)

By definition, $\vec{\mbox{\boldmath{$E$}}}_{\rm R\star}$
[$\vec{\mbox{\boldmath{$E$}}}_{\rm L\star}$] rotates in a right-handed [left-
handed] sense about $\vec{\mbox{\boldmath{$B$}}}_{0}$. Since
$\vec{\mbox{\boldmath{$B$}}}_{0}$ is in the positive
$\hat{\mbox{\boldmath{$e$}}}_{z}$ direction, we can write

$\vec{\mbox{\boldmath{$E$}}}_{\rm R\star}\ =\ {\textstyle\frac{1}{2}}\,E_{0}\
\left[\cos(\omega\,t)\ \hat{\mbox{\boldmath{$e$}}}_{x}+\sin(\omega\,t)\
\hat{\mbox{\boldmath{$e$}}}_{y}\right]$ (44)

and

$\vec{\mbox{\boldmath{$E$}}}_{\rm L\star}\ =\ {\textstyle\frac{1}{2}}\,E_{0}\
\left[\cos(\omega\,t)\ \hat{\mbox{\boldmath{$e$}}}_{x}-\sin(\omega\,t)\
\hat{\mbox{\boldmath{$e$}}}_{y}\right]\ \cdot$ (45)

At a distance $s$ from the source, the electric field vector of the right
[left] mode, $\vec{\mbox{\boldmath{$E$}}}_{\rm R}$
[$\vec{\mbox{\boldmath{$E$}}}_{\rm L}$], is given by Eq. (44) [Eq. (45)], with
time, $t$, replaced by the retarded time,
$\Big{(}t-{\displaystyle\int_{0}^{s}}\frac{\displaystyle
ds^{\prime}}{\displaystyle V_{\phi,{\rm R}}}\Big{)}$
[$\Big{(}t-{\displaystyle\int_{0}^{s}}\frac{\displaystyle
ds^{\prime}}{\displaystyle V_{\phi,{\rm L}}}\Big{)}$], where
$V_{\phi}=\frac{\displaystyle\omega}{\displaystyle k}$ is the phase velocity.
This substitution is equivalent to replacing $(\omega\,t)$ by the phase,
$\phi_{\rm R}$ [$\phi_{\rm L}$], defined through Eq. (4), so that

$\vec{\mbox{\boldmath{$E$}}}_{\rm R}\ =\ {\textstyle\frac{1}{2}}\,E_{0}\
\left[\cos\phi_{\rm R}\ \hat{\mbox{\boldmath{$e$}}}_{x}+\sin\phi_{\rm R}\
\hat{\mbox{\boldmath{$e$}}}_{y}\right]$ (46)

and

$\vec{\mbox{\boldmath{$E$}}}_{\rm L}\ =\ {\textstyle\frac{1}{2}}\,E_{0}\
\left[\cos\phi_{\rm L}\ \hat{\mbox{\boldmath{$e$}}}_{x}-\sin\phi_{\rm L}\
\hat{\mbox{\boldmath{$e$}}}_{y}\right]\ \cdot$ (47)

The vector sum of Eqs. (46) and (47) yields

$\displaystyle\vec{\mbox{\boldmath{$E$}}}_{\ell}$ $\displaystyle=$
$\displaystyle\vec{\mbox{\boldmath{$E$}}}_{\rm
R}+\vec{\mbox{\boldmath{$E$}}}_{\rm L}$ (48) $\displaystyle=$
$\displaystyle{\textstyle\frac{1}{2}}\,E_{0}\ \left[\left(\cos\phi_{\rm
R}+\cos\phi_{\rm
L}\right)\,\hat{\mbox{\boldmath{$e$}}}_{x}+\left(\sin\phi_{\rm
R}-\sin\phi_{\rm L}\right)\,\hat{\mbox{\boldmath{$e$}}}_{y}\right]$
$\displaystyle=$ $\displaystyle E_{0}\ \cos\bar{\phi}\
\left(\cos{\textstyle\frac{\Delta\phi}{2}}\
\hat{\mbox{\boldmath{$e$}}}_{x}+\sin{\textstyle\frac{\Delta\phi}{2}}\
\hat{\mbox{\boldmath{$e$}}}_{y}\right)\ ,$

where $\bar{\phi}=\frac{\displaystyle\phi_{\rm R}+\phi_{\rm L}}{\displaystyle
2}$ is the average phase between the right and left modes and
$\Delta\phi=\phi_{\rm R}-\phi_{\rm L}$ is the phase difference between them.
Hence, the right and left modes still superpose to create a linearly polarized
wave, but the polarization orientation has rotated by an angle
$\frac{\Delta\phi}{2}$ from the polarization orientation at the source (taken
here to be along $\hat{\mbox{\boldmath{$e$}}}_{x}$; see Eq. 42). This is
exactly what we found in Sect. 2.2, where we also derived the expression of
the Faraday rotation angle, $\frac{\Delta\phi}{2}$ (Eq. 8).

For completeness, we now derive the expression of the phase $\bar{\phi}$ of
the linearly polarized wave. From Eq. (4), it follows that

$\bar{\phi}\ =\ \omega\,t-\int_{0}^{s}\bar{k}\,ds^{\prime}\ ,$ (49)

where $\bar{k}=\frac{\displaystyle k_{\rm R}+k_{\rm L}}{\displaystyle 2}$ is
the average wavenumber between the right and left modes. The latter can be
inferred from Eq. (3) rewritten as an equation for the wavenumber:

$k\ =\ \frac{\omega}{c}\ \left(1-\frac{\omega_{\rm
e}^{2}}{2\,\omega^{2}}\mp\frac{\omega_{\rm e}^{2}\,|\Omega_{\rm
e}|}{2\,\omega^{3}}\right)\ ,$ (50)

where the upper [lower] sign in the right-hand side pertains to the right
[left] mode:

$\bar{k}\ =\ \frac{\omega}{c}\ \left(1-\frac{\omega_{\rm
e}^{2}}{2\,\omega^{2}}\right)\ \cdot$ (51)

Thus, the wavenumber $\bar{k}$ and the phase $\bar{\phi}$ of the linearly
polarized wave are just the wavenumber and the phase of an electromagnetic
wave with the same $\omega$ in an unmagnetized plasma ($\Omega_{\rm e}=0$).