An accurate analysis of the radiation characteristics of a plane piston transducer with phase apodization for focusing.

A plane piston transducer can be focused by a continuous variation of the excitation signal phase (or time delay) over the transducer surface. Prior analyses of this scheme used the Fresnel approximation, thereby limiting the validity. Using the angular spectrum method, an accurate radiation model of such a transducer has been developed that includes amplitude and phase apodization. The derivation includes the effects of diffraction and evanescent waves without using the Fresnel approximation. Moreover, this model develops insights into radiated field characteristics, including: (a) the spatial frequency bandwidth is constant over axial depth, suggesting that spatial resolution can be improved away from the focus; (b) the phase of the angular spectrum determines the spatial resolution for a given transducer configuration-a constant phase is optimal on any observation plane; (c) focusing can significantly increase the spatial frequency bandwidth; (d) the velocity potential on a plane parallel to the transducer is the Hankel convolution of the transducer surface velocity with the Green's function; and (e) evanescent waves decay both with increasing spatial frequency and axial depth. The analytical model and associated insights enhance understanding of the radiated field characteristics, which can be of value in the development of signal processing techniques for image enhancement.