Patterns of primary beam non-redundancy in close-packed 21 cm array observations

Radio interferometer arrays such as HERA consist of many close-packed dishes arranged in a regular pattern, giving rise to a large number of `redundant' baselines with the same length and orientation. Since identical baselines should see an identical sky signal, this provides a way of finding a relative gain/bandpass calibration without needing an explicit sky model. In reality, there are many reasons why baselines will not be exactly identical, giving rise to a host of effects that spoil the redundancy of the array and induce spurious structure in the calibration solutions if not accounted for. In this paper, we perform a wide range of simulations for a small HERA-like array to understand how different types of non-redundancy manifest in the observed interferometric visibilities and their resulting frequency (delay-space) power spectra. We focus in particular on differences in the primary beam response between antennas, including variations in the main lobe, sidelobes, ellipticity, and orientation. We find that different types of non-redundancy impart characteristic patterns into the redundant gain solutions, which in turn introduce additional structure into the calibrated visibilities and therefore the delay spectra. We show that the most severe effects of primary beam non-redundancy are induced by the brightest sources passing through the beam, while diffuse emission has a lesser (but non-negligible) effect. We also find that redundant baseline groups with `outlier' antennas (where only one antenna deviates from perfect redundancy) sustain the largest gain errors, while even $\mathcal{O}(1)$ non-redundancies in the sidelobes seem to have a relatively minor impact in comparison.