Limited Diffraction Beams Research Articles

Second-harmonic generation of limited diffraction beams

This paper presents an analytical result concerning second-harmonic generation of a general limited diffraction beam in nonlinear media. The result is based on quasilinear solutions of the Khokhlov?Zabolotskaya?Kuznetsov nonlinear wave equation. It is shown that second-harmonic generation by the nonlinear interaction of two arbitrary Bessel beams gives a nearly radially limited diffraction beam. For an arbitrary limited diffraction beam from the linear superposition of basis mode Bessel beams, results also indicate that the second-harmonic beam remains an almost radially or transversely limited diffraction beam.

Study on application of complementary Golay code into high frame rate ultrasonic imaging system

The high frame rate (HFR) ultrasonic imaging system, which is developed with limited diffraction beams, constructs images at a high frame rate. However the rectangular imaging area, to some extent, restricts the far field imaging information. At the same time, by virtue of one transmission event for constructing image, the system suffers from low SNR. In this paper we present a computationally efficient method to construct sector mode image and to increase the SNR in HFR system. The method uses orthogonal complementary Golay coded excitation to realize two transmit and receive events. Each emission simultaneously transmits two plane waves with different transmission angle. Then according to Golay code orthogonality, the received echo signals related to different angles are isolated and used to construct two images of different imaging area by HFR method. Finally the two images are synthesized to one frame of sector mode image.

Extended high-frame rate imaging method with limited-diffraction beams

Fast three-dimensional (3-D) ultrasound imaging is a technical challenge. Previously, a high-frame rate (HFR) imaging theory was developed in which a pulsed plane wave was used in transmission, and limited-diffraction array beam weightings were applied to received echo signals to produce a spatial Fourier transform of object function for 3-D image reconstruction. In this paper, the theory is extended to include explicitly various transmission schemes such as multiple limited-diffraction array beams and steered plane waves. A relationship between the limited-diffraction array beam weighting of received echo signals and a 2-D Fourier transform of the same signals over a transducer aperture is established. To verify the extended theory, computer simulations, in vitro experiments on phantoms, and in vivo experiments on the human kidney and heart were performed. Results show that image resolution and contrast are increased over a large field of view as more and more limited-diffraction array beams with different parameters or plane waves steered at different angles are used in transmissions. Thus, the method provides a continuous compromise between image quality and image frame rate that is inversely proportional to the number of transmissions used to obtain a single frame of image. From both simulations and experiments, the extended theory holds a great promise for future HFR 3-D imaging.

Electrical and Acoustical Characterization of the Bessel Transducers

Ultrasonic Bessel transducers produce limited diffraction beams which make them optimal for medical images applications. They satisfy the condition of improved lateral resolution and large depth penetration of the generated ultrasonic field. In this work Bessel transducers using conventional ceramic disks were constructed following a direct poling method, with a center frequency of 2 MHz. The piezoelectric elements contain three concentric electrode rings, which were poled following a Bessel function. Transducers were backed with low impedance and matched to water with a front matching layer for characterizations. The mechanical vibration characteristics of these transducers were investigated through the analysis of electric impedance measurements of their radial and thickness vibration modes. The experimental results were compared with the simulations made by the finite element method (FEM). The experimental resonance frequencies of the Bessel transducer are in agreement with the FEM simulated results in the low and high frequency range. For Bessel transducers, results show that some mechanical resonant modes (i.e. radial fundamental and several harmonics modes) are suppressed when an in phased excitation of the rings is applied. That is not the case for uniformly poled disk.

Fast Reconstruction of the Ultrasonic Bessel Transducer Field by Using Schlieren Techniques

Ultrasonic transducers uniformly poled have been used for clinical diagnostic applications, and usually medical imaging use focused beams that present a short depth of ultrasonic field. With ultrasonic transducers that produce limited diffraction beams (Bessel transducers), high-resolution images can be achieved over a large depth of field compared with the focused beams. In this paper we use the Schlieren method in order to obtain a 2-dimensional image of the acoustic field. By assuming axi-symmetrical proprieties of the field, we develop a fast algorithm that allows as reconstructing the 3-dimensional field of acoustical power. We compare its diffraction field with one emitted by a piston-like transducer with the same diameter and excited with the same frequency. We see that the diffraction effect in the last transducer produces a power spread greater than the Bessel one.

Field Computation for Two-Dimensional Array Transducers with Limited Diffraction Array Beams

A method is developed for calculating fields produced with a two-dimensional (2D) array transducer. This method decomposes an arbitrary 2D aperture weighting function into a set of limited diffraction array beams. Using the analytical expressions of limited diffraction beams, arbitrary continuous wave (cw) or pulse wave (pw) fields of 2D arrays can be obtained with a simple superposition of these beams. In addition, this method can be simplified and applied to a 1D array transducer of a finite or infinite elevation height. For beams produced with axially symmetric aperture weighting functions, this method can be reduced to the Fourier-Bessel method studied previously where an annular array transducer can be used. The advantage of the method is that it is accurate and computationally efficient, especially in regions that are not far from the surface of the transducer (near field), where it is important for medical imaging. Both computer simulations and a synthetic array experiment are carried out to verify the method. Results (Bessel beam, focused Gaussian beam, X wave and asymmetric array beams) show that the method is accurate as compared to that using the Rayleigh-Sommerfeld diffraction formula and agrees well with the experiment.

Engineering applications of limited diffraction beams

Limited diffraction beams (LDBs) are non-divergence and non-dispersive solutions to isotropic-homogeneous wave equations. These beams have a common characteristic of X-shaped branches and thus are also called X waves. Because of their highly directional propagation property, they have potential applications in medicine, underwater acoustics, and nonlinear optics (Charles Day, Phys. Today, October, 2004, pp. 25–26). In this talk, an overview of the development of LDBs will be given. This includes the conversion of any existing solutions to homogeneous or non-homogeneous wave equations to LDB solutions using Lorentz-related transformation, and a discussion of the orthogonal properties of the X wave transformation pair in representing any physically realizable waves. Experiments on applications of LDBs using our newly developed general-purpose ultrasound system will also be reported. (The system is capable of 128-channel simultaneous ultrasound data acquisitions at 12-bit/40 MHz rate and can hold real-time radio-frequency (RF) data up to 64 GB in one acquisition operation. Arbitrary ultrasound waveforms can be simultaneously produced by 128 12-bit/40 MHz D/A converters and then be linearly amplified to drive 75-Ohm loads or an array transducer at about ±150 V.)

A study of angular spectrum and limited diffraction beams for calculation of field of array transducers

Angular spectrum is one of the most powerful tools for field calculation. It is based on linear system theory and the Fourier transform and is used for the calculation of propagating sound fields at different distances. In this report, the generalization and interpretation of the angular spectrum and its intrinsic relationship with limited diffraction beams are studied. With an angular spectrum, the field at the surface of a transducer is decomposed into limited diffractions beams. For an array transducer, a linear relationship between the quantized fields at the surface of elements of the array and the propagating field at any point in space can be established. For an annular array, the field is decomposed into limited diffraction Bessel beams [P. D. Fox and S. Holm, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 49, 85–93 (2002)], while for a two-dimensional (2-D) array the field is decomposed into limited diffraction array beams [J-y. Lu and J. Cheng, J. Acoust. Soc. Am. 109, 2397–2398 (2001)]. The angular spectrum reveals the intrinsic link between these decompositions. [Work supported in part by Grant 5RO1 HL60301 from NIH.]

Logic design and implementation of FPGA for a high frame rate ultrasound imaging system

Recently, a method has been developed for high frame rate medical imaging [Jian-yu Lu, ‘‘2D and 3D high frame rate imaging with limited diffraction beams,’’ IEEE Trans. Ultrason. Ferroelectr. Freq. Control 44(4), 839–856 (1997)]. To realize this method, a complicated system [multiple-channel simultaneous data acquisition, large memory in each channel for storing up to 16 seconds of data at 40 MHz and 12-bit resolution, time-variable-gain (TGC) control, Doppler imaging, harmonic imaging, as well as coded transmissions] is designed. Due to the complexity of the system, field programmable gate array (FPGA) (Xilinx Spartn II) is used. In this presentation, the design and implementation of the FPGA for the system will be reported. This includes the synchronous dynamic random access memory (SDRAM) controller and other system controllers, time sharing for auto-refresh of SDRAMs to reduce peak power, transmission and imaging modality selections, ECG data acquisition and synchronization, 160 MHz delay locked loop (DLL) for accurate timing, and data transfer via either a parallel port or a PCI bus for post image processing. [Work supported in part by Grant 5RO1 HL60301 from NIH.]

Design and construction of a high frame rate imaging system

A new high frame rate imaging method has been developed recently [Jian-yu Lu, ‘‘2D and 3D high frame rate imaging with limited diffraction beams,’’ IEEE Trans. Ultrason. Ferroelectr. Freq. Control 44, 839–856 (1997)]. This method may have a clinical application for imaging of fast moving objects such as human hearts, velocity vector imaging, and low-speckle imaging. To implement the method, an imaging system has been designed. The system consists of one main printed circuit board (PCB) and 16 channel boards (each channel board contains 8 channels), in addition to a set-top box for connections to a personal computer (PC), a front panel board for user control and message display, and a power control and distribution board. The main board contains a field programmable gate array (FPGA) and controls all channels (each channel has also an FPGA). We will report the analog and digital circuit design and simulations, multiplayer PCB designs with commercial software (Protel 99), PCB signal integrity testing and system RFI/EMI shielding, and the assembly and construction of the entire system. [Work supported in part by Grant 5RO1 HL60301 from NIH.]

Modeling of CW annular arrays using limited diffraction Bessel beams.

We present a method for characterizing the linear field of any flat, continuous-wave annular array in terms of a set of known limited-diffraction Bessel beams. The technique uses Fourier-Bessel series to model the surface pressure of the transducer surface, with each term in the series giving rise to a Bessel beam with known propagation parameters. The analysis is applied numerically to two different transducers discussed previously in the literature. In both cases, a deeper understanding of the field emitted than was previously available is gained. Brief outlines for extending the technique to pulsed wave and non-annular arrays are also given.

New X-wave solutions of free-space scalar wave equation and their finite size realization.

Based on the method proposed by Donnelly and Ziolkowski [1], [2], a new general solution has been obtained for the isotropic/homogeneous scalar wave equation in cylindrical coordinates. It is shown that well-known limited diffraction beams such as Durnin's Bessel beams [4], Lu and Greenleaf's X-wave [15], localized waves of Donnelly and Ziolkowski [1], [2], and limited-diffraction, band-limited waves of Li and Bharath [19], [20] can be obtained from this generic solution as particular cases. In addition, we have obtained new X-wave solutions and have calculated the field characteristics for one of them using a finite aperture realization. It is shown that with a proper choice of the free parameter values, well-behaved X-waves with narrow beamwidths and large depths of field can be achieved. For similar source spectra, the results are compared with Lu and Greenleaf's zeroth-order X-wave, and it is shown that the depth of field and beamwidth are very comparable.

An X wave transform

Limited diffraction beams such as X waves can propagate to an infinite distance without spreading if they are produced with an infinite aperture and energy. In practice, when the aperture and energy are finite, these beams have a large depth of field with only limited diffraction. Because of this property, limited diffraction beams could have applications in medical imaging, tissue characterization, blood flow velocity vector imaging, nondestructive evaluation of materials, communications, and other areas such as optics and electromagnetics. In this paper, a new transform, called X wave transform, is developed. In the transform, any well behaved solutions to the isotropic-homogeneous wave equation or limited diffraction beams can he expanded using X waves as basis functions. The coefficients of the expansions can be calculated with the properties that X waves are orthogonal. Examples are given to demonstrate the efficacy of the X wave transform. The X wave transform reveals an intrinsic relationship between any well behaved solutions to the wave equation and X waves, including limited diffraction beams. This provides a theoretical foundation to develop new limited diffraction beams or solutions to the wave equation that may have practical usefulness.

A study of simulation methods of limited diffraction beams

Limited diffraction beams such as Bessel beams and X waves have been studied extensively because of their potential applications in many areas such as medical imaging and optical communications. Computer simulations have been powerful tools for understanding the fundamentals of these beams. In this study, limited diffraction beams, specifically, Bessel beams and X waves, were simulated. The simulations were based on the frequency-domain method (Rayleigh–Sommerfeld diffraction formula), time-domain method, and finite-element or finite-difference method. For Bessel beams, a circular transducer divided into multiple rings was assumed. For pressure distributions on axial axis, the simulations were simplified due to the symmetry. In the case of X waves, the differential property of acoustic waves was used. The results of the simulation methods were compared with each other. Advantages and disadvantages of these methods will be reported. In all cases, the results are very close to those predicted by theory. [This work was supported in part by Grant HL 60301 from the National Institute of Health of the U.S.A.]

Sidelobe reduction of limited diffraction beams with Chebyshev aperture apodization

A limited diffraction beams (LDB) imaging system with Chebyshev weighting is presented. The objective of the paper is to reduce the sidelobes of the LDB without impacting on main-lobe performance and increase the contrast-resolution of the imaging system. The Chebyshev weighting is applied to the LDB and an analytic description and the simulation results are obtained. Theoretical analysis and simulation results show that the LDB with Chebyshev weighting can reduce sidelobes, and improve imaging system performances.

Beams with large depth of field

Acoustic bullets, also known as limited diffraction beams, are solutions of the scalar wave equation with the property of maintaining their shape upon propagation, without spreading in space and in time. A necessary and sufficient condition for the generation of acoustic bullets is presented. In particular, it is proved that a necessary condition for acoustic bullets is that all the wave vectors characterizing the beam are disposed over a cone with a given vertex angle. Impulse responses for the case of axial symmetry, and for the situation in which axial symmetry is lacking, are then given. The results obtained for acoustic bullets are then contrasted with those of other types of beams with large depth of field recently presented in literature (an example is the so-called ‘‘sinc wave’’). Finally, the topic of the realization of practical approximation of these kind of beams, which, in principle, requires infinite apertures, is addressed. Numerical simulations are presented in order to analyze the departures from the ideal case when using a finite-dimension aperture.

Increasing field of view of high frame rate ultrasonic imaging

A high frame rate imaging method has been developed recently based on the theory of limited diffraction beams for medical diagnosis. This method constructs images in a direction normal to the transducer surface. In this report, the high frame rate method is extended so that it will not only construct images in the normal direction, but also in any oblique directions. This increases the image construction area because images from several oblique angles can be combined. A new equation has been derived for this extension. A computer simulation was performed. Results show that the field of view of images constructed with the extended method is increased from about 15.74 deg to about 96.57 deg with a linear array of a width (foot print) of about 18.288 mm after combining images from only seven angles (the depth of the images is assumed to be 66.12 mm). This work is significant because a large field of view with a small transducer footprint is desirable for high frame rate heart imaging. [Work supported in part by Grant No. HL60301 from NIH.]

Decomposition of acoustic fields in quantised Bessel beams

Bessel beams are non-diffracting solutions to the wave equation, which become limited diffraction beams when implemented on finite apertures. In previous work we have applied Fourier–Bessel theory to deduce theoretically that the quantization of the Bessel beam profile on annular arrays results in a field which is a sum of limited diffraction fields. Here we demonstrate this for a five-ring, 20-mm diameter, 2.5-MHz transducer. The quantized field comprises a weighted sum of a main component corresponding to the desired field, along with three other major components and 28 lesser components representing undesired field components. The three major components correspond to limited diffraction beams with narrower beamwidths and shorter depths of field than the desired beam, and we show that these account for most of the discrepancies between the desired field and actual quantized field. An estimate and an interpretation of the number of field components as a function of the wavenumber are also given.

Human uses of ultrasound: ancient and modern

For untold millennia certain animals have used ultrasound to probe places where light is unavailable, echo-locating bats being among the most adept. With ultrasonics, bats can quickly and safely ‘see’ at night in pursuing insects or flying in dark caves. Unable to hear ultrasound, humans have nevertheless made use of it. They did this anciently by taming wolves, with their keen ultrasonic hearing, for aiding in the hunt. Currently, they are doing this by developing technology to detect, generate and process ultrasound for searching in air or other gases, in water or other liquids, and in solids. The story of these technological developments is a large and fascinating mirror of human history involving the advent of such discoveries and inventions as magnetostriction, piezoelectricity, sonar, ultrasonic microscopy, etc. – the list is long. By now we are skilled in probing for underwater objects, the internal structure in materials, organs inside the human body, etc. – again the list is long. A number of different ultrasonic systems can be categorized into one of three key generic approaches: pulse–echo exploration, intensity mapping, and phase–amplitude measurement. In addition, each of these categories can be combined with the others to produce hybrid systems for which an unambiguous categorization is difficult or impossible. Challenging problems remain but solutions are being found. New principles and techniques are being discovered that will improve the use of ultrasound. Employing tomo-holographic techniques to reduce ambiguity in probing three-dimensional objects, near-field techniques to boost resolution and using limited-diffraction beams to provide image construction with ultra high frame rates are cases in point.

Effects of phase aberration on high frame rate imaging

A high frame-rate (HFR) imaging method (about 3750 frames/s for imaging of biological soft tissues at a depth of 200 mm) has been developed recently with limited diffraction beams. This method uses the fast Fourier transform (FFT) and inverse fast Fourier transform (IFFT) to construct images, and can be implemented with simple and inexpensive hardware, compared to the conventional delay-and–sum method where a digital beam former is usually used. In this paper, phase aberration effects are studied for both the high frame rate and the conventional methods by adding random phase shifts to echo signals obtained from an experiment. In the study, two broadband linear arrays were used to construct images of an ATS 539 tissue-equivalent phantom that has a frequency-dependent attenuation of about 0.5 dB/MHz/cm. The first array has 48 elements, a central frequency of 2.25 MHz, an aperture of 18.288 mm, and a width of 12.192 mm in elevation. The second has 64 elements, a central frequency of 2.5 MHz, and a dimension of 38.4 mm × 10 mm. The-6dB pulse-echo bandwidth of both arrays is about 40% of their center frequencies. Radiofrequency (RF) signals were digitized at 20 mega samples/s at a 12-bit resolution to construct images. Results show that phase aberration has about the same effect on both methods in terms of image resolution and contrast, although the high frame-rate method can be implemented with a simpler system.