Semi-analytical computation of the acoustic field of a segment of a cylindrically concave transducer in lossless and attenuating media

Conventional ultrasound transducers used for medical diagnosis generally consist of linearly aligned rectangular apertures with elements that are focused in one plane. While traditional beamforming is easily accomplished with such transducers, the development of quantitative, physics-based imaging methods, such as tomography, requires an accurate, and computationally efficient, model of the field radiated by the transducer. The field can be expressed in terms of the Helmholtz-Kirchhoff integral; however, its direct numerical evaluation is a computationally intensive task. Here, a fast semianalytical method based on Stepanishen's spatial impulse response formulation [J. Acoust. Soc. Am. 49, 1627-1638 (1971)] is developed to compute the acoustic field of a rectangular element of cylindrically concave transducers in a homogeneous medium. The pressure field, for, lossless and attenuating media, is expressed as a superposition of Bessel functions, which can be evaluated rapidly. In particular, the coefficients of the Bessel series are frequency independent and need only be evaluated once for a given transducer. A speed up of two orders of magnitude is obtained compared to an optimized direct numerical integration. The numerical results are compared with Field II and the Fresnel approximation.