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.     

Producing bowtie limited diffraction beams with synthetic arrayexperiment

Limited diffraction beams have a large depth of field and could have applications in medical ultrasound and other wave related areas such as electromagnetics and optics. However, these beams have higher sidelobes than conventional focused beams at their focuses. Recently, a new type of beam, called bowtie limited diffraction beams, was developed. These beams can achieve both low sidelobes and a large depth of field in medical imaging. In this paper, the production of bowtie beams in water with a synthetic array experiment is reported. A broad-band PZT ceramic/polymer composite transducer of about 1 mm diameter and 2.5 MHz central frequency was scanned in a raster format and placed at the centers of elements of an equivalent two-dimensional array of 50 mm diameter aperture. A polyvinylidene fluoride (PVDF) needle hydrophone of 0.5 mm diameter was used to receive the waves produced by the transducer. Proper weighting functions were applied to the received signals to produce various beams. Results show that the bowtie beams produced with the synthetic array experiment are in good agreement with those derived from theory and obtained by computer simulations. The depth of field of these beams is about 216 mm and sidelobes of a tenth derivative bowtie X wave in pulse-echo imaging are about 30 dB lower than those of rotary symmetric limited diffraction beams such as the zeroth-order X wave discovered previously     

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.     

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.     

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     

Experimental verification of nondiffracting X waves

The propagation of acoustic waves in isotropic/homogeneous media and electromagnetic waves in free space is governed by the isotropic/homogeneous (or free space) scalar wave equation. A zeroth-order acoustic X wave (axially symmetric) was experimentally produced with an acoustic annular array transducer. The generalized expression includes a term for the frequency response of the system and parameters for varying depth of field versus beam width of the resulting family of beams. Excellent agreement between theoretical predictions and experiment was obtained. An X wave of finite aperture driven with realizable (causal, finite energy) pulses is found to travel with a large depth of field (nondiffracting length).     

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     

Interaction of shear waves with a griffith crack situated in an infinitely long elastic strip

This paper deals with the propagation of shear waves in a wave guide which is in the form of an infinite elastic strip with free lateral surfaces. This strip contains a Griffith crack. An integral transform method is used to find the solution of the equation of motion from the linear theory for a homogeneous, isotropic elastic material. This method reduces the problem into an integral equation. It has been observed that only shear waves with frequencies less than a parameter-value, depending on the width of the wave guide, can propagate. The integral equation is solved numerically for a range of values of wave frequency and the width of the strip. These solutions are used to calculate the dynamic stress intensity factor, displacement on the surface of the crack and crack energy. The results are shown graphically.     

Nondiffracting X waves-exact solutions to free-space scalar wave equation and their finite aperture realizations

The authors report families of generalized nondiffracting solutions of the free-space scalar wave equation, and specifically, a subset of these nondiffracting solutions, which are called X waves. These nondiffracting X waves can be almost exactly realized over a finite depth of field with finite apertures and by either broadband or bandlimited radiators. With a 25-mm diameter planar radiator, a zeroth-order broadband X wave will have about 2.5-mm lateral and 0.17-mm axial -6-dB beam widths with a -6-dB depth of field of about 171 mm. A zeroth-order bandlimited X wave was produced and measured in water by a 10 element, 50-mm diameter, 2.5-MHz PZT ceramic/polymer composite J (0) Bessel nondiffracting annular array transducer with -6-dB lateral and axial beam widths of about 4.7 mm and 0.65 mm, respectively, over a -6-dB depth of field of about 358 mm. Possible applications of X waves in acoustic imaging and electromagnetic energy transmission are discussed.     

The sequence Fourier transform method for the analysis of a periodic viaduct subjected to non-uniform seismic waves

This study is concerned with the analysis of the dynamic response of an infinite periodic viaduct to non-uniform seismic waves. The periodic viaduct is assumed to be composed of an infinite number of spans, and each span is supposed to consist of a pier, two longitudinal beams (left and right beams) and three linking springs. The sequence Fourier transform method is used to decompose a non-uniform seismic wave into a set of spatially harmonic waves. By using the periodic condition of the viaduct and the transfer matrix method, the wavenumber domain response of the viaduct to each component of the non-uniform seismic wave is obtained. The space-domain response of the periodic viaduct is retrieved by applying the inverse sequence Fourier transform to the wavenumber domain solutions. Numerical results for energy bands of the periodic viaduct and those for the dynamic responses of the periodic viaduct to spatially harmonic waves as well as to non-uniform seismic waves are presented. It is found that there exist three kinds of characteristic waves for both the in-plane and out-of-plane wave vibrations of the periodic viaduct. Also, proposed numerical results show that when the periodic viaduct is subjected to traveling waves, serious resonance may occur. Resonance may occur at the bounding frequencies of the passbands for the characteristic waves. Also, due to the coincidence effect between traveling seismic waves and the characteristic waves, resonance may occur at frequencies in the passbands of the characteristic waves.     

Generation of sinc wave by a one-dimensional array for applicationsin ultrasonic imaging

A new solution to the 2-D scalar wave equation is presented which describes an ultrasonic beam maintaining the lateral field response expressed by the sinc function over a finite depth of field. This new beam is realizable with a linear array transducer, and less subject to diffraction spreading than conventional focused beams, physically, it is a superposition of plane waves having the same wavelength, but traveling at different angles. It is shown by numerical simulation that the beam can provide more uniform lateral beamwidth and smoother on-axis field magnitude over a greater depth of field than the rectangular transducers and Gaussian apodized transmitters which have been used to increase the limited depth of field of conventional focused beams. Compared with currently developed limited diffraction beams which must be generated by 2-D array transducers, the beam has a wider lateral beamwidth but with lower sidelobe levels. In ultrasonic medical imaging, the beam enables one to obtain a line focus using a 1-D array transducer and to eliminate the diffraction correction required in some applications such as tissue characterization     

Elastic analysis of torsion crack problems of a bar with ring segment sections

In an earlier paper [6] we have studied the case of interaction of shear waves with a crack centrally situated in an infinite elastic strip; we, in this paper, extend the study to the case of two coplanar Griffith cracks. An integral transform method is used to find the solution of the equation of motion from the linear theory for a homogeneous, isotropic — elastic material. This method resolves the problem into an integral equation. It has been observed that shear waves with frequencies less than a parameter depending on the width of the wave guide can only propagate. The integral equation is solved numerically for a range of values of wave frequency, width of strip and the inter-crack distance. These solutions are used to calculate the dynamic stress intensity factor. The results are shown graphically.     

A new synthetic aperture focusing method to suppress the diffraction of ultrasound

Spatial resolution of an ultrasound image is limited by diffraction of ultrasound as it propagates along the axial direction. This paper proposes a method for reducing the diffraction spreading effect of ultrasound by using a synthetic aperture focusing (SAF) method that uses plane waves instead of spherical waves. The new method performs data acquisition and beamforming in the same manner as conventional SAF methods. The main difference is that all array elements are used on each firing to generate a plane wave, the traveling angle of which varies with the position of a receive subaperture. On reception, each scan line is formed by synthesizing RF samples acquired by relevant receive subapertures with delays to force the plane waves to meet at each imaging point. Theoretical analysis and computer simulation with infinite transmit aperture show that the proposed method is capable of suppressing the diffraction of ultrasound and especially causing the lateral beam width to remain unchanged beyond a certain depth determined by the size of a receive subaperture and the maximum traveling angle of plane waves. It is demonstrated that the proposed method is realizable using a linear array transducer. It is also shown that the lateral radiation pattern produced by the proposed method has smaller beam width than that using conventional SAF methods in the region of interest because it suppresses the diffraction of ultrasound.     

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.     

Quasi-diffraction-free beams.

A diffraction-free beam is obtained by the superposing of plane waves whose wave vectors make an angle with the propagation axis. These plane waves are realized with point sources that are distributed uniformly around a circle and an infinitely large aperture lens. After the field passes through the lens it has nondiffracting properties and is described by the zero-order Bessel function. Relaxing these conditions makes the beam diffraction free within only a limited region. The beam generated from such a geometry is referred to as a quasi-diffraction-free beam. The effects of the width of the annular source on the beam spread are discussed and compared with those for a Gaussian beam. Approximate expressions for quasi-diffraction-free beams are also obtained.     

Optical generation of focus wave modes

Localized wave solutions of free-space wave equation can be used in numerous applications where the localized transmission of electromagnetic energy is of major importance. However, an optical implementation of localized wave fields has not been accomplished yet, except for an ultrashort version of the Bessel beams or the so called Bessel-X pulses. We propose an approach to constructing realizable optical schemes for generation of localized wave fields. We show that wavelength dispersion of the cone angle of axicons and circular diffraction gratings can be used to generate good approximation to focus wave modes.     

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.     

On diffraction of waves by the infinite grating of circular dielectric cylinders at oblique incidence: Floquet representation

Abstract

We investigate the diffraction of plane E-polarized electromagnetic waves by an infinite array of infinitely long circular dielectric cylinders for the most generalized case of oblique incidence. Employing the Sommerfeld integral representation of the Hankel function, and exploiting Poisson's summation formula in separation-of-variables solution, we have acquired a new representation for the diffraction of waves by the infinite grating of the circular dielectric cylinders at oblique incidence. The exact solutions for the external electric and magnetic field intensities have been derived in terms of the diffraction angles of the grating which are obtained by solving the grating equation. In addition, the transmitted and reflected fields of the infinite grating have been presented in terms of propagating and evanescent Floquet modes.     

Propagation of limited-diffraction X-waves in dissipative media

Diffractionless solutions of the wave equation in the form of X-waves have previously been obtained based on the inviscid form of the wave equation. A new general solution to the cylindrically symmetric wave equation for a medium with classical viscous losses is obtained. Particular solutions called dissipative Arcsin X-waves have been derived from this general solution. The properties of these waves are discussed for both infinite and finite size transducers and for different viscous liquids. To calculate the field produced by a finite transducer diameter, we have derived a dissipative form of the Rayleigh integral