A Sine-Space, Mixed-Coordinate Polarization Representation for Rotated Phased Arrays: Using a linear algebra calculus to transform radiated fields

The industry-standard phased-array pattern coordinate system, sine space, leads to a representation of polarized fields which can be misleading. This is especially true at the array’s boresight, where the polarization vector degenerates and becomes multivalued. This article presents a mixed-coordinate polarization representation for phased-array fields, with position components in sine space and vector components in a selectable, rotated-coordinate-system basis. The transforms required to move to this representation,
${E}_{El'}{(}{u'}{,}\,{v'}{)}$
and
${E}_{Az'}{(}{u'}{,}\,{v'}{)}$
, from the fields described in a local spherical basis,
${E}_{\mathit{\theta}}{(}\mathit{\theta}{,}\mathit{\phi}{)}$
and
${E}_{\mathit{\phi}}{(}\mathit{\theta}{,}\mathit{\phi}{)}$
, are given. Although the formulation focuses on phased arrays, the transforms are general to the fields generated from any source (e.g., single-antenna and phased, conformal, and distributed arrays). Several applications of the developed basis transformations are provided to demonstrate their use and validity. Circular-polarization purity degradation based on a cross-Hertzian dipole pair is considered and quantified for a dual-polarized communication link. Additionally, the impact of different measurement approaches on the observed polarization is also examined. Finally, platform rotation with respect to a static polarization frame is investigated to quantify increases in the cross-polarization coupling of an antenna array over the observation angle, which may negatively impact a linearly polarized communication link due to an incurred polarization mismatch loss.