Efficient storage of embedded element patterns for low frequency radio telescopes

The use of spherical modes offers an efficient solution for storing embedded element patterns with significant angular structure for large scale arrays, such as the Square Kilometer Array (SKA)-Low radio telescope. These patterns are required for calibration of the numerous stations comprising the telescope, each containing several hundred elements, and operating over a 7:1 bandwidth. However, implementation is significantly complicated by the many differences in the notation used in the literature for the Legendre special functions. The differing phasor conventions used in electrical engineering and physics further complicate this. This paper synthesizes much of the existing literature on this topic, paying special attention to these issues. Mathematical implementation issues are also addressed. A number of suitable tests using canonical dipole radiators to verify correct implementation are outlined. The paper concludes with tests on an individual SKALA4 antenna and a full-scale SKA-Low prototype station comprising 256 of these antennas. The storage saving afforded is some three orders of magnitude; this is very significant for a full SKA-Low station. Supporting material summarizes differing formulations and conventions encountered in the literature.