The correct sense of Faraday rotation

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)

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

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):

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 ($t=0$).

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).

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.

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

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$}}}_{\ell}$:

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

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.

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):

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

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

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

where

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

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).

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.

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.

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

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

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.

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.

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

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.

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 ($s=d$):

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:

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

which is equivalent to

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

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

(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

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.