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.     

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.     

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 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     

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     

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.     

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.     

Time of flight diffraction imaging for double-probe technique

Due to rapid progress in microelectronics and computer technologies, the system evolving from analog to digital, and a programmable and flexible synthetic aperture focusing technique (SAFT) for the single-probe pulse-echo imaging technique of ultrasonic nondestructive testing (NDT) becomes feasible. The double-probe reflection technique usually is used to detect the nonhorizontal flaws in the ultrasonic NDT. Because there is an offset between the transmitter and receiver, the position and size of the flaw cannot be directly read from the image. Therefore, a digital signal processing (DSP) imaging method is proposed to process the ultrasonic image obtained by double-probe reflection technique. In the imaging, the signal is redistributed on an ellipsoid with the transmitter and receiver positions as focuses, and the traveltime sum for the echo from the ellipsoid to the focuses as the traveltime of signal. After redistributing all the signals, the useful signals can be constructively added in some point in which the reflected point is; otherwise, the signals will be destructively added. Therefore, the image resolution of the flaw can be improved and the position and size of the flaw can be estimated directly from the processed image. Based on the experimental results, the steep flaw (45°) cannot be detected by the pulse echo technique but can be detected by the double-probe method, and the double-probe B-scan image of 30° tilted crack is clearer than the pulse echo B-scan image     

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.     

Spot size, depth-of-focus, and diffraction ring intensity formulas for truncated Gaussian beams

Simple polynomial formulas to calculate the FWHM and full width at 1/e2 intensity diffraction spot size and the depth of focus at a Strehl ratio of 0.8 and 0.5 as a function of a Gaussian beam truncation ratio and a system f-number are presented. Formulas are obtained by use of the numerical integration of a Huygens-Fresnel diffraction integral and can be used to calculate the number of resolvable spots, the modulation transfer function, and the defocus tolerance of optical systems that employ laser beams. I also derived analytical formulas for the diffraction ring intensity as a function of the Gaussian beam truncation ratio and the system f-number. Such formulas can be used to estimate the diffraction-limited contrast of display and imaging systems.     

Multiplex localization imaging and sub-diffraction limited measurement

A novel technique, multiplex localization imaging, is proposed to enhance the image reconstruction of point sources beyond the diffraction limit for an incoherent remote sensing system. The technique utilizes position localization of point sources to create an image of a scene and to allow sub-diffraction limited measurement. Two types of algorithms, mean calculations and chi-square statistics, are implemented to analyze the limits of position localization. Three different multiplex localization imaging methods, time-, color-, and polarization-multiplexing, were investigated experimentally. The resolution of multiplex localization imaging was found to be 14 times better than the diffraction limit of the optics. The multiplex localization imaging technique has applications in remote sensing and astronomy, such as position measurement, multiple targets tracking, and image enhancement.     

Reconstructive elasticity imaging for large deformations

A method is presented to reconstruct the elastic modulus of soft tissue based on ultrasonic displacement and strain images for comparatively large deformations. If the average deformation is too large to be described with a linear elastic model, nonlinear displacement-strain relations must be used and the mechanical equilibrium equations must include high order spatial derivatives of the displacement. Numerical methods were developed to reduce error propagation in reconstruction algorithms, including these higher order derivatives. Problems arising with the methods, as well as results using ultrasound measurements on gel-based, tissue equivalent phantoms, are given. Comparison to reconstructions using a linear elastic model shows that equivalent image quality can be produced with algorithms appropriate for finite amplitude deformations.     

An iterated IRS technique for cross-sectional damage modelling and identification in beams using limited sensors measurement

This paper presents an effective method for cross-sectional damage localization and quantification in beams. First, a new strategy is suggested for cross-sectional damage modelling by means of Iterated Improved Reduction System (IIRS) approach. Then, a novel damage localization index is proposed employing Grey System Theory (GST) as a geometrical criterion for quantifying the amount of correlation between vectors of the calculated curvatures for the diagonal members of the flexibility matrices in the damaged and undamaged states. Since the method employs only the modal data of the translational degrees of freedom, it can be interpreted as damage identification method by utilizing incomplete modal data or installing a limited number of sensors. After detecting the damage location, to estimate the exact parameters of the cross-sectional damage, the problem is defined as a finite element model-updating problem which is solved with a new evolutionary optimization approach named Imperialist Competitive Algorithm (ICA). The applicability of the method is demonstrated by studying different damage patterns on two numerical examples of beams. In addition, its robustness is investigated in the presence of random noises and modelling errors. Obtained results emphasize the high accuracy and promising performance of the method, especially when noisy incomplete modal data are used.     

Two digital X-ray imaging systems for applications in X-ray diffraction

Two digital X-ray imaging systems developed at the Rutherford Appleton Laboratory are described: the Mark I and the Mark II. Both use a bidimensionally sensitive multiwire proportional counter (MWPC) as the basic X-ray image transducer coupled, in the case of the Mark I to a Digital LSI 11–23 microcomputer system via CAMAC, and in the case of the Mark II to a Digital LSI 11–73 microcomputer system via custom-built data acquisition hardware mounted directly on the Q-bus of the microcomputer. The Mark I system provides the advantages of high speed, high sensitivity digital imaging directly into the computer with the potential for software control of the sample orientation and environment. The Mark II system adds the novel features of signal averaging and multiframe exposures. The dedicated digital memories have a resolution of 512×512 pixels of 16 bits, matching well to the spatial resolution of the xenon-filled MWPC (0.5 mm fwhm over an aperture of 200 mm×200 mm). A 512×512×4 bit video graphics system displays the images in grey scales or colour.     

Zonal diffraction efficiencies and imaging of micro-Fresnel lenses

Abstract

The dependence of the zonal efficiency of a transmission Fresnel lens on the number of quantization levels and on the polarization of the light, without and with an antireflection (AR) overcoating is investigated. The height of the grooves with a blaze structure is locally optimized for maximum efficiency. A decreasing structure height over the zonal radius from the centre to the outer zones improves the efficiency because of a better-adapted blaze. An additional increase in the efficiency can be obtained by increasing the AR coating thickness over the zonal radius from the centre outwards. Furthermore, the point spread functions (PSFs) for several quantization levels are compared for both polarization cases. While the AR overcoating has nearly no effect on the shape of the x polarization PSF, the y polarization PSFs show differences between both AR coating and no coating and is reduced with an increasing number of levels. Finally, the comparison between the measured efficiency of coated silicon Fresnel lenses and a calculation lead to acceptable results. All calculations are performed with the integral equation system method with parametrization of the grating profile, an exact electromagnetic grating diffraction method for arbitrary profile shapes.