Maximum focal depth

An array focuses by delaying the outer elements so that all contributions arrive in phase at one point. That only works where the aperture is still large compared with the spreading of the beam, which is inside the near field. Beyond N the delays required become vanishingly small and the focal law degenerates into plain steering: the beam is unchanged.

The hard limit is therefore F < N. In practice a focus is only worth setting where F is below roughly 0.8 N; closer to N the gain over the unfocused beam is negligible while the depth of field becomes very long.

If a deeper focus is needed the first remedy is a larger aperture, because N grows with the square of the aperture: doubling the number of elements in the group multiplies the maximum focal depth by four. Raising the frequency also lengthens the near field - N is inversely proportional to wavelength, so it rises in direct proportion to frequency - as far as the extra attenuation in the material allows. Lowering the frequency does the opposite: it shortens N and brings the maximum focal depth closer.

Instruments will happily accept a focal depth beyond the near field and display a focused-looking beam. Nothing physical happens. Always check the requested focus against N for the aperture actually in use, not for the whole probe.

N = k · A_long² / (4λ)
wedge equivalent = L_wedge · c_wedge / c
requirement: F + wedge equivalent < N
S_F = (F + wedge equivalent) / N
A_required ≈ √(4·λ·(F + wedge equivalent) / k)