Nonlinear change of on-axis pressure and intensity maxima positions and its relation with the linear focal shift effect

Abstract

A comprehensive experimental, analytical and numerical study of the true focal region drift relative to the geometrical focus (focal shift effect) in acoustic focused beams and its nonlinear evolution is presented. For this aim, the concept of Fresnel number, proportional to the linear gain, is introduced as a convenient parameter for characterizing focused sources. It is shown that the magnitude of the shift is strongly dependent on the Fresnel number of the source, being larger for weakly focused systems where a large initial shift occurs. Analytical expressions for axial pressure distributions in linear regime are presented for the general case of truncated Gaussian beams. The main new contribution of this work is the examination of the connection between the linear and nonlinear stages of the focal shift effect, and its use for the estimation of the more complicated nonlinear stage. Experiments were carried out using a continuous-wave ultrasonic beam in water, radiated by a focused source with nominal frequency f=1 MHz, aperture radius a=1.5 cm and geometrical focal distance R=11.7 cm, corresponding to a Fresnel number N(F)=1.28. The maximum measured shifts for peak pressure and intensity were 4.4 and 1.1cm, respectively. The evolution of the different maxima with the source amplitude, and the disparity in their axial positions, is interpreted in terms of the dynamics of the nonlinear distortion process. Analytical results for the particular case of a sound beam with initial Gaussian distribution are also presented, demonstrating that the motion of peak pressure and peak intensity may occur in opposite directions.

Nonlinear focal shift beyond the geometrical focus in moderately focused acoustic beams

“…Averkiou and Hamilton 12 (1997) observed this phenomenon experimentally in a moderately focused piston (linear gain G ¼ p/p 0 ¼ 10.36; where p is the value of the pressure in the geometrical focus and p 0 the pressure at the surface of the transducer). The nonlinear focal shift phenomenon was reported by Makov et al 13 in low gain transducers and discussed in terms of the harmonics nonlinearly generated during the propagation of a finite amplitude wave. They provided also experimental evidence of the nonlinear shift in slightly focused transducers (G ¼ 4).…”

“…The curve has been performed by simulating different transducer geometries, from low amplitudes (linear regime, $20 kPa in the focus) to sufficiently large amplitudes ($4-5 MPa in the focus) to reach saturation in the maximum pressure shift, according to the procedure followed in Ref. 13. A global agreement can be observed, even considering that the experiments were not optimized for the observation of the nonlinear focal shift.…”

“…Although the nonlinear focal shift phenomenon has been observed and discussed in previous studies [8][9][10][11][12][13][14] for slightly focused beams, a specific study with the objective to analyze, experimentally and numerically, the focal region of moderate Fresnel number transducers (4 < N F < 8) and the magnitude of this shift is absent, as can be seen in Fig. 1.…”

“…As discussed in Ref. 13, the Fresnel number is proportional to the linear gain (G ¼ pN F ); however, because, due to the linear focal shift phenomenon, the real gain (G r ¼ p max /p 0 where p max is the on-axis pressure maximum and p 0 the pressure in the surface of the transducer) is different from G (linear gain), we adopt N F to characterize the focusing of the sound beam in this work. Note that G only estimates the magnification of the beam in the absence of focal shift.…”

“…This is a focusing region of special interest because selfrefraction plays a more important role than in highly focused beams (like HIFU devices, where the volume of the focus is too small to produce significant self-refraction effects) and in weakly focused beams (where high voltages have to be applied to the transducer to reach the amplitudes necessary to observe nonlinear effects). Also, numerical simulations 13,14 of moderate Fresnel number transducers predict that the on-axis pressure maximum position could surpass the geometrical focal point due to the effect of nonlinearity. We present the first experimental demonstration and explanation of this phenomenon in the current study.…”