  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], [2], [3], [4], [5], [6], and a description of
how to acquire coherent field maps for direct detector arrays was demonstrated
in Davis et. al. 2017. This technique has the ability to determine both
amplitude and phase radiation patterns from individual pixels on an array.
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].
  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.
