Polarization Pipeline for Fast Radio Bursts Detected by CHIME/FRB

The polarization of an electromagnetic wave relates to the preferred geometric orientation of its oscillating electric and magnetic fields. By convention, the polarization of an electromagnetic wave is determined by the direction of the electric field. In the case of fully linearly polarized radiation, oscillations in the electric field occur entirely along a single direction that, combined with the axis of propagation, defines the plane of polarization of the emission. Circular polarization, meanwhile, refers to the case where the fields rotate in the plane perpendicular to the direction of propagation, with the direction of rotation determining the "handedness" of the polarization. In this way, unlike linear polarization, circular polarization can have either a negative or positive sign.

A convenient way of representing the different geometries of the polarized emission is to transform the complex electric field into Stokes parameters. The geometry of CHIME's feed design is consistent with the IAU/IEEE convention ¹⁶ where the X and Y linear feeds point toward the east and north, respectively. In this convention, Stokes I, Q, U, and V parameters can be obtained by applying the transformations,

$$\begin{eqnarray}\begin{array}{rcl}I & = & \langle | X{| }^{2}+| Y{| }^{2}\rangle \\ Q & = & \langle | X{| }^{2}-| Y{| }^{2}\rangle \\ U & = & \langle 2\,\mathrm{real}({\mathrm{XY}}^{* })\rangle \\ V & = & \langle -2\,\mathrm{imag}({\mathrm{XY}}^{* })\rangle .\end{array}\end{eqnarray} \tag{ 1 }$$

Here, Stokes I refers to the total intensity of the emission, Stokes Q and U correspond to the linearly polarized component and Stokes V refers to the circularly polarized component.

The observed polarization angle, ψ, can be expressed in terms of Stokes Q and U parameters, such that,

$$\begin{eqnarray}&&\psi (t,\nu )=\displaystyle \frac{1}{2}{\tan }^{-1}\displaystyle \frac{U(t,\nu )}{Q(t,\nu )}\quad [\mathrm{rad}].\end{eqnarray} \tag{ 2 }$$

Units here and elsewhere are denoted by [ ].

Equation (2) has been expressed in terms of time (t) and frequency (ν). This allows for the possibility of a change in ψ over the burst envelope (see Section 2.3) or across the spectrum that can either be intrinsic to the source or introduced later as a propagation effect, such as Faraday rotation (see Section 2.2). Intrinsic variations in ψ may be produced by a radius-to-frequency mapping (RFM) similar to what has been posited for pulsars where emission occurs at different altitudes within the magnetosphere (e.g., Thorsett 1991; Mitra & Rankin 2002; Noutsos et al. 2015). Although there has been some work done exploring the applicability of RFM in describing certain FRB phenomena (e.g., FRB frequency drifts; Lyutikov 2020), the validity of such a model remains uncertain.

Faraday rotation (quantified by the rotation measure, RM) is a magneto-optical propagation effect observed as a rotation of the plane of polarization that is linearly proportional to the square of the wavelength such that,

$$\begin{eqnarray}&&\mathrm{RM}=\displaystyle \frac{d\psi }{d{\lambda }^{2}}\quad [\mathrm{rad}\,{{\rm{m}}}^{-2}].\end{eqnarray} \tag{ 3 }$$

Here, RM, ψ, and λ are the rotation measure, polarization angle, and observing wavelength, respectively. The RM is proportional to the magnetic field parallel to the line of sight (LOS) weighted by the free electron density and integrated along the path between the source and observer. Specifically, for an FRB located at a redshift z = z_i , the RM in the observer's frame is,

$$\begin{eqnarray}&&\mathrm{RM}={C}_{R}{\int }_{{z}_{i}}^{0}\displaystyle \frac{{n}_{e}(z){B}_{\parallel }(z)}{{\left(1+z\right)}^{2}}\displaystyle \frac{{dl}}{{dz}}{dz}\quad [\mathrm{rad}\,{{\rm{m}}}^{-2}],\end{eqnarray} \tag{ 4 }$$

where C_R = 811.9 rad m⁻²/(μG pc cm⁻³), z is redshift, n_e is the free electron density, B_∥ is the magnetic field strength parallel to the LOS, and dl(z) is the LOS line element at z.

The RM, therefore, is an integrated quantity that when combined with the DM, can be used to estimate the average magnetic field strength of intervening plasma (e.g., Akahori et al. 2016). The extragalactic nature of FRBs implies contributions to the RM from not only the Milky Way's interstellar medium (ISM) and the surrounding Galactic halo but also the intergalactic medium (IGM), intervening systems such as individual galaxies and/or groups/clusters, and finally, the host galaxy and local circumburst environment.

The polarization position angle (PPA) corresponds to the polarization angle of the emission at the source as a function of time. The PPA is commonly measured in radio pulsars where a characteristic S-shaped PPA curve is often observed and interpreted within the popular rotating vector model of pulsar emission (Radhakrishnan & Cooke 1969). In this way, it is different from the observed polarization angle (see Equation (2)) in that it characterizes the geometry of the polarized signal prior to being modulated by Faraday rotation. The effect of Faraday rotation can be removed by using the measured RM to de-rotate the spectrum through a multiplicative phase factor such that,

$$\begin{eqnarray}&&\begin{array}{l}{[Q+{iU}]}_{\mathrm{int}}(\lambda ,t)={[Q+{iU}]}_{\mathrm{obs}}(\lambda ,t)\\ \quad \times \exp [2i(\mathrm{RM}({\lambda }^{2}-{\lambda }_{0}^{2})+{\psi }_{0}(t))]\end{array}\end{eqnarray} \tag{ 5 }$$

Here, [Q + iU]_obs is the observed spectrum and [Q + iU]_int is the intrinsic polarization vector at the source, while RM and ψ₀ are fitted parameters. ψ₀ is the polarization position angle at a reference wavelength λ₀ (i.e., at infinite frequency or zero wavelength). In the case of calibrated polarized observations, ψ₀ is often referenced at infinite frequency where Faraday rotation is zero. In principal, any time dependence of ψ₀ can be determined by fitting the polarized signal through the burst duration. In practice, S/N limitations complicate this time-resolved analysis and are, in any case, unsuitable for an automated pipeline where robust methods of characterizing the polarized signal take precedence. An alternative method for characterizing time dependence in ψ₀ is to apply Equation (2) to the burst profiles of the de-rotated Stokes Q, U parameters such that,

$$\begin{eqnarray}&&{\psi }_{0}(t)=\displaystyle \frac{1}{2}{\tan }^{-1}\left(\displaystyle \frac{{U}_{\mathrm{derot}}(t)}{{Q}_{\mathrm{derot}}(t)}\right)\quad [\mathrm{rad}].\end{eqnarray} \tag{ 6 }$$

Here, Q_derot and U_derot are integrated over frequency to optimize the signal-to-noise of the ψ₀ measurement under the assumption that there is no frequency dependence in the intrinsic polarization angle at the source. Calculating the ψ₀(t) curve in this way makes it less sensitive to measurement errors associated with Stokes Q and U, yielding a more stable curve through the burst duration.

RM synthesis (Burn 1966; Brentjens & de Bruyn 2005) is a robust technique for measuring Faraday rotation that amounts to a Fourier-like transformation, such that,

$$\begin{eqnarray}&&{\boldsymbol{F}}(\phi )={\int }_{-\infty }^{\infty }P({\lambda }^{2}){e}^{-2i\phi {\lambda }^{2}}d{\lambda }^{2}.\end{eqnarray} \tag{ 7 }$$

Here, ϕ is referred to as the Faraday depth and is an extension of RM for scenarios where the polarized signal is Faraday-rotated by different amounts. ∣F(ϕ)∣ is the total linearly polarized intensity across the bandpass after de-rotating the complex vector representing the observed linearly polarized intensity, P(λ²) = Q(λ²) + iU(λ²).

Carrying this procedure over multiple ϕ values results in a Faraday dispersion function (FDF) representing the polarized intensity at different trial values. Applying RM synthesis to emission that occurs over an extended region of space often yields a complex FDF with substantial polarized emission at multiple Faraday depths (e.g., Anderson et al. 2016; Dickey et al. 2019). With FRB emission, the short millisecond timescales strongly limit the amount of differential Faraday rotation that can occur within such a small emitting volume. In cases such as this, the FDF will appear as a single peak in polarized signal at a single Faraday depth. This regime is known as "Faraday thin," where ϕ and RM are interchangeable terms.

Figure 1 shows a simulated burst with RM = +100 rad m⁻². The effect of Faraday rotation can clearly be seen in the plot, showing the burst as a function of frequency and time (waterfall plot) for Stokes Q and U of panel (a). Applying RM synthesis to this spectrum produces the FDF of panel (b). The orange curve is the "dirty" FDF for the event and includes both contributions from the signal as well as side lobes introduced by the bandpass limitations of the observation, known as the rotation measure transfer function (RMTF). Side lobes can be cleaned by applying an RM-CLEAN algorithm (Heald et al. 2009) that deconvolves the RMTF from the observed FDF in a manner analogous to the CLEAN deconvolution routines applied in aperture synthesis radio imaging (Högbom 1974; Clark 1980). The blue curve of panel (b) corresponds to the clean FDF. In the case where polarized emission is well described by a single RM, the best estimate of the RM will correspond to the ϕ value at which the FDF peaks.

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An alternative method for extracting an RM value is to invoke a model that can fit the oscillations in Stokes Q and U introduced by Faraday rotation. In the case where all polarized emission is Faraday-rotated by a single RM value, the methods of Stokes QU-fitting and RM synthesis are essentially identical. This is highlighted in Figure 1, where the RM determined by fitting the Stokes spectrum of panel (c) results in a fitted RM value, shown in panel (d), that is consistent with that independently determined via RM synthesis. Panel (c) shows the Stokes I, Q, U, and V spectra along corresponding models fits, obtained from a modified version of the RM-tools ¹⁷ software (Purcell et al. 2020) that implements a nested sampling algorithm (Skilling 2004) to find the best-fitting parameters. Parameters in this simple benchmark model are the RM, the polarization angle at infinite frequency, ψ₀, and the linear polarization fraction, p. In this simple model, Stokes V is assumed to be zero while Stokes I is fit by a fifth-order polynomial (${I}_{\mathrm{mod}}$) and is used as input in the fitting procedure applied to Stokes Q and U. Models for Stokes Q and U can therefore be expressed as,

$$\begin{eqnarray}\begin{array}{rcl}{Q}_{\mathrm{mod}} & = & {I}_{\mathrm{mod}}p\cos (\alpha ),\\ {U}_{\mathrm{mod}} & = & {I}_{\mathrm{mod}}p\sin (\alpha ),\end{array}\end{eqnarray} \tag{ 8 }$$

where α = 2(RMλ² + ψ₀) corresponds to the frequency-dependent phase introduced by Faraday rotation.

Optimal values are determined numerically through nested sampling, a Monte Carlo method for Bayesian analysis that simultaneously calculates both Bayesian evidence and posterior samples. This method benefits from more efficiently sampling the parameter space than conventional Markov Chain Monte Carlo (MCMC) methods and is particularly useful for degenerate, multi-modal likelihoods. Parameter estimation seeks to optimize the likelihood function given a model and the data. Each data point in the fit is weighted by the inverse square of the RMS noise of the frequency channel. In this sense, model and data are compared such that ${Q}_{i}={Q}_{\mathrm{mod},i}+{n}_{i}$ and ${U}_{i}={U}_{\mathrm{mod},i}+{n}_{i}$, where n_i is the Gaussian noise for channel i. Following the prescription of O'Sullivan et al. (2012), the prior likelihood of particular RM and ψ₀ values for an observation of a single channel, d_i , under the assumption of Gaussian noise is,

$$\begin{eqnarray}&&\begin{array}{c}\begin{array}{l}{P}_{i}({d}_{i}| {\rm{RM}},{\psi }_{0})=\displaystyle \frac{1}{\pi {\sigma }_{{Q}_{i}}{\sigma }_{{U}_{i}}}\exp \\ \,\times \left(-\displaystyle \frac{{\left({Q}_{i}-{Q}_{{\rm{mod}},i}\right)}^{2}}{2{\sigma }_{{Q}_{i}}^{2}}+\displaystyle \frac{{\left({U}_{i}-{U}_{{\rm{mod}},i}\right)}^{2}}{2{\sigma }_{{U}_{i}}^{2}}\right).\end{array}\end{array}\end{eqnarray} \tag{ 9 }$$

Here, σ_Q,U is the single channel RMS. For N frequency channels, the prior likelihood becomes,

$$\begin{eqnarray}&&P(d| \mathrm{RM},{\psi }_{0})=\prod _{n=1}^{N}{P}_{i}({d}_{i}| \mathrm{RM},{\psi }_{0}).\end{eqnarray} \tag{ 10 }$$

This formulation of the likelihood function ensures that parameter estimation is carried out by comparing absolute values of Q and U to model predictions. This results in more robust fit values compared to fitting the fractional polarization (i.e., Q/I, U/I), particularly for low S/N events where normalizing by total intensity can introduce substantial departures from Gaussianity in the noise.

Parameter estimation is done through the Multinest software (Feroz et al. 2009), which numerically searches for optimal parameter values that optimize the log-likelihood function. The resulting posterior distributions are shown in panel (d) of Figure 1. Along the diagonal are histograms of the estimated 1D marginal posterior probability distribution for each parameter. The best-fit value and 1σ uncertainty region for each parameter are indicated by vertical blue lines and black dotted lines, respectively.

A Faraday thin model, expressed mathematically in Equation (8), is generally adequate for describing the polarized signal of an FRB and is employed in the automated polarization analysis pipeline of CHIME/FRB. Indeed, as with pulsars, FRBs are not likely to display Faraday complexity due to the small presumed size of the emission region over which very little internal Faraday rotation is likely to occur. However, in certain scenarios this may not be the case and the parametric nature of Stokes QU-fitting can be leveraged to fit for effects not contained in the simple Faraday rotation model. These excursions from a simple Faraday model can be produced by astrophysical or instrumental effects. The application of QU-fitting to astrophysical excursions from a Faraday simple model are further discussed in Section 7. More relevant to the automated pipeline are the effects introduced by instrumental systematics, which strongly affect a significant fraction of FRBs detected by CHIME. The specifics of the QU-fitting implementation in the polarization pipeline and contaminant systematics are discussed further in Section 5.

In cases where ∣RM∣ values are large, a significant change in the polarization angle, ψ, can occur within a single frequency channel such that (see Equation 4.12 of Burke & Graham-Smith 2014),

$$\begin{eqnarray}&&\delta \psi =\displaystyle \frac{-2{\mathrm{RM}}_{\mathrm{obs}}{c}^{2}\delta \nu }{{\nu }_{c}^{3}}\quad [\mathrm{rad}].\end{eqnarray} \tag{ 11 }$$

Here, δ ψ corresponds to the degree of intra-channel Faraday rotation, RM_obs is the observed RM, δ ν is the channel width, and ν_c is the central frequency of the channel. Due to the strong frequency dependence in Equation (11), this effect becomes more pronounced at longer wavelengths. If ∣RM∣ and δ ν are large enough, ψ undergoes a large rotation within a frequency channel. The net effect is depolarization within each channel, with the level of depolarization dependent on observing frequency. This effect, known as intra-channel or bandwidth depolarization, limits the range of RM values to which any instrument is sensitive, with detections of larger ∣RM∣ values either requiring finer frequency resolution or higher observing frequencies. The fractional depolarization can be approximated within each channel using (Schnitzeler & Lee 2015; Michilli et al. 2018), ¹⁸

$$\begin{eqnarray}&&{f}_{\mathrm{depol}}=1-\left(\displaystyle \frac{\sin (\delta \psi )}{\delta \psi }\right).\end{eqnarray} \tag{ 12 }$$

In the case of CHIME, the relatively low observing band of 400–800 MHz and its modest frequency-channel resolution of δ ν = 390 kHz (i.e., 1024 channels) limits sensitivity to RM detections of several thousand rad m⁻², with the precise value depending on the S/N and spectrum of the event. Figure 2 shows the expected fractional depolarization as a function of ∣RM∣. At ν_c = 600 MHz there is an approximately 50% drop in sensitivity to polarized emission at RM ≈ 5000 rad m⁻², effectively putting an upper limit on the RM range detectable at the native spectral resolution of CHIME/FRB baseband data. The exact level of bandwidth depolarization is dependent on the precise spectrum of each burst. This frequency dependence is highlighted by the depolarization shown in Figure 2 where lower frequencies (ν = 400 MHz) are seen to be generally more significantly depolarized than higher frequencies (ν = 800 MHz) for a given RM.

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Fortunately, baseband data retain the phase information of the incident electric field. This allows the limitations imposed by the native spectral resolution to be overcome, by re-sampling such that time resolution can be swapped for enhanced frequency resolution, a process we have dubbed "up-channelization." Alternatively, the electric field phase allows us to correct for the frequency-dependent phase offsets introduced by Faraday rotation. Formulating Faraday rotation as the result of the differing group velocities of the left and right circular polarization states allows us to express it as an additional dispersive effect operating differentially on the two circular bases. Expressed in this form, the correction for Faraday rotation is analogous to coherent de-dispersion (Hankins 1971), in which a transfer function is invoked that corrects for the phase change within frequency channels (van Straten 2002).

This method of coherently correcting for Faraday rotation amounts to a frequency-dependent phase factor that is applied to the circular polarization basis pair (∣R〉, ∣L〉) such that,

$$\begin{eqnarray}\begin{array}{rcl}| {R}^{{}^{{\prime} }}\rangle & = & {e}^{-i\beta }| R\rangle \\ | {L}^{{}^{{\prime} }}\rangle & = & {e}^{i\beta }| L\rangle \end{array}\end{eqnarray} \tag{ 13 }$$

where $| {R}^{{}^{{\prime} }}\rangle ,| {L}^{{}^{{\prime} }}\rangle$ are the right and left circular polarized components, respectively, after correcting for the phase offset, β, introduced by Faraday rotation,

$$\begin{eqnarray}&&\beta =\mathrm{RM}\displaystyle \frac{{c}^{2}}{{\nu }^{2}}.\end{eqnarray} \tag{ 14 }$$

Figure 3 shows an example of a simulated burst with a very large RM of +(200, 000) rad m⁻². As expected, when omitting the increased RMS noise introduced by the burst, there is an absence of polarized signal in the Stokes Q and U waterfall plot (panel a) due to the extreme level of intra-channel depolarization. In addition, the burst appears to split into the two circular bases at the bottom of the band, a product of the differing group velocities of the two bases that are "resolved out" for sufficiently narrow bursts with extreme RMs (Suresh & Cordes 2019). Since the RM is a priori known for this simulated burst, coherent de-rotation can be trivially applied to this burst by first transforming the simulated baseband data from a linear to a circular basis, applying Equations (13) and (14), and then transforming back to linear basis. Panel (b) shows the Stokes waterfall plots after correcting for the deleterious effects of the intra-channel Faraday rotation and rotating all of the recovered polarized signal into Stokes Q.

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A comparison of the depolarization-corrected FDF (gray line) and its uncorrected counterpart (green line) is shown in panel (c). The method of coherent de-rotation effectively extends our sensitivity range to RM values far beyond what would be predicted from the native spectral resolution. Much like coherent de-dispersion, this method is resource-intensive. This prevents a naive search over many RM trials since each trial requires the computationally costly procedure of re-sampling the channelized voltages. In light of this, a semi-coherent method has been implemented in the CHIME/FRB pipeline that consists of coherent de-rotation to a sparse grid of RM trials followed by an incoherent search at neighboring RM values. The details of this semi-coherent RM search method is presented in Section 5.1.4.

5.1.1. Preprocessing

5.1.2. Stokes Parameter Extraction

5.1.3. RM Detection

RM detections are made through a modified version of the RM-tools package (Purcell et al. 2020) that contain implementations for both RM synthesis and QU-fitting. The equation,

$$\begin{eqnarray}&&| {\mathrm{RM}}_{\max }| \approx \displaystyle \frac{\sqrt{3}}{\mathrm{median}(\delta {\lambda }^{2})}\end{eqnarray} \tag{ 15 }$$

(Brentjens & de Bruyn 2005) is used to determine the approximate RM at which 50% bandwidth depolarization occurs where δ λ² is the channel width in units of wavelength squared (m²). This value is used to set the RM search limits of both RM synthesis and QU-fitting methods. For values exceeding this, RM search methods must be preceded by a coherent de-rotation operation (see Section 5.1.4 for details).

RM Synthesis:

RM synthesis is a robust method for obtaining an initial RM detection. It is well suited for implementation in an automated pipeline where low S/N events or residual RFI may stymie a parametric method that is sensitive to initial guesses of model parameters. Moreover, the resulting FDF produced by RM synthesis is an ideal diagnostic tool for parsing an astrophysical signal from instrumental effects. In light of this, RM synthesis is applied first in the pipeline to obtain an initial estimate of the RM that is then further refined by QU-fitting.

Performing RM synthesis on the extracted Stokes spectrum produces a "dirty" FDF that is then cleaned of artifacts introduced by the RMTF (see Section 4). This cleaning procedure amounts to modeling the intrinsic FDF of the source by discrete Dirac delta functions in ϕ space that are then convolved with the RMTF of the observation as a best attempt at reconstructing the observed FDF (see Heald et al. 2009 for details). The level of cleaning is determined by the threshold relative to the RMS noise, such that ϕ bins where the FDF exceeds this value are modeled as delta functions. Cleaning is generally advantageous in scenarios where Faraday complexity is present. This is generally not the case for FRBs, making cleaning a somewhat superfluous step for the purposes of RM determination. Instead, cleaning is implemented in the pipeline for diagnostic reasons, helping determine if complex structure in the "dirty" FDF is an artifact of the RMTF or some other unknown systematic. For the automated pipeline, FDFs are cleaned conservatively to a level of 8σ. Here, σ refers to the noise in the FDF and is estimated from a quadratic sum of the RMS in Stokes Q and U across all frequency channels (i.e., $\sigma ={\sum }_{i=0}^{N}\sqrt{{\sigma }_{{Q}_{i}}^{2}+{\sigma }_{{U}_{i}}^{2}};$ N = number of channels) over a time interval preceding the burst.

An RM is obtained from the clean FDFs by applying a parabolic fit to the FDF peak. Measurement uncertainties are estimated in a manner analogous to radio imaging (Condon 1997), using the relation σ = FWHM/(2 S/N). Here, the FWHM characterizes the width of the peak in Faraday depth space and S/N corresponds to the signal-to-noise ratio of the peak-polarized intensity in the FDF. In the idealized scenario of Figure 1, RM synthesis and QU-fitting are effectively equivalent methods. The limitations of RM synthesis become apparent when additional polarized signal is introduced by instrumental effects. In the case of CHIME, polarized observations are dominated by two systematics: a delay in the beam-formed voltages between the two polarizations and, to a much lesser extent, a differential response between them. Appendix A illustrates the effect of these two systematics, highlighting how RM values obtained by RM synthesis are vulnerable to certain systematic biases. This is in contrast to the QU-fitting, for which the model provided in Equation (8) can be extended to fit for additional instrumental effects.

QU-fitting:

QU-fitting is applied to refine the initial RM detection made by RM synthesis or to confirm a non-detection. Models that simultaneously capture the polarized astrophysical and instrumental signal are implemented into the nested sampling QU-fitting framework outlined in Section 4. The default mode of the pipeline is to fit for the astrophysical parameters of the linear polarized fraction, p, RM, ψ₀, and the physical delay between the two linear polarizations, τ (cable delay). This amounts to fitting a revised model that accounts for the Stokes U–V leakage introduced by a non-zero τ,

$$\begin{eqnarray}\begin{array}{rcl}{Q}_{\mathrm{mod}}^{{\prime} } & = & {Q}_{\mathrm{mod}}\\ {U}_{\mathrm{mod}}^{{\prime} } & = & {U}_{\mathrm{mod}}\cos (2\pi \nu \tau )-{V}_{\mathrm{mod}}\sin (2\pi \nu \tau )\\ {V}_{\mathrm{mod}}^{{\prime} } & = & {U}_{\mathrm{mod}}\sin (2\pi \nu \tau )+{V}_{\mathrm{mod}}\cos (2\pi \nu \tau )\end{array}\end{eqnarray} \tag{ 16 }$$

Here, ${Q}_{\mathrm{mod}}$, ${U}_{\mathrm{mod}}$, and ${V}_{\mathrm{mod}}$ refer to the models for the astrophysical polarized signal described in Equation (8). ${Q}_{\mathrm{mod}}^{{\prime} }$, ${U}_{\mathrm{mod}}^{{\prime} }$, and ${V}_{\mathrm{mod}}^{{}^{{\prime} }}$, meanwhile, are models for the observed Stokes parameters after being modified by the frequency-dependent phase difference between X and Y voltages introduced by a non-zero τ. Assuming that the polarized signal is dominated by the linear component, we set ${V}_{\mathrm{mod}}=0$. This condition can be relaxed to allow for non-negligible circular polarization that is intrinsic to the source and is further explored in Appendix B. Modifying the likelihood function Equation (10) to account for the leaked signal found in Stokes V allows us to once again estimate best-fitting model parameters by maximizing the modified likelihood function. In all cases, uniform priors are assumed on the fitted parameters.

5.1.4. Semi-coherent Search

The semi-coherent RM search implemented in the pipeline is a two-stage process involving a coherent routine to correct for Faraday rotation over a sparse grid of trial RMs, followed by an incoherent search at neighboring RM values. Possible RM detections at neighboring values are probed by performing RM synthesis on a coherently de-rotated spectrum, producing an FDF for each trial RM. If the resulting FDF does not produce above a detection threshold, the routine moves to the next trial RM, performing the identical operations of coherent de-rotation and incoherent search until detection is made or a full range of RM values has been explored. A rather stringent detection threshold of S/N > 6 is used to avoid triggering false detections on artificial FDF peaks introduced by systematics.

A schematic summarizing this routine is shown in Figure 5. While this routine can, in principle, be performed on arbitrarily large ∣RM∣ values, we limit the automated pipeline to search within the range −10⁶ ≤ RM ≤ 10⁶ rad m⁻² to ensure the automated pipeline processes incoming events on a reasonable timescale. This amounts to several thousand coherent de-rotation operations to explore the entire RM range. Coherent de-rotation operations over the sparse grid of trial RMs is by far the most resource-intensive stage of the polarization pipeline, taking roughly 20 minutes to complete a search out to ∣RM∣ = 10⁶ rad m⁻² when running on a single-core CPU. The redundant nature of the operation makes it well suited for parallelization which is a focus of ongoing work.

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The RM step size between coherent operations, δRM ∼ 700 rad m⁻², is determined as the 10% depolarization level, referenced at the bottom of the CHIME band (ν = 400.390625 MHz). While this omits the spectral dependence of intra-channel depolarization (i.e., bandwidth depolarization is frequency-dependent) it is sufficiently conservative that RM detections from bright, highly polarized bursts are unlikely to be missed. That said, a phase space does exist over which polarized events will evade detection under current configuration of the semi-coherent search. These problematic events include fainter bursts with intrinsically low linear polarized fractions or bursts with an ∣RM∣ value that exceeds the search limits of the automated pipeline. Rather than be treated by the automated pipeline, these problematic events are left to be manually processed with a tighter, more extensive grid of trial RM values. While RM detection can, in principle, exist out to arbitrarily large values, upper bounds on the maximum possible ∣RM∣ can be deduced by the absence of a burst-splitting morphological imprint (Suresh & Cordes 2019). For a 1 ms burst, this morphological imprint begins to manifest at ∣RM∣ ≳ 2 × 10⁶ rad m⁻² as an apparent widening of the burst at the bottom of the CHIME band relative to the top. Meanwhile, events that continue to evade RM detection despite manual processing can be used to infer upper bounds on the linear polarized fractions given their S/N.

In 2019 December, CHIME/FRB detected a bright burst (S/N > 170) from FRB 20191219F across six of the 1024 formed sky beams of the real-time intensity data. A trigger was initiated by the system that successfully captured baseband data for this event. ¹⁹ Running the baseband localization pipeline resulted in a refined localization of R. A., Decl. = (2262034 ± 00441, 854168 ± 00037) (J2000) and an S/N-optimizing DM = 464.558 ± 0.003 pc cm⁻³. The DM measurement is obtained at an earlier stage in the baseband pipeline by dedispersing to a reference value and then incoherently dedispersing over a range of trial DM values. The small DM uncertainty quoted here is a product of the brightness of the event and the $\tfrac{1}{{\rm{S}}/{\rm{N}}}$ scaling in the Gaussian fit of the peak in S/N, DM phase space.

The polarization pipeline was then run on the beam-formed data, resulting in the diagnostic plots shown in Figure 6. The waterfall plot has been rebinned to a time and frequency resolution of 10.24 μs bin⁻¹ and 1.56 MHz bin⁻¹, respectively. Evidence of Faraday rotation is seen in the frequency-dependent modulation of Stokes Q and U in addition to an apparent leaked polarized signal in Stokes V. Running RM synthesis on the Stokes Q, U spectrum, uncorrected for leakage, yields a clear RM detection at RM = −5.614 ± 0.001 rad m⁻². This initial detection is refined by fitting Equation (16), applying a univariate spline to Stokes I to obtain a smoothed model of the burst spectrum, ${I}_{\mathrm{mod}}$. ²⁰

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Implementing this model into the QU-fitting routine yields a best-fit value for the cable delay of τ = − 0.8686 ± 0.0001 ns. Interestingly, the fitted RM = + 6.370 ± 0.002 rad m⁻² is of opposite sign to the initial detection. This sign ambiguity is introduced by the effects of the uncorrected cable delay (see Appendix A for details). Correcting for the cable delay amounts to a multiplicative phase factor that scales with τ and ν, such that,

$$\begin{eqnarray}&&{Y}^{{\prime} }=Y\exp (-2\pi i\nu \tau ).\end{eqnarray} \tag{ 17 }$$

Reconstructing the Stokes spectrum from the corrected $(X,{Y}^{{\prime} })$ polarizations ²¹ successfully removes the frequency-dependent modulation seen in the Stokes V waterfall and re-performing RM synthesis with this corrected spectrum yields RM and ψ₀ values that are in agreement with those measured from QU-fitting along with a ∼5% boost in signal. Table 1 summarizes the fit results. An ionospheric RM contribution of RM_iono = 0.35 ± 0.05 rad m⁻² is calculated using ionFR5 ²² (Sotomayor-Beltran et al. 2013, see Section 7.3). Using this value to correct for the ionospheric contribution leaves us with a measurement of RM = 6.020 ± 0.002 ± 0.050 rad m⁻², where the errors represent statistical and ionospheric uncertainties, respectively.

Figure 7 shows the burst profile for the total intensity (black) along with the linear (red) and circular (blue) components at the native baseband time resolution of 2.56 μs. The burst is highly linearly polarized (L/I > 80%) with a slight upward trend in the linear polarized fraction that suggests some tempo-spectral evolution in the polarized component. Interestingly, the linearly polarized fraction is highest at the trailing edge of the burst, suggesting an evolution in the polarized signal across the burst. Substantial residual Stokes V signal is present even after correcting for the cable delay. It is uncertain from this analysis whether this residual Stokes V signal is a result of some as yet unknown systematic or is intrinsic to the source. In Appendix B, we extend the analysis of this FRB by incorporating additional parameters characterizing the intrinsic properties of the linearly and circularly polarized components, finding evidence for a significant circular component. Finally, the P.A. is remarkably flat over the burst profile, but does appear to display some interesting correlated structure on very short timescales.

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In Section 4.3 we used simulated baseband data to demonstrate how the coherent de-rotation algorithm successfully retrieves bandwidth depolarized signal. While this is an encouraging check, it is possible that unknown systematics introduced by the telescope optics (e.g., coupling) or further downstream in the signal chain (e.g., channelization, spectral leakage) may pose practical limitations on the reliable application of our coherent de-rotation algorithm. The only foolproof method of ruling out deleterious effects introduced by CHIME-specific systematics is to perform a similar analysis on real FRBs with RM values far outside the nominal sensitivity range of the baseband data (see Figure 2). Unfortunately, running a semi-coherent search on the limited subsample of apparently unpolarized events has not returned any detections at large RMs. ²³ While this could indicate the ineffectiveness of the coherent de-rotation algorithm on CHIME/FRB baseband data, we show here that an FRB with a moderate ∣RM∣ still suffers from partial depolarization and can therefore be used to validate our algorithm.

In this section, we perform coherent de-rotation on FRB 20200917A. Discovered in 2020 September, this event triggered a successful recording of baseband data. A single beam was formed in the direction of best localization, R. A., Decl. = (3151217 ± 00544, 758036 ± 00090), and dispersed to an S/N-optimizing DM of 883.3 ± 0.1 pc cm⁻³. Figure 8 shows the resulting Stokes waterfall plot rebinned with a time and frequency resolution of 164.84 μs bin⁻¹ and 1.56 MHz bin⁻¹, respectively. An initial RM detection near +1300 rad m⁻² was made via RM synthesis. This detection was subsequently refined by QU-fitting. As was the case in the previous example, accounting for cable delay in the QU-fitting results in a best-fit RM of opposite sign to the initial detection, RM = − 1294.3 ± 0.1 rad m⁻². The validity of the sign was confirmed by observing a boost in the FDF peak intensity after re-performing RM synthesis on the cable-delay-corrected spectrum. Table 2 summarizes the fit results. An ionospheric RM contribution of RM_iono = 0.17 ± 0.05 rad m⁻² was determined and used to correct for the ionospheric contribution, leaving us with a measurement of RM = −1294.47 ± 0.10 ± 0.05 rad m⁻².

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The fact that path length differences in CHIME, an interferometer with 1024 dual feeds, can be well characterized by a single delay (τ) is a product of X and Y polarization being independently calibrated of one another. There is a significant difference between the best-fit values for τ of FRB 20200917A and FRB 20191219F. The source of this disagreement is associated with thermal expansion of the instrument, as previously noted in the context of CHIME/FRB localization (Michilli et al. 2021). Interestingly, this temperature-dependent effect captured in polarized data offers an alternate means of characterizing the thermal expansion of the dish independent of similar efforts through systematic offsets in the localization of known sources.

Coherently de-rotating this cable-delay-corrected spectrum by the RM value determined from QU-fitting and re-performing RM synthesis results in a depolarization-corrected FDF shown in Figure 9. Similar to that previously shown for a simulated burst (i.e., Figure 3), panel (a) highlights the S/N boost achieved by the depolarization-corrected FDF (green line) over its uncorrected counterpart (gray line). Calculating the ratio of the uncorrected and corrected FDF peak intensities yields a value of 0.87. The lower plot of panel (b) compares this value, indicated by the red marker, to the depolarization curve (black line) for the burst. The depolarization curve is constructed by integrating the frequency-dependent depolarization across the burst sub-band and using a Stokes I fit (cubic spline) to obtain weights of the depolarization contribution at each frequency (top panel). The co-incidence of the red marker with the theoretical depolarization curve at the fitted RM indicates that the coherent de-rotation routine is indeed retrieving all of the bandwidth depolarized signal.

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The coherently de-rotated polarized burst profile of FRB 20200917A is shown in panel (b) of Figure 7. The burst is linearly polarized (L/I > 60%) with no significant circular component. Like FRB 20191219F, FRB 20200917A displays a slight increase in the linear polarized fraction at the trailing edge of the burst. Meanwhile, evolution in the P.A. is apparent across the burst phase, displaying small but significant substructure similar to that seen in other FRBs at high time resolution (e.g., Day et al. 2020; Luo et al. 2020; Nimmo et al. 2021). This structure can possibly be explained by a time dependence of ψ₀ or as an artifact of a slight frequency dependence of ψ₀, manifesting as structure in the PPA curve from a changing spectrum through the burst phase. An additional complication is potential effects introduced by scattering, which is non-negligible for this event. Here, the well known flattening property of scattering on the P.A. curve (e.g., Li & Han 2003), combined with the strong frequency dependence, can give rise to artificial P.A. structure by more strongly affecting lower frequencies. This explanation is somewhat at odds with the secular increase in linear polarized fraction at later times where the deleterious effects of scattering are most significant leading to partial or complete depolarization (e.g., Sobey et al. 2021). A systematic method for probing the observed P.A. structure and relating this analysis to different emission models and propagation effects is left for future work.

For both FRB 20191219F and FRB 20200917A, we obtain an estimate of the RM contribution of the Galactic foreground, RM_MW (Hutschenreuter et al. 2021). In the case of FRB 20191219F, we estimate RM_MW = − 20 ± 7 rad m⁻², implying a modest excess RM (∣RM_excess∣ ∼ 20–30 rad m⁻²) from extragalactic sources of Faraday rotation. This is contrasted by FRB 20200917A, which has a Galactic RM contribution of RM_MW = − 12.0 ± 18 rad m⁻², implying a large excess RM of ∣RM_excess∣ ∼ 1260–1300 rad m⁻². These results are summarized in Figure 10 which compares the ∣RM_excess∣ values of these two bursts to the corresponding sample from the published FRB catalog and the Galactic pulsar RM sample. While the ∣RM_excess∣ value of FRB 20191219F is consistent with the published FRB sample, FRB 20200917A displays an ∣RM_excess∣ that is substantially greater than most other FRBs, with the exception of FRB 20121102A (Michilli et al. 2018). Moreover, ∣RM_excess∣ of FRB 20200917A significantly exceeds typical values of the pulsar RM sample ²⁴ (Manchester et al. 2005), ²⁵ suggesting a supplemental source of Faraday rotation other than the host galaxy's diffuse ISM. We emphasize here that the comparison to the Galactic pulsar sample is used to inform upper limits on the RM contribution of the host galaxy and not the Galactic RM contribution, which is more readily estimated from extragalactic sources (e.g.; Hutschenreuter et al. 2021).

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The significance of the offset between ∣RM_excess∣ of FRB 20200917A with that of the published FRB sample is less significant than it might appear due to a strong selection/publication bias. Specifically, this burst was selected from a much larger sample of CHIME-detected FRBs for its high RM, such that the efficacy of our analysis on high-RM sources could be demonstrated. In fact, the sample from which it was selected contains more sources than the previously published RMs shown in Figure 10. Hence, it is somewhat unsurprising that we should observe an FRB with such an ∣RM_excess∣ value. Moreover, the comparison of this measurement to those of the published FRB sample omits cosmological dilation, which dilutes the RM contribution of the host Galaxy by a factor 1/(1 + z)² (Equation (4)). Accounting for this effect adds significant ambiguity in assessing the significance of RM_excess of FRB 20200917A relative to other FRBs. Indeed, as noted by Connor et al. (2020), the local RM of FRB 20160102A (Caleb et al. 2018) could be as large as − 2400 rad m⁻² if its DM = 2596.1 ± 0.3 pc cm⁻³ is dominated by the IGM.

We follow the analysis applied to the recently discovered repeating source, FRB 20200120E (Bhardwaj et al. 2021), to probe for intervening structures that could produce supplemental Faraday rotation. We rule out Galactic sources of Faraday rotation, finding the sightline of FRB 20200917A to be unassociated with any known foreground structures, including Hii regions (Anderson et al. 2014), star-forming regions, and stellar clusters (Avedisova 2002). For extragalactic sources of Faraday rotation, we do not find any nearby galaxies or galaxy clusters within 1 square degree of the localization region ²⁶ (Wen et al. 2018). Following a similar line of reasoning to that of Connor et al. (2020) in the analysis of FRB 20191108A, we conclude that the substantial ∣RM_excess∣ observed from FRB 20200917A likely originates within the host galaxy itself. This Faraday rotation includes a contribution from the smoothly distributed component of the diffuse ISM as well as possible contributions from intervening discrete structures displaying enhanced electron column densities and/or magnetic field strengths.

Discrete structures may be related to the central engine, as in the case of the dense, magnetized plasma of a supernova remnant. This possibility has recently been put forth to describe the large but decreasing ∣RM∣ observed from FRB 20121102A (Hilmarsson et al. 2021). Alternatively, the excess ∣RM∣ may reflect an environmental preference of the population, such as the proximity of the Galactic center magnetar, PSR J1745-2900, to Sagittarius A⋆ (e.g., Bower et al. 2003; Desvignes et al. 2018) or a manifestation of a fortuitous alignment of the FRB sightline with a galaxy's large-scale magnetic field. Indeed, a Galactic analog of this latter scenario would be the strong Faraday rotation (several thousand rad m^–2) observed from extragalactic sources intersecting the Sagittarius arm tangent and attributed to the diffuse ISM rather than any discrete structures (Shanahan et al. 2019). While this scenario is in principle possible, it is disfavored by the strong scattering and dispersion signatures imparted on the emission at such low inclinations angles. With only one observation from this source, it remains difficult to distinguish among these possibilities.

In the absence of additional information, the exact RM contribution of discrete, overdense regions of the ISM and its diffuse counterpart remain entirely degenerate in describing the observed Faraday rotation. One method for distinguishing these competing sources is to incorporate additional information contained in the scattering properties of the burst. Scintillation, the variation of intensity with frequency due to multi-path interference, can be used to determine the nature and geometry of the scattering medium. Masui et al. (2015), in their analysis of FRB 20110523A, used the scattering/scintillation properties to conclude that the observed Faraday rotation originated from a dense, magnetized plasma near (≲40 Kpc) the source. Carrying out similar analysis here for FRB 20200917A is promising given the strong evidence for scattering but is beyond the scope of this paper and is left for future work. ²⁷

Substantial uncertainty in CHIME's primary beam (PB) model, which seeks to describe the frequency and polarization-dependent response as a function of position on the sky, remains. The absence of an accurate model complicates not only the calibration of important FRB observables such as fluence and spectral properties but also other science done with CHIME such as pulsar monitoring (CHIME/Pulsar Collaboration et al. 2020) and the 21 cm line emission survey between 0.8 ≤ z ≤ 2.5 (Newburgh et al. 2014). ²⁸ In the case of the latter, differences in the radiation patterns of the two polarized beams result in conversion of unpolarized signal into polarized, greatly complicating the process of isolating the unpolarized 21 cm signal from polarized contaminant signal and motivating methods for refinement of CHIME's PB (Singh et al. 2021, in preparation; Wulf et al. 2021, in preparation). In the case of CHIME/FRB, uncertainties in the PB complicate the analysis of polarized spectra by adding artificial features in the constructed Stokes parameters. These features result from the differential gain and phase between the X and Y polarizations. For CHIME, the phase errors are secondary to the much larger instrumental polarization produced by the differences in the absolute sensitivities of the two polarizations.

In Appendix A we illustrate the effect of differential gain, showing a simulated burst (RM = +100 rad m⁻²) where the Y polarization sensitivity is 50 of the X polarization. In such cases, Stokes QU-fitting can be extended by invoking a parameter characterizing the differential gain, η, between the two polarizations. Unfortunately, a realistic PB model for CHIME cannot be characterized by a single η value due to the chromaticity of the two polarizations. This is particularly true at large angular excursions from the main lobe where differences in polarized gains are greatest and change significantly with frequency. In the absence of an accurate PB, corrections for the instrumental leakage introduced by differential gain of the two polarizations is a challenging problem. Thankfully, FRBs for which this effect is significant can be easily identified by an FDF leakage artifact that peaks near RM ≈ 0 rad m⁻². While this instrumental polarized signal may lead to a sample of FRBs with incorrect RM detections near 0 rad m⁻², for the vast majority of cases the intrinsic polarized signal greatly exceeds the instrumental polarization.

Future refinements of the polarization pipeline will include an accurate beam model that captures the frequency-dependent leakage. This can be done either by using the PB to produce bandpass-corrected channelized voltages prior to forming the Stokes parameters or by including the beam model directly in the Stokes QU-fitting procedure. The latter method has the advantage of retaining flexibility, potentially allowing PB model refinements in the fitting procedure. This refined fitting procedure could also be extended to the polarization analysis of the daily pulsar monitoring program of CHIME/Pulsar (CHIME/Pulsar Collaboration et al. 2020) which tracks 400 pulsars, covering declinations down to decl. ≈ − 20°. The combined FRB/pulsar data set would greatly benefit ongoing efforts to map CHIME's primary beam (Berger et al. 2016) by extending the number of usable calibrator sources. While uncertainties in the intrinsic spectra of FRBs, and to a lesser extent pulsars, prevent their use as calibrators of the absolute gain of the PB, model fits to the polarized leakage from these sources can nonetheless be used to determine the relative gain between the two polarizations.

At the moment, the CHIME/FRB polarization pipeline makes no attempt to correct for the RM contribution of Earth's ionosphere, RM_iono. In general, ∣RM_iono∣ will be of order a few rad m⁻², with the precise value dependent on several factors including the direction in the sky, geographic location, time of day, and activity cycle of the Sun (Mevius 2018a). Variability of this magnitude represents a substantial contribution to the systematic error on any RM measurement. If left uncorrected, FRBs detected by CHIME will be biased by the preferential coverage of the northern hemisphere such that Earth's bipolar magnetic field will skew the resulting RM distribution. The size of this systematic bias is likely not sufficient to effect the interpretation of overall RM distribution but may be important for certain science questions predicated on a population of FRBs with low ∣RM_extra∣ values (e.g., see Hackstein et al. 2019, 2020).

In general, ionospheric contributions will be much more important for the interpretation of individual sources rather than the sample as a whole. Specifically, the significance of RM variability observed in bursts from repeating sources will need models that accurately estimate and correct for RM_iono. Accurate models allow correlations in the polarized observables and other burst properties to be probed; namely, the correlation between DM and RM can be used to constrain magnetization of the local circumburst medium as has been done for the Vela and Crab pulsars, for example (Hamilton et al. 1985; Rankin et al. 1988). Accurate ionospheric modeling will only become more relevant as CHIME continues to detect more repeating sources and captures events covering a larger time span where ionospheric conditions are likely to change significantly. Moreover, the recent establishment of periodic activity from repeating sources FRB 20180916B and FRB 20121102A (CHIME/FRB Collaboration et al. 2020b; Cruces et al. 2020) has motivated consideration of whether these periodicities is replicated in variability of certain burst properties like polarization.

Estimates of RM_iono are generally obtained from combining a model for Earth's magnetic field with The IONosphere Map EXchange (IONEX) ²⁹ maps, describing the ionized turbulent plasma layer in the upper atmosphere. There are numerous software packages available that attempt to accurately describe various ionospheric contributions (e.g., ionFR, RMextract; Sotomayor-Beltran et al. 2013; Mevius 2018b). Another package, ALBUS, ³⁰ developed at the Dominion Radio Astrophysical Observatory (DRAO) which hosts the CHIME telescope, uses readings from local GPS stations. This allows for a higher cadence of calculations and a better sampling of the local variability in the ionosphere that, in theory, should lead to more reliable RM_iono estimates.

A systematic comparison of the performance of these software packages is planned using CHIME/Pulsar (CHIME/Pulsar Collaboration et al. 2020) and has already led to improved RM measurements for 80 pulsars (Ng et al. 2020). Preliminary testing has shown RMextract and ALBUS to be in reasonable agreement at elevations greater than 45 degrees but somewhat discrepant at lower elevations (A.G. Willis, private communication). Tracing the source of this discrepancy will be important for CHIME/FRB where the instrument design and the tiling of the 1024 formed sky beams of the real-time search pipeline yield a non-negligible fraction of FRBs detected at lower elevations.

In Appendix A, we provide details on how the QU-fitting routine is modified to successfully fit for additional parameters that characterize instrumental systematics. Here, we explore how the simple Faraday model of Figure 1 can be extended to fit for additional features that are intrinsic to the polarized signal. Figure B4 shows the fitted spectrum of FRB 20191219F, where the model has been extended to fit for spectral parameters that define the linear and circular polarized signal over the CHIME band. In particular, a power-law spectrum is assumed for the two polarized components such that,

$$\begin{eqnarray}\begin{array}{rcl}p(\nu ) & = & {p}_{0}{\left(\displaystyle \frac{\nu }{{\nu }_{0}}\right)}^{{\gamma }_{L}},\\ {p}_{V}(\nu ) & = & {p}_{V,0}{\left(\displaystyle \frac{\nu }{{\nu }_{0}}\right)}^{{\gamma }_{V}}.\end{array}\end{eqnarray} \tag{ B3 }$$

Here, p₀ and p_V,0 are the linear and circular polarized fractions at the bottom of the burst sub-band.

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Unlike the model currently implemented in the pipeline, this model allows for a non-zero circularly polarized component that is intrinsic to the source and allows both circular and linear polarized fractions to vary across the burst sub-band. Comparing Figure 6 with Figure B4, we see that the refined model results in a substantially improved fit, particularly at frequencies above 600 MHz, where the default model does a poor job of simultaneously fitting Stokes U and V.

The 2D posterior distributions for the fit parameter of the refined model show substantial degeneracy between (p₀, p_V,0) and their respective indices, (γ_L , γ_V ). As is the case for other model parameters, uniform priors are assumed here. Inspection of the corner plot reveals that QU-fitting of this refined model leads to p₀ converging on an unrealistic value, p₀ > 1. This is likely an artifact of coupling between the X, Y polarizations in individual feeds leading to mixing between linear and circular polarized signal. This effect is likely only noticeable for extremely bright events such as the one analyzed here.

Figure B5 shows the total (black), linear (red), and circular (blue) polarized fractions across the burst sub-band. Solid and dashed lines represent the intrinsic model fits before and after convolution with cable delay. Simple power-law models for the linear and circular components do a remarkable job fitting the observed spectrum. Looking at the model fits for the intrinsic spectrum, the burst appears to be 100% linearly polarized at the 400 MHz. The steady decrease in the linearly polarized fraction toward higher frequencies seems to indicate that this is intrinsic to the source itself and not a result of differential Faraday rotation through a scattering foreground medium. Interestingly, this loss of linear polarized component at higher frequencies is partially offset by an increase in the circular component, and possibly suggests some relation either through Faraday conversion (e.g., Gruzinov & Levin 2019, Vedantham) or some other process.

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