Propagation Through Inhomogeneous Media Using the Angular Spectrum Method

Abstract

This paper presents a technique for analyzing the acoustic fields generated by ultrasonic transmitters when radiating into layered, inhomogeneous media. The technique is based on the angular spectrum method of wavefield analysis, which is used to forward- or backpropagate acoustic fields between two parallel planar surfaces. Wave propagation is modelled as shift invariant filtering in the spatial frequency domain, with each spatial frequency component multiplied by the appropriate phase propagation factor. By modifying the phase propagation factors, the effects of attenuation, dispersion, refraction, and phase distortion may be modelled. Simulation results demonstrate several features of this approach, and experimental results show the ability of the technique to determine the pressure and velocity fields from different transducers. Wideband propagation is considered as an extension to the basic monochromatic model. 1 .O INTRODUCTION The design of ultrasonic imaging transducers has evolved over the last thirty-five years [l] to the point that there are standard design approaches employed to produce a transducer with specified imaging properties [2]. These imaging properties are typically stated in terms of resolution in a homogeneous medium such as water. Unfortunately, tissue is not homogeneous, and the resolving power of the transducer is affected by the attenuation, refraction, and phase distortion present in tissue. The goal of our work has been to develop a prediction technique which could simultaneously account for several of these tissue propagation properties. In addition, the technique was to be flexible, computationally efficient, and suitable for the analysis of practical transducer geometries, including phased linear arrays. This latter requirement stemmed from recent results [3,4], which note that the inhomogeneous nature of tissue is the limiting factor in the development of higher resolution diagnostic ultrasound equipment. These requirements formed the basis for the approach which was taken. The technique was based on the angular spectrum, or Fourier decomposition method [5], which is one of the more powerful techniques for predicting acoustic field distributions in homogeneous media. The angular spectrum method decomposes the pressure distribution over a plane surface into a two-dimensional spectrum of plane waves. Propagation is then modelled by multiplying each Fourier spectral component by the appropriate phase factor, effectively performing a linear filtering operation. The advantages of this technique include: high computational efficiency using the FFT algorithm, high spatial resolution even in the nearfield of the acoustic source, and the ability to predict acoustic fields over an entire plane in a single two-dimensional FFT operation. The method has been used to analyze transducer performance [6], to predict transducer beam patterns [7], and to image scattering objects [a], all in homogeneous media. For completeness, the basic derivation of the angular spectrum method is presented in Section 2. One of the underlying assumptions in the development of the angular spectrum method is that the medium is