Widefield Interferometric Imaging

• Widefield interferometry is a method that exploits mutual coherence of electromagnetic fields to reconstruct spatially extended scenes across large fields of view.
• It employs advanced beam modeling and Fourier-based mapmaking to achieve full-polarization calibration, optimizing signal fidelity and computational efficiency.
• The approach enables lossless, horizon-to-horizon imaging critical for next-generation radio arrays and cosmic surveys, while mitigating polarization leakage.

A widefield interferometric approach refers to a suite of methodologies in which mutual coherence (or interference) of electromagnetic fields is exploited to reconstruct information about spatially extended scenes—often over fields of view much larger than the classical isoplanatic or small-angle regime. Such approaches are central to modern radio astronomical imaging, hyperspectral microscopy, precision metrology, and quantum-enhanced sensing. They are characterized by advanced measurement equations, beam (or transfer function) models, computationally efficient mapmaking, and often involve optimal polarization and calibration strategies to manage both signal fidelity and computational tractability.

1. Measurement Equation and Widefield Mapmaking Formalism

The foundation of widefield interferometric imaging is the generalized van Cittert–Zernike measurement equation that relates the observed visibilities to the sky brightness or object structure. For a coplanar array with fully polarized signal, the measurement equation is: $$v_{jk}(\nu, t) = \int_{\Omega} d^2\hat{s} \left[J_j(\hat{s}) \otimes J_k^*(\hat{s})\right]\,S(\hat{s})\,e^{2\pi i \mathbf{u}_{jk} \cdot \hat{s}}$$ where:

• $v_{jk}$ is the 4-vector of measured visibilities for all polarization products $[v_{pp}, v_{qq}, v_{pq}, v_{qp}]^T$,
• $J_j(\hat{s})$ is the (direction-dependent) Jones matrix of antenna $j$,
• $S(\hat{s})$ is the polarized sky coherency vector,
• $\mathbf{u}_{jk}$ is the baseline vector in wavelengths.

The optimal widefield mapmaking procedure, as implemented in A-projection frameworks (such as the FHD pipeline), seeks a lossless and horizon-to-horizon reconstruction by:

• Gridding each baseline’s visibility onto the Fourier plane with a kernel given by the Fourier transform of the polarized beam (i.e., $\widetilde{B}_{jk,ab}(u)$), yielding the "apparent sky" for each instrumental polarization.
• Inverting the beam response to recover the instrumental coherency, followed by transformation into the true sky basis and Stokes parameters.

This full-field mapmaking rigorously propagates all direction-dependent gains, leakages, and widefield geometric effects, and provides a mathematically optimal estimator if the instrumental response is exactly known.

2. Polarized Beam Modeling and the Instrumental Basis

In widefield contexts, the instrumental (Jones) response generally varies across the field of view, and the choice of basis strongly affects computational efficiency and calibration quality. The Jones matrix is: $$J_j^{ZA}(\hat{s}) = \begin{pmatrix} J_{Zp} & J_{Ap} \ J_{Zq} & J_{Aq} \end{pmatrix}$$ for a zenith angle/azimuth basis, and rotated to equatorial (RA/Dec) coordinates using a parallactic-angle rotation.

A crucial strategy is the decomposition: $J_j^{RD}(\hat{s}) = R(\hat{s})\,D(\hat{s})$ where $D(\hat{s})$ is a diagonal matrix giving per-polarization beam amplitudes and $R(\hat{s})$ rotates the sky basis into the principal instrumental polarizations, which need not be orthogonal. In this "instrumental" basis, each observed polarization couples only to its corresponding component in the measurement, and the 4×4 beam matrix becomes diagonal in the $\{pp, qq, pq, qp\}$ ordering. This eliminates off-diagonal coupling and ensures beam, gain, and polarization leakage corrections are handled in a single linear system.

This instrumental basis is optimal in arrays with uniform (though potentially non-orthogonal or direction-dependent) antenna alignments.

3. Efficiency and Accuracy from Non-Orthogonal Polarization Bases

Configuring the analysis in the non-orthogonal instrumental basis leads to major improvements:

• Each visibility contributes to the uv plane exactly once, whereas in any orthogonal basis each measurement is redundantly gridded up to four times.
• The basis tracks the true polarization eigenmodes of the antennas, eliminating calculation of artificial mixing due to algebraic but unphysical projections.
• All widefield, direction- and frequency-dependent effects are simultaneously corrected in a single inversion, and the system can handle arbitrary beam patterns and polarization leakage without requiring ad hoc post-processing.

Formally, the polarization coherency in the instrumental basis is: $S_{\rm inst}(\hat{s}) = R(\hat{s})\,S(\hat{s})$ and polarized mapmaking proceeds solely in this basis with diagonalized beam matrices.

4. Fully Polarized Calibration in Widefield Regimes

Widefield calibration must handle the coupling between all polarization products over the field. The measured visibility for each cross-correlation is: $$v_{jk,ab} = g_{ja}\,g_{kb}^*\,m_{jk,ab} + n_{jk,ab}$$ where $g_{ja}$, $g_{kb}$ are direction-independent complex gains for each antenna and polarization, and $m_{jk,ab}$ is the model visibility.

FHD's approach:

1. Solves for independent $pp$ and $qq$ gains from the self-polarized visibilities.
2. Determines the cross-pol phase by least-squares fitting the $pq$ and $qp$ data, leveraging the fact that leakage of unpolarized emission into these products allows phase determination even in the absence of intrinsic cross-polarized sky signal.

This procedure is robust and avoids the degeneracies associated with cross-pol calibration in narrow-field or strictly orthogonal bases, fully exploiting the information present in widefield leakage patterns.

5. Distinctions from Conventional Widefield Polarimetry

The widefield interferometric approach embodied in FHD presents several distinct advantages:

• Enables full horizon-to-horizon (i.e., ultra-widefield) image reconstruction, with exact treatment of beam non-orthogonality and polarization effects. No faceting, w-stacking, or explicit source-by-source "peeling" is required.
• Guarantees lossless imaging to the theoretical thermal noise floor, contingent on beam model fidelity.
• Computational efficiency: minimal redundant calculations (each visibility mapped once); fast, built-in mosaicking normalized by the beam weights across snapshots.
• Linear, rigorous correction of direction-dependent polarization leakage and gain, yielding high-quality Stokes I, Q, U, V maps.
• Unifies mapmaking, beam correction, and fully polarized calibration in a mathematically rigorous and scalable framework. Comparative analysis versus pipelines for LOFAR (A-projection), wsclean, and MWA's RTS demonstrates superior computational performance and accuracy.

6. Significance, Limitations, and Broader Impact

The widefield interferometric approach, particularly when adopting a non-orthogonal instrumental basis and optimal mapmaking, achieves unprecedented efficiency and accuracy in fully polarized, horizon-to-horizon imaging. It is directly motivated by the requirements of next-generation radio facilities (MWA, LOFAR, SKA) and is operationally validated on large datasets (e.g., MWA).

Principal limitations are set by the accuracy of beam and gain models. Inexact beam modeling can introduce residual polarization leakage and limit dynamic range, emphasizing the need for comprehensive electromagnetic simulations and cross-validation with calibrators.

Wider implications include:

• Enabling high dynamic range, full-polarization sky surveys over extremely wide fields.
• Serving as a benchmark for calibration and imaging methods in next-generation arrays.
• Providing a path for real-time, fully self-consistent calibration strategies, as required for EoR and cosmic dawn science, precision Galactic magnetic field mapping, and transient detection over the widest possible FoVs.

In summary, the widefield interferometric approach—rooted in rigorous mapmaking, instrumental basis selection, and advanced calibration—defines the current state of the art in full-polarization, computationally scalable imaging for large radio arrays and offers a general paradigm for complex, multi-dimensional inverse problems in interferometric astronomy (Byrne et al., 2022).