On the stability of the effective apodization of the nonlinearly generated second harmonic with respect to range.

The concept of an effective apodization was introduced to describe the field pattern for the nonlinearly generated second harmonic (2f) within the focal zone using a linear propagation model. Our objective in this study was to investigate the validity of the concept of an effective apodization at 2f as an approach to approximating the field of the second harmonic over a wide range of depths. Two experimental setups were employed: a vascular imaging array with a water path and an adult cardiac imaging array with an attenuating liver path. In both cases the spatial dependencies of the ultrasonic fields were mapped by scanning a point-like hydrophone within a series of planes orthogonal to the propagation direction. The sampling distances were located before, within, and beyond the focal zone. The signals were Fourier transformed and the complex values at 2f were linearly backpropagated to the transmit plane in order to obtain an effective apodization. The measured results demonstrated a relatively constant effective apodization at 2f as a function of propagation distance. Finite amplitude computer simulations were found to be in agreement with these measurements. Thus the measure of the effective apodization at 2f provides an approximation to the second harmonic field outside the focal zone.

Spatial coherence of backscatter for the nonlinearly produced second harmonic for specific transmit apodizations.

To be successful, correlation-based, phase-aberration correction requires a high correlation among backscattered signals. For harmonic imaging, the spatial coherence of backscatter for the second harmonic component is different than the spatial coherence of backscatter for the fundamental component. The purpose of this work was to determine the effect of changing the transmit apodization on the spatial coherence of backscatter for the nonlinearly generated second harmonic. Our approach was to determine the effective apodizations for the fundamental and second harmonic using both experimental measurements and simulations. Two-dimensional measurements of the transverse cross sections of the finite-amplitude ultrasonic fields generated by rectangular and circular apertures were acquired with a hydrophone. Three different one-dimensional transmit apodization functions were investigated: uniform, Riesz, and trapezoidal. An effective apodization was obtained for each transmit apodization by backpropagating the values measured from within the transmit focal zone using a linear angular spectrum approach. Predictions of the spatial coherence of backscatter were obtained using the pulse-echo Van Cittert-Zernike theorem. In all cases the effective apodization at 2f was narrower than the transmit apodization. We demonstrate that certain transmit apodizations result in a greater spatial coherence of backscatter at the second harmonic than at the fundamental.

Spatial coherence of the nonlinearly generated second harmonic portion of backscatter for a clinical imaging system.

Correlation-based approaches to phase aberration correction rely on the spatial coherence of backscattered signals. The spatial coherence of backscatter from speckle-producing targets is predicted by the auto correlation of the transmit apodization (Van Cittert-Zernike theorem). Work by others indicates that the second harmonic beam has a wider mainlobe with lower sidelobes than a beam transmitted at 2f. The purpose of this paper is to demonstrate that the spatial coherence of backscatter for the second harmonic is different from that of the fundamental, as would be anticipated from applying the Van Cittert-Zernike theorem to the reported measurements of the second harmonic field. Another objective of this work is to introduce the concept of the effective apodization and to verify that the effective apodization of the second harmonic is narrower than the transmit apodization. The spatial coherence of backscatter was measured using three clinical arrays with a modified clinical imaging system. The spatial coherence results were verified using a pseudo-array scan in a transverse plane of the transmitted field with a hydrophone. An effective apodization was determined by backpropagating these values using a linear angular spectrum approach. The spatial coherence for the harmonic portion of backscatter differed systematically and significantly from the auto correlation of the transmit apodization.