Advanced Diagnostics for the Study of Linearly Polarized Emission. I: Derivation
Linearly polarized emission is described, in general, in terms of the Stokes parameters $Q$ and $U$, from which the polarization intensity and polarization angle can be determined. Although the polarization intensity and polarization angle provide an intuitive description of the polarization, they are affected by the limitations of interferometric data, such as missing single-dish data in the u-v plane, from which radio frequency interferometric data is visualized. To negate the effects of these artefacts, it is desirable for polarization diagnostics to be rotationally and translationally invariant in the $Q$-$U$ plane. One rotationally and translationally invariant quantity, the polarization gradient, has been shown to provide a unique view of spatial variations in the turbulent interstellar medium when applied to diffuse radio frequency synchrotron emission. In this paper we develop a formalism to derive additional rotationally and translationally invariant quantities. We present new diagnostics that can be applied to diffuse or point-like polarized emission in any waveband, including a generalization of the polarization gradient, the polarization directional curvature, polarization wavelength derivative, and polarization wavelength curvature. In Paper II we will apply these diagnostics to observed and simulated images of diffuse radio frequency synchrotron emission.