Complex field measurements are increasingly becoming the standard for state-of-the-art astronomical instrumentation. Complex field measurements have been used to characterize a suite of ground, airborne, and space-based heterodyne receiver missions,1-6 and a description of how to acquire coherent field maps for direct detector arrays was demonstrated in Davis et. al. 20177. This technique has the ability to determine both amplitude and phase radiation patterns from individual pixels on an array for direct comparison to optical simulations. Phase information helps to better characterize the optical performance of the array (as compared to total power radiation patterns) by constraining the fit in an additional plane.4 This is a powerful technique to diagnose optical alignment errors through the optical system, as a complex field scan in an arbitrary plane can be propagated either forwards or backwards through optical elements to arbitrary planes along the principal axis. Complex radiation patterns have the advantage that the effects of optical standing waves and alignment errors between the scan system and the instrument can be corrected and removed during post processing. Here we discuss the mathematical framework used in an analysis pipeline developed to process complex field radiation pattern measurements. This routine determines and compensates misalignments of the instrument and scanning system. We begin with an overview of Gaussian beam formalism and how it relates to complex field pattern measurements. Next we discuss a scan strategy using an offset in z along the optical axis that allows first-order optical standing waves between the scanned source and optical system to be removed in post-processing. Also discussed is a method by which the co-and cross-polarization fields can be extracted individually for each pixel by rotating the two orthogonal measurement planes until the signal is the co-polarization map is maximized (and the signal in the cross-polarization field is minimized). We detail a minimization function that can fit measurement data to an arbitrary beam shape model. We conclude by discussing the angular plane wave spectral (APWS) method for beam propagation, including the near-field to far-field transformation.