The element-gain paradox for a phased-array antenna

In a phased-array antenna with a very large number of regularly-spaced radiating elements, the gain realized at the beam peak is equal to the number of elements times the gain realized in the same direction when only one typical element is excited. The ideal radiation pattern of one such element in a large planar array has a cosine variation of gain with angle when the elements are closely spaced, and has a peak value of gain equal to
4\pi A/\lambda^{2}
where
A
is the area allotted to each element. The active impedance of each element in a practical phased array varies with scan angle, because of mutual coupling between the elements. The associated mismatch causes power to be returned to the generators, thereby reducing the gain realized by the array and by the element. The element pattern, measured in the proper environment of surrounding elements, deviates from the ideal pattern in proportion to this effect. Mutual coupling is inherently unavoidable in a closely-spaced infinite array of elements; for example, in a square array with less than
\lambda/\sqrt{2}
spacing. There is a loss of element efficiency caused by the coupling, and since coupling increases with closer spacing, this accounts for the lower gain expected from ideal elements with reduced allotted area. Grating lobes can exist when the elements are not closely spaced; for example, in a square array with more than
\lambda/2
spacing. In this case, the ideal pattern is truncated to discriminate against grating lobes; this gives the higher gain expected from ideal elements with larger allotted area. It is concluded that in a phased-array antenna having a very large number of regularly-spaced radiating elements, perfect impedance match for all scan angles can be postulated for every typical element without encountering any real discrepancy in the determination of element gain. In the absence of grating lobes, such an antenna would realize the greatest possible gain for all scan angles.