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Nonlinear pulsed pressure field from focused rectangular apertures: experimental and numerical simulation results

In this paper the nonlinear pulsed acoustic pressure fields from the focused square and rectangular apertures are considered. Experimental results in water of the 4D nonlinear sound field radiated from a 2.8 MHz focused (focal distance F = 80 mm) transducer of a square (20 x 20 mm) and rectangular (10 x 25 mm) geometry for various excitation levels (producing the average pressure P0 at the source equal to 0.045, 0.225 and 0.45 MPa) are presented. The measurement results are compared with the numerical calculations obtained for the same boundary conditions. The novel, developed by the second author, numerical algorithm was used to simulate the 4D nonlinear pressure field propagation from nonaxisymmetric focused acoustic sources radiating pulsed waves in the attenuating media. Our theoretical model is based on the Time-Averaged Pressure Envelope (TAPE) method, recently developed in our lab. This method employs the representation of the propagated pulsed pressure waveform as a quasi-Fourier series, i.e. as a superposition of sinusoidal wavelet- like packets with carrier frequencies equal to harmonics of the initial tone burst, and with envelopes (being functions of space coordinates and time) determined by the TAPE method. The spectrum of the propagating pressure waveform is considered to be a sum of contributions of each wavelet spectrum. Our model uses the second-order operator splitting approach with an incremental propagation scheme whereby the effects of combined diffraction and absorption are computed separately from the effects of nonlinear harmonic interactions over incremental steps. The proposed model is free from paraxial approximation, is computationally efficient and capable of predicting the 4D ultrasound field in nonlinear and attenuating media with the arbitrary absorption from pulsed, arbitrary shaped, plane and focused sources (including phased arrays with an angular beam deflection). Using a computational power of a standard PC a computation time required for the full 4D nonlinear field simulation depends on the source dimensions, radiated frequency and excitation level as well as on the medium absorption and nonlinearity strength and can vary from few minutes to few hours. Comparison between experimental and simulation results clearly show that our numerical model predicts well the structure of the nonlinear ultrasound field for all boundary conditions considered. I. INTRODUCTION The theoretical and experimental studies of the finite amplitude acoustic waves propagation in attenuating media from nonaxisymmetric sources rather rarely can be found in literature in spite of the fact that probes of the rectangular geometry (such as linear phased arrays) are commonly used in clinical practice for medical ultrasonic imaging purposes. The main reason of such situation is a lack in simpler theoretical models and in computationally efficient numerical algorithms that are able to predict accurately the nonlinear effects in 4D ultrasound fields from pulsed, arbitrarily shaped sources (plane and focused) in biological media with arbitrary frequency- dependent absorption. In recent years the only study describing the computationally efficient numerical model that is able to simulate accurately the 4D nonlinear ultrasound field in water and in biological tissues from pulsed nonaxisymmetric sources was developed by Zemp et al. (1). Their model is based on the second order operator-splitting method, proposed by Tavakkoli et al., with the modified fractional step scheme whereby the combined effects of diffraction and absorption are accounted for over half-steps and the effects of nonlinear harmonic interactions over full incremental steps. The computation of diffraction and absorption sub-steps was based on the angular spectrum technique with modified sampling method (to obtain computational savings due to larger axial propagation steps) while the computation of nonlinear steps was based on the time-domain solution to Burgers' equation. There are not reports yet describing an experimental confirmation of an agreement between the simulated nonlinear acoustic pulsed fields in water or in soft tissues from nonaxisymmetric focused sources (obtained by using the numerical model proposed) and nonlinear field from realistic probes. In this work the experimental measurement results of the 4D nonlinear field from the both square and rectangular ultrasonic focused transducer radiating the pulsed pressure wave in water are presented. The realistic beam patterns are compared with the simulation results obtained by using our novel numerical algorithm.