Focal law delays

A focal law is a list of transmit and receive delays, one per element, that makes every element's contribution arrive at the target point at the same instant. Geometrically the delay is simply a path difference divided by velocity: element i sits at x_i from the centre of the aperture, the focal point sits at range F and angle θ, and the distance between them is r_i = √(F² + x_i² − 2·F·x_i·sin θ).

Because a delay cannot be negative, the law is referenced to the element with the longest path, which fires first at zero delay. Every other element is delayed by (r_max − r_i)/c. The result is the familiar curved delay profile: a symmetric bowl for a pure focus, a straight ramp for pure steering, and a tilted bowl for both together.

The total delay span matters practically. It must fit inside the instrument's delay range and its delay resolution - typically 2.5 to 10 ns - because quantising the delays quantises the beam. Coarse delay resolution shows up as raised side lobes and as angular jitter in a sectorial scan.

These are contact or immersion laws, with no wedge. A wedge law also needs the refraction point on the wedge face solved element by element with Snell's law, which the instrument's ray tracer does; the wedge contribution is then added to each delay. Use this calculator to sanity check the shape and the span, not to hand-type wedge laws.

x_i = (i − (n+1)/2) · p
r_i = √(F² + x_i² − 2·F·x_i·sin θ)
t_i = (r_max − r_i) / c
delays in ns for mm and m/s: × 10⁶