Faraday Depth Resolution

• Faraday depth resolution is the measure of discriminating distinct magneto-ionic structures along the line of sight using the Fourier inversion of polarization data.
• It is determined by the λ² coverage, spectral channelization, and the properties of the RMSF, which together set the FWHM and sensitivity of the reconstruction.
• Emerging techniques like wavelet transforms, Gaussian process imputation, and 3D deconvolution offer super-resolution capabilities that extend beyond classical limits.

Faraday depth resolution quantifies the ability of broadband polarimetric radio observations to distinguish structures at different Faraday depths along the line of sight, providing a critical measure of tomographic fidelity in studies of cosmic magneto-ionic media. The practical resolution in Faraday depth space is fundamentally controlled by the frequency coverage, spectral sampling, and weighting of observations. The formalism underlying Faraday depth resolution extends from the RM-synthesis framework, in which the observed complex polarization as a function of wavelength squared, $P(\lambda^2)$, is Fourier-inverted to recover the Faraday dispersion function $F(\phi)$, which describes the polarized emission as a function of Faraday depth.

1. Principles and Formalism of Faraday Depth Resolution

The concept of Faraday depth resolution arises from the properties of the Rotation Measure Transfer Function (RMTF) or Rotation Measure Spread Function (RMSF), which acts as the point-spread function in Faraday space. When $P(\lambda^2)$ is only sampled over a finite and discrete range of $\lambda^2$, the reconstructed Faraday spectrum $\tilde F(\phi)$ is given by the convolution of the true Faraday dispersion function $F(\phi)$ with the RMTF: $$\tilde{F}(\phi) = \int F(\phi') \, \mathrm{RMTF}(\phi - \phi')\, d\phi'$$ For an observation covering $\lambda^2_{\min}$ to $\lambda^2_{\max}$ and uniform weighting, the FWHM of the main lobe of the RMTF, $\delta\phi$,—often identified as the Faraday depth resolution—is approximated by

$$\delta\phi \approx \frac{2\sqrt{3}}{\Delta\lambda^2}$$

where $$\Delta\lambda^2 = \lambda^2_{\max} - \lambda^2_{\min}$$ (Ordog et al., 10 Oct 2025, O'Sullivan et al., 2012, Andrecut et al., 2011, Kim et al., 2016, O'Sullivan et al., 2018).

This relationship encapsulates the essential trade-off: maximizing $\Delta\lambda^2$ (i.e., broadening the wavelength-squared coverage) directly improves Faraday depth resolution, allowing finer discrimination of separate RM components.

2. Maximum Recoverable Faraday Thickness and Related Scales

The sensitivity to Faraday-thick structures—continuous or extended distributions in $\phi$—is limited by the shortest wavelength (highest frequency) in the dataset. Structures broader than (Ordog et al., 10 Oct 2025): $$\phi_{\max\text{-scale}} \approx \frac{\pi}{\lambda^2_{\min}}$$ are strongly depolarized and effectively filtered out; observed power in $F(\phi)$ from such broad regions drops precipitously. This scale sets the largest Faraday-thick feature that can be robustly characterized. The maximum measurable Faraday depth, before bandwidth depolarization or grating lobes arise, is governed by the channel width in $\lambda^2$: $$|\phi_{\mathrm{max}}| \approx \frac{\sqrt{3}}{\delta(\lambda^2)_{\mathrm{chan}}}$$ (O'Sullivan et al., 2012, Andrecut et al., 2011, Beck et al., 2012).

A table synthesizing generic values for characteristic Faraday depth scales in broadband polarimetric surveys is shown below:

Survey	$\Delta\lambda^2$ (m$^2$)	$\delta\phi$ (rad m$^{-2}$)	$\phi_{\max\text{-scale}}$ (rad m$^{-2}$)
DRAGONS (350–1030 MHz)	0.645	$\sim 6$	$\sim 38$
GMIMS-HBN (1280–1750 MHz)	0.0258	$\sim 150$	$\sim 110$
LoTSS-HBA (120–168 MHz)	3.05	$\sim 1.1$	$\sim 0.98$

Respective values are cited directly in (Ordog et al., 10 Oct 2025, Wolleben et al., 2021, O'Sullivan et al., 2018), with all parameters computed from their exact band specifications.

3. Impact of Frequency Coverage, Channelization, and Sidelobe Structure

The fundamental determinant of Faraday depth resolution remains the total $\lambda^2$-span. The performance, however, is impacted by other factors:

• Spectral Channelization: Finer channels push $|\phi_{\mathrm{max}}|$ higher, suppressing bandwidth depolarization and maximizing dynamic $\phi$-range, but increase per-channel noise (Ordog et al., 10 Oct 2025, O'Sullivan et al., 2012, Andrecut et al., 2011).
• RFI Excision: Gaps in $\lambda^2$-coverage due to RFI flagging introduce sidelobes in the RMTF. Use of interpolation or imputation (e.g., Gaussian Process filling) can partially restore resolution and lower RMTF sidelobes (Ndiritu et al., 2021).
• Weighting and Spectral Windowing: Uniform weighting in $\lambda^2$ minimizes sidelobes but is sensitive to RFI. Tapered or apodized windows suppress sidelobes at the cost of broader RMSF and hence coarser $\delta\phi$ (O'Sullivan et al., 2012, Andrecut et al., 2011).
• Restoring Beam Definitions: The adoption of a "nominal" ($\lambda_0^2 = \langle \lambda^2 \rangle$) vs. "full" ($\lambda_0^2 = 0$) reference in RM synthesis leads to different beamwidths: "full" beams ($$\phi_{\text{full}} \simeq 2/(\lambda^2_{\max} + \lambda^2_{\min})$$) offer sharper separation of multiple components than the standard "nominal" ($$\phi_{\text{nom}} \simeq 3.8/(\lambda^2_{\max}-\lambda^2_{\min})$$) (Rudnick et al., 2023).

4. Recoverability of Faraday Structures and Survey Design Constraints

Discrimination of multiple or continuous Faraday components is limited not only by $\delta\phi$ but also by $\phi_{\max\text{-scale}}$ and the finer structure of the RMTF. Detectability degrades rapidly once component separation falls below $\sim\delta\phi$; two delta-function components closer than $\delta\phi$ will blend, though advanced deconvolution or sparsity methods can sometimes extract partial information (Andrecut et al., 2011). The sensitivity to broad, continuous Faraday distributions is characterized not just by $\phi_{\max\text{-scale}}$, but also by the empirical half-power width $W_{\text{max}}$—the Faraday thickness at which detected power drops by $>50\%$: $$W_{\text{max}} \approx 0.67 (\lambda^{-2}_{\min} + \lambda^{-2}_{\max})$$ for Gaussian profiles (Rudnick et al., 2023). Surveys optimized for complexity must match $\delta\phi$ and $W_{\text{max}}$ to the astrophysical scales of interest, e.g., interstellar medium filaments or cluster relics exhibiting internal Faraday structure at $10$–$30$ rad m$^{-2}$.

A distinction arises in the requirements for different science goals:

• High-resolution Faraday tomography (e.g., Galactic turbulence, thin ISM features) demands both fine $\delta\phi$ and sensitivity to low $\phi_{\max\text{-scale}}$ via coverage at low frequencies and large $\Delta\lambda^2$ (O'Sullivan et al., 2018, Eck et al., 2016).
• Detection of extreme-RM sources (e.g., cluster cores, AGNs) depends critically on narrow channel widths to avoid bandwidth depolarization.

5. Advances in Faraday Tomography and Resolution Enhancement Techniques

While the classical $\delta\phi$ limit is dictated by Fourier-domain coverage, emerging algorithms leverage sparsity, wavelet transforms, and nonparametric modeling to push beyond this nominal boundary:

• Wavelet and Multi-scale Approaches: Wavelet-based RM synthesis and algorithms such as CRAFT+WS (CRAFT with Wavelet Shrinkage) enable super-resolution, providing improved reconstruction of complex Faraday profiles and detection below the classical $\delta\phi$, as validated by lower NRMSE and enhanced recovery of multi-component FDFs (Cooray et al., 2021, Beck et al., 2012).
• Gaussian Process Imputation: Probabilistic interpolation over RFI-flagged channels (GP models) demonstrably restore both effective $\Delta\lambda^2$ and reduce RMTF sidelobes, sometimes improving the measured Faraday depth resolution by nearly a factor of two under high flagging rates (Ndiritu et al., 2021).
• 3D Faraday Synthesis and Direction-Dependent Deconvolution: Full 3D algorithms coupled with direction-dependent corrections (DDFSCLEAN in DDFACET) supply not only formal realization of the theoretical $\delta\phi$, but also reduced imaging artifacts and unbiased recovery at high $|\phi|$ due to accurate modeling of frequency-dependent PSF and bandwidth depolarization (Gustafsson et al., 31 Mar 2025).

6. Practical Recommendations and Limitations

Survey planning for optimized Faraday depth resolution requires maximizing $\Delta\lambda^2$, maintaining continuous spectral coverage, and minimizing instrumental gaps and RFI (Ordog et al., 10 Oct 2025, Andrecut et al., 2011, Wolleben et al., 2021). Spectral channel widths should be chosen such that the corresponding $|\phi_{\mathrm{max}}|$ is well above anticipated astrophysical RMs or Faraday dispersions. Trade-offs include increased data volume (especially at lower frequencies), more complex calibration (e.g., ionospheric RM corrections scaling as $\lambda^2$), and frequency-dependent instrumental beams requiring homogenization for consistent synthesis (Ordog et al., 10 Oct 2025). Clean bias—systematic underestimation of amplitudes after deconvolution—must be calibrated for each survey using forward simulations (Rudnick et al., 2023).

A practical recipe for targeting a desired resolution $\delta\phi_{\mathrm{target}}$ involves selecting observing bands such that

$$\Delta\lambda^2 \gtrsim \frac{2\sqrt{3}}{\delta\phi_{\mathrm{target}}}$$

while ensuring that the channel width is sufficiently fine to avoid bandwidth depolarization over the expected RM range (O'Sullivan et al., 2012).

7. Astrophysical and Methodological Implications

Faraday depth resolution underpins the ability to disentangle line-of-sight structures in magneto-ionic media. High-resolution surveys (e.g., DRAGONS) enable mapping of Galactic foregrounds, calibration for other telescopes, and study of Faraday complexity on $\lesssim$ tens of rad m$^{-2}$ (Ordog et al., 10 Oct 2025). For extragalactic sources, insufficient resolution leads to blended RM components and misinterpretation of source structure or foreground contributions (O'Sullivan et al., 2012, Kim et al., 2016). Emerging super-resolution techniques are essential for SKA-class and next-generation surveys, where the complexity of Faraday spectra is expected to be significantly underestimated under classical limits (Cooray et al., 2021). The breadth of scales accessed is directly determined by the ratio $(\lambda_{\max}/\lambda_{\min})^2$, which controls not only the range of $\phi$ but the dynamic range of scales and precision in intrinsic polarization angle recovery (Beck et al., 2012).

In summary, Faraday depth resolution is determined by the FWHM of the RMTF, set by the total $\lambda^2$ coverage of the observation. Wide, continuous spectral bands and fine channelization yield superior resolution and maximize the detectability of both thin and moderately thick features in Faraday space, whereas new computational methodologies are required to achieve and interpret super-classical resolutions in complex and noisy datasets.