Extended angular spectrum method for calculation of higher harmonics

National audienceUltrasound imaging in the superharmonics band is getting important in the ultrasound community, since all the advantages of the second harmonic imaging modality are further increased at higher harmonics (3rd, 4th, ...). Then, a fast, reliable and relatively accurate modelling and estimation tool of ultrasound field at higher harmonics is of clear interest. Different numerical solutions (in time or frequency domain) of KZK equation are mostly used. Although the KZK simulators are largely used to understand the nonlinear propagation, they are not appropriated to simulate steered beams, propagating through inhomogeneous media. Moreover, simulations based on KZK equation are time consuming. We propose an alternative method by solving the Westervelt equation based on the Angular Spectrum Method (ASM). The idea consists in separating the equation in n equations (n is number of considered harmonics). For each harmonic component, a nonlinear wave equation is deduced, where the n-th harmonic depends on the previously calculated harmonics. Since the solution of the nonlinear wave equation is a known mathematical problem, the calculation is fairly simple. Coordinate transformation can be used to simulate steered propagation. Furthermore, modelling the solution of the nonlinear equation in the frequency domain (hence the name angular spectrum method) facilitates the introduction of the attenuation in the model. The solution of the nonlinear wave equation was implemented in Matlab software. The calculation of ultrasound field up to third harmonic showed good agreement with experimental measurements performed in a water tank. The main beam width and side lobe levels differ between simulation and measurements by less than 5%. The calculation of the pressure field was about 15 times faster than the KZK solution