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/sub 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. >

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

Systems of two-dimensional (2-D) imaging arrays and apertures are considered from the point of view of their performance in the imaging of spatially incoherent as well as coherent source distributions. Such systems find applications in radar, sonar, and ultrasound imaging, as well as in applications such as seismology and radio astronomy. For linear imaging techniques related to beamforming and based on the Fourier transform relationship between the source distribution and the aperture plane measurements, the point spread function of the system completely characterizes its performance. This function is determined by the geometry of the physical aperture or array as well as the weighting that can be applied to measurements. It is shown that the introduction of the concept of coarray, both for receive apertures in incoherent imaging and for transmit/receive systems in reflection-mode coherent imaging, provides a convenient and elegant framework within which many apparently isolated techniques for point-spread function or aperture synthesis can be understood. In addition to this unifying role, coarray concept gives new insight into the aperture synthesis process, which allows interesting new imaging techniques to be developed, especially in coherent imaging. >

The unifying role of the coarray in aperture synthesis for coherent and incoherent imaging

Real-time ultrasonic imaging systems have been available for more than thirty years. During this time much has occurred to the basic architecture and functions of these clinical systems and their beamformers, which are, in many ways, the most important components of these systems. This talk will review some of the changes that have occurred and will discuss current trends in beamformer design. Throughout most of the 30 years of real time imaging, analog beamformers have been the mainstay of all instruments. The common methods for the implementation of analog beamformers for annular, linear/curved arrays, and phased arrays will be reviewed and the distinguishing characteristics identified. At the present time the industry is undergoing a major shift toward digital beamformation with the introduction of several commercial systems. Given that the earliest digital systems were available roughly 15 years ago, it is fair to say that the introduction of digital beamformers has been relatively slow. Today this shift seems to be gaining momentum. Some of the factors causing these shifts will be reviewed along the the common types of designs and implementations. The ongoing search for improved imaging performance will continue to introduce new challenges for beamformer designers. Among already proposed imaging methods and techniques are elevation focusing (1.5D arrays), synthetic apertures, 2D and sparse arrays, phase aberration correction, and others. The most common complication introduced by these is a significant increase in channel count. Further, there will also be beamformer design issues related to signal processing. These include not only Doppler processing but also the processing of signals due to novel contrast agents. Finally, with advances in computer and microelectronics technologies, the way we view beamformation may have to undergo sizeable changes. These topics will be reviewed not only as technical challenges but also in light of the constraints introduced by today's marketplace.

Evolution of ultrasound beamformers

Concentric-ring phased arrays subdivided into sectors (radial slices) can, with appropriate phasing, produce power absorption patterns useful for hyperthermia cancer therapy. The ability of a concentric-ring array to move a focal region along the central axis of the transducer is well known. Less well known is the ability of such an array to produce variable diameter annular (or ring) focal regions. Such focal rings can be effective in heating some tumors if directed around the tumor periphery. These focal rings have been produced in the past by fixed annular focus lenses, or effectively by mechanical scanning of "point" focus ultrasonic transducers. Production of these focal rings by a concentric-ring phased array has the advantage of allowing the focal ring diameter and focal length to be easily changed and scanned by phasing providing much greater heating flexibility. However, under some conditions such arrays produce very large secondary focus effects along the central axis of the amay. Concentric-ring arrays can also provide only patterns of circular symmetry. These problems can be partially solved by dividing the disk of the array into sectors. By appropriate phasing of the sectors, the intensity along the central axis can be greatly reduced. Moreover, appropriate phasing of the rings and sectors can produce patterns that are circularly asymmetric. By controlling these asymmetries, nonspherical tumors can be heated more optimally. Power absorption patterns in Iossy media for this class of applicators are analyzed numerically allowing a quantitative evaluation of both advantages and limitations of this approach. A thermal model based on the bioheat equation is also used to predict temperature distributions in volumes where important thermal parameters, particularly blood flow, are varied.

Concentric-Ring and Sector-Vortex Phased-Array Applicators for Ultrasound Hyperthermia

The transducer is probably the single most important component of any ultrasonic imaging system. A basic introduction to the problems and paradoxes of transducer design is given. After introducing the piezoelectric equations and discussing important transducer material such as lead zirconate titanate and polyvinylidene difluoride, the techniques for modeling the electromechanical impulse response are reviewed. Quarter-wave matching and short pulse techniques are discussed. The prediction of the ultrasound field of plane, spherical, and conical transducers is reviewed with emphasis on the spatio-temporal impulse response technique. Finally, the use of the above approaches is illustrated in a very practical fashion for three interesting transducer geometries: 1) a split aperture device with two focal lengths, 2) a five-element annular array, and 3) a 37.5 degree conical/annular array hybrid transducer.

Ultrasound Transducers for Pulse-Echo Medical Imaging

The B‐scan technique, which is used in ultrasonic medical diagnosis and in materials testing, suffers from a low lateral resolution because of the large diameter of the ultrasound beam. The present paper describes a beamforming device which focuses the beam over a large depth, i.e., the well‐known depth of focus problem is circumvented. An experimental device is described which gives a lateral resolution of 2 to 3 mm at a frequency of 2 MHz (λ = 0.75 mm in water).

Ultrasound Axicon: a device for focusing over a large depth

A method for producing cross-sectional images using an ultrasonic imaging unit operating on the pulse-echo principle, which unit includes a transducer system having a stationary elongated array of adjacent transducer elements with transverse electrode segments adjacent one another on one side of the array Successively and cyclically selected groups of adjacent elements are energized to produce an ultrasonic beam as a result of pulsed signals being applied to the corresponding electrode segments Focussing of the ultrasonic beam produced by each group of the transducers is effected by having the transmitter signals applied to the electrode segments or subgroups thereof, and/or the echo-signals derived by the electrode segments from the reflected energy, time shifted relative to one another, in which each transmitter or echo-signal has a time shift associated thereto which is a function of the distance between the corresponding electrode segments (or subgroups of segments) and the center of the group of transducers

Method and apparatus for producing cross-sectional images using ultrasound

B-scan displays a picture of a cross-section through the body and is widely used in ultrasonic diagnosis. The main deficiency of B-scan is the low lateral resolution, which has its origin in the large diameter of the ultrasound beam.

Methods for Increasing the Lateral Resolution of B-Scan

An alternative method is described for focussing ultrasound over a large depth. The method is useful in B-scan. The final echo signal is obtained as the s u m of two partial echo signals and different segments of an annular transducer are used for transmission and reception of the two partial echo signals. The advantages with respect o a method described previously are simplicity and a higher dynamic range. A lateral resolution of 2 mm is obtained which is almost an order of magnitude better than conventional B-scan. Scans of fetuses in virfo are shown. THE INTRODUCTION 4 E-scan is a technique which is widely used in medical ultrasonic diagnosis and which gives a picture of a cross section of the body (for a short review of B-scan operation see, e.g., [1,2,3]). B-scan has a good longitudinal resolution but a bad lateral resolution, because an unfocussed or only weakly focussed beam is used. It is possible to focus the beam with a large aperture and obtain good Iateral resolution but only over a limited depth of focus. It was shown in Ref. [S] that focussing wit’h an annular t,ransducer gives good lateral resolution over a large depth. Basically this is so because on every point of the axis the contributions of all elements of the annulus add up in phase and give a large amplitude, whereas outside the axis they do not add up in phase. When the beam is focussed with an annular transducer one obtains rather large sidelobes. A method was demonstrated in Ref. [3] for reducing these sidelobes. This method works as follows. The annular transducer is built out of several segments. Manusrript received April 10. 1974; revised June 17, 1974. The authors are with F. Hoffmann-La Roche & Co., AG, Bssel, Switzerland. For a first ultrasound pulse all segments transmit and receive in phase. The echoes so received are stored. For a second pulse, the phase of the pulse transmitt’ed by each segment is made proportional to the angular position of the segment on the ring. On reception the signal received by each segment is multiplied by a phase factor which has the same magnitude but opposite sign as the phase factor of the transmitted pulse. The echoes received from the second pulse are then subtracted from the stored echoes of the first pulse. It. was shown that the following point spread function b ( r ) can be obtained with this method where Jo and Jz are Bessel functions of the zeroth and second order, respectively, R is the radius of the annular transducer, r is the distance of the point reflector from the axis, X is the sound wavelength, and d is the distance between transducer and reflector. The arrangement is illustrated in Fig. 1. In the present paper we will describe a different method for reducing the sidelobes and obtaining the same point spread function as in Eq. ( 1 ) . The new method has the advantages of greater simplicity and of a larger dynamic range. These advantages will be elaborated upon at the end of the next sect’ion. NEW METHOD FOR SIDELOBE REDUCTION The new method will be explained for an annular transducer built out of eight segments and is illustrated in Fig. 2. The segments drawn black transmit sound whereas 12 IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, JANUARY 1975 the Fourier transform of the amplitude distribution on to the width of a segment shown in Fig. 2 and not with respect to the diameter of t,he ring. Additional comment on this is given in Ref. [S].) The contribution ak(r,cp) due to the term exp ( jkcp) is [3,4] NNULAR TRANSDUCER the ring. (Note hat we are only in the far field with respect adr,cp) = exp [jk 6 + v)] Jk (d) . (3) 2 ~ R r FOINT R E R E C r O R J k ( S ) is the Bessel function of the kth order and the other 1. symbols are the same as in Eq. ( 1 ) . \\ Fig. 1. Perspective view of the annular transducer. We therefore obtain for the total amplitude 1. PULSE 2 . PULSE a(r,cp) = +Jo (7) ; Jt (y) cos 2p 21rRr 2 27rRr + Js (y) cos6cp + (4) 2sRr 3* because J k ( S ) = Jr ( m ) for k even. In an analogous way the sensitivity s(r,(p) to a sound source at r and cp for the segment’s shown as receiving in Fig. 2. Transmitting and receiving segments for the first and Fig. 2 is computed as second pulse. Black: transmitting, White: receiving. s(r,cp) = ~ J o (7) + ; J Z (7) cos 2cp 21rRr 2 2aRr 2 27rRr 3* J J S (7) cos6cp f * S * . (5) 0 d 2 n %l2 2n c4 The echo amplitude bl(r,cp) of a point reflector is obtained Fig. 3. Transmitted amplitude a t l (q ) for the first pulse a functhe product’ Of Eqs. (4) and (5) tion of angle ‘p on the ring. 2xRr 4 2aRr bl(r,cp) = 2Jo2 (F) ,z J2’ (7) cos2 2cp the segments drawn white receive sound. Two pulses are used for the sidelobe reduction and between the two pulses J Z (T) J S (7) cos 6cp cos 2cp the configuration is “rotated” by 45’ (by electronically switching on and off appropriate segments), as is seen + . . . . . . ( 6 ) in Fig. 2. The partial echo signals of two pulses are added in order to btain the final echo signal. For the second pulse we obt,ain the echo amplitude We will now analyze this method. The transmitted amplitude atl(cp) on the ring during the first pulse is shown bz ( r , cp ) = 1 ~ ~ 2 (T) 2rRr 4 J ~ Z (T) sin2 zcp in Fig. 3 as a function of angle cp. The maximum amplitude 2 ~ R r 21rRr 3??

Focussing Ultrasound Over a Large Depth with an Annular Transducer-An Alternative Method

An apparatus for focusing ultrasonic waves in a focal line by an ultrasonic optical system having at least one rotation-symmetrical, non-spherical acoustical reflection surface which together with an ultrasonic transducer element transmits an at least partially convergent ultrasonic field having an annular cross-section.

Christoph Benedikt Burckhardt

Papers

Ultrasound Axicon: a device for focusing over a large depth

Method and apparatus for producing cross-sectional images using ultrasound

Methods for Increasing the Lateral Resolution of B-Scan

Focussing Ultrasound Over a Large Depth with an Annular Transducer-An Alternative Method

Method of and apparatus for focusing ultrasonic waves in a focal line