Kirchhoff Approximation Revisited—Some New Results for Scattering in Isotropic and Anisotropic Elastic Solids

Through a series of numerical studies that compare the Kirchhoff approximation to more exact scattering theories, it is demonstrated that the Kirchhoff approximation can accurately predict the pulse–echo peak-to-peak responses of spherical pores and circular cracks in isotropic media over a very wide range of cases that extend well beyond the limits normally associated with this approximation. The reason for this good agreement is shown to lie in the ability of the Kirchhoff approximation to model accurately the very early time response of the flaw. It is also shown that in the Kirchhoff approximation the pulse–echo response of an arbitrary traction-free scatterer in an isotropic elastic solid is identical to the same response obtained using a scalar (fluid) scattering model. This leads to simple analytical expressions for the pulse–echo far-field scattering amplitude of some canonical geometries (circular cracks, spherical voids, cylindrical holes) and to simplified numerical expressions for more general scatterers. For general anisotropic volumetric flaws in a anisotropic elastic solid, it is shown that a high-frequency asymptotic evaluation of the Kirchhoff approximation yields an explicit analytical expression for the pulse–echo leading-edge response of the flaw. Explicit expressions are also given for the pitch–catch response of an elliptical-shaped flat crack in a general anisotropic solid.     

SAW focusing by circular-arc interdigital transducers on YZ-LiNbO(3)

Using the angular spectrum theory and experimental velocity data of surface acoustic waves (SAW) on YZ-LiNbO(3), the focusing characteristics of a circular-arc interdigital transducer have been demonstrated. The calculated results show that the depth of focus is long and the compressed acoustic beam width is very narrow. The concept of a caustic is shown to be an excellent way of characterizing SAW focusing by a circular-arc interdigital transducer on YZ-LiNbO(3). Comparison between theoretical and experimental results shows good agreement.     

聚焦超声声场中的散射体成像方法

在实验室条件下,研究了聚焦超声声场中的散射体成像问题。将一个纽扣作为散射体置于一聚焦换能器所产生的声场中,以频率为1.17MHz的脉冲信号经功率放大后激励聚焦换能器,声波作用于纽扣后发生散射,借助于三维自动扫描系统和PVDF膜片水听器,测量得到了散射声场中前三次谐波的声场分布,膜片水听器敏感元件有效尺寸为1mm。利用角谱法反演了散射体所在处的声场,通过与散射体实际的形状特征进行比较,发现反演的结果与散射体特征吻合良好,由此得出结论,在信噪比足够大的情况下,利用聚焦声场所产生出的高次谐波,可以得到分辨率较高的成像效果。     

Verification of the Westervelt equation for focused transducers

This study investigates the validity of the Westervelt equation for focused transducers. The angular spectrum method is employed to analyze the second-harmonic acoustic field under the weakly nonlinear approximation. Although it is well known that the Westervelt equation is accurate for the case of quasi-plane waves, the present work demonstrates accurate solution for the highly focused case of a spherically-curved ultrasound transducer having an aperture angle of 80°. It is further found that the solution error is inversely dependent on the nonlinearity coefficient.     

Ultrasonic Beam Models for Angle Beam Surface Wave Transducers

ABSTRACT

Explicit three-dimensional (3D) point source and multi-Gaussian beam models are obtained for the Rayleigh waves generated by a surface wave angle beam transducer using an angular plane wave spectrum approach. Simulations show that the multi-Gaussian surface wave beam model agrees well with the point source model while being computationally more efficient. The theoretical predictions obtained with the models are also compared to the experimental measurement results where good agreement with the models is found for both on-axis and off-axis field comparisons.     

Propagation and backpropagation for ultrasonic wavefront design

Wave backpropagation is a concept that can be used to calculate the excitation signals for an array with programmable transmit waveforms to produce a specified field that has no significant evanescent wave components. This concept can also be used to find the field at a distance away from an aperture based on measurements made in the aperture. For a uniform medium, three methods exist for the calculation of wave propagation and backpropagation: the diffraction integral method, the angular spectrum method, and the shift-and-add method. The boundary conditions that are usually implicitly assumed by these methods are analyzed, and the relationship between these methods are explored. The application of the angular spectrum method to other kinds of boundary conditions is discussed, as is the relationship between wave backpropagation, phase conjugation, and the time-reversal mirror. Wave backpropagation is used, as an example, to calculate the excitation signals for a ring transducer to produce a specified pulsatile plane wave with a limited spatial extent.     

双传感器振强测量及误差分析

简述了双传感器振强测量的原理，重点对该测量方法的有限差分误差、相位误差以及近场误差进行了理论分析和讨论，同时给出了该测量方法的使用条件和近场误差的补偿方法。利用声强测量设备对一模拟无限大板结构进行了实验和分析，获得了满意的测量结果。     

P(VDF-TrFE) transducer with a concave annular structure for the measurement of layer thickness

A new type of transducer that measures the layer thickness at a small spot by spectroscopy is developed. It is made of piezoelectric copolymer film P(VDF-TrFE) with a concave annular structure. This transducer converges an ultrasonic wave on the specimen surface with oblique incidence. The layer thickness is estimated by means of a dip in the frequency spectrum of a reflected wave. Theoretical analysis using the angular spectrum theory and experimental evaluation of the transducer are described. The transducers operate in a frequency range from 7 MHz to 90 MHz. The output variation when the transducer scans a thin wire is in good agreement with a calculation of a line spread function. The lateral and axial resolutions estimated by the calculation of a point spread function are 0.61 lambda and 8.05 lambda, respectively.     

基于超声相控技术的船舶吃水测量方法

为了解决内河船舶因超吃水引起的通航阻塞以及船舶安全问题,提出一种基于超声相控技术的船舶吃水测量方法。以超声相控技术为理论基础,通过控制一维线型超声相控阵列换能器发射聚焦声波,实现对目标船舶的相控扫描,获取各扫描点的回波信号;利用匹配滤波算法对回波信号进行滤波,改善信噪比;利用阈值法提取回波信号的时延;利用渡越时间法,计算出扫描点到各发射振元中心的距离,利用双曲交汇法计算出扫描点的空间坐标;分析回波信号幅值和扫描点位置坐标即可得到船舶吃水深度。为验证方法的可靠性,搭建了小比尺船模吃水测量实验系统,分析了110~140 mm不同吃水深度下的实验结果,计算了实际吃水与测量吃水间的相对误差。实验结果表明,使用匹配滤波法处理后的回波信号信噪比从15.26 dB提高至36.39 dB,实验时,最大相对误差出现在船舶实际吃水130 mm时,绝对误差为2.3 mm,相对误差为1.7%。     

Quantum-well enhancement of the Goos-Hänchen shift for p-polarized beams in a two-prism configuration

It is predicted that the Goos-H?nchen effect can be resonantly enhanced by placing a metallic quantum well (ultrathin film) at the dielectric-vacuum (air) interface. We study the enhancement of the phenomenon, as it appears in frustrated total internal reflection with p-polarized light, both theoretically and numerically. Starting from boundary conditions for the electromagnetic field, which in a self-consistent manner take into account the quantum-well dynamics, we derive new expressions for the amplitude reflection and transmission coefficients of light, and from these the stationary phase approximation to the Goos-H?nchen shifts is obtained. It is shown that large peaks appear in the Goos-H?nchen shift below the critical angle in reflection, and these are located at the minima for the energy reflection coefficient. Both positive and negative shifts may occur, and the number of peaks depends on the gap width. To determine the accuracy of the simple stationary phase approximation, we carry out a rigorous stationary energy-transport calculation of the Goos-H?nchen shift. Although the overall agreement between the two approaches is good, the stationary phase approach mostly overestimates the peak heights. For a Gaussian incident beam, the resonance displacement of the reflected beam can be as large as the Gaussian width parameter. It is suggested that the possible relation between the Goos-H?nchen effect and the optical tunneling phenomenon in the two-prism configuration should be reinvestigated by depositing quantum wells on the glass-vacuum interfaces to obtain a better spatial photon localization.     

Predictions of the Gauss-Hermite beam model and finite element methodf or ultrasonic propagation through anisotropic stainless steel

The predictions of the Gauss-Hermite beam model are compared to those obtained by the finite-element method for a model problem. This is motivated by the desire to examine the trade-offs between computational speed and accuracy in the Gauss-Hermite model. In the model problem, a contact strip transducer radiates through an isotropic layer of ferritic steel into an anisotropic layer of austenitic stainless steel with various directions of the preferred axis of columnar grain alignment. Comparisons are made of time-domain waveforms in a common observation axis in the austenitic material. The predictions of the two models are found to be in good agreement near the center of the beam, with deviations developing as one moves away from the central ray. These are interpreted to be a consequence of the Fresnel approximation made in the Gauss-Hermite model. However, the region that contains most the energy is in the vicinity of the central ray, where there is excellent agreement between the two models. This loss in accuracy is accompanied by a several orders of magnitude increase in computation time.     

Generalized Analytical Solutions for Planar and Non-planar Focal Curves Reconstruction

Abstract

A generalized solution in the scope of the Fresnel approximation for diffractive optical elements producing focal curves is derived in this paper. The method includes planar and non-planar focal lines parametrized by differentiable functions and different types of holograms (Fourier, Fresnel). The integral describing the complex amplitude distribution generated by such a focal curve is evaluated with the help of the stationary phase method and the points in the hologram plane contributing to a particular point of the focal curve are found to be circles or straight lines. As the result equations for stationary points turn out to be solvable analytically only in some specific cases, an approximate approach is proposed additionally. In this way the class of analytic solutions for non-planar curves can be broadened into cases for which there exists a solution related to their projection on to the hologram plane. The paper is completed by presentation of non-paraxial ray tracing diagrams for elements designed with the help of both methods.     

Far-field vectorial structure of a higher-order cosh-Gaussian beam

Based on the vector angular spectrum representation and the method of stationary phase, the vectorial structural characteristics of a higher-order cosh-Gaussian (HOCG) beam in the far field are investigated. Analytical expressions for the energy flux distributions of the TE term, TM term and the whole beam are derived and illustrated by numerical examples. Influences of beam orders and cosh parameters on energy flux distributions are analyzed, respectively. Furthermore, the discrepancy for the total energy flux distributions between the non-paraxial and paraxial cases is presented and analytical explanations for the phenomenon are also given.     

An improved moving‐least‐squares reconstruction for immersed boundary method

The current work presents an improved immersed boundary method based on the ideas proposed by Vanella and Balaras (M. Vanella, E. Balaras, A moving‐least‐squares reconstruction for embedded‐boundary formulations, J. Comput. Phys. 228 (2009) 6617–6628). In the method, an improved moving‐least‐squares approximation is employed to build the transfer functions between the Lagrangian points and discrete Eulerian grid points. The main advantage of the improved method is that there is no need to obtain the inverse matrix, which effectively eliminates numerical instabilities caused by matrix inversion and reduces the computational cost significantly. Several different flow problems (Taylor‐Green decaying vortices, flows past a stationary circular cylinder and a sphere, and the sedimentation of a free‐falling sphere in viscous fluid) are simulated to validate the accuracy and efficiency of the method proposed in the present paper. The simulation results show good agreement with previous numerical and experimental results, indicating that the improved immersed boundary method is efficient and reliable in dealing with the fluid–solid interaction problems. Copyright © 2015 John Wiley & Sons, Ltd.     

Shear elasticity probe for soft tissues with 1-D transient elastography

Important tissue parameters such as elasticity can be deduced from the study of the propagation of low frequency shear waves. A new method for measuring the shear velocity in soft tissues is presented in this paper. Unlike conventional transient elastography in which the ultrasonic transducer and the low frequency vibrator are two separated parts, the new method relies on a probe that associates the vibrator and the transducer, which is built on the axis of the vibrator. This setup is easy to use. The low frequency shear wave is driven by the transducer itself that acts as a piston while it is used in pulse echo mode to acquire ultrasonic lines. The results obtained with the new method are in good agreement with those obtained with the conventional one.     

Characterization of changes in therapeutic ultrasound transducer performance over time using the angular spectrum method

Strongly focused large aperture transducers used in high-intensity focused ultrasound treatments are prone to manufacturing defects and degradation. Current methods for evaluating transducer quality measure only bulk physical changes of transducers. We have determined the pressure distribution at the transducer surface, using the angular spectrum method, to detect defects of the transducer. Three therapeutic transducers were investigated. The pressure distribution at the focal plane of each transducer was measured and input into a back-projection algorithm to calculate the pressure distribution at the transducer surface. A number of scan window sizes were used for the pressure distribution measurement at the focal plane to determine the effect on the resolution of the calculated pressure distribution at the transducer surface. Results showed that one transducer might have suffered manufacturing defects. The second transducer degraded over 1 year of use with one half of the transducer suffering a partial loss of efficiency. The third transducer remained unchanged over 1 year. The scan window of 40 mm X 40 mm at the focal plane was required to identify defects 6 mm in diameter on the transducer surface. The results demonstrate that the angular spectrum method could be a useful tool for evaluating transducer quality.     

Backscattering measurement from a single microdroplet

Backscattering measurements for acoustically trapped lipid droplets were undertaken by employing a P[VDF-TrFE] broadband transducer of f-number = 1, with a bandwidth of 112%. The wide bandwidth allowed the transmission of the 45 MHz trapping signal and the 15 MHz sensing signal using the same transducer. Tone bursts at 45 MHz were first transmitted by the transducer to hold a single droplet at the focus (or the center of the trap) and separate it from its neighboring droplets by translating the transducer perpendicularly to the beam axis. Subsequently, 15 MHz probing pulses were sent to the trapped droplet and the backscattered RF echo signal received by the same transducer. The measured beam width at 15 MHz was measured to be 120 μ m. The integrated backscatter (IB) coefficient of an individual droplet was determined within the 6-dB bandwidth of the transmit pulse by normalizing the power spectrum of the RF signal to the reference spectrum obtained from a flat reflector. The mean IB coefficient for droplets with a 64 μ m average diameter (denoted as cluster A) was -107 dB, whereas it was -93 dB for 90-μm droplets (cluster B). The standard deviation was 0.9 dB for each cluster. The experimental values were then compared with those computed with the T-matrix method and a good agreement was found: the difference was as small as 1 dB for both clusters. These results suggest that this approach might be useful as a means for measuring ultrasonic backscattering from a single microparticle, and illustrate the potential of acoustic sensing for cell sorting.     

Vectorial structure of Ince–Gaussian beam in the far field

By means of the vector angular spectrum representation of the electromagnetic beam and the method of stationary phase, the analytical vectorial structure of the Ince–Gaussian beam has been presented in the far field. The amplitude distributions of the Ince–Gaussian beam and its TE and TM terms are investigated in the far field. The extreme cases of the ellipticity parameter tending to infinity or zero are also considered. Although the vectorial structures of different Ince–Gaussian beams are apparently distinct, the ratios of the amplitude distributions of the TE and TM terms to the whole beam amplitude are independent of the parity and the values of the radial and angular elliptic mode numbers. This research reveals the abundant and interesting internal details of the Ince–Gaussian beam in the far field.     

Sandwiched piezoelectric ultrasonic transducers oflongitudinal-torsional compound vibrational modes

Sandwiched piezoelectric ultrasonic transducers of longitudinal-torsional compound vibrational modes were studied. The transducers consist of coaxially segmented, longitudinally and tangentially polarized piezoelectric ceramic rings, a back metal cylinder, and a front exponential solid metal horn. Based on the plane-wave approximation, the equivalent circuits of the longitudinal and torsional vibrations in the sandwiched transducer were obtained and the resonance frequency equations of the transducer in longitudinal and torsional vibrations were derived. By means of choosing the radius decay coefficient of the front exponential horn, the longitudinal and torsional vibrations are made to resonate at the same frequency in the transducer. Sandwiched piezoelectric ultrasonic transducers of longitudinal-torsional compound modes were designed and fabricated according to the frequency equations. It is demonstrated that the measured resonance frequencies of the transducers are in good agreement with the theoretical results, and the measured resonance frequencies of the transducers in longitudinal and torsional vibration modes are also in good agreement with each other. Theoretical and experimental results show that this kind of transducer can be used in ultrasonic welding, ultrasonic machining, ultrasonic motors, and other ultrasonic applications which need large displacement amplitudes     

The chirp-Z transform applied to adhesively bonded structures

A signal-processing technique based on the chirp-Z transform is presented to evaluate the echo signals of longitudinal ultrasonic transducers in contact with bonded materials. Both a simulated glass-glass interface of variable thickness and a realistic double lap bonded aluminum sample were tested. The observed frequency dips and peaks in the transducer spectrum from pulse-echo and through-transmission modes were recorded and related to the condition of zero reflection coefficient at the interfacial layer. Resulting thickness predictions for different transducer center frequencies ranging from 5-20 MHz are in excellent agreement with experimental measurements