Limited Diffraction Beams Research Articles

Tunable liquid-based lenses for ultrasonic beamforming

Quasi-planar stepped lenses (QSL’s) based on phasing have proven to perform closely to conventional refractive lenses for the generation of both focused and limited diffraction beams. However, since QSLs are passive and made from solid materials, each must be designed with a specific frequency and beam type in mind. In this presentation, we report on a tunable lens with a planar aperture that employs liquid channels to perform phase-based beamforming. The desired phase patterns are produced using specific speed-of-sound profiles and can be flexibly configured for a range of frequencies and beam types without modification. In this talk, we report on the results of both experimental and numerical results for this class of lenses and discuss their potential applications.

Super-resolution imaging with modulation of point spread function

The spatial resolution of an imaging system using waves is limited by the spatial bandwidth of the point spread function (PSF) of the system, which is related to the wavelength. However, when the PSF is modulated either in amplitude or phase or in both, the resulting spatial bandwidth of the PSF is increased. In this study, the PSF-modulation method is used to obtain super-resolution imaging of objects and to distinguish wave sources that are closely located in space and are not normally separable due to diffraction limit. In imaging using waves, such as ultrasound, acoustics, optics, electromagnetics, radar, and sonar, the PSF can be modulated in their respective fields. For example, in ultrasound, shear wave in biological soft tissues has a low wave speed and thus has a small wavelength. A ring-shaped shear wave can be generated locally (remotely) deep in the tissue by the radiation force of a focused Bessel beam, X wave, or other limited-diffraction beam at their focuses [Lu, 2021 IEEE IUS, 2021 ASA POMA] to produce a sharp peak at the center of the ring (due to shear wave focusing with a small wavelength). This sharp peak of the shear wave modulates the center of a conventional focused beam transmitted after the shear wave ring is produced. The modulated focused beam is then used to scan through an object to obtain a super-resolution image after removing the contribution of the original beam. In this talk, the theory, computer simulation, and experiment results of the super-resolution method will be presented.

High-frame-rate limited-diffraction beam imaging

Limited-diffraction beams such as Bessel beam, X wave, and array beam are a class of beams that in theory do not spread as they propagate to infinite distance. When realized with a finite aperture, these beams have a large depth of field. In this paper, limited-diffraction beams and their applications to high-frame rate imaging are reviewed and further studied. Using a limited-diffraction beams (in either transmission or reception, or both, or with spatial Fourier transform of received signals), images can be reconstructed in Fourier domain. This significantly reduces the amount of computation in 3D imaging since FFT can be used. As the bandwidth of the transducer is increased, the image reconstructed can cover a larger field of view since a frequency in limited-diffraction beam corresponds to one angular direction of the beam (similar to the “frequency encoding” in the magnetic resonance imaging or MRI). Using the angular dependence with frequency, focused limited-diffraction beams can be used for high-frame-rate and high-resolution tissue property imaging. When limited-diffraction beams are used in transmissions, no time delay (only amplitude weighting) is needed, which simplifies imaging system hardware. Also, flow vector imaging can be obtained with limited-diffraction beams using sine and cosine (quadrature) aperture weighting.

A phase-shifting method for computation reduction for high-frame-rate imaging

High-frame rate (HFR) imaging using steered plane wave (SPW) or limited-diffraction beam has a high-temporal resolution and thus has found many applications. To further study the HFR imaging methods, computer simulations were performed. However, the simulations require a large number of computations, especially for 3D imaging with 2D array transducers for a large imaging volume. In this paper, a phase-shifting method was developed to reduce the number of computations. In the method, the grid points of the transmit and receive beams were calculated at 1-mm interval in the depth direction that is perpendicular to the transducer surface. The interval is much larger than the 1/4 of the 0.58-mm wavelength required for an accurate interpolation for millions of random scatterers in pulse-echo response without aliasing. Since the HFR imaging uses either SPW or LDB, the wave vectors of these beams are fixed at each frequency. Due to the fact that the amplitude of ultrasound beams changes very little over a couple of wavelengths, the interpolation in the depth direction was replaced with a phase shift. Results show that images reconstructed with the phase-shifting method removed the artifacts caused by aliasing when conventional tri-linear interpolations were used for 3D imaging.

A focused limited-diffraction beam for high-resolution imaging

Unfocused limited-diffraction beams such as Bessel beams and X waves have been studied extensively to obtain a large depth of field for medical imaging. In this study, a focused zeroth-order Bessel beam j 0(αr), where α = 1202.45 m−1 and r is the radius, produced with a 50 mm diameter and 2.5-MHz transducer and with an acoustic lens of 50-mm focal length (f-number = 1) was used to obtain a C-mode image of an object in water (0.6mm wavelength). The results of the study showed that at the focal distance, the lateral resolution of the image obtained with an unapodized transducer in both transmission and reception was lower than that obtained with an unapodized beam in transmit but the j 0 Bessel beam was used in receive. Beam plots of the unapodized beam and the j 0 Bessel beam at the focal distance showed that the lateral beamwidths were about 0.78 and 0.59 mm, respectively. The 0.59-mm beamwidth of the focused Bessel beam is better than that of the diffraction limit (1.22 × wavelength × f-number = 0.732 mm). The decreased beamwidth of the j 0 Bessel beam results in a higher lateral image resolution while the sidelobes of the beam were suppressed by the focused unapodized transmit beam.

Extraordinary trapping by Gorkov potential of ordinary and vortex limited-diffraction beams

Trapping behaviors by using standing waves or focused beams have been investigated and widely used for acoustic levitation and tweezers. It is commonly understood that dense and stiff particles are repelled from pressure anti-nodes (pressure maximum) where the Gor'kov potential has a maximum. Here, the trapping behaviors of a small particle by using Gor'kov potential of ordinary and vortex non-diffracting beams are examined. Unlike the standing waves or strongly focused beams, the ordinary non-diffracting beam with proper parameters is found to have a Gor'kov potential minimum at its pressure maximum to trap a relatively dense and stiff particle therein [Fan and Zhang, Phys. Rev. Appl. 11(1), 014055 (2019)]. For a vortex beam to trap a dense and stiff particle at the pressure null, the paraxiality of the beam has to be larger than some values. These seemly to be extraordinary behaviors are interpreted by the contribution of axial velocity to the Gor'kov potential. The results are applicable to beams with weak diffraction or weak focusing. Following from the results, ways to simplify methods for particle manipulation and development of acoustic tweezers are suggested.

2-D and 3-D Reconstruction Algorithms in the Fourier Domain for Plane-Wave Imaging in Nondestructive Testing.

Time-domain plane-wave imaging (PWI) has recently emerged in medical imaging and is now taking to nondestructive testing (NDT) due to its ability to provide images of good resolution and contrast with only a few steered plane waves. Insonifying a medium with plane waves is a particularly interesting approach in 3-D imaging with matrix arrays because it allows to tremendously reduce the volume of data to be stored and processed as well as the acquisition time. However, even if the data volume is reduced with plane wave emissions, the image reconstruction in the time domain with a delay-and-sum algorithm is not sufficient to achieve low computation times in 3-D due to the number of voxels. Other reconstruction algorithms take place in the wavenumber-frequency (f-k) domain and have been shown to accelerate computation times in seismic imaging and in synthetic aperture radar. In this paper, we start from time-domain PWI in 2-D and compare it to two algorithms in the f-k domain, coming from the Stolt migration in seismic imaging and the Lu theory of limited diffraction beams in medical imaging. We then extend them to immersion testing configurations where a linear array is facing a plane water-steel interface. Finally, the reconstruction algorithms are generalized to 3-D imaging with matrix arrays. A comparison dwelling on image quality and algorithmic complexities is provided, as well as a theoretical analysis of the image amplitudes and the limits of each method. We show that the reconstruction schemes in the f-k domain improve the lateral resolution and offer a theoretical and numerical computation gain of up to 36 in 3-D imaging in a realistic NDT configuration.

Focusing Leaky Waves: A Class of Electromagnetic Localized Waves with Complex Spectra

Focusing light's energy in the near field, beyond the diffractive limit, is key to progress in fields as diverse as microscopy, electronics, telecommunication, wireless power transfer, and ablation and tomography in medicine. The authors address the problem by considering a specific class of partially guided waves, known as ``leaky'' waves. Interestingly, only the subclass of backward leaky waves is found to be useful here. Qualitative and quantitative assessments furnish general design rules, and a simple but efficient leaky-wave device is proposed for generating limited-diffraction beams and pulses in the millimeter-wave range, where few experiments have been performed.

Sector-scanning 3D ultrasound imaging in frequency domain with 1D array transducer

Sector-scanning is a conventional scanning mode in ultrasound imaging, which can increase the area to observe. In three-dimensional (3D) ultrasound, freehand imaging system is usually used. This method uses a moving elevation focused one-dimensional (1D) transducer to construct a series of B-mode slice images, and then these B-mode slice images are combined to form a 3D volume image. When sector-scanning is used to acquire the B-mode slices, the elevation resolution is poor because of the elevation resolution of the probe and the interpolations between the slices. In this paper, based on the linear scanning method in our previous work, a sector-scanning 3D imaging method is proposed. In this imaging method, a linear array transducer without elevation focusing is also used, and the 1D transducer transmits limited diffraction beams and receives echo signals repeatedly when rotated around an axis parallel to the transducer. After finishing the scanning, all the received signals are combined to construct the 3D image in frequency domain with Fourier transform. Simulation results show that the new method can construct the 3D image effectively. Compared with the imaging method based on B-mode slices, the new method can improve the elevation resolution significantly. The elevation resolution can be promoted to less than 2 mm with the imaging depth 100 mm by one transmission at each position. Besides, because only one transmission is needed at each position, the frame rate can be increased to some extent.

3D ultrasound imaging in frequency domain with 1D array transducer

Freehand three-dimensional (3D) ultrasound imaging usually uses a moving elevation focused one-dimensional (1D) transducer to construct a series of B-mode slice images, and then these B-mode slice images are combined to form a 3D volume image. This method has poor elevation resolution and low frame rate. In this paper, a novel 3D imaging method with a moving 1D array transducer is proposed. Different from the freehand 3D reconstruction, a linear array transducer without elevation focusing is used in the proposed method, and the 1D transducer transmits limited diffraction beams and receives echo signals repeatedly when moving along the elevation direction. After finishing the scanning, all the received signals are combined to construct the 3D image in Frequency domain with Fourier transform. Simulation results show that compared with the freehand imaging method, the new method can improve the elevation resolution significantly, and the resolution keeps almost fixed in the axial direction. The elevation resolution can be promoted to less than 2mm with the imaging depth 100mm by one transmission at each position. Besides, because only one transmission is needed at each position, the frame rate can be increased to some extent.

200 ps pulse generation at 1180 nm with a SrWO4 Raman crystal pumped by a sub-nanosecond MOPA laser system

We are reporting Raman conversion experiments with a SrWO4 crystal in a single-pass traveling-wave setup. As a pump source we employed a master oscillator power amplifier (MOPA) laser system at 1064 nm, delivering 200 µJ, 500 ps long pulses at a 9 kHz maximum repetition rate with limited diffraction beam quality. At 2 kHz we obtained up to 45 µJ pulse energy and 200 ps pulse duration at 1180 nm, with a conversion efficiency of 22%. Up to the maximum incident average pump power of 1.35 W no serious beam quality degradation was observed, and we measured a factor M2 ⩽ 1.5 for the Raman shifted beam.

A Way to Improve the Efficiency of High Frame Rate Imaging Algorithm with Limited Diffraction Beams

Jian-yu Lu has studied a high frame rate (HFR) ultrasound imaging system since 1997. In the system, weighting echo signal by limited diffraction beam spends most of the computing time of imaging algorithm. In the paper the spectrum of HFR ultrasound imaging is analyzed. Combining limited bandwidth properties of image spectrum and the plane wave characteristics of limited diffraction wave, a method, which can improve the efficiency of weighted processing algorithms, is put forward. Simulation results show that it effectively reduces the calculation amount of weighting, meanwhile maintaining image quality.

Generation of a two-dimensional limited-diffraction beam with self-healing ability by annular-type photonic crystals

In this work, we present the design of a photonic structure for the generation of in-plane two-dimensional (2D) limited-diffraction beam. We have numerically investigated the characteristics of the light propagation passing through a 2D square-lattice annular-type photonic crystal shaped in an axicon configuration. Careful selection of the operating frequency as well as the optimization of the apex rod position creates a less diffracted beam whose transverse intensity profile closely resembles a zero-order Bessel function. The created beam dramatically resists against the spatial spreading over a propagation distance of 50 μm, after focusing with a spot size of ∼0.23 μm. The self-healing capability of the generated limited-diffraction beam is demonstrated by placing obstacles with different sizes and shapes along the optical axis. The two features that accompany with such beams, i.e., diffraction-limited propagation and reconstruction ability after encountering obstructions, may strengthen its usage in manipulation of light propagation in various environments.

Quantitative assessment of effects of phase aberration and noise on high-frame-rate imaging

The goal of this paper is to quantitatively study effects of phase aberration and noise on high-frame-rate (HFR) imaging using a set of traditional and new parameters. These parameters include the traditional −6-dB lateral resolution, and new parameters called the energy ratio (ER) and the sidelobe ratio (SR). ER is the ratio between the total energy of sidelobe and the total energy of mainlobe of a point spread function (PSF) of an imaging system. SR is the ratio between the peak value of the sidelobe and the peak value of the mainlobe of the PSF. In the paper, both simulation and experiment are conducted for a quantitative assessment and comparison of the effects of phase aberration and noise on the HFR and the conventional delay-and-sum (D&S) imaging methods with the set of parameters. In the HFR imaging method, steered plane waves (SPWs) and limited-diffraction beams (LDBs) are used in transmission, and received signals are processed with the Fast Fourier Transform to reconstruct images. In the D&S imaging method, beams focused at a fixed depth are used in transmission and dynamically focused beams are used in reception for image reconstruction.The simulation results show that the average differences between the −6-dB lateral beam widths of the HFR imaging and the D&S imaging methods are −0.1337mm for SPW and −0.1481mm for LDB, which means that the HFR imaging method has a higher lateral image resolution than the D&S imaging method since the values are negative. In experiments, the average differences are also negative, i.e., −0.2804mm for SPW and −0.3365mm for LDB. The results for the changes of ER and SR between the HFR and the D&S imaging methods have negative values, too. After introducing phase aberration and noise, both simulations and experiments show that the HFR imaging method has also less change in the −6-dB lateral resolution, ER, and SR as compared to the conventional D&S imaging method. This means that the HFR imaging method is less sensitive to the phase aberration and noise.Based on the study of the new parameters on the HFR and the D&S imaging methods, it is expected that the new parameters can also be applied to assess quality of other imaging methods.

A tale of two beams: an elementary overview of Gaussian beams and Bessel beams

An overview of two types of beam solutions is presented, Gaussian beams and Bessel beams. Gaussian beams are examples of non-localized or diffracting beam solutions, and Bessel beams are example of localized, non-diffracting beam solutions. Gaussian beams stay bounded over a certain propagation range after which they diverge. Bessel beams are among a class of solutions to the wave equation that are ideally diffraction-free and do not diverge when they propagate. They can be described by plane waves with normal vectors along a cone with a fixed angle from the beam propagation direction. X-waves are an example of pulsed beams that propagate in an undistorted fashion. For realizable localized beam solutions, Bessel beams must ultimately be windowed by an aperture, and for a Gaussian tapered window function this results in Bessel-Gauss beams. Bessel-Gauss beams can also be realized by a combination of Gaussian beams propagating along a cone with a fixed opening angle. Depending on the beam parameters, Bessel-Gauss beams can be used to describe a range of beams solutions with Gaussian beams and Bessel beams as end-members. Both Gaussian beams, as well as limited diffraction beams, can be used as building blocks for the modeling and synthesis of other types of wave fields. In seismology and geophysics, limited diffraction beams have the potential of providing improved controllability of the beam solutions and a large depth of focus in the subsurface for seismic imaging.

Estimation of two-dimensional strain rate based on high frame rate ultrasound imaging method

Strain rate (SR) measurement of heart tissue based on ultrasound images provides useful information of tissue hardness for diagnosing heart diseases. However, current SR estimation methods utilizing speckle tracking technique are based on conventional delay and sum (DS) imaging method, which causes skewed heart image resulting in inaccurate SR. To overcome the problem, a method to combine high frame rate (HFR) imaging method with speckle tracking technique was proposed. Using only one or a few transmissions for each image; compared with 91 for DS, new method can get a snapshot of moving targets, avoiding the skewing problem in DS. Two studies, with simulated and experimental echo data, respectively, were performed to verify the method. Both plane wave and limited diffraction beam (LDB) were studied for HFR. Both SR estimations in lateral and axial directions were calculated for the new and conventional ultrasound imaging methods. Results show that the new method has comparable or lower velocity errors than DS and more accurate SR, especially in lateral direction. Moreover, it can measure high velocity for other applications such as blood flow measurement. With the full view of heart image, SR of interest can be localized and then accurately estimated.

Low sidelobe limited diffraction beams in the nonlinear regime

In linear propagation, sidelobe levels of Bessel limited diffraction beams are only about 8 dB down relative to the mainlobe. In the nonlinear regime, these beams will have a region near the source where the side-lobe level of the second harmonic is 16 dB down, but this region has usually been considered to be so small that it is of little practical interest. In this paper it is shown that when there are only 1 to 3 sidelobes in a finite aperture Bessel beam, the second harmonic field will have low sidelobes for distances up to half of the depth of field. This result is backed up by simulations. In a medium with absorption, previous theory has shown that the sidelobes of the Bessel beam will also be reduced but only for absorption that was too high to be of practical use. Simulations presented here show that for breast tissue, which only has about 10% of the absorption of previous criteria, one will still get sidelobes which are comparable to that of a rectangular aperture even when the sidelobes would be high in a non-absorbing medium.

Limited-diffraction light propagation with axicon-shape photonic crystals

Electromagnetic beams are subject to spatial spreading as they propagate. I have investigated the light propagation passing through a finite-aperture, which is obtained by two-dimensional square-lattice photonic crystals (PCs). It is found that the beam that is coupled to the free-space by exiting the axicon-shape PC resists considerably against the diffraction. The inspection of the beam profile in the transverse to the propagation direction reveals the appearance of the side-lobes, and I have attributed the limited-diffraction beam propagation to these artificially created lobes. I optimize the length of the aperture while keeping the width constant and show that an order of magnitude improvement for beating the diffraction length is achievable. The advantages of the presented PC-based axicon over the bulk refractive axicons are the compactness and integrated nature of the former one, in addition to the flexibility of engineering individual unit cells of PC structure.

Nonlinear ultrasound fields simulation of harmonics from exponential and bessel beams sources

It's well known that the harmonic imaging quality can be improved by using sources that radiate narrower and attenuated sidelobe beams. Hence we try to enhance the harmonics' cartography by studying different source's power distributions. We developed a numerical code, using the spectral method, in order to resolve the parabolic wave equation. The numerical results are compared to those given by other researchers in order to validate our algorithm. Two source's power distributions (exponential and Bessel beams) are studied and compared to the uniform case. The use of exponential source leads to harmonics diagrams without sidelobes neither nearfield oscillations. But the beam width is increasing with propagation. The Bessel source presents a limited diffraction beam. The beam width is almost constant throughout the nearfield and the transition zone. The sidelobes have a weak level in the fundamental curves and they don't appear in the second harmonic ones.

High frame rate ultrasonic imaging through Fourier transform using an arbitrary known transmission field

The high frame rate (HFR) method, derived from the limited diffraction beams [Jian-yu Lu. 2D and 3D high frame rate imaging with limited diffraction beams. IEEE Trans Ultrason Ferroelectrics Freq Control 1997;44(4):839–56; Jian-yu Lu. Experimental study of high frame rate imaging with limited diffraction beams. IEEE Trans Ultrason Ferroelectrics Freq Control 1998;45(1):84–97; Hu Peng, Jianyu Lu, XueMei Han. High frame rate ultrasonic imaging system based on the angular spectrum principle. Ultrason 2006;44(Suppl):e97–9] theory, has a drawback that its imaging area, as wide as its array beam transmission field, can generally not be wider than the transducer is long; and that the only way to widen the area is to steer the transmission beams several times from different angles, which lowers the frame rate. Besides, the array beam field is quite demanding to produce. To widen the imaged area, we developed an extended, more general HFR method for 2D imaging. It allows any form of transmission field as long as the field is known, and it achieves the frame per transmission imaging rate. For verification, computer simulations use a linear transducer array and a cylindrical diffraction transmission field defined by the zero order Hankle function. The output images are in the sector format and have low sidelobes. The simulation results are consistent with our theory.