274 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 48, no. 1, january 2001 New X-Wave Solutions of Free-Space Scalar Wave Equation and Their Finite Size Realization Nikolai V.Sushilov, JahangirTavakkoli, Associate Member, IEEE, and Richard S. C. Cobbold, Life Member, IEEE Abstract—Based on the method proposed by Donnelly and Ziolkowski [1], [2], a new general solution has been ob- tained for the isotropic/homogenous scalar wave equation in cylindrical coordinates. It is shown that well-known lim- ited diffraction beams such as Durnin’s Bessel beams [4], Lu and Greenleaf’s X-wave [15], localized waves of Don- nelly 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 aper- ture 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. I. Introduction X -waves are a special class of so-called diffractionless beamsthataresolutionstotheisotropic/homogeneous scalar wave equation. Such beams, which must be cre- ated with an infinite aperture and energy, have the im- portant property that there is no beam divergence even as they propagate to an infinite distance [3]. When pro- duced with a finite aperture and energy, they are usu- ally described as limited diffraction beams, in which case they have beam profiles that remain approximately un- changed over large distances. Because of these properties, thelimited-diffractionbeamshavemanypotentialmedical and industrial applications, both in optics and acoustics. In recent years, many different types of limited diffrac- tion beams have been studied both theoretically and ex- perimentally. Some important examples are Bessel beams and Bessel bullets [4]–[8], Bessel-X pulse [9], X-shaped Sinc wave [10], localized waves [1], [2], [11]–[14], X-waves of zeroth order [15]–[18], and non-zeroth order [19], [20], mth derivatives of nth order X-waves (bowtie beams) [21]–[23], and limited-diffraction array beams [24]. Practi- cal applications of limited-diffraction beams include high frame rate imaging, nondestructive evaluation, Doppler Manuscript received January 26, 2000; accepted May 11, 2000. The Medical Research Council and the Natural Sciences and Engineering Research Council of Canada partially supported this work. The authors are with the Institute of Biomaterials and Biomedi- cal Engineering, University of Toronto, Toronto, Ontario M5S 3G9, Canada (e-mail: cobbold@ecf.utoronto.ca). velocity estimation, pulse-echo imaging, high-speed trans- mission of images, and optical communications [24]–[31]. Some of the practical problems associated with the use of limited-diffraction beams are the higher sidelobes and lower lateral resolution as compared with conventional beams [3]. Lu and Greenleaf have proposed several meth- ods for reducing the sidelobes. One solution [32], the summation-subtraction method, consists of using three transmit bursts, summing the echoes produced by two second-order X-wave beams that are rotated around the beamaxiswithrespecttoeachother,andthensubtracting the result from that obtained with a zeroth-order beam. Unfortunately, this method reduces frame rate signifi- cantly. They have also proposed modified X-waves, which are the regular X-waves with a radially dependent free parameter [33]. For a properly chosen parameter, the re- sultantbeamcanhavelowersidelobesatgreaterdistances fromthetransducersurfacethanthoseoforiginalX-waves and better lateral resolution closer in. These modified X- waves are a good compromise between the depth of field and lateral resolution. When applied to pulse-echo sys- tems, bowtie X-waves [21]–[23] can have lower sidelobes than the zeroth-order X-wave and almost the same depth offield.LiandBharath[19],[20]haveshownthatthefirst, second, and third derivatives of the zeroth-order X-wave arealsoX-waves,butwithlowersidelobesthanoriginalX- waves and with frequency spectra that no longer tend to infinityasthefrequencyapproacheszero.Hence,theseX- waves can be produced approximately with a practical fi- niteaperturetransducer.Furthermore,inaseriesofpapers [34]–[36], a new method of calculations of both localized wavesandX-wavesgeneratedbyfiniteaperturetransduc- ers has been proposed. In these papers, the authors have explored the possibilities of using a finite-time excitation of a dynamic aperture to generate a finite-energy approx- imation to a limited-diffraction pulse. The dynamic aper- ture method was also applied to solve the Klein-Gordon equation, which accounts for the dispersive nature of the propagation media [37]. In this paper, new diffractionless solutions to the isotropic/homogeneousscalarwaveequationincylindrical coordinates are derived. From a general solution obtained using the method described by Donnelly and Ziolkowski for an infinite aperture, known diffractionless solutions as well as new solutions are obtained. We then apply one of ournewsolutionstoconsiderthecaseofafinitesizeplane 0885–3010/$10.00 c  2001 IEEE