Limited Diffraction Beams Research Articles

Limited diffraction beams and their applications on signal processing

Limited diffraction beams such as Bessel beams and X waves have been studied recently. These beams have many potential applications in medical imaging, tissue identification, blood flow velocity vector estimation, nondestructive evaluation of industrial materials, optical communications, and other physics related areas such as electromagnetics and optics. In this talk, fundamentals of limited diffraction beams will be presented. Results of applications of X wave theory on high-frame rate medical imaging will be shown. Applications of these beams in other areas, especially, in signal processing will be discussed. [This work was supported in part by grant HL 60301 from the National Institutes of Health.]

Comparison of high-frame rate and delay-and-sum imaging methods

Recently, a high-frame rate imaging method has been developed with limited diffraction beams to construct either two-dimensional (2-D) or three-dimensional (3-D) images (up to 3750 frames or volumes/s for biological soft tissues at a depth of about 200 mm). In this talk, the new method is compared with the conventional delay-and-sum (dynamic focusing) method. Both computer simulation and experiment results show that the quality of images constructed with the two methods are virtually identical when the maximum Axicon angle of X waves in the high-frame rate method is approaching to 90 degrees. Theoretical analysis is carried out to confirm the results. This is significant because the high-frame rate method requires thousands of times less computations while achieving the same high-imaging quality as the conventional delay-and-sum method. [This work was supported in part by grant HL 60301 from the National Institutes of Health.]

Bessel and conical beams and approximation with annular arrays

The Bessel beam is one of the relatively new limited-diffraction beams that have been discovered. It is compared with the conical transducer, which also gives an approximate limited-diffraction solution to the wave equation. The conical transducer's field deviates from the predicted field in the nearfield, where it is wider. Therefore, the Bessel beam is better for use in a hybrid system where a limited-diffraction beam is used for transmission and a dynamically focused beam for reception. The limited-diffraction Bessel beam of order zero can be excited on an annular transducer with equal-area division of elements and with a fixed prefocus, i.e., conventional transducers used in commercial medical imaging equipment. The element division implies that the scaling parameter must be chosen to contain the first lobe of the Bessel function in the first element. In addition, the prefocus must be such that the array is steerable to infinite depth with minor loss. Even when the Bessel beam yields a larger depth of field than that of an unfocused transducer, it has the advantage of a narrower beam. Simulated examples are shown where the approximate Bessel beam compares favorably with a spherically focused beam with a fixed focus, an unfocused beam, and a conical transducer.

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.

2D and 3D high frame rate imaging with limited diffraction beams

A new 2D (two-dimensional) and 3D (three-dimensional) pulse-echo imaging method (Fourier method) has been developed with limited diffraction beams. In this method, a plane wave pulse (broadband) is used to transmit and limited diffraction beams of different parameters are used to receive. Signals received are processed to obtain spatial Fourier transform of object functions and images are constructed with an inverse Fourier transform. Because only one transmission is required to construct images, this method may achieve a high frame rate (up to 3750 frames/s for biological soft tissues at a depth of 200 mm). To demonstrate the efficacy of the method, both 2D C-mode and 3D images have been simulated using conditions that are typical for medical ultrasound. Results show that images of high resolutions (about 6 wavelengths at 200 mm) and low sidelobes (around -60 dB) can be constructed over a large depth of interest (30 to 200 mm) with a 50 mm diameter aperture. Experiments with the new method have also been carried out. 2D B-mode images have been constructed with conventional linear arrays. In the experiment, an ATS 539 tissue-equivalent phantom and two linear arrays were used. The first array had a center frequency of 2.25 MHz, dimension of 18.288 mm/spl times//spl times/12.192 mm, and 48 elements. The second had a center frequency of 2.5 MHz, 38.4 mm/spl times/10 mm in dimension, and 64 elements. Images of different fields of views were constructed from RF data acquired with these arrays using both the new and conventional dynamic focusing (delay-and-sum) methods. Results show that qualities of images constructed are almost identical with the two methods in terms of sidelobes, contrast, and lateral and axial resolutions. Phase aberration has also been assessed for the two methods, and results show that its influence is about the same on both methods. In addition, a practical imaging system to implement the new method is suggested and potential applications of the method are discussed.

Characteristics of a Conical Piezocomposite Transducer with an Apodized Vibration Area

A conical transducer with an apodized vibration area has been designed and fabricated using a piezocomposite material with 1–3 connectivity for creating limited-diffraction beams. The large electromechanical coupling coefficient k of the piezocomposite material increases the transducer sensitivity. It is shown that an almost uniform -6 dB beamwidth of 6–9 wavelengths is obtained in the axial range of z = 20 mm to 170 mm using a fabricated transducer 25 mm in diameter operating at 3.6 MHz. It is also demonstrated through preliminary experiments with B-mode imaging that high resolution is maintained over a large depth.

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.

Limited diffraction array beams

Limited diffraction beams have a large depth of field and could have many applications. In this article, new limited diffraction beams are developed. They are composed of multiple parallel beams and are thus called array beams. A broadband synthetic array experiment is used to produce these beams of a finite aperture, and results are in excellent agreement with theory over a large depth of field. In addition, potential applications of the new beams to real-time volumetric imaging and measurement of transverse velocity of blood flow are described. © 1997 John Wiley & Sons, Inc. Int J Imaging Syst Technol, 8, 126–136, 1997

Conical type ultrasonic transducer with apodisedvibration area for generation of limited diffraction beams

Conical transducers with a velocity distribution varying in accordance with the envelope function of the Bessel function J0 can easily be used to create Bessel-type limited diffraction beams. The authors present a simple and practical method of realising weighted conical transducers. The method uses apodisation on the vibration area by forming thin electrode strips of a uniform width spreading radially on the transducer. An almost uniform –6 dB beamwidth of 6–9 wavelengths has been obtained in the axial range of z = 10 to 180 mm using a fabricated transducer of 25 mm in diameter operating at 3.6 MHz.

Producing bowtie limited diffraction beams with synthetic array experiment

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.

Characteristics of a Discretely Weighted Conical Transducer for Generation of Limited Diffraction Beams

We proposed discretely weighted conical transducers as a simple and practical method of generating Bessel-type limited diffraction beams. In this paper we present the design and fabrication of a discretely weighted conical transducer and examine its capability of generating narrow-beam ultrasound. The transducer has eight annular electrodes, to which position-dependent discrete voltages are applied to create approximately the desired velocity distribution of vibration. It is shown experimentally that an almost uniform -6 dB beamwidth of 6–9 wavelengths is obtained in the axial range of z=10 mm to 200 mm using a transducer 25 mm in diameter operating at 3.6 MHz.

Generation of sinc wave by a one-dimensional array for applications in 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.

Bowtie limited diffraction beams for low-sidelobe and large depth of 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/spl deg/ 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.

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. >

Application of Bessel beam for Doppler velocity estimation

Limited-diffraction beams have a large depth of field and could be applied to medical imaging, tissue characterization, and nondestructive evaluation of materials. This paper reports the application of limited-diffraction beams, specifically, the Bessel beam, to Doppler velocity estimation. The Bessel beam has the advantage that velocity estimation is less subject to the depth of moving objects and the Doppler spectrum has distinct shoulders that increase the accuracy of velocity (both magnitude and Doppler angle) estimation in noisy environments. The shoulders of the Doppler spectrum might also help in solving the inverse problem, e.g., estimation of the velocity distribution in vessels.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

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.

Sidelobe reduction for limited diffraction pulse-echo systems

Conventional focused transducers have a sharp focal spot with low sidelobes, but they also have a short depth of field. Commercial medical scanners obtain increased depth of field by combining several images in a montage, each obtained at a different focal depth. Therefore, to get low sidelobes over a large depth of field several transmits must be used, which decrease the frame rate. Limited diffraction beams such as Bessel beams and X waves obtain good resolution over very large depth of field, but they have high sidelobes. A summation-subtraction method developed for decreasing the sidelobes is presented. The method requires three transmits, decreasing the frame rate to one third. The authors apply this technique to limited diffraction beams and obtain an analytic description.

Producing Deep Depth of Field and Depth-Independent Resolution in Nde with Limited Diffraction Beams

Limited diffraction beams, such as Durnin's J0 Bessel beam, are a class of nonspreading solutions to the isotropic/homogeneous scalar wave equation. These beams can be approximately produced with finite aperture and energy over a deep depth of field. In this paper, we report the application of a broadband J0 Bessel beam to nondestructive evaluation (NDE) of materials. Pulse-echo images of a stainless steel block phantom were obtained with both the J0 Bessel beam and a conventional focused Gaussian beam. Results show that uniformly high resolutions were obtained with the J0 Bessel beam over a large distance. In addition, the lateral resolution of the J0 Bessel beam is almost independent of the speed of sound of the materials inspected. In contrast, the lateral resolution of images obtained with the conventional focused Gaussian beam changes dramatically with the distance and the focal length of the beam in water is greatly reduced by the steel block. Therefore, limited diffraction beams could be useful for nondestructive evaluation of materials of different speeds of sound. Restoration of pulse-echo images obtained with these beams could be simplified.

Producing Deep Depth of Field and Depth-Independent Resolution in NDE with Limited Diffraction Beams

Limited diffraction beams, such as Durnin"s J0 Bessel beam, are a class of nonspreading solutions to the isotropic/homogeneous scalar wave equation. These beams can be approximately produced with finite aperture and energy over a deep depth of field. In this paper, we report the application of a broadband J0 Bessel beam to nondestructive evaluation (NDE) of materials. Pulse-echo images of a stainless steel block phantom were obtained with both the J0 Bessel beam and a conventional focused Gaussian beam. Results show that uniformly high resolutions were obtained with the J0 Bessel beam over a large distance. In addition, the lateral resolution of the J0 Bessel beam is almost independent of the speed of sound of the materials inspected. In contrast, the lateral resolution of images obtained with the conventional focused Gaussian beam changes dramatically with the distance and the focal length of the beam in water is greatly reduced by the steel block. Therefore, limited diffraction beams could be useful for nondestructive evaluation of materials of different speeds of sound. Restoration of pulse-echo images obtained with these beams could be simplified.

In vitro and in vivo real-time imaging with ultrasonic limited diffraction beams

Recently, there has been great interest in a new class of solutions to the isotropic/homogeneous scaler wave equation which represents localized waves or limited diffraction beams in electromagnetics, optics, and acoustics. Applications of these solutions to ultrasonic medical imaging, tissue characterization, and nondestructive evaluation of materials have also been reported. The authors report a real-time medical imager which uses limited diffraction Bessel beams, X-waves, Axicons, or conventional beams. Results (in vitro and in vivo) show that the images obtained with limited diffraction beams have higher resolution and good contrast over larger depth of field compared to images obtained with conventional focused beams. These results suggest the potential clinical usefulness of limited diffraction beams.