Designing limited diffraction beams

Theoretically, limited diffraction beams can only be produced with an infinite aperture. In practice, they can be closely approximated with a finite aperture over a large depth of field. Because of this property, these beams could have applications in medical imaging, tissue characterization, Doppler velocity estimation, and nondestructive evaluation (NDE) of materials, as well as other physics-related areas such as electromagnetics and optics. In this paper, a new method is developed to design limited diffraction beams of desired beam shapes within a finite aperture of interest. It uses previously discovered limited diffraction beams such as Bessel beams and X waves as basis functions, and constructs new beams with linear superpositions of the bases. To construct a new beam of a desired shape, coefficients of the basis functions in the linear superposition are chosen so that the difference between the new beam and a desired beam is minimized under the criterion of least-squares error within the aperture. This procedure is implemented by digitizing both the basis beams and desired beams in the aperture and solving a system of linear equations from its normal equation. The method is applied to several desired beams that are limited diffraction beams known previously. Results show that the designed beams and the desired beams are virtually identical. If the desired beams are not solutions to the wave equation, the designed beams are new limited diffraction beams that are similar in shapes to the desired beams. This suggests that the method may be a powerful and practical tool for developing new limited diffraction beams of desired properties.     

A new approach to obtain limited diffraction beams

Limited diffraction beams could have applications in medical imaging, tissue characterization, and nondestructive evaluation, as well as other wave related areas such as electromagnetics and optics. In this paper, we develop a novel approach that can convert any diffracting solution of the isotropic-homogeneous wave equation to a limited diffraction solution. As an example, this approach was applied to an n-dimensional wavelet solution that we generalized from the three-dimensional solution obtained by Kaiser et al. (1992). This example establishes a relationship between localized limited diffraction beams and the wavelet theory. The resulting limited diffraction beam was compared with those discovered previously     

Experimental study of high frame rate imaging with limited diffraction beams

Limited diffraction beams have a large depth of field and have many potential applications. Recently, a new method (Fourier method) was developed with limited diffraction beams for image construction. With the method and a single plane wave transmission, both 2D (two-dimensional) and 3D (three-dimensional) images of a very high frame rate (up to 3750 frames/s for a depth of 200 mm in biological soft tissues) and a high signal-to-noise ratio (SNR) can be constructed with relatively simple and inexpensive hardware. If limited diffraction beams of different parameters are used in both transmission and reception and transducer aperture is shaded with a cosine function, high-resolution and low-sidelobe images can be constructed with the new method without montage of multiple frames of images [the image quality is comparable to that obtained with a transmit-receive (two-way) dynamically focused imaging system]. In this paper, the Fourier method was studied with both experiment and computer simulation for 2D B-mode imaging. In the experiment, two commercial broadband 1D array transducers (48 and 64 elements) of different aperture sizes (18.288 and 38.4 mm) and center frequencies (2.25 and 2.5 MHz) were used to construct images of different viewing sizes. An ATS539 tissue-equivalent phantom of an average frequency-dependent attenuation of 0.5 dB/MHz/cm was used as a test object. To obtain high frame rate images, a single plane wave pulse (broadband) was transmitted with the arrays. Echoes received with the arrays were processed with both the Fourier and conventional dynamic focusing (delay-and-sum) methods to construct 2D B-mode images. Results show that the quality (resolution and contrast) of constructed images is virtually identical for both methods, except that the Fourier method is simpler to implement. Both methods have also a similar sensitivity to phase aberration distortions. Excellent agreement among theory, simulation, and experiment was obtained.     

A study of two-dimensional array transducers for limited diffraction beams

The newly developed limited diffraction beams such as the Bessel beams and X waves have a large depth of field and approximate depth-independent property. They have possible applications in medical imaging, color Doppler imaging, tissue characterization, and nondestructive evaluation of materials, and in other wave related physical branches such as electromagnetics and optics. However, limited diffraction beams are currently produced with an annular array transducer that has to be steered mechanically. In this paper, we study the feasibility of steering these beams with a two-dimensional array, and show that there will be almost no distortion of beams if the effective aperture reduction of the array is properly compensated so that the beams have a constant transverse profile as they are steered. In addition, methods for reducing the complexity of the electronic multiplexing of the array elements are proposed. We also investigate the influences of the interelement distance and the size of array elements on the sidelobes and grating lobes of limited diffraction beams as the beams are steered. They are similar to those previously reported for conventional beams.     

Modeling of CW annular arrays using limited diffraction Bessel beams

Presents 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     

Bowtie limited diffraction beams for low-sidelobe and large depthof field imaging

Limited diffraction beams such as Bessel beams and X waves have a large depth of field and thus could have many applications. However, these beams have higher sidelobes as compared to conventional focused beams in their focal planes. In this paper, a new class of limited diffraction beams is developed. These beams are termed bowtie limited diffraction beams because they have bowtie shapes in a plane perpendicular to the beam axis. To obtain pulse-echo images of low sidelobes and a large depth of field, a bowtie limited diffraction beam is used in transmission and its 90° rotated response (around the beam axis) is used in reception. Unlike the summation-subtraction method developed previously, this method does not reduce image frame rate or dynamic range of signals and is not motion sensitive. The theory of the bowtie limited diffraction beams is developed. Computer simulation of the theoretical beams under practical conditions, such as finite aperture, finite bandwidth, and causal excitation, is performed with the Rayleigh-Sommerfeld diffraction formula. The simulated beams are very close to those predicted analytically over a large depth of field     

High-efficiency multibeam acousto-optic diffraction of pulsed light with nonequidistant beams

Acousto-optic diffraction of pulsed laser radiation on a multifrequency acoustic wave has been studied. It is established that the high efficiency of diffraction (i.e., the formation of a multibeam optical field with low losses) can be also achieved for a nonequidistant set of frequencies and, hence, nonuniform distribution of beams in the angular space. Results of experimental verification of the proposed approach are presented.     

Parallel beamforming using synthetic transmit beams

Parallel beamforming is frequently used to increase the acquisition rate of medical ultrasound imaging. However, such imaging systems will not be spatially shift invariant due to significant variation across adjacent beams. This paper investigates a few methods of parallel beam-forming that aims at eliminating this flaw and restoring the shift invariance property. The beam-to-beam variations occur because the transmit and receive beams are not aligned. The underlying idea of the main method presented here is to generate additional synthetic transmit beams (STB) through interpolation of the received, unfocused signal at each array element prior to beamforming. Now each of the parallel receive beams can be aligned perfectly with a transmit beam--synthetic or real--thus eliminating the distortion caused by misalignment. The proposed method was compared to the other compensation methods through a simulation study based on the ultrasound simulation software Field II. The results have been verified with in vitro experiments. The simulations were done with parameters similar to a standard cardiac examination with two parallel receive beams and a transmit-line spacing corresponding to the Rayleigh criterion, wavelength times f-number (lambda x f#). From the results presented, it is clear that straightforward parallel beamforming reduces the spatial shift invariance property of an ultrasound imaging system. The proposed method of using synthetic transmit beams seems to restore this important property, enabling higher acquisition rates without loss of image quality.     

受光阑限制贝塞耳-高斯光束的衍射特性

对贝塞耳-高斯光束经不同几何构形光阑的衍射作了比较研究。结果表明,在选取合适的光阑参数情况下,锯齿光阑能减弱所关心区域(F=20)轴上光强的衍射调制,而超高斯光阑能抑制大部分菲涅耳区(F>3.7)的衍射调制。锯齿光阑和超高斯光阑都有较大的填充因子,使用合适的超高斯光阑可得到有较大填充因子(如fSuper-Gaussian=0.25)又更平滑的横向光强分布。数值计算说明,为有效抑制光强的衍射调制,适当选取超高斯光阑的阶数n和半宽ws(如n=4,ws=0.4mm)是必要的。     

Effective schema for the rigorous modeling of grating diffraction with focused beams

Most modal diffraction methods are formulated for incident plane waves. In practical applications, the probing beam is focused. Usually, this is simulated by means of numerical integration where Gaussian quadrature formulas are most effective. These formulas require smooth integrands, which is not fulfilled for gratings due to Rayleigh singularities and physical resonances. The violation of this condition entails inaccurate integration results, such as kinks and other artifacts. In this paper, a methodology for the efficient treatment of the numerical integration with improved accuracy is presented. It is based on the subdivision of the aperture along the lines of Rayleigh singularities, mapping of these subapertures into unit squares, and separate application of the Gaussian cubature formulas for each subarea.     

Single-pulse tissue doppler using synthetic transmit beams

Tissue Doppler imaging (TDI) is a common technique for investigating myocardial function. Typically, B-mode data and TDI data are recorded using separate acquisitions and combined into a single, color overlaid image. In this work, we present a novel method for TDI imaging, where both TDI and B-mode are created from the same acquisition. Velocities are calculated from the phase shift between neighboring transmit events in the B-mode scan; hence the name singlepulse tissue Doppler (SPTD). Using a novel transmit beam interleaving pattern, this method provides TDI and B-mode at the same high frame rate with an adjustable Nyquist velocity limit. Through simulations and measurements, this work investigates the bias and variance of the SPTD velocities and compares the estimates to those of the conventional TDI autocorrelation estimation method. The results showed that the method introduces an additional bias and variance in the velocity estimates compared with conventional TDI. However, by applying bias compensation, the SPTD velocity estimates were close to those of regular TDI. Using SPTD, the whole left ventricle was imaged within a 65-degree sector at a frame rate of 110 frames per second (43 transmissions per frame).     

An iterative synthetic aperture imaging algorithm with correction of diffraction effects

In this paper we present an iterative version of the synthetic aperture imaging algorithm extended synthetic aperture technique (ESAFT) proposed recently. The algorithm is based on a linear model that accounts for the distortions effects of an imaging system used for acquisition of ultrasonic data. Improved resolution (both lateral and temporal) in the reconstructed image is obtained as a result of minimizing the reconstruction mean square error. In this work, the minimization is extended to parameters that characterize expected amplitudes of each image element in the area of interest. An iterative optimization scheme is proposed, which in each step performs minimization of the reconstruction error based on the parameter matrix found in the previous step. Comparing to ESAFT, the proposed approach yields a significant improvement in resolution and a high degree of robustness with regard to initial choice of the parameter matrix. Performance of the proposed algorithm is evaluated using both real and simulated ultrasonic data.     

Subwavelength diffraction in a limited number of metal waveguide arrays

Subwavelength diffraction in a limited number of nanostructured metal waveguide arrays has been investigated. We use a new method to derive the coupling constant. Perturbation approach and supermode theory are employed to obtain the solution for propagation constants and wave functions. The distributions of the field intensity with propagation distance are obtained accurately. Our theories are verified by the finite-difference time-domain method. Numerical simulation results show a good agreement with the theoretical predictions.     

Image reconstructions in diffraction tomography from limited transmitted field data sets

Diffraction tomography reconstructions of objects from limited transmitted field data sets are discussed together with theoretical analyses and results of numerical experiments. It is shown that limited data sets, representing only a small part of the complete data sets, can be used for reconstructions in diffraction tomography with satisfactory accuracy. We also find that, in diffraction tomography based on the hybrid filtered backpropagation and the first-Rytov approximation, the use of limited data sets can provide a larger range of validity than the use of complete data sets, the reason being that limited data sets pose less-severe phase-unwrapping problems.     

Recording of concave diffraction gratings in counterpropagating beams using meniscus blanks

The most technological way of recording blazed concave holographic gratings is by using the direct- and back-reflected beams. Usually plano-concave blanks are used for diffraction grating fabrication. To compensate the refraction at the plane back surface of the blank, one has to use additional elements or two-step recording mountings. Another solution is to use diffraction grating blanks having zero optical power. A number of gratings with different groove frequencies have been fabricated using concave—convex blanks. The theoretical investigation of the aberration properties of the recording mountings, and the experimental study of the grating properties are discussed. Using the CODE V program it is shown that a significant reduction in the aberration size of the virtual recording source can be achieved by optimization of the radius of curvature of the blank back surface. The experimental results confirm the possibility of achieving gratings that can be used in flat-field spectrometers with a limit of resolution of about 2–3 nm even in the case where an aberrated virtual recording source is used.     

High frame rate imaging system for limited diffraction array beam imaging with square-wave aperture weightings.

A general-purpose high frame rate (HFR) medical imaging system has been developed. This system has 128 independent linear transmitters, each of which is capable of producing an arbitrary broadband (about 0.05-10 MHz) waveform of up to +/- 144 V peak voltage on a 75-ohm resistive load using a 12-bit/40-MHz digital-to-analog converter. The system also has 128 independent, broadband (about 0.25-10 MHz), and time-variable-gain receiver channels, each of which has a 12-bit/40-MHz analog-to-digital converter and up to 512 MB of memory. The system is controlled by a personal computer (PC), and radio frequency echo data of each channel are transferred to the same PC via a standard USB 2.0 port for image reconstructions. Using the HFR imaging system, we have developed a new limited-diffraction array beam imaging method with square-wave aperture voltage weightings. With this method, in principle, only one or two transmitters are required to excite a fully populated two-dimensional (2-D) array transducer to achieve an equivalent dynamic focusing in both transmission and reception to reconstruct a high-quality three-dimensional image without the need of the time delays of traditional beam focusing and steering, potentially simplifying the transmitter subsystem of an imager. To validate the method, for simplicity, 2-D imaging experiments were performed using the system. In the in vitro experiment, a custom-made, 128-element, 0.32-mm pitch, 3.5-MHz center frequency linear array transducer with about 50% fractional bandwidth was used to reconstruct images of an ATS 539 tissue-mimicking phantom at an axial distance of 130 mm with a field of view of more than 90 degrees. In the in vivo experiment of a human heart, images with a field of view of more than 90 degrees at 120-mm axial distance were obtained with a 128-element, 2.5-MHz center frequency, 0.15-mm pitch Acuson V2 phased array. To ensure that the system was operated under the limits set by the U.S. Food and Drug Administration, the mechanical index, thermal index, and acoustic output were measured. Results show that higher-quality images can be reconstructed with the square-wave aperture weighting method due to an increased penetration depth as compared to the exact weighting method developed previously, and a frame rate of 486 per second was achieved at a pulse repetition frequency of about 5348 Hz for the human heart.