Use of the Generalized Vector Addition Theorem for Antenna Position Translation for Spherical Mode-Filtering-Based Reflection Suppression

Monochromatic mode-filtering-based scattering suppression techniques have been shown to be applicable to all commonly used forms of far- and near-field antenna and RCS measurement techniques. Traditionally, the frequency-domain mode-filtering technique takes a far-field pattern, either measured directly or obtained using a suitable near-field to far-field transformation, as its starting point. The measurement is required to be conducted such that the antenna under test (AUT) is positioned offset from the origin of the measurement coordinate system. This physical offset introduces a phase taper across the AUT pattern and results in far greater interference occurring between the direct and indirect parasitically coupled spurious scattered signals. The method is very general and can be applied to all forms of near- or far-field measurements. However, for the case of a spherical near-field measurement (SNF) approach, it is somewhat cumbersome and tedious as first we must perform a probe-corrected spherical near-field to far-field transformation, which itself involves the computation of a complete set of spherical mode coefficients, and then after the displacement has been applied to the far-electric-fields, a second spherical wave expansion and summation is required to implement the mode-filtering procedure. While this data processing chain has been widely deployed and exhaustively validated, it requires passing through the asymptotic far-field, which inevitably results in additional computational effort, as well as incurring some loss of information, which can impose limitations on further near-field applications. This paper introduces an alternative, novel, rigorous algorithm that applies the displacement of the AUT directly using the vector addition theorem for spherical waves. An efficient implementation has been developed, and it is shown that the new, rigorous algorithm for the translation and filtering can be easily implemented directly within the data processing chain of any standard spherical near-field transformation algorithm, avoiding the need to first transform to the asymptotic far-field and also removing the need for a secondary spherical mode expansion and secondary spherical mode summation. While the vector addition theorem required for the spherical near-field to far-field transformation (SNFFFT) algorithm has been described in detail in the open literature, its implementation has been limited to the case of impinging waves and positive <i>z</i>-directed translations where the magnitude of the displacement is necessarily larger than the minimum sphere radius (MRE). In the current paper, the addition theorem will be derived in a new form that allows the translation to be applied in <i>any</i> desired direction, without the need for additional rotations, as well as being valid for solutions for waves transitioning through the sphere and applicable for the case where the magnitude of the translation is smaller <i>or</i> larger than the radius of the minimum sphere.