Angular spectrum method for the estimation of the lateral profile of the ultrasound pressure field of the third harmonic

Nonlinear propagation is an important physical phenomenon, which is exploited in ultrasound harmonic imaging. Second harmonic imaging, is most often used in clinical practice. By using higher harmonics image quality and tissue characterization may be further improved. There is a clear need to model and understand the ultrasound field of the higher harmonics. Numerical simulations, often based on KZK equation, for studying nonlinear wave propagation are time consuming. In this paper, we present a closed-form solution for the higher harmonics pressure fields and propose a light-weighted procedure, based on the angular spectrum method (ASM), to compute them. Using the successive approximation method, we have derived an analytical expression, which predicts the change of the shape of the transmitted signal during propagation in a lossless medium. The theoretical findings were corroborated on the lateral beam profiles of a phased-array ultrasound transducer. The relative difference in the main lobe width between measurement and simulations at -6 dB was 5.5%, 6.5%, and 2.9% for 1^st, 2^nd, and 3^rd harmonic, respectively. The same parameters at -10 dB were of the same order of magnitude, with the exception of the 3^rd harmonic difference being 0.7%.