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.

Compression or shear

Worked example

Probe frequency	5 MHz
Wave mode	shear
Material velocity	3240 m/s
Required steering angle (half range)	30 deg
Actual probe pitch	0.6 mm
Wavelength	0.648 mm
Maximum grating lobe free pitch	0.432 mm
Grating lobe free steering for the actual pitch	4.59 deg
Margin (p_max − actual pitch)	-0.168 mm

λ = 3240/(5 × 1000) = 0.648 mm. sin 30° = 0.5, so p_max = 0.648/1.5 = 0.432 mm. A 0.6 mm pitch probe gives λ/p = 1.08, so grating lobe free steering only reaches asin(0.08) = 4.59°, and the margin is 0.432 − 0.600 = −0.168 mm.

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