Planar phased array antenna relationships for FFT and Butler matrix generated beamlets with graphical display of results

In many present day communication systems it is desirable to have multiple beam illumination of the coverage area. This is certainly true when the goal is ubiquitous earth coverage from an orbiting satellite constellation. For previous geosynchronous satellite applications where a limited number of beams were required in a limited-scan configuration an array feed and reflector were well suited. For low earth orbit (LEO) applications, where wide scan is necessary and many beams are required, a direct-radiating phased array with orthogonal beamforming (Butler matrix and FFT beamformers are examples) is more appropriate. This paper defines and graphically depicts the relationship between the beamlets (diffraction limited beams) created by an FFT or Butler matrix beamformer and (1) scan angles containing the primary beamlet set, (2) grating lobe positions (scan angles) in real space, (3) resulting grating lobe patterns, and (4) the fraction of useable beamlets (i.e. the primary beamlets which are not in imaginary space nor would result in grating lobes in real space if used). The above conditions are defined and graphically portrayed for both rectangular (square) element spacing and triangular (equilateral) element spacing. The results are graphically displayed for selected element spacings between one-half and one wavelength. Relationships are defined to allow an extrapolation of results to lattices with greater than one wavelength element spacing to accommodate limited-scan antenna design.