A transducer can be focused by curving the crystal to conform to the shape of a sphere, as shown in Figure P10.1. To analyze the field pattern, however, it is convenient to assume that the transducer is flat but that we can independently control the time delay of each element on its face. Assume that the point (x_m, y_m, 0) is the point farthest out on the flat face.

(a) Show that the time delay for a point (x₀, y₀, 0) on the flat face is approximately

where  m and d is the desired focal depth.

(b) Use the steady-state approximation and show that at z = d the Fresnel field pattern is

where S(u, v) is the 2-D Fourier transform of s(x, y)

(c) Discuss the merits of focusing.

Figure P10.1 Geometry of curved transducer for Problem 10.16.