Full Resolution Deconvolution of Complex Faraday Spectra

Abstract: Polarized synchrotron emission from multiple Faraday depths can be separated by calculating the complex Fourier transform of the Stokes' parameters as a function of the wavelength squared, known as Faraday Synthesis. As commonly implemented, the transform introduces an additional term $\lambda_0^2$, which broadens the real and imaginary spectra, but not the amplitude spectrum. We use idealized tests to investigate whether additional information can be recovered with a clean process restoring beam set to the narrower width of the peak in the real ``full" resolution spectrum with $\lambda_0^2=0$. We find that the $\lambda_0^2$ choice makes no difference, except for the use of a smaller restoring beam. With this smaller beam, the accuracy and phase stability are unchanged for single Faraday components. However, using the smaller restoring beam for multiple Faraday components we find a) better discrimination of the components, b) significant reductions in blending of structures in tomography images, and c) reduction of spurious features in the Faraday spectra and tomography maps. We also discuss the limited accuracy of information on scales comparable to the width of the amplitude spectrum peak, and note a clean-bias, reducing the recovered amplitudes. We present examples using MeerKAT L-band data. We also revisit the maximum width in Faraday depth to which surveys are sensitive, and introduce the variable $W_{max}$, the width for which the power drops by a factor of 2. We find that most surveys cannot resolve continuous Faraday distributions unless the narrower full restoring beam is used.