  Large radio telescopes in the 21st century such as the Low-Frequency Array
(LOFAR) or the Murchison Widefield Array (MWA) make use of phased aperture
arrays of antennas to achieve superb survey speeds. The Square Kilometer Array
low frequency instrument (SKA1-LOW) will consist of a collection of non-regular
phased array systems. The prediction of the main beam of these arrays using a
few coefficients is crucial for the calibration of the telescope. An effective
approach to model the main beam and first few sidelobes for large non-regular
arrays is presented. The approach exploits Zernike polynomials to represent the
array pattern. Starting from the current defined on an equivalence plane
located just above the array, the pattern is expressed as a sum of Fourier
transforms of Zernike functions of different orders. The coefficients for
Zernike polynomials are derived by two different means: least-squares and
analytical approaches. The analysis shows that both approaches provide a
similar performance for representing the main beam and first few sidelobes.
Moreover, numerical results for different array configurations are provided,
which demonstrate the performance of the proposed method, also for arrays with
shapes far from circular.
