# Towards solving the origin of circular polarisation in FRB 20180301A

Pavan Uttarkar,1 Ryan M. Shannon1, Marcus E. Lower2, Pravir Kumar3, Danny C.
Price4,5, A. T. Deller1, K. Gourdji1

1Centre for Astrophysics and Supercomputing, Swinburne University of
Technology, Hawthorn, VIC
2Australia Telescope National Facility, CSIRO, Space and Astronomy, PO Box 76,
Epping, NSW 1710, Australia
3Department of Particle Physics and Astrophysics, Weizmann Institute of
Science, 76100 Rehovot, Israel
4Square Kilometre Array Observatory (SKAO)
5International Centre for Radio Astronomy Research (ICRAR)
E-mail<EMAIL_ADDRESS>0000-0002-2346-6853 0000-0002-7285-6348
0000-0001-9208-0009 0000-0003-1913-3092 0000-0003-2783-1608
0000-0001-9434-3837 0000-0002-0152-1129

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Fast Radio Bursts (FRBs) are short-timescale transients of extragalactic
origin. The number of detected FRBs has grown dramatically since their
serendipitous discovery from archival data. Some FRBs have also been seen to
repeat. The polarimetric properties of repeating FRBs show diverse behaviour
and, at times, extreme polarimetric morphology, suggesting a complex magneto-
ionic circumburst environment for this class of FRB. The polarimetric
properties such as circular polarisation behaviour of FRBs are crucial for
understanding their surrounding magnetic-ionic environment. The circular
polarisation previously observed in some of the repeating FRB sources has been
attributed to propagation effects such as generalised Faraday rotation (GFR),
where conversion from linear to circular polarisation occurs due to the non-
circular modes of transmission in relativistic plasma. The discovery burst
from the repeating FRB 20180301A showed significant frequency-dependent
circular polarisation behaviour, which was initially speculated to be
instrumental due to a sidelobe detection. Here we revisit the properties given
the subsequent interferometric localisation of the burst, which indicates that
the burst was detected in the primary beam of the Parkes/Murriyang 20-cm
multibeam receiver. We develop a Bayesian Stokes-Q, U, and V fit method to
model the GFR effect, which is independent of the total polarised flux
parameter. Using the GFR model we show that the rotation measure (RM)
estimated is two orders of magnitude smaller and opposite sign ($\sim$28 rad
m-2) than the previously reported value. We interpret the implication of the
circular polarisation on its local magnetic environment and reinterpret its
long-term temporal evolution in RM.

###### keywords:

polarisation – fast radio bursts – statistics – methods:signal processing

††pubyear: 2015††pagerange: Towards solving the origin of circular
polarisation in FRB 20180301A–A

##  1 Introduction

Fast radio bursts (FRBs) are microsecond to millisecond duration dispersed
bursts of radio emission of extragalactic origin. The first FRB was reported
by Lorimer et al. (2007) in a 20-cm multibeam receiver archival dataset from
the Parkes/Murriyang radio telescope. Since the first reported burst, the
number of FRBs discovered has grown to $>$2000 FRBs to date (Xu et al., 2023).
A large fraction of the new bursts detected have been reported by instruments
commissioned during the last decade, such as the Australian Square Kilometer
Array Pathfinder (ASKAP; Bannister et al., 2017), Canadian Hydrogen Intensity
Mapping Experiment (CHIME; CHIME/FRB Collaboration et al., 2021), Deep
Synoptic Array (DSA; Sherman et al., 2024). Even with a rapid increase in the
number of detected FRBs, the progenitor of FRBs remains elusive, with much of
the physics (e.g., emission mechanism, energy distribution, surrounding
magnetic field) yet to be fully explained.

Among the sample set of FRBs discovered, $\sim$65 FRBs ($\sim$2% of the
population; CHIME/FRB Collaboration et al., 2021) have been observed to
repeat. Whether all FRBs repeat (eventually) is unclear. Soon after the
discovery of a population of repeating FRBs, it became evident that there were
differences in the spectro-temporal properties (e.g., pulse morphology,
spectral extent) of repeaters and (apparent) non-repeaters (CHIME/FRB
Collaboration et al., 2021), with the linear frequency drift, or "sad
trombone" being a distinct spectral behaviour seen in some of the repeating
FRB sources (e.g., Hessels et al., 2019; Luo et al., 2020; CHIME/FRB
Collaboration et al., 2021). Additionally, smaller spectral extents are
observed in repeating FRB sources compared to non-repeaters, with the spectral
and temporal widths in repeating FRB sources found to be a function of
frequency (Bethapudi et al., 2022). Further, some repeaters such as FRB
20180916B, have been shown to have burst activity windows that are frequency-
dependent. (e.g., Pleunis et al., 2021; Bethapudi et al., 2022).

In addition to the spectro-temporal characteristics, the polarisation
properties of FRBs provide crucial inferences on the FRB emission mechanism,
circumburst media, and potential progenitor. A linearly polarised radio wave
when travelling through a dispersive, cold magnetised plasma such as
interstellar medium (ISM), or intergalactic medium (IGM) can induce typical
nanosecond scale delay between the left and right handed circular polarisation
modes, resulting in Faraday rotation (FR). The effect of FR can be
quantitatively described using the Rotation Measure (RM), a measure of the
quadratic variation in linear polarisation position angle (PPA) with the
frequency. In general, FRBs have been observed to display a plethora of
additional polarisation properties, such as the PPA swing seen in FRB
20180301A (Luo et al., 2020) relative to a flat PPA seen in most other FRBs
(e.g., Bannister et al., 2019; Day et al., 2020; Kumar et al., 2021). Further,
a significantly large variation in linear polarisation fraction has been
observed between different one-off bursts, and also from an active repeating
source (e.g., Day et al., 2020; Kumar et al., 2022). In addition to the
conventional FR effect, which is caused by the circularly polarised natural
transmission modes, some repeaters such as FRB 20201124A have shown to have
significant frequency-dependent circular polarisation fraction (Kumar et al.,
2022). The frequency-dependent circular polarisation has been attributed to
Faraday conversion or generalised Faraday rotation (GFR) effect (Kumar et al.,
2023a). GFR can induce circular polarisation by the conversion of the linear
Stokes components to circular polarisation. The propagation of polarised
radiation through a medium having an elliptical or linear transmission mode
results in such conversion (Kennett & Melrose, 1998). Like conventional
Faraday rotation, GFR is a chromatic process. Hence, it will result in strong
frequency dependence for the circular polarisation fraction.

Previous investigations of pulsars (e.g., Melrose, 2003; Melrose & Luo, 2004)
point to three main causes of circular polarisation: (a) a process intrinsic
to the emission; (b) a favorable natural transmission mode; (c) propagation
effects (e.g., in relativistic plasma). Additionally, some studies such as
Gruzinov & Levin (2019) have also proposed the generation of circular
polarisation through conventional cold plasma - without relativistic electrons
- due to the magnetic field reversals along the line of sight.

There has also been a rapid increase in repeating FRBs showing extreme
polarisation behaviour (e.g., 20121102A, 20190520B, and 20201124A). Two FRBs,
20190520B and 20180301A have been observed to undergo a change in the sign of
the RM, suggesting a reversal in the parallel magnetic field component (Anna-
Thomas et al., 2023; Kumar et al., 2023b). Additionally, some repeaters have
been seen to have spectral depolarisation towards lower frequencies,
suggesting a multipath propagation effect (Feng et al., 2022b). In general,
the magnitude of circular polarisation observed in repeating FRBs is
relatively smaller than in non-repeating FRB sources. Some sub-components of
(apparent) non-repeaters, such as FRB 20190611B, have been seen to have a high
circular polarisation fraction ($\sim$57%; Day et al., 2020). The circular
polarisation in FRBs can be used as a unique probe to understand the
progenitor environment and its local complex magneto-ionic environment if the
Faraday conversion occurs in its circumburst medium.

The circular polarisation observed in several repeating FRB sources (e.g., FRB
20201124A, 20220808A) has been interpreted as being caused by GFR. FRB
20201124A was one of the first repeaters to have been observed to have
significant and notable frequency dependent circular polarisation ($\sim$57%;
Kumar et al., 2022). While FRB 20180301A was the first repeater to have been
seen to have a significant frequency-dependent circular polarisation; it was
initially speculated that the circular polarisation was induced due to
instrumental leakage due to a sidelobe detection (Price et al., 2019).
However, the milli-arcsecond localisation by Bhandari et al. (2022) has
provided precise localisation information which can now be leveraged to study
the Parkes/Murriyang discovery burst spectro-polarimetric characteristics.

In this paper, we revisit the spectro-polarimetric properties of FRB 20180301A
using GFR modelling. We summarise the data used in this analysis, and the GFR
modelling used to explain the circular polarisation behaviour in Section 2. We
discuss the results and their implications from the GFR modelling in Sections
3 and 4, respectively. We conclude our analysis in Section 5.

Figure 1: The dynamic Stokes spectra of FRB 20180301A. The dynamic Stokes I,
Q, U, and V spectra are shown in Panels I, Q, U, and V, respectively. The
lighter regions show higher flux relative to the darker regions in the
spectra. The dotted line shows the integrated region used to generate the
time-averaged spectra for GFR modelling. The frequency dependent behaviour of
Stokes-V can be seen in the banding across frequency in Panel V. The dynamic
spectra are smoothed with a Gaussian filter having a frequency and temporal
width of 0.11 MHz and 15.24$\upmu$s, respectively.

##  2 Methods

###  2.1 Data description and preparation

We use data presented in Price et al. (2019) for our analysis. Here we briefly
summarise the properties of the data. FRB 20180301A was discovered using the
21-cm multibeam receiver on the Parkes/Murriyang radio telescope (Staveley-
Smith et al., 1996), using the Berkeley–Parkes–Swinburne Recorder (BPSR)
system in a Breakthrough Listen (BL) observation (Price et al., 2019). Repeat
bursts from FRB 20180301A were reported using the Five Hundred Meter Aperture
Radio Telescope (FAST) by Luo et al. (2020), confirming the source to be a
repeating FRB. The initial burst was reported to have an RM of -3156 rad
m-2(Price et al., 2019) and a dispersion measure (DM) of 522 pc cm-3. The
dynamic Stokes spectrum from the multibeam receiver is shown in Figure 1. We
use methods described in Price et al. (2019) to flux calibrate the data on the
existing polarisation calibrated dataset between the frequencies 1359.8 MHz
and 1513.7 MHz, for the 154 MHz bandwidth over which the the FRB emission is
apparent. We do not include the lower 154 MHz in our analysis, where there is
no detectable burst emission. Following Price et al. (2019), we use a Gaussian
filter with a frequency width of 0.109 MHz and a time width of $\sim$60
$\upmu$sec to smooth the spectra.

The burst was discovered in beam $3$ of the multibeam receiver. The lack of
precise localisation after the initial discovery led to the speculation that
the burst was detected in a sidelobe (Price et al., 2019). Hence, to explain
frequency-dependent circular polarisation, polarisation leakage due to the
beam 3 sidelobe detection was considered as one of the likely scenarios.
However, the subsequent $\sim$milli-arcsec localisation by Bhandari et al.
(2022), and the pointing position of the Parkes/Murriyang observation confirm
a primary-lobe detection of the burst. Considering a full-width half-power
(FWHP) beamwidth of $\sim$14.5’ (Staveley-Smith et al., 1996) for beam 3, the
multibeam detection shows a radial offset of $\sim$7’. Figure 2 shows the FWHP
of beam 3 in the blue-shaded region and the full-width (FW) of beam 1 in the
grey-shaded region to show the general extent of the FW of individual beams.
Additionally, the cross-polarisation leakage analysis by Carozzi & Woan (2011)
shows that the leakage of linear polarisation into circular polarisation is
smaller than 15-20 dB (a factor of 30-100), indicating low polarisation
leakage in the multibeam receiver. Hence, the frequency-dependent circular
polarisation structure in the burst is likely intrinsic to the source and not
anthropogenic or an instrumental artifact, as speculated by Price et al.
(2019).

We model the spectral behaviour of the polarisation structure in the burst
using a phenomenological GFR model described in Lower (2021). We extend this
model to include the modelling of RM from the conventional FR effect. We use
this FR-GFR model to investigate the polarised behaviour of the burst. To
verify the model, we use data from the Parkes/Murriyang multibeam receiver
observations of known pulsars J1644$-$4559 and J0835$-$4510, which have
reported measurements of RM of
-626.9111https://www.atnf.csiro.au/research/pulsar/psrcat/ and 31.381,
respectively. The data for pulsars J0833$-$4559 and J0835$-$4510 were taken in
$2018$ and recorded using the Parkes Digital Filterbank Backend Mark-4. The
centre frequency for both observations is 1369 MHz, with a bandwidth of 256
MHz, with a frequency resolution of 125 KHz. We calibrate the pulsar data for
flux and polarisation using a standard Parkes flux calibrator and a noise
diode source for polarisation calibration using psrchive.

Figure 2: Localisation of FRB 20180301A. The VLA localisation of a repeat
burst reported by Bhandari et al. (2022) is shown by the blue cross. The FWHP
of beam 3 relative to the localisation is shown by the blue shaded region. The
grey shaded region shows the extent of a single beam. FRB 20180301A was
radially offset by $\sim$7’ from the beam center. The localisation information
of the burst and the Parkes beam pointing information suggest a beam 3 primary
lobe detection of the burst.

The data projected on the Poincare sphere can provide useful visualisation of
the polarimetry, in particular for GFR analysis. We show the projection of the
RM uncorrected Stokes components on the Poincare sphere across frequency, with
measurements having a fractional polarised intensity greater than 5% in Figure
3. The polarisation vector of a purely FR-dominated polarisation source does
not transverse different latitudes on the Poincare sphere. However, a
frequency-dependence of Stokes-V could result in the polarisation vector
moving to different latitudes. The projection of the polarisation vector on
the Poincare sphere for FRB 20180301A shows a clear frequency-dependent
structure across the frequency band, suggesting linear to circular
polarisation conversion across frequency.

Figure 3: Poincare sphere representation of the spectropolametric properties
of FRB 20180301A. The data are time averaged and smoothed using a Gaussian
filter with a frequency width of 0.109 MHz and a time constant of $\sim$60
$\upmu$sec. The colour map shows the time averaged measurements at different
frequencies. The polarisation information is shown for the data points having
an S/N greater than 4.5 in individual frequency channels of 0.109 MHz
bandwidth.

###  2.2 Faraday rotation - Generalised Faraday rotation (FR-GFR) model

####  2.2.1 Modelling Faraday rotation

We use the FR-GFR model to estimate both RM and GRM for the data. In the FR-
GFR model we calculate the PPA for FR and GFR cases simultaneously. The PPA
for the case of the conventional FR model (index $\alpha$=2) is given by

$\psi=\psi_{o}+\rm RM(\lambda^{2}-\lambda_{o}^{2}),$ (1)

where RM quantifies the magnitude of phase change induced by the intervening
cold magnetised plasma, $\lambda$ is the wavelength, $\lambda_{o}$ is the
wavelength associated with a reference frequency. The RM can be related to
physical quantatites using

${\rm RM}=\frac{e^{3}}{2\pi
m_{e}^{2}c^{4}}\int_{d}^{0}\frac{n_{e}B_{||}}{(1+z)^{2}}dl,$ (2)

where $e$ is the electron charge, $z$ is the redshift of the electrons,
$m_{e}$ is the mass of the electron, $c$ is the speed of light, $n_{e}$ is the
free electron density, and $B_{||}$ of the magnetic field component parallel
to the line of sight.

####  2.2.2 Modelling non-circular modes of transmission

In addition to the conventional FR, where the natural transmission modes are
circularly polarised, conversion from linear to circular polarisation can be
induced if the modes of transmission of the radio waves are either non-
circular (elliptical) or completely linear. The non-circular modes of
transmission can also lead to a non-$\lambda^{2}$ dependency of PPA, with the
chromaticity dependent on the properties of the plasma and underlying
theoretical assumptions. These non-circular modes can be modelled by using an
arbitrary spectral scaling index $\alpha$. Hence, Equation 1 can be
generalised for the case of GFR to be

$\Psi(\lambda)=\Psi_{o}+\rm GRM(\lambda^{\alpha}-\lambda_{o}^{\alpha}),$ (3)

where $\Psi_{o}$ is the intrinsic PPA, GRM is generalised rotation measure
(GRM) an analogue of RM for the non-circular modes of transmission.

The polarisation vector due to the effects of cold plasma and GFR it modelled
to be

$\bf{P}_{\rm GFR}=\bf{R}{\psi}\bf{R}_{\theta\phi}\bf{P}(\lambda),$ (4)

where Rψ is the FR-induced shift in the polarisation vector, with a fixed
latitude on the Poincare sphere, Rθϕ is the rotation matrix describing the
arbitrary rotation of the polarisation vector on Poincare sphere over
different latitudes (a characteristic of the GFR; see Lower (2021) for a
detailed discussion) and P($\lambda$) is the Stokes vector, which is a
function of PPA and the elipticity angle (EA).

The rotation matrices Rϕ, Rθϕ, and P($\lambda$), which are used to model the
angle of polarisation vector on a Poincare sphere can be described by

$\textbf{R}_{\rm\psi}=\begin{pmatrix}\cos 2\psi&-\sin 2\psi&0\\\ \sin
2\psi&\cos 2\psi&0\\\ 0&0&1\\\ \end{pmatrix},$ (5)
$\textbf{R}_{\rm\theta\phi}=\begin{pmatrix}\cos(\theta)\cos(\phi)&-\cos(\theta)\sin(\phi)&\sin(\theta)\\\
\sin(\phi)&\cos(\phi)&0\\\
-\sin(\theta)\cos(\phi)&\sin(\theta)\sin(\phi)&\cos(\theta)\\\ \end{pmatrix},$
(6)

and

$\bf{P}(\lambda)=P\begin{pmatrix}\cos(2\Psi)\cos(2\chi)\\\
\sin(2\Psi)\cos(2\chi)\\\ \sin(2\chi)\\\ \end{pmatrix},$ (7)

where $\theta$ describes the angle of the axis about which the polarisation
vector rotates, $\phi$ is the rotation about the Stokes-Q axis on the Poincare
sphere, $\Psi_{o}$ is the intrinsic polarisation angle, and $\chi$ is the EA
describing the angle made by the polarisation vector with respect to the
Stokes-V axis, given by

$\chi=\frac{1}{2}\arctan\left(\frac{V}{\sqrt{Q^{2}+U^{2}}}\right).$ (8)

####  2.2.3 Bayesian estimation of the GFR parameters

We perform a Bayesian parameter inference using bilby (Ashton et al., 2019)
with the DYNESTY nested sampler to infer the posterior probability
distribution for our models. We expand the method described in Bannister et
al. (2019) to infer Faraday Rotation to include GFR-induced circular
polarisation. Assuming the noise in the data can be described by a Gaussian
distribution (valid if the thermal noise dominates the data), we can infer the
posterior probability distribution for the GFR to be

$\textit{L}(P(\lambda)|P_{M}(\lambda),\sigma)=\prod_{i}^{N}\frac{1}{\sqrt{2\pi\sigma^{2}}}\,\,\exp\left(-\frac{(P_{i}(\lambda)-P_{M,i}(\lambda))^{2}}{2\sigma^{2}}\right),$
(9)

where $P(\lambda)$ is the polarisation vector of the data consisting of all
the Stokes components, $P_{M}(\lambda)$ is the modeled polarisation vector
with all the Stokes parameters, $\sigma$ is the standard deviation of the
Stokes spectrum, and $\lambda$ is the wavelength. The posterior in the basis
is

$\begin{split}&\textit{L}(P(\lambda)|P_{M}(\lambda),\sigma)=\prod_{i}^{N}\frac{1}{(2\pi)^{3/2}\sigma^{3}}\\\
&\int_{0}^{\infty}\exp\left[-\frac{(Q-Q_{m})^{2}}{2\sigma^{2}}\right]\exp\left[-\frac{(U-U_{m})^{2}}{2\sigma^{2}}\right]\exp\left[-\frac{(V-V_{m})^{2}}{2\sigma^{2}}\right]\,dP,\end{split}$
(10)

where Q, U, and V are the Stokes components from the data, and $Q_{m}$,
$U_{m}$, and $V_{m}$ the modelled Stokes components.

Following Bannister et al. (2019), we marginalise the above Equation over the
total polarisation fraction $P$ to make the estimation more robust to changes
in the polarised flux in any of the frequency channels. We modify the
integration intervals from 0 to $\infty$ from -$\infty$ to $\infty$ used in
Bannister et al. (2019). We modify the integration interval as the latter
leads to a bi-model distribution in the likelihood function for PPA and EA. We
consider two plausible scenarios: either the noise in all the Stokes
components is identical, or they are all different, which we term, cases (a)
and (b), respectively.

For the case (a), the marginal posterior distribution function is

$\begin{split}&\textit{L}(P(\lambda)|P_{M}(\lambda),\sigma)=\prod_{i}^{N}\frac{1}{2\pi\sigma^{2}}\\\
&\exp\left[{-\frac{1}{2\sigma^{2}}\left((Q^{2}+U^{2}+V^{2})-(QQ_{m}+UU_{m}+VV_{m})^{2}\right)}\right]\\\
&\rm\left[erf\left({\frac{\sigma}{\sqrt{2}}\left(QQ_{m}+UU_{m}+VV_{m}\right)}\right)+1\right],\end{split}$
(11)

where $Q_{m}$, $U_{m}$, $V_{m}$ are the modeled Stokes vectors from Equation
7.

For case (b), the marginal posterior distribution is

$\begin{split}&\textit{L}(P(\lambda)|P_{M}(\lambda),\sigma)=\frac{1}{4\pi\sqrt{C_{2}}}\exp{\Biggr{[}\frac{C_{1}}{2N_{T}}\Biggr{]}}\\\
&\exp{\Biggr{[}\frac{C_{3}^{2}}{2C_{2}N_{T}}\Biggr{]}}\rm\Biggr{[}erf\left(\frac{C_{3}}{\sqrt{2N_{T}C_{2}}})\right)+1\Biggr{]},\end{split}$
(12)

where the factors $N_{T}$, $N_{1}$, $N_{2}$, $N_{3}$, $C_{1}$, $C_{2}$ and
$C_{3}$ are

$\begin{split}&N_{T}=\sigma_{Q}^{2}\sigma_{U}^{2}\sigma_{V}^{2},\\\
&N_{1}=\sigma_{U}^{2}\sigma_{V}^{2},\\\
&N_{2}=\sigma_{Q}^{2}\sigma_{V}^{2},\\\
&N_{3}=\sigma_{Q}^{2}\sigma_{U}^{2},\\\
&C_{1}=Q^{2}N_{1}+U^{2}N_{2}+V^{2}N_{3},\\\
&C_{2}=Q_{m}^{2}N1+U_{m}^{2}N_{2}+V_{m}^{2}N_{3},\textnormal{and}\\\
&C_{3}=QQ_{m}N_{1}+UU_{m}N_{2}+VV_{m}N_{3}.\\\ \end{split}$ (13)

Figure 4: Maximum likelihood GFR model for FRB 20180301A. The QUV fit to the
FRB 20180301A from the FR-GFR model for Stokes Q, U, and V fit are shown in
Panels A, B, and C, respectively. The data from the burst are shown in green
coloured points, and the solid black line shows the model fit to the data. The
EA and PA angles derived from the data are shown as blue points in the Panels
D and E, respectively. The model fit for the EA and PA angles are shown in
black solid lines. The residuals after subtracting the data from the model are
shown in red points for EA and PA angles in Panels D and E.

We sample the uncertainties in the Stokes spectrum in the parameter estimation
and calculate the uncertainties in the PPA and the EA using

$\sigma_{\psi}=\sqrt{\frac{Q_{\rm de-rot}^{2}\sigma^{2}+U_{\rm de-
rot}^{2}\sigma^{2}}{4(Q_{\rm de-rot}^{2}+U_{\rm de-rot}^{2})}},$ (14)

and

$\sigma_{\chi}=\frac{\sqrt{(VQ_{\rm de-rot})^{2}\sigma^{2}+(VU_{\rm de-
rot})^{2}\sigma^{2}(Q_{\rm de-rot}^{2}+V^{2})^{2}\sigma^{2}}}{4(Q_{\rm de-
rot}^{2}+U_{\rm de-rot}^{2}+V^{2})\sqrt{Q_{\rm de-rot}^{2}+U_{\rm de-
rot}^{2}}}.$ (15)

Bayesian parameter estimation is undertaken on a complete dataset without
excising any frequency channels since the posterior probability distribution
function described by Equation 11 is agnostic to the total polarisation
fraction of the burst. Hence, it is immune to any spurious change in total
polarised amplitude in the channels, making it more robust relative to the GFR
likelihood function described in Lower (2021). We use uniform priors for all
the input parameters in our estimation.

##  3 Results

###  3.1 Model verification on known sources

We verify the FR-GFR model on known pulsars from archival
observations222https://data.csiro.au/ from the Parkes/Murriyang multibeam
receiver using the model described by Equation 4 and the likelihood function
described by Equation 12. The posterior probability distributions for
J1644-4559 and J0835-4510 are shown in Figure 6. We show the model fit for
Stokes Q, U, and V from the estimated parameters in Figure 8. The FR-GFR model
estimates the RM for J1644-4559 and J0835-4510 to be -626.31${}_{-1.2}^{+1.2}$
rad m-2and 48.36${}_{-0.9}^{+0.9}$ rad m-2, respectively. The estimated
parameters for the FR-GFR model, rmfit, and FR model are listed in Table 1.
The FR-GFR polarisation parameters were estimated by averaging 154 and 174
bins over the on-pulse region for J1644-4559 and J0835-4510, respectively. The
RM for the pulsars J1644-4559 and J0835-4510 were computed by psrchive$-$rmfit
using a resolution of 0.5 rad m-2over the range -1000 to 0 rad m-2 and 0 to 50
rad m-2 , respectively. The posterior probability distribution for parameters
GRM and $\alpha$ return one of the prior boundaries for the case of
J1644-4559, indicating a near constant Stokes-V across the observation
frequency band. Similarly, the index $\alpha$ for J0835-4510 also returns one
of the prior boundaries, indicating a constant Stokes-V across the frequency
band for J0835-4510, even though the GRM is inferred to be 71${}_{-15}^{+20}$.
We further verify the FR-GFR model on magnetar XTE J1810-197 from Lower et al.
(2024) to confirm the recovery of the expected GRM and $\alpha$ values by
comparing the results to Lower et al. (2024). We show the GRM and $\alpha$ and
values of our fits in Figure 9 in the appendix. The results are in broad
agreement with those derived in Lower et al. (2024). We attribute the
difference in the recovered GRM and $\alpha$ values to be due to the
simultaneous RM measurement in our approach.

Pulsar | method |  RM (rad m-2) |  GRM (rad m-α) |  $\psi$ (deg) |  $\chi$ (deg) |  $\phi$ (deg) |  $\alpha$ |  $\theta$ (deg)
---|---|---|---|---|---|---|---|---
J1644-4559 | FR-GFR | -626.3${}_{-1.2}^{+1.2}$ | < 433.49 | 54.6${}_{-0.3}^{+0.3}$ | 42.0${}_{-0.1}^{+0.1}$ | -62.2${}_{-3.4}^{+3.2}$ | < 1.98 | 73.6${}_{-0.2}^{+0.2}$
| rmfit | -621.0${}_{-2.3}^{+2.3}$ | - | - | - | - | - | -
| FR | -623.1${}_{-1.3}^{+1.3}$ | - | 54.0${}_{-0.3}^{+0.3}$ | 8.3${}_{-0.1}^{+0.1}$ | - | - | -
J0835-4510 | FR-GFR | 48.4${}_{-0.9}^{+0.9}$ | 71${}_{-15}^{+20}$ | 82.0${}_{-0.2}^{+0.2}$ | -42.4${}_{-0.1}^{+0.1}$ | -86.7${}_{-5.9}^{+5.9}$ | 1.2${}_{-0.7}^{+0.3}$ | 78.8${}_{-0.3}^{+0.3}$
| rmfit | 46.8${}_{-0.01}^{+0.01}$ | - | - | - | - | - | -
| FR | 47.0${}_{-0.6}^{+0.5}$ | - | -7.0${}_{-0.1}^{+0.1}$ | -5.7${}_{-0.1}^{+0.1}$ | - | - | -
Table 1: Comparison of Faraday conversion measurements for PSRs J0835-4510 and
J1644-4559. The table lists estimated polarisation parameters from FR-GFR,
rmfit, and FR methods.

###  3.2 Modelling GFR in FRB 20180301A

In contrast to the two test pulsars described in the previous section, FRB
20180301A returns constrained posterior parameters. The posterior probability
distribution of FRB 20180301A for the FR-GFR mode is shown in Figure 7. The RM
and GRM value estimated using the FR-GFR model is 27.7${}^{+5.22}_{-5.19}$ rad
m-2and 4351.7${}^{+340.40}_{-313.61}$ rad m-α, respectively. The RM derived
from the FR-GFR model is significantly smaller in magnitude than the previous
RM estimation of -3163 rad m-2(Price et al., 2019). The RM derived by Price et
al. (2019) relied on QU fit without considering the Stokes-V fit to the data.
Moreover, the parameter $\theta$, which described the modes of transmission
converges to $\sim$104∘, which indicates non-circular and nearly elliptical
mode of transmission. The converged model parameters RM, GRM, $\psi$, $\chi$,
$\phi$, $\alpha$ and $\theta$ for FRB 20180301A are listed in Table 2. The
model fit to the data is shown in Figure 4. The index $\alpha$ converges to a
$2.29\pm 0.07$. The EA and PA angles calculated from the polarisation data
across the frequency are shown in panels D and E in Figure 4. Additionally,
the FR-GFR model has a log10 Bayes evidence of 1389.6 relative to the FR-only
model, indicating a strong preference (Trotta, 2008) to FR-GFR model.

Source | FRB 20180301A
---|---
RM (rad m-2) | 27.7${}_{-5.1}^{+5.2}$
GRM (rad m-α) | 4351.7${}_{-313.6}^{+340.4}$
$\psi$ (deg.) | -87.3${}_{-0.7}^{+0.8}$
$\chi$ (deg.) | -0.1${}_{-0.4}^{+0.4}$
$\phi$ (deg.) | 76.3${}_{-1.0}^{+1.1}$
$\alpha$ | 2.3${}_{-0.07}^{+0.07}$
$\theta$ (deg.) | 104.2${}_{-0.9}^{+1.0}$
Table 2: The converged GFR parameters for FRB 20180301A. The listed parameters
are derived from the FR-GFR model using the likelihood Equation 12. The
analysis was performed on a data set having frequency resolution of 0.109 MHz.
The Stokes spectra were averaged using a simple boxcar.

##  4 Discussion

###  4.1 Rationale for the polarimetric reanalysis of FRB 20180301A

  1. 1.

The subsequent localisation from (Bhandari et al., 2022) and the beam pointing
information from BL detection of FRB 20180301A Price et al. (2019) shows that
the detection of the burst was in the FWHP of beam 3 of the multibeam
receiver. Hence, the circular polarisation behaviour in the burst is not
likely the result of instrumental leakage.

  2. 2.

Follow-up observation from the Parkes/Murriyang ultra wide band low (UWL) and
FAST (Luo et al., 2020) has shown that FRB 20180301A has complex polarisation
behaviour e.g., a reported switch in the sign of RM, tentative evidence of
anti-correlation in the variation of DM and RM (Kumar et al., 2023b).

  3. 3.

Since the first discovery of the burst, other repeaters have been shown to
have significant circular polarisation in some of their bursts (Feng et al.,
2022a, 2023). Hence, it is plausible that the circular polarisation is
astrophysical. Such circular polarisation behaviour in their repetitions can
be leveraged to study the surrounding media, and answer some of the
outstanding questions related to the class of FRBs.

###  4.2 Implications of the GFR modelling on FRB 20180301A

Bursts from FRB 20180301A have been observed to have complex spectral,
temporal, and polarimetric properties, such as the secular variations in the
RM in the previous Parkes and FAST observations (e.g., Luo et al., 2020; Kumar
et al., 2022). Among the known repeater sample, only $\sim$3 FRBs (FRBs
20190520B, 20201124A, and 20221912A) have been observed to have a relatively
high circular polarisation fraction (Kumar et al., 2022; Feng et al., 2022a).
In addition to a high circular polarisation fraction, FRB 20201124A was seen
to have significant frequency-dependent circular polarisation (Kumar et al.,
2022), suggesting linear to circular polarisation conversion due to
relativistic plasma (Kumar et al., 2023a). Similar frequency-dependent
circular polarisation behaviour was first observed in FRB 20180301A (Price et
al., 2019). However, because of an uncertain localisation of the FRB, it was
speculated to be due to the instrumental polarisation leakage, potentially
because of a sidelobe detection (Price et al., 2019). The available
localisation information points to the origin of the circular polarisation to
be non-instrumental. Hence, the polarisation modelled using the GFR explains
the origin of the Stokes-V emission in the source.

An extensive follow-up campaign from Parkes/Murriyang UWL and FAST has shown
that the FRB 20180301A has undergone significant RM evolution since its
discovery (Luo et al., 2020; Kumar et al., 2023b), including an apparent
switch in sign in RM (Kumar et al., 2023b). The FRB 20190520B was the first
repeater to have been seen to have such reversal in the sign of RM Anna-Thomas
et al. (2023), with a flip from an extreme RM of $\sim$-104 rad m-2to
$\sim$104 rad m-2. In general, such RM reversal due to the reorientation of
the magnetic field parallel component has been speculated to be due to various
progenitor models, such as a magentar-Be star binary system (Wang et al.,
2022) (such as the B1259-63 binary system Johnston et al. (1996)), or even due
to the FRB progenitor around an intermediate mass black-hole (Anna-Thomas et
al., 2023). In the case of a binary system, as suggested by Wang et al.
(2022), the RM evolution should be periodic, which is yet to be observed.
Further, complex polarisation behaviour has been observed from active
repeaters, hence the circular polarisation behaviour in this burst is not
surprising in itself but provides additional evidence that repeating FRB
progenitors likely reside in a more complex magneto-ionic environment.

Using the FR-GFR model, we find that the RM of the discovery burst is
27.7${}_{-5.1}^{+5.2}$ rad m-2, less extreme than the previously reported RM
of $-3163$ rad m-2 (Price et al., 2019). The sign of the RM is opposite,
suggesting there has only been one observed flip in the sign of the burst
between the discovery of the burst from the Parkes/Murriyang multibeam
receiver and additional follow-up from Parkes/Murriyang UWL; between May 2021
and June 2022. The QUV fit for the data shown in Figure 4 shows good agreement
with the Stokes U and V. However, we see a relatively larger residual with
Stokes Q. We attribute this to the differential gain in the two linearly
polarisation feeds from the multibeam receiver. The Stokes-Q parameter with
linear feeds is a differential parameter, compared to the Stokes U and V,
which are cross correlation parameters. Hence, Stokes-Q is less immune to
variations in differential gain.

Assuming the surrounding media to be dominated by relativistic gas plasma, we
would expect the spectral index $\alpha=3$. From Kennett & Melrose (1998), the
GRM or relativistic rotation measure (RRM) for this case can be defined by GRM
= $10^{9}$LB4 rad m-α, where B is the magnetic field in Gauss and L is the
distance scale in parsec. If we set $\alpha=3$, we estimate the GRM to be
5121.84${}^{+24.45}_{-25.59}$ rad m-3. Assuming a length scale of $\sim$1 pc
(the fiducial scale used in Kennett & Melrose (1998)), the magnetic field
strength for the media is $\sim$2.2 mG. This is comparable to measurements
reported by Johnston et al. (2005) for the 2004 periastron passage of B1259-63
system, albeit in a much more compact environment.

In the recent observation of FRB 20180301A from Parkes/Murriyang UWL (Kumar et
al., 2023b), no burst was observed to have circular polarisation $>15\%$, and
there was no evidence for frequency-dependent circular polarisation333If there
was a change in sign in $V$ it would be possible to have frequency dependent
circular polarisation with small band-averaged total circular polarisation..
This would indicate that the circular polarisation could be a transient
characteristic due to changing circumburst environment of the FRB progenitor,
similar to the behaviour seen in the FRB 20201124A, where significant
frequency dependent circular polarisation was only seen in some of the bursts
during its active phase in April 2021 (Kumar et al., 2022). Additionally,
observational bias could lead to missing bursts showing significant circular
polarisation fraction during this transient phase. The FRB 20180301A provides
a unique window into studying the progenitor environment of FRBs using its
complex polarimetric properties. Hence, a high cadence monitoring of this
source is essential to confirm any periodic polarimetric behaviour in FRB
20180301A.

##  5 Conclusions

We use GFR to model the Stokes-V behaviour of FRB 20180301A. We extend the RM
estimation method using QU-fit described in Bannister et al. (2019) and Lower
(2021) to QUV-fit for GFR modelling and derive a combined RM-GFR estimation
method. We verify this method using known pulsars J1644$-$4559 and
J0835$-$4510.

We use arcsec localisation from Bhandari et al. (2022) and the pointing
information from the observation to conclude that the burst was detected
within the FWHP of the multibeam receiver, with a radial offset of $\sim$7’
from the beam 3 centre. The precise localisation information suggests that the
frequency-dependent Stokes-V burst behavior was likely non-instrumental. The
combined estimation of the RM and GRM suggests a less extreme RM of $\sim$27
rad m-2, relative to the previously reported RM of -3163 rad m-2(Price et al.,
2019). Our results provide another example in an ever-growing subset of FRBs
showing signature of passage through relativistic plasma. This confirms
another promising way to use FRBs as probes of fundamental physics.

## Acknowledgements

PAU and RMS acknowledges support through Australia Research Council Future
Fellowship FT190100155. RMS also acknowledges support through Australian
Research Council Discovery Project DP220102305. We acknowledge the Wiradjuri
people as the Traditional Owners of the Observatory site. Murriyang, the
Parkes radio telescope, is part of the Australia Telescope National Facility
(https://ror.org/05qajvd42) which is funded by the Australian Government for
operation as a National Facility managed by CSIRO. This work was performed on
the OzSTAR national facility at Swinburne University of Technology. This work
makes use of OzSTAR supercomputing facility. The OzSTAR program receives
funding in part from the Astronomy National Collaborative Research
Infrastructure Strategy (NCRIS) allocation provided by the Australian
Government, and from the Victorian Higher Education State Investment Fund
(VHESIF) provided by the Victorian Government.

This work makes use of BILBY (Ashton et al. 2019), MATPLOTLIB (Hunter 2007)
and NUMPY (Harris et al. 2020) software packages.

## Data Availability

This work used publicly available data. The codebase used for the GFR analysis
is available on https://github.com/pavanuttarkar/GFR_codebase.

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##  Appendix A Circular polarisation likelihood equation

The likelihood function for the GFR model described by Equation 4 can be
described by

$\begin{split}&\textit{L}(P(\lambda)|P_{M}(\lambda),\sigma)=\frac{1}{\sigma_{Q}\sqrt{2\pi}}\exp{\frac{-(Q-Q_{m})^{2}}{2\sigma_{Q}^{2}}}\\\
&\frac{1}{\sigma_{U}\sqrt{2\pi}}\exp{\frac{-(U-U_{m})^{2}}{2\sigma_{U}^{2}}}\frac{1}{\sigma_{V}\sqrt{2\pi}}\exp{\frac{-(V-V_{m})^{2}}{2\sigma_{V}^{2}}},\end{split}$
(16)

where $P(\lambda)$ is the polarised flux across wavelength $\lambda$,
$P_{M}(\lambda)$ is the model polarised flux across wavelength, Q, U, and V
are the Stokes components, $Q_{m}$, $U_{m}$, and $V_{m}$ are the model Stokes
parameters, $\sigma_{Q}$, $\sigma_{U}$, and $\sigma_{V}$ are the rms value of
the individual Stokes components. Using model parameters from Equation 7 the
above Equation simplifies to

$\begin{split}&\textit{L}(P(\lambda)|P_{M}(\lambda),\sigma)=\\\
&\frac{1}{(2\pi)^{3/2}\sigma^{3}}\exp{-\frac{1}{2\sigma^{2}}(Q^{2}+U^{2}+V^{2})}\\\
&\exp{\Biggr{[}\frac{1}{2\sigma^{2}}\left[P^{2}-2P(QQ_{m}+UU_{m}+VV_{m})\right]\Biggr{]}},\end{split}$
(17)

where P is the total polarised flux, following Bannister et al. (2019) we
integrated the total polarised from Equation 17. We also note that Bannister
et al. (2019) used integration from -$\infty$ to $\infty$ which induces bi-
modality in the likelihood distribution. We correct for this by integrating
the above Equation from 0 to $\infty$, to avoid bi-modality

$\begin{split}&\textit{L}(P(\lambda)|P_{M}(\lambda),\sigma)=\\\
&\frac{1}{(2\pi)^{3/2}\sigma^{3}}\exp{\frac{-1}{2\sigma^{2}}(Q^{2}+U^{2}+V^{2})}\\\
&\int_{0}^{\infty}\exp{\Biggr{[}\frac{1}{2\sigma^{2}}\left[P^{2}-2P(QQ_{m}+UU_{m}+VV_{m})\right]\Biggr{]}}dP.\end{split}$
(18)

Further, Equation 18 reduces to Equation 11. The posterior probability
distributions obtained from the likelihood Equation 11 for test pulsars
J1644$-$4559 and J0835$-$4510 are shown in Figure 6. The posterior
probablility distribution for FRB 20180301A is shown in Figure 7. The model
fit to pulsars J1644$-$4559 and J0835$-$4510 is shown in Figure 8. A
comparison of the GFR parameters for the magnetar XTE J1810$-$197 can be found
in Figure 9.

(a) (a) Figure 6: The posterior distribution for the test pulsars J1644-4559
and J0835-4510. We use archival multibeam data for the verification of the
model. The corner plots shows the posterior parameters as listed in Table 1.
We use uniform priors for all the parameters. The dashed lines show 1-$\sigma$
confidence interval for all the parameters. The inner and outer contours show
1-$\sigma$ and 3-$\sigma$ intervals. (a) Figure 7: The posterior probability
distribution of the FR-GFR model for FRB 20180301A. The modelled posterior
distribution is from the FR-GFR model described by Equation 11. The estimated
parameters from the posterior distribution are listed in Table 2. We use
uniform priors for all the parameters shown in the posterior probability
distribution. The dashed lines shows the 1-$\sigma$ confidence interval for
all the parameters. The three contours represent 1-$\sigma$, 2-$\sigma$, and
3-$\sigma$ confidence intervals.

(a) (b)

Figure 8: The QUV fit for pulsars J0835$-$4510 and J1644$-$4559\. Figures (a)
and (b) show the model fit derived for the polarisation calibrated dataset for
pulsars J0835$-$4510 and J1644$-$4559, respectively. The model fits are
derived using the FR-GFR model. Stokes Q, U, and V fit are shown in Panels A,
B, and C, respectively, for each figure. We use archival data (project ID
P595) from the Parkes/Murriyang radio telescope. The data for the time
averaged on-pulse region are shown in green coloured points, and the solid
black line shows the model fit to the data. The EA and PA angles derived from
the data are shown in blue points in Panels D and E, respectively, for Figures
(a) and (b). The model fit for the EA and PA angles are shown in black solid
lines. The residual after subtracting the data from the model is shown in red
points for EA and PA angles in Panels D and E, respectively. Figure 9: The FR-
GFR model comparison with results from Lower et al. (2024) for magnetar
J1810$-$197\. The panels A and B show the GRM and $\alpha$ values recovered
for J1810$-$197 for different pulse phase bins. The black and blue points show
the recovered values from Lower et al. (2024) and the FR-GFR model. The
errorbars on individual points show 1-$\sigma$ error range of recovered
values.