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.