Grating lobe free pitch

A phased array is a sampled aperture. Sampling a wavefront at intervals of p produces the wanted main lobe plus repeats of it at the spatial-frequency aliases, the grating lobes. They appear at angles satisfying sin θ_g = sin θ_s ± m λ/p. A grating lobe only exists as a real beam when that sine has magnitude of one or less; otherwise the alias is evanescent and stays trapped at the aperture.

Setting the first order alias exactly at grazing incidence gives the design rule p ≤ λ / (1 + sin θ_max). For an array that only fires straight ahead, a pitch of one wavelength is enough. To steer to 30 degrees the pitch must fall to about two thirds of a wavelength, and to steer to 60 degrees to about half a wavelength.

Grating lobes matter because they are almost as strong as the main lobe and they arrive at a different angle, so a reflector they illuminate is plotted at the wrong place. On a sectorial scan they show as a curved ghost that moves the opposite way to the real indication as the probe is moved.

For a probe on a wedge the sampling happens in the wedge, so the wavelength that controls grating lobes is the wedge wavelength (2730 m/s in Rexolite or Perspex), not the wavelength in steel. Wedge wavelengths are shorter, which is why wedge probes need finer pitch for the same steering range.

p_max = λ / (1 + sin θ_max)
λ = c / f
θ_free = asin(λ/p − 1)   (grating lobe free steering for the actual pitch)