Main beam modeling for large irregular arrays

Abstract

Large radio telescopes in the 21^st 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.

Appendix I

We consider a 2D aperture in the x y −plane with an equivalent current distribution $I^{e} \equiv ({I^{e}_{x}},{I^{e}_{y}})$, with a Fourier transform given by f _t ≡ (f _{t x} , f _{t y} ). The radiation pattern G ≡ (G _x , G _y , G _z ) of the aperture can be expressed as the following projection [19]

$$ \bar{G} = C \left( \bar{f_{t}} - \hat{u} (\bar{f_{t}}.\hat{u}) \right) $$

(16)

where C = −j k η/4π with η is the free-space impedance. The inverse Fourier transform of G _x corresponds to the “pseudo-current” I _x (see (13)). Likewise I _y and G _y are linked by a Fourier transform. From (16), the G _x and G _y components of the pattern are obtained as

$$\begin{array}{@{}rcl@{}} G_{x} &=& C \left( f_{tx} - u_{x}(f_{tx}u_{x} + f_{ty}u_{y}) \right) \end{array} $$

(17)

$$\begin{array}{@{}rcl@{}} G_{y} &=& C \left( f_{ty} - u_{y}(f_{tx}u_{x} + f_{ty}u_{y}) \right) \end{array} $$

(18)

It is noted that the G _z pattern is automatically obtained from G _x and G _y patterns via the relation $\bar {G}.\hat {u} = G_{x}u_{x} + G_{y}u_{y} + G_{z}u_{z} = 0$. The above expression is consistent with Equation (11) in [18], noting that $\hat {u} \equiv (u_{x},u_{y},u_{z}) = (\sin \theta \cos \phi ,\sin \theta \sin \phi ,\cos \theta )$.

From (17) and (18), one can easily express (f _{t x} , f _{t y} ) in terms of (G _x , G _y ) as

$$ \left[\begin{array}{l} f_{tx} \\ f_{ty} \end{array}\right] =\frac{1}{{C\,{u^{2}_{z}}}} \left[\begin{array}{cc} 1-{{u^{2}_{y}}} \,\,\,\,\,\,\,\,\, {u_{x}u_{y}} \\ \,\, {u_{x}u_{y}} \,\,\,\,\,\,\,\,\, \,\,\, 1-{{u^{2}_{x}}} \end{array}\right] \left[\begin{array}{l} G_{x} \\ G_{y} \end{array}\right] $$

(19)

The left-hand side of (19) is the Fourier transform of equivalent currents, while [G _x G _y ]^T is the Fourier transform of the “pseudo current” used in this paper. (19) provides the link between these two currents. It would also be possible to start from physical or equivalent currents (if available), and to model the pattern (f _{t x} , f _{t y} ) using the technique presented in this paper. The radiation pattern G _x , G _y are then obtained from (17) and (18). For arrays of complex antennas with the available radiation pattern, the “pseudo-current” probably is the most straightforward way, because it only involves a Fourier transform link without any projection.

As a reminder, equivalent electric $\bar {I}^{e}$ and magnetic currents $\bar {M}^{e}$ [19, Section 4.3; 21], on the surface enclosing electromagnetic sources correspond to $\hat {n}\times \bar {H}$ and $\bar {E}\times \hat {n}$, where $\bar {E}$ and $\bar {H}$ are electric and magnetic fields, $\hat {n}$ is the normal unit vector to the surface. When the surface corresponds an infinite plane, the magnetic equivalent current $\bar {M}^{e}$ can be omitted if the equivalent electric currents are doubled [21].