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A study of localized waves and their application to medical ultrasound imaging Bie, Linda 1994

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A STUDY OF LOCALIZED WAVES AND THEIR APPLICATION TOMEDICAL ULTRASOUND IMAGINGByLinda BieB. A. Sc. (Engineering Physics) University of British Columbia, 1991B. Sc. (Mathematics/Physics) University of British Columbia, 1989A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF ELECTRICAL ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAApril 1994© Linda Bie, 1994In presenting this thesis in partial fulfilment of the requirements for an advanced degree atthe University of British Columbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission for extensive copying of thisthesis for scholarly purposes may be granted by the head of my department or by hisor her representatives. It is understood that copying or publication of this thesis forfinancial gain shall not be allowed without my written permission.Department of Electrical EngineeringThe University of British Columbia2075 Wesbrook PlaceVancouver, CanadaV6T 1W5Date:2J/4AbstractA study of localized waves and their potential for application to medical ultrasoundimaging is conducted using analytical and numerical simulation techniques. Simulatedfocused ultrasound fields representing approximations to the focus wave modes and modified power spectrum pulses are generated from synthesized, two-dimensional arrays. Theresults in terms of attenuation and diffraction are compared with previously publisheddata for continuous waves and X waves, which are another localized wave solution.Through the course of simulations different array sizes and source element densitiesare examined to determine the consequent effects on the generated localized waves. Arraysize determines the distance to which the wave will propagate while source element densityaffects the smoothness or amount of error in the reconstruction. In addition, the effect of‘folding’ is examined and found to reduce attenuation but to have little effect on beamwaist width within the depth of interest for medical ultrasound imaging (up to O.4m).Introducing folding terms increases substantially the complexity and magnitude of therequired source waveforms.Pure focus wave mode beams are found to differ insignificantly from the modifiedpower spectrum pulse when generated by a finite array. Simulations show that for asquare source array as small as nine by nine elements with element spacing 5.5mm, thelateral beam half-width (at the 1/e point) at 30cm depth penetration in water is only10mm, and the depth of field, defined as the distance where the wave falls to half its original magnitude, is 405mm. This is a substantial improvement over continuous waves—onthe order of 23% for the beam half-width—and is comparable to results reported for Xwaves launched from a circular array with the same area.HTable of ContentsAbstractTable of Contents iiiList of Figures VAcknowledgment vii1 Introduction2 Background2.1 Localized Waves2.1.1 Launching2.2 Ultrasound3 Localized Wave Performance3.1 Launching Localized Waves: The MPS3.1.1 Array Considerations3.2 The Focus Wave Mode3.2.1 Launching FWM Pulses .3.2.2 Pulse Behavior as it Decays3.3 Current Arrays4 Comparison With Other Waves4.1 Continuous WavesPulse5511131616173032363741411114.1.1 Making a Fair Comparison 414.1.2 Results 444.2 X Waves 485 Conclusions and Future Directions 52ivList of Figures2.1 The FWM at the origin 72.2 The MPS pulse at the origin 92.3 Propagation characteristics of the MPS pulse 103.4 Reconstructions of the MPS pulse for arrays of different size and elementspacing 193.5 The driving functions as a function of p’1 for (a) v1 = 27r and (b) v1 = 207r 213.6 Evolution of reconstructed pulse at the pulse center as a function of v1 foran array of size 1.6 x 1.6 with a) 21 x 21, b) 41 x 41, and c) 81 x 81 elements 223.7 Evolution of reconstructed pulse at the pulse center as a function of v1 foran array of size 16 x 16 with a) 21 x 21, b) 41 x 41, and c) 81 x 81 elements 233.8 Normalized spectra of driving functions for v1 = 2ir and v1 = 2Oir 253.9 Evolution of the reconstructed MPS pulse at the pulse center as a functionof v1 with and without the folded term 273.10 Reconstructions of the MPS pulse for an array using the folded sourcefunctions 293.11 Time evolution of the non-folded and folded terms of the source function 303.12 Evolution of reconstructed pulse at the pulse center as a function of v1 foran array of size 1.6 with 41 x 4lelements: a) the FWM; b) the MPS pulse;c) the difference between the two 343.13 Reconstructions of the FWM at a) v1 = 2ir and b) v1 = 4r, and the MPSpulse at c) v1 = 2ir and d) v1 = 4ir 35v3.14 The difference between the normalized FWM and MPS pulse at a) v1 = 2irandb)vi=47r 363.15 FWM normalized pulse as a function of u1 at p = 0 for pulse centers a)v1 = 2ir, b) v1 = 47r, c) v1 = 6ir, and d) v1 = 87r 383.16 FWM normalized pulse as a function of P1 at u1 = 0 for pulse centers a)= 2ir, b) v1 = 4ir, c) v1 = 6?r, and d) vi = 87r 394.17 FWM spectrum for parameters z = 9 x i0 and k = 2.5m’ 424.18 w, vs k for p = 0.95 434.19 Beam half-width as a function of propagation distance for a Gaussian beamand FWM beams generated from realistic arrays 464.20 Beam half-width as a function of propagation distance for a Gaussian beamand FWM beams with and without the folded term 474.21 Lateral and axial plots of the reconstructed FWM with and without thefolded term 494.22 Evolution of Ic for different arrays 50viAcknowledgmentA debt of gratitude is owed to my thesis supervisor, Dr. Matthew R. Palmer, for hiscontribution to this work. I would also like to thank everybody else who read and offeredcomments on my work in progress. Finally, I wish to thank my friends and family,especially my parents Lyle and Faye Bie, for their support throughout this endeavor.viiChapter 1IntroductionUltrasound imaging is used in many areas. In industry it is used to perform nondestructive testing, geologists use it to examine subterranean structures, but probablythe most widely-known use is in medicine. Over the past few years, medical applicationshave expanded considerably. A major advantage ultrasound has over many other typesof medical imaging is that it does not use ionizing radiation, which is believed to make itrelatively safe. So far, exposure to ultrasound at diagnostic levels has not been linked toany harmful effects (Hykes et al., 1992, p. 204). Diagnostic, as opposed to therapeutic,uses of ultrasound include obstetrics, cardiac imaging, detection of mid-line shift following head trauma, localization of foreign bodies in the eye, and evaluation of abdominalmasses.In the most common ultrasound modes’, the image is composed of reflected ratherthan transmitted waveforms. The incident wave is reflected back to the transducer atreflective interfaces. The intensity of the reflected wave is a measure of the reflectivity ofthe interface, and the time required for the round trip is a measure of the depth of theinterface. Soft tissue interfaces are weakly reflective, so the amplitude of the returningwaves is small. Also, waves are attenuated due to divergence, absorption, diffraction,scattering, and refraction. Diffractive spreading is responsible for a major reductionin resolution, making it one of the most limiting factors in attaining a high quality1The different configurations used in ultrasound imaging are referred to as ‘modes’. For a descriptionof these modes see Ultrasound Physics and Instrumentation (Ilykes et al., 1992) or Medical ImagingSystems (Macovski, 1983).1Chapter 1. Introduction 2image. Attempts have been made to compensate for diffraction, such as mechanicallyor electronically focusing the wave, but these measures bring a different set of problems.The poor image quality, especially the problem with resolution, means that ultrasoundis restricted to qualitative use. The introduction of a more suitable, robust wave whichdoes not suffer diffractive spreading would alleviate some of the image quality problems.Specifically, if resolution can be improved, it may be possible to use ultrasound forquantitative analysis, thereby making it a more powerful diagnostic tool.There have been attempts made to use transmitted rather than reflected waves inultrasound imaging, where the transmitting and receiving transducers rotate around thepatient opposite each other, much like x-ray computed tomography (CT) scanning. Theimage is reconstructed using the same techniques as for CT scans. For imaging in manyareas of the body, this type of scanning is not feasible, since highly reflective interfacessuch as tissue-bone or tissue-gas result in a transmitted wave with very low intensity,which causes detection problems. In bodily structures that are more uniform, such asin the detection of breast cancer, tomographic ultrasound could be useful (Hykes et al.,1992, p. 78). It is with this type of ultrasound imaging that the greatest success withquantitative imaging has been experienced (Greenleaf, 1983).In designing an ultrasound wave, desirable qualities are that the wave be unidirectional, uniform in intensity, and of limited spatial dimensions for good spatial resolution.Over the past decade, new solutions to the scalar wave equation have been reportedwhich suggest the existence of highly focused waves. These waves retain their compact,focused shape over great distances, that is, they don’t suffer diffractive spreading like thecontinuous waves now used. They exhibit an extended depth of field, which makes themideal for ultrasound imaging (Lu and Greenleaf, 1992b; Fatemi and Arad, 1992). Theycan be created using a two-dimensional array of transducers, with each element in thearray being independently addressable. The size of the array and the spacing betweenChapter 1. Introduction 3the array elements are critical in the generation of these waves. If an array of suitablesize and element spacing, with independently addressable elements, can be manufactured, localized waves could be launched and applied to ultrasound imaging, potentiallyproducing a better, higher resolution image than is currently available.The focus of this work will be to study localized waves and their potential application to medical ultrasound imaging. Attention will be focused on two versions oflocalized waves, the original focus wave mode (FWM) introduced by Brittingham, (Brittingham, 1983), and the modified power spectrum (MPS) pulse proposed by Ziolkowski,(Ziolkowski, 1985). The FWM has finite energy density but infinite energy, and willpropagate to infinity without decay. It was believed that the infinite energy nature ofthe FWM was a barrier to launching these waves, so researchers set out to find finiteenergy variations of this solution. The MPS pulse is a finite energy pulse derived fromthe FWM which has the addition of a multiplicative term that causes 1/r decay after anextended period of propagation.Ziolkowski later proposed a method to launch the MPS pulse using a two-dimensionalarray of transducers (Ziolkowski, 1989). This launching scheme is based on an integralformulation over an infinite surface, so to create a perfect pulse using this method, anarray of infinite size with infinitely fine (effectively continuous) element spacing is required, although approximations to the pulse can be launched from a finite array withdiscrete elements. With the introduction of the array launching scheme came researchinto overcoming shortcomings produced by truncating the array and discretizing the elements. Ziolkowski introduced a mapping which ‘folded’ points in the source plane whichare outside the array to points inside the array (Ziolkowski, 1989). When the elementsare excited with the folded source function, the array should behave like one which islarger.The characteristics and limitations of localized waves generated from a finite, discreteChapter 1. Introduction 4array of transducers will be examined through computer simulation. The effects of different array properties will be examined, including array size and element spacing. Itis expected that a large, dense array will produce a better approximation to a localizedwave than a smaller, sparse array. A description of exactly how truncation and samplingrate affect the reconstruction is sought. The use of Ziolkowski’s folding scheme will beinvestigated. Although folding makes the array act like one which is larger, the effective element spacing between the folded elements increases, which will exaggerate thedetrimental effects of a non-continuous sampling rate. Both the benefits and the cost ofincorporating this scheme will be explored.The FWM and MPS pulses will be used in these simulations. The same array launching method proposed to generate the MPS pulse will be applied to the FWM formulation.The infinite energy nature of the FWM should not be a problem, because it is beinglaunched from a finite array. The waves generated by a finite, discrete array are onlyapproximations to the intended wave, and it is the array which determines the energyand propagation characteristics. The finiteness of the array should override the infiniteenergy nature of the FWM pulse. Since the array probably determines the propagationcharacteristics, it seems sensible to start with the ‘perfect’ FWM rather than the MPSpulse which has its own decay behavior built in. There are limitations caused by thearray, so there should be no need to have limitations in the wave function.Comparisons will be made between the localized waves discussed in this paper andthe currently used continuous wave, as well as another limited diffraction beam, the Xwave. The localized waves should perform better than the continuous waves, in that theyshould propagate farther without diffracting, and comparably to the X waves.Chapter 2Background2.1 Localized WavesFocus wave modes (FWM) were first proposed by Brittingham (Brittingham, 1983).He was searching for electromagnetic pulses which were solutions to the homogeneousMaxwell’s equations, continuous and non-singular with a three-dimensional pulse structure, and nondispersive for all time. They also were to move at light velocity in straightlines and carry finite energy. His solutions met all these criteria but the last. Althoughthey had finite energy density, they carried infinite energy. Brittingham’s work spurredactivity in finding packet-like solutions, resulting in theories for nondiffracting beams(Durnin et al., 1987; Durnin, 1987), electromagnetic missiles (Wu, 1985; Wu et al.,1987), focus wave modes (Hillion, 1986), X waves (Lu and Greenleaf, 1992a; Lu et aL,1993), and the modified power spectrum pulse (Ziolkowski, 1985; Ziolkowski, 1989).An elegant relation between the original FWM’s and solutions of the homogeneous,free-space wave equation, bh was put forth by Palmer and Donnelly (Palmer and Donnelly, 1993), and will be reiterated here. The governing wave equation is:1 a2(V2—= (2.1)where c is the wave propagation speed. Assuming the direction of propagation is alongthe z axis, and substituting the characteristic variables u = z — ct and v = z + ct, gives5Chapter 2. Background 6the following form for Equation (2.1):+= 0 (2.2)where V is the transverse differential operator, in this case02/0x +92/0y. A class offunctions which solve Equation (2.2) is given by:U0( + P2) (2.3)where f is an arbitrary twice differentiable function, p is the transverse distance, andno is an arbitrary constant. Consider the case f() = where k is a real constant.Substituting into Equation( 2.3) yields:=(2.4)Setting the constant u0 = iz0, where z0 is real gives, except for a multiplicative constant,the FWM solution introduced by Ziolkowski which represents a modulated, moving Gaussian pulse (Ziolkowski, 1985).1 1 k2 1 _k2= = . eei = . eikt)ezo+(_ct (2.5)47r 4’n(zo+zu) 4rz[zo+z(z—ct)]The value of kz0 must be positive, in order for the solutions to be bounded as p —* oc.The FWM with parameters ZO = 4.5 x 104m and k 2m’ is shown in Figure 2.1.At the pulse center, that is, for z = d, the intensity of the Gaussian pulse-like FWMdecreases with the distance from the axis of propagation as So, the parametersk and z0 determine the localization to the z axis. Small values of k/zo produce pulseswhich look like transverse plane waves while large values yield highly localized particle-likesolutions. The pulse is also localized along the axis of propagation, as away from the pulsecenter on this axis its amplitude decays as 1/(z + (z — ct)2). As with Brittingham’ssolutions, these pulses have finite energy density but infinite energy. They will propagatewithout decay at the pulse center infinitely, oscillating with period ir/k.Chapter 2. Background 7IFigure 2.1: The FWM at the origin for parameters z0 = 4.5 x 1Om and k = 2m’.It was thought that in order to be useful, these theoretical pulses must be madephysically realizable and therefore of finite energy. The infinite energy solutions of Equation( 2.5) can be used as a basis to construct finite energy solutions (Ziolkowski, 1985).The following function is a superposition over k with weighting function F(k), so it alsosolves the wave equation.f(,t)= W(k)dk= . J F(k)e_c8dk (2.6)47rz[zo-f-z(z—ct)J owheres = s(p,z,t)= zO+iz— Ct)—i(z+ct) (2.7)Chapter 2. Background 8This superposition will produce a finite energy solution if F(k) satisfies:j F(k)Idk < 00 (2.8)as shown in (Ziolkowski, 1989).The function f(i, t) in Equation( 2.6) has the form of a Laplace transform, leadingto the conclusion that further interesting solutions can be found by consulting a table ofLaplace transforms. In this way, Ziolkowski’s modified power spectrum (MPS) pulse wasdeveloped. It is so named because the chosen spectrum is a scaled and truncated versionof the power spectrum,F5(k) =(/3k — b)_le_a(_b) b4irz/3 k>—Fmps(k) = F(a) /3b (2.9)0 0k<-1The form of the MPS pulse is:f(,t) = . e (2.10)z0 + z(z — Ct) (s/3 + a)awhere .s = s(p, z, t) is given in Equation( 2.7). Figure 2.2 shows the MPS pulse at theorigin for parameters a = 1.Om, a = 1.0, b = 600m’, 3 = 300, and z0 = 4.5 x 10m.The speed of sound in water is c = 1.5 x 103m/s.The values of the parameters a, b, a, /3, and z0 can be altered to produce a wavewith the desired characteristics. For example, if z0 is decreased, the center frequency ofthe pulse is increased, but the localization to the z-axis is decreased. The decrease inlocalization can be countered by increasing the value of b or decreasing /3 since all threeparameters affect the degree of localization.To examine the behavior of the wave along the direction of propagation, , at thepulse center, consider the case where a = 1, /9>> 1 (1//3z0 << 1), p = 0, and z = ct. TheChapter 2. Background 9Figure 2.2: The MPS pulse at the origin for parameters a = 1.Om, o = 1.0, b = 600m’,= 300, and z0 = 4.5 x 104m.real part of the MPS pulse is given by:J?{f(p = 0,z = ct)}= —+ ()2 (cos () — sin (v)) (2.11)In the region where z <</3/2b and z <a3/2, the value of the function at the pulse centeris constant at 1/azo. When z > /3/2b and z < a/3/2, the cos(2bz//3) term dominates,so the pulse becomes oscillatory with period of oscillation 7r/3/b. When z > a/3/2, thefunction decays as 1/z. By choosing a very large value for /3, this decay behavior canbe made to occur arbitrarily far from the origin. The behavior of the pulse along thedirection of propagation is illustrated in Figure 2.3.The MPS pulse is like the FWM but with the addition of the term which brings aboutthe 1/z decay, so, in the range z < a/3/2 at least, the localization properties for the twoChapter 2. Background 10(a) (b)(c) (d)Figure 2.3: The MPS pulse at distances (a) 7r/3/b, (b) 5r/3/4b, (c) 3irj9/2b, and (d)it a13 (/3/b)Chapter 2. Background 11are the same.2.1.1 LaunchingPart of the appeal of the MPS pulse is that a two-dimensional array launching schemehas already been proposed (Ziolkowski, 1989) and implemented (Ziolkowski and Lewis,1990). Through use of Huygen’s representation, a scalar field, such as the MPS pulse,can be defined at an observation point ‘ within a closed surface as (Jones, 1964):r .-. lôf -.01?f(r,t) = I g(r,r’) -ç-- + —-— g(r,r’)-—JS Ofl(,t_)C at (?,t_.a) Ofl- f (,t- ) -g(,,’)] ds’ (2.12)where n’ is the inward pointing normal to the surface, S, ? represents the source coordinates on S, and 1? is the distance from the observation point to the source coordinates,1?— .The function g(, i) is the propagator 1/4irR. So, the value of the functionf(, t) can be determined at any point within the surface as long as the function and itsderivatives are known over all points of the surface.Consider the surface to be a hemisphere connected to an x’y’ plane centered at z’,with the radius of the hemisphere —* cc. In this case, every observation point withz > z’ is contained within the surface. Assuming no contribution from the hemisphere,the integral over the surface becomes a double integral over the x’y’ plane, with inwardpointing normal —z’. If the driving functions on the x’y’ plane are defined as:—= az’L,_)— z_RZ0ct(?,t_)— Z_Z’f(?,t — (2.13)then the function can be expressed as:R 1‘ = —L f (r’, t — —) —dx’dy’ (2.14)Chapter 2. Background 12Unfortunately, a continuous source over an infinite plane is not feasible so some approximations must be made. In Ziolkowski’s proposed launching scheme (Ziolkowski,1989), the integral is expressed as a summation over an (2N + 1) x (2M + 1) array oftransducers as:N M D 1f(,t) = — (n/x,mzy,z’,t_ -ui)R(2.15)n——Nm=—M C nmwhere Zx and zy are the element spacings in the x’ and y’ directions, and Rnm[(x — n/x)2 + (y — my)2+ (z — zl)2}h/2.The array of transducers is both finite and discrete, so errors due to truncation anddiscretization are inevitable. The errors manifest themselves as a decrease in smoothnessof the pulse and a decrease in the distance to which the pulse can be reconstructed, aswill be demonstrated in Section 3.1.1.Efforts have been made to minimize errors due to truncation and sampling while stillmaintaining a realistic array. One way to make a small array (radius rma) act like alarger array is to ‘fold’ exterior points (p’ > rmax) onto interior points using the conformalmap p’ i—* rax/p’ (Ziolkowski, 1989).For the planar array considered previously, all points x’2 + y’2 > rax will be mappedso that x’ ‘—p [rax/(x’2+y’2)]x’ and y’ i—* [rax/(x’2+y’2)]y’ . The Huygen’srepresentation then becomes:f(, t) = — f L i’ (x’, y’, z’, t — ) —----dx’dy’C 4’rRfXmax fYmax / R\ 1= —j j ‘I’ (x’,y’,z’,t — —) —dx’dy’Xmam Ymax \ C J 4irRXmax Ymax r2 2 R 1— LXmax Lmax (c22) (c z’,t — )4Rfd (2.16)where Cf = C[rL/(C2+2)J, i1j = [r/( +q2)], and Rf = [(x — Cf)2 + (y — 77f)2 +(z — zI)2]h/.Chapter 2. Background 13Because the propagator term (1/4irR) has the folded value for the distance fromthe source to the reconstruction point, Rf, rather than the actual distance from thetransducer to the point, R, this is still an infinite array representation. To overcomethis, Ziolkowski approximates the distance Rf by R, and introduces the appropriate timeoffsets for the retarded time in the folded term driving functions. The array now acts asa larger array, however the signals required by the sources are much more complicated.The folded portion is responsible for recreating the pulse at great distances from thearray, while the non-folded term causes the reconstruction close to the array.A potential problem arises in that the driving functions, ‘Jt(i, t — R/c), contain Rand z in the multiplicative terms (z — z’)/R and (z — z’)/R2,which implies that the pulsecan only be created at one point. This problem can be overcome with some assumptions.If the reconstruction point is relatively far from the generating array, relatively largecompared to the size of the array, and close to the z—axis, then (z—z’) R and 1/R —* 0.In this case, the driving functions can be approximated by ‘I! 28f (Ziolkowski, 1991),or equally well by ‘I’ 28’f.Experiments have been conducted to verify the production of localized waves, andcomparisons have been made with conventional continuous waves (Ziolkowski et al., 1989;Ziolkowski and Lewis, 1990). They have concluded that, using a two-dimensional array,localized waves can be launched which outperform their conventional counterparts inboth beam quality and energy efficiency.2.2 UltrasoundUltrasound1is widely used for non-invasive imaging of soft tissue. Over the past few years,the medical applications for this imaging modality have grown considerably. Reasons for‘Most of the information about ultrasound beams and imaging presented in this section is taken fromUlfrasound Physics and Insfrumenaiion, (Hykes et al., 1992)Chapter 2. Background 14this growth include the apparent harmlessness of ultrasound as well as advances in realtime instrumentation. So far, use of ultrasound has mainly been qualitative. Quantitativeanalysis capabilities would significantly enhance the diagnostic uses of the instrument.The ideal ultrasonic beam would be unidirectional, uniform in intensity, and of limiteddimensions for good spatial resolution. Both lateral and axial resolution are important,so the beam should be localized in both directions. Diffractive spreading is a seriousproblem with ultrasonic waves because the beams diverge at an angle proportional totheir wavelength, and ultrasonic waves have a relatively large wavelength. It is thelimiting factor in determining the resolution limits of this imaging modality.It is possible to improve resolution through use of focusing techniques. Both mechanical and electronic methods of focusing are used. An acoustic lens or acoustic mirrorcan be used to mechanically focus the lens, however the most common technique of mechanical focusing for frequencies less than five MHz is to use a curved transducer crystal.Electronic focusing is accomplished by superimposing ultrasound waves from linear ormatrix arrays of transducers. The signal from each transducer in the array is offset byan appropriate time delay so that the generated wave front arrives at a specific point atthe same time, in phase. The result of this is a focused beam at that point.Unfortunately, these methods of focusing have a very limited depth of field. Tocompensate for this, dynamic focusing and multiple pulses which are focused at differentdepths are employed. This, in turn, leads to low frame rates and unclear images formoving objects.There are problems associated with using arrays of transducers and electronic focusing. Secondary lobes of ultrasonic energy are formed which cause artifacts to appear inthe image. These lobes come in two varieties, side and grating. Side lobes are presentwith all transducers, but are reduced when the density of similar elements is increased.Grating lobes present a larger problem. They are caused by the regular, periodic spacingChapter 2. Background 15of elements in the array and are especially prominent at strong interfaces.Beams which have an extended depth of field, that is they will propagate withoutdiffracting thereby not suffering loss of resolution, would be ideal for ultrasonic imaging.Both the FWM and MPS pulse fall into this category. Reduced resolution due to diffractive spreading is no longer an issue, and they don’t suffer from limited depth of fieldor low frame rates like mechanically or electronically focused continuous waves. On thedownside, the dense, independently addressable element arrays required to launch thesewaves could cause an increase in equipment cost.Over the past few years, some emphasis has been placed on developing two dimensional arrays in order to improve further the diagnostic capabilities of ultrasound imaging.(Goldberg et al., 1992; Smith and Light, 1992). These arrays are being developed to improve launching and reception of continuous waves, however it is possible that they couldbe configured to launch localized waves.Chapter 3Localized Wave PerformanceThis chapter contains a study of the characteristics and limitations of localized wavesgenerated from a finite, discrete array of transducers. The study is conducted throughcomputer simulations of the array launching procedure proposed by Ziolkowski, (Ziolkowski, 1989), described in Section 2.1.1. The effects of array size and element spacingwill be explored, along with the use of folding to expand an array. The possibility ofgenerating a FWM pulse is considered, and the FWM is compared to the MPS pulse.Finally, research into development of two-dimensional arrays will be considered in orderto decide what array dimensions may be considered for realistic application to launchinglocalized ultrasound waves.3.1 Launching Localized Waves: The MPS PulseFor the purposes of illustration, dimensionless coordinates will be introduced, along withthe characteristic variables u z — ci and v = z + ci. To non-dimensionalize the variablesin the equation describing the MPS pulse, (2.10), they will be multiplied by b//3. Thenew coordinates will be (pt, u1,v1) = b/,8(p, z — ci, z + ci) where, as before, p2 = x2 +y2.Using these coordinates, along with the new parameters z = bzo//3 and a1 = ab, andassigning a = 1, the form of the MPS pulse is:b2 1 1 b2f(pi,ui,vi) = — . e 1 = —fi(pi,ui,vi) (3.17)3z0+zu5ai /316Chapter 3. Localized Wave Performance 17wheresi si(pi,ui,vi) =, .— iv1 (3.18)zo + miThe source functions for the following simulations will use the approximation ‘I,29f, as the results are virtually identical to those when the full function (Equation (2.13))is used. Using the dimensionless coordinates and parameters, and ignoring the multiplicative constant because it can be adjusted later, the source functions are:S(p,u+Ri,v—Ri)= + (3.19)u1(p ,u’ +Ri ,v—R1)V1(p ,u’1 +Ri ,v—R1)wheref1 ‘9fi —i / 1 ‘ los1 Os’+= z + iu — i\5 + ai+ } + fi (3.20)andOs1 Os.p2+ —z ! ! 2 + 1 (3.21)9U1 8v1 (zo+ mu1)R1 is the distance from the source to the reconstruction point. For an array of Ntransducers, the reconstructed value of the function fi is:nrrN 1fi(pi,ui,vi) = — S(p,u +R1,v— Rl)A dA (3.22)n1where p’1 is the distance from the origin to the th source in the source plane, and dA isthe area represented by that source. For an evenly spaced square array, dA = Zx1Ly,where Lx1 and Ly1 are the element spacings in the x and y directions.For the simulations that follow, the summation over the sources given by Equation (3.22) was calculated using original C code. The real part of the reconstructed waveis displayed using Xmath software.3.1.1 Array ConsiderationsThere are many considerations when deciding upon an optimal array, such as the shapeof the array, size, element spacing, and whether to utilize the folded array scheme. OfChapter 3. Localized Wave Performance 18these factors, array shape is probably the least important. Although it makes sense to usea circular array in that the magnitudes of the driving functions decrease radially fromthe origin, it is easier to construct and manipulate an evenly spaced square array. Itseems that the arrays that are actually being developed are of the square or rectangularvariety (Smith and Light, 1992), so that is what will be used here. The rest of the factorsdeserve further, in depth consideration and will be discussed in the following sections.Array Size and Element SpacingArray size and element spacing both greatly affect the quality of the reconstructed pulse.Element spacing seems to have its greatest effect on the smoothness of the reconstructedpulse, especially in the trailing wake. Array size determines the distance to which thepulse can be reconstructed. These effects are demonstrated in Figure 3.4 for the MPSpulse at v1 = 2K with parameters a1 600 and z = 9.0 x 1O. The first waveform,Figure 3.4a, is the MPS pulse. Next, Figure 3.4b, is the MPS pulse as reconstructed froma 21 x 21 square array, with element spacing zx = = 0.02. The next reconstruction,Figure 3.4c, is from a 21 x 21 square array which is made twice as large by doubling theelement spacing to zx = 0.04. The reconstruction appears to be more accurate,in that the fundamental shape of the pulse is more like that of Figure 3.4b. The largerarray has produced the leading wake which was missing in the reconstruction from thesmaller array. The courser element spacing, however, has increased the choppiness in thetrailing wake. Finally, Figure 3.4d is a reconstruction from an array which was doubledin size (compared to Figure 3.4b) by increasing the number of elements to 41 x 41 andleaving the spacing at /x’1 = = 0.02. In this pulse, the reconstruction error causedby the course spacing has been smoothed significantly. In all cases, the pulses have beennormalized to their maximum values at p = 0 and u1 = 0.The above discussion is of a qualitative nature, and while it can be used to get anChapter 3. Localized Wave Performance 19A/ o.6 1,00,v(a),1(c)(b)-/6’:••1 o.SN... O,4P0 r/(d)Figure 3.4: The MPS pulse and reconstructions for parameters a1 = 600 andz = 9.0 x 10; a) the MPS pulse at v1 = 2ir; b) reconstruction from a 21 x 21array with spacing /x’1 = = 0.02; c) reconstruction from a 21 x 21 array withspacing /x’1 = = 0.04; and d) reconstruction from a 41 x 41 array with spacingzx = zy = 0.02.Chapter 3. Localized Wave Performance 20idea as the effects of array size and element spacing, a quantitative description of theseeffects would be more informative. Following is an attempt to separate and quantify theeffects of these two factors.Array Size The maximum distance to which the pulse can be reconstructed is completely determined by array size. This can be seen through examination of either thedriving functions for different values of v1, which is representative of twice the reconstruction distance, or the reconstruction at the pulse center for different array sizes as afunction of v1. The driving functions will be investigated first.Since the driving functions are circularly symmetric, any line through the origin isrepresentative of their evolution. For u1 = 0 and varying values of v1, the required sourcefunction, as given in Equation (3.19) is charted as a function of p’1. As v1 is increased, thedistance from the origin at which there are significant contributions also increases. Theincrease in required array size varies approximately linearly with reconstruction distance,as illustrated in Figure 3.5. Figure 3.5a is the source function for v1 = 2ir and Figure 3.5bis for v1 = 20K. The value of p’1 to which the driving function has significant amplitudeis approximately ten times greater for the reconstruction at v1 = 207r than for v1 = 2ir.The values of the other parameters are a1 600 and 4 = 9 x iO.Reinforcing the observation that array size determines reconstruction distance is theinvestigation of the reconstruction at the pulse center as a function of array size. As thearray size increases, the distance to which the pulse can be reconstructed also increases.The evolution of the pulse at the pulse center is the same for a given array size, regardlessof the number of elements in the array, as illustrated in Figures 3.6 and 3.7. Figure 3.6shows the evolution of the reconstructed pulse at the pulse center (ui = 0, p = 0) as afunction of v1 for three different arrays, all of the same size but with different numbersof elements. The total array size is 1.6 x 1.6 for all three cases, but in the first, theChapter 3. Localized Wave Performance 21Figure 3.5: The driving functions as a function of p’1 for (a) v1 27r and (b) vi = 20irarray is 21 x 21 elements with element spacing 0.08, the second is from a 41 x 41 arraywith element spacing 0.04, and the third features an 81 x 81 array with element spacing0.02. The distance to which the pulse can be reconstructed does not improve with theincrease in source elements and accompanying decrease in element spacing. Figure 3.7shows the evolution of the reconstructed pulse at the pulse center as a function of vi forarrays which are ten times the size of those in Figure 3.6. The first plot is from a 21 x 21,array with element spacing 0.80, next is from a 41 x 41 array with element spacing 0.40,and the third demonstrates the reconstruction due to an 81 x 81 array with elementspacing 0.20. As before, the number of elements, and therefore element spacing, does2001 000—100—2000C-)ci-)(-)Cl)tZEEE0.1 0.2 0.3 0.4p,(0)0.5 0.62010.czcDC-)=5ci-)C-)ESC’,).. . . . --V0 1 2 3 4 5 6p1(b)Chapter 3. Localized Wave PerformanceO.5>D5? -°0—ci,>5?:3c -0.5....—.— I-Ici,022Figure 3.6: Evolution of reconstructed pulse at the pulse center as a function of ii1 foran array of size 1.6 x 1.6 with a) 21 x 21, b) 41 x 41, and c) 81 x 81 elements.(a)600..______(b)60>5?:30— I-,ci,0o. \f\f\z z:..EEEEsEE10 20 30 40V(c)50 600.5>c-o.50-00Figure 3.7: Evolution of reconstructed pulse at the pulse center as a function of v1 foran array of size 16 x 16 with a) 21 x 21, b) 41 x 41, and c) 81 x 81 elements.Chapter 3. Localized Wave Performance 23II ‘1iUIImiHI,--! ilitii’’’’100 200 300 400 500V(a)600ww0 100 2000.5>c?0:3c -0.500.5>-00300 400V(b)500..ijk1600600100 200 300 400V(c)500Chapter 3. Localized Wave Performance 24not affect the maximum reconstruction distance. However, in this case, the propagationdistance before the decay behavior is observed is approximately ten times that for thearrays which are ten times smaller. As before, the values of the other parameters area1 =600 and z.=9 x 1O.Element Spacing Analysis of the driving functions, as a function of p’1 for differentvalues of reconstruction distance, v1, also yields information about the effects of elementspacing. Through application of a Fourier transform, the spectrum of a function canbe found. If the transform is approximately zero for frequencies greater than a cut-offfrequency, f, then the function is bandlimited. If a bandlimited function is sampled atintervals less than or equal to 1/2f, it can be accurately reconstructed from the samples.Sampling at larger intervals results in aliasing, which degrades the reconstruction. So,the required array element spacing can be determined by taking the spatial Fourier transform of the driving functions. To investigate this, the evolution of the required sourcefunction, given in Equation (3.19), as a function of x, with y = 0, was used. The spatialFourier transform of the source function was found numerically for different values of v1.The resulting spectra for v1 = 2r and v1 = 20’r are illustrated in Figure 3.8. In thefigure, the transforms are normalized to their maximum values. Although increasing thereconstruction distance increases the distance from the origin at which there are significant contributions by the driving functions, the spectrum remains mostly unchanged. Ifa cut-off frequency is defined as the frequency after which the spectrum does not rise toabove one per cent of it’s maximum value, then the cut-off frequency for both values ofv1 are similar. For v1 = 27r, the cut-off frequency is around 35, while for v1 = 207r it isabout 25. There is a slight decrease, but both are of the same order of magnitude. Thismeans that the spacing required to produce a good reconstruction does not change. Inshort, to increase the distance to which the pulse can be reconstructed, the array needs toChapter 3. Localized Wave Performance 25:: :::::,::::::::.:::: : .::::::::::•.,,:::•.:::::0. 1E \.+ ...j ..S .;. i. .i.\ .,001I[H l0001...c I I I..) + ....•...jS .. I >1 ‘ I I(i Ic n nnni. : V :L).I..J’J’.JI I I0.1 1 10 100spatial frequency (dimensionless units)______ __________(a)______oi\/Th0.0 10.00 1S I 11 II I I III1 I I \I I Io.oooi fl U0.1 1 10 100spatial frequency (dimensionless units)(b)Figure 3.8: Normalized spectra of driving functions for v1 2ir and v1 = 2Oir.be made larger. To accomplish this, the number of elements should be increased ratherthan simply increasing the element spacing.The Folded Array ConfigurationWhether or not to use the folded array configuration is another important consideration.The driving functions for the folded array can produce a reconstructed pulse at furtherdistances from the generating array, but they are much more complicated and are ofgreater amplitude than the non-folded driving functions. These factors may be importantin terms of transducer limitations.Chapter 3. Localized Wave Performance 26For comparison purposes, reconstructions using both the regular and the folded sourcefunctions from the same array were simulated. The value of the reconstructed pulse as afunction of propagation distance is compared, as well as the driving functions required.The source functions used to drive the elements in the folded array configurationconsist of the regular, non-folded term (S, as given in Equation (3.19)) along with asecond term representing the folded elements. For an array in which the farthest elementfrom the origin is at p’1 Tma, folded coordinates can be defined as = x [r/p}and = y [r,/p]. Using these folded coordinates, the folded portion of the sourcefunctions is:Sf = ()S(x1,yu+ Rf,v — Rf) (3.23)where Rf = [(x1 — z4f)2 + (yi — y’)2 + (z1 — z)2]h/2 is the distance from the foldedcoordinates to the reconstruction point. Adding this folded term to the non-folded termmakes the source element at (xi, y, z) behave like it is also at Yf z). Whenthe folded term is incorporated into the summation formulation for the value of thereconstructed function, Equation (3.22), the value for R used in the propagator (1/4irR)is R1 rather than Rf, since the distance to the reconstruction point is really R1. Thefunction, then, is given by:n=N 1fi(pi,ui,vi)=— (S+Sf) dA (3.24)n=1 47rRiFigure 3.9 shows the evolution of the reconstructed MPS pulse at the pulse center as afunction of v1 for source functions both with and without the folded term. The generatingarray in both cases has 21 x 21 elements with element spacing 0.01, making the arraysize 0.2 x 0.2. The values of the other parameters are a1 = 600 and z = 9 x iO,as in the preceding sections. When very close to the array, the two reconstructionsare indistinguishable, but the pulse generated without the folded term decays quickly,while the pulse generated by the folded array appears to propagate at least an order ofChapter 3. Localized Wave Performance 27cII _n- ‘.J.)c-10.5CII _n•J”-’I —Figure 3.9: Evolution of the reconstructed MPS pulse as a function of v1 from a 21 x 21element array of size 0.2 x 0.2 for source functions (a) without and (b) with the foldedterm.magnitude farther, experiencing only a slight drop where the non-folded pulse decaysbefore recovering. This drop is probably because the folding is intended for a roundarray, so when incorporated for a square array that fits inside the corresponding roundarray, there are gaps that will be most noticeable at the point where the folded term firststarts to dominate.The graph depicting the evolution at the pulse center for the folded array looksencouraging, but this may be deceiving. The reconstruction error due to element spacingmust be considered as well. The effective spacing for the folded elements increases as theR ................00123456R7. p....f\[. .. ./0 10 20 30 40 50 60 70V(b)Chapter 3. Localized Wave Performance 28square of the distance from the origin to the represented point. So, the effective elementspacing becomes very course for the distant folded elements. Since these elements areresponsible for recreating the pulse at greater distances from the array, it is worthwhile toexamine the actual pulse reconstructions for some different values of the reconstructiondistance that are due to the folded term.Figure 3.10 shows the reconstructed pulse at distances of v1 = 27r, 6ir, 107r, and 20?rfor the same pulse as in the preceding figure. Comparing these reconstruction distancesto the plot of the evolution of pulse center, Figure 3.9, shows that at v1 = 27r, the foldedterm is starting to dominate, while for the greater distances the reconstruction is dueentirely to the folded term. At v1 = 2ir, the reconstruction is not bad, with the pulsebeing easily distinguishable from the surrounding noise, but it deteriorates as the distanceincreases, with the pulse being virtually indistinguishable from the reconstruction errorat v1 = 207r. This deterioration is due to the increased effective spacing. So, althoughthe pulse has not yet decayed to half its maximum value, the quality of the reconstructedpulse is such that it is of not much use. This means that although the folded sourcefunctions can produce a pulse to a greater distance, this distance is not nearly as greatas the evolution at the pulse center seems to promise.While application of the folded source functions can generate a reconstructed pulseat greater distances from the array, there is a cost involved. Figure 3.11 shows the time(actually (b//3)ct) evolution of the source function required of a transducer located at adistance of p’1 = 0.04 from the origin in the source plane, for an array of size 0.2 x 0.2.The source function is divided into the non-folded and folded terms. The folded term ismuch more complicated than the non-folded component, and three orders of magnitudelarger. Also, the folded term grows increasingly more complicated as the element involvedapproaches the center of the array. The driving function considered in the figure is closeto the center, but for an array with the element spacing considered earlier, it is not theChapter 3. Localized Wave Performance 29/IFigure 3.10: The MPS pulse as reconstructed from a 21 x 21 element array of size 0.2 x 0.2using the folded source functions at distances of v1 = 2r, 6ir, 10r, and 2Oir.(a) (b)//4;,,’,(c) (d)Chapter 3. Localized Wave Performance 30Figure 3.11: The time evolution of the non-folded (top) and folded (bottom) terms ofthe driving function at p = 0.04 in an array of size 0.2 x 0.2.closest, therefore not the most complicated.3.2 The Focus Wave ModeThe expected propagation characteristics for the MPS pulse are never achieved in areconstruction from a finite array. For v1 < a1, the pulse is supposed to oscillate but notdecay, which does not occur. Rather, as is apparent in Figures 3.6 and 3.7, the height ofthe pulse at the pulse center starts to decay almost immediately. This leads to a couple400300200100C.-)=—100—200C) —300—400______—0.03 —0.02 —0.01 0 0.01 0.02 0.03ct(a)400003000020000100000—10000-—20000—30000—40000AR==c)C,)1ll-:::jjiiiiit1iI- .lIllhI Il 1 1\ i—::::::!!—03 —0.2 —0.1 0 0.1 0.2 0.3ct(b)of points which require further investigation.Chapter 3. Localized Wave Performance 31The first point is that although the source functions used to drive the array determinethe pulse shape, it seems that the array actually determines the propagation distance.It was previously thought that in order for a pulse to be launched, it had to be of finiteenergy. For this reason, the MPS pulse was introduced, rather than trying to launch theoriginal FWM. The MPS pulse has a term which causes the pulse to decay as 1/v in theregion v1 a1, whereas if the FWM is charted, it never decays. If, however, the decaybehavior caused by the array limitations overrides the natural propagation characteristicsof the wave in question, there is no longer a problem. The fact that the pulse is launchedfrom a finite array guarantees that it is of finite energy.The difference between the waves can be seen most easily through examining theexpressions for both pulses at their pulse centers, Pi = 0 and u1 = 0. The real part ofthe MPS pulse evolves as:1 1 / V1. ‘\= 0,u1 = 0,v1)} = 2 jcosv1 — s1nVi) (3.25)a1zo1+Y a1while the real part of the FWM evolves as:= O,u1 = 0,vi)} =4cosv1 (3.26)In the region where v1 <a1, the value ofv1/a is insignificant, making the MPS evolutioncos vi/(a1z), which is the same as for the FWM. At great distances from the array,v1 > a1, the value of vi/ai grows significant and the MPS pulses behavior becomessinv1/(zvi)The previous paragraph establishes the main difference between the two waveforms asthe multiplicative term which causes the MPS pulse to decay as 1/v in the region v1 > a1.The fact that decay due to the finite size of the array occurs long before this region makesthat term of limited importance. For this reason, it may be interesting to try to launcha FWM pulse using the same method as described for producing the MPS pulse, thatChapter 3. Localized Wave Performance 32is, try driving an array of transducers using source functions2Oz(P, U + R, V —where describes the FWM solution of the wave equation, as given in (2.5).The second point is that the effect of the premature decay behavior on the localizationof the pulse needs to be investigated. It is important to know if the decay is accompaniedby spreading of the pulse. The decay is a problem, in that it decreases the strength ofthe signal, but spreading is a greater problem because it causes a decrease in resolution.3.2.1 Launching FWM PulsesThe equation describing the FWM, (2.5), is non-dimensionalized by multiplying thevariable by the parameter k. As with the MPS pulse earlier, the characteristic variablesu = z — ct and v = z + et will also be used. Using the variables (pi, u1,v1) = k(p, u, v)and the parameter z = kz0, Equation (2.5) is:k 1 k(pi, u1,v1) — . ezo+A e” = —qi(pi, u1, vi) (3.27)47rzz0+zu1 47rzThe same launching method and approximations to the driving functions as describedfor the MPS pulse will be used to create a FWM pulse. The source functions are:SFWM(P’1,U’ +R1,v — R1)= + (3.28)(p1 ,u1 +Ri ,v1—R1) (p1 ,u1 +R, ,v1—Ri)where+q=[Z)2+-z+iuç](3.29)These source functions are used to excite an array of transducers, as with the MPS pulse,thereby launching a FWM pulse.To compare the FWM results with the MPS results, both pulses were simulated usingthe same value for the parameter z, arrays of the same size and number of elements,and at the same distances from the generating array. The results for both pulse typesare remarkably similar, as will be demonstrated.Chapter 3. Localized Wave Performance 33Comparison to MPS PulseWhen the evolution of the reconstructed FWM pulse at the pulse center is examined, itappears exactly the same as that for the MPS pulse. Figure 3.12 shows the evolutionof both reconstructed pulses, normalized to their maximum values at v1 2ir, and thedifference between them for a 41 x 41 element array with element spacing LI 0.04. Thereconstructed MPS, Figure 3.12b, is the same as that from Figure 3.6b. Both the FWM,Figure 3.12a, and the MPS pulse appear to decay at the same rate, which reinforces theopinion that it is actually the array that determines propagation distance. The differencebetween the two, Figure 3.12c, is two orders of magnitude smaller than the pulse height,so although it exists, it is not immediately evident.Both pulses look the same when reconstructions in an area around the pulse center areexamined. Figure 3.13 shows reconstructions of the FWM pulse at v1 27r and v1 = 4ir,Figures 3.13a and 3.13b, and reconstructions of the MPS pulse at the same distances,Figures 3.13c and 3.13d, for the same parameters and array given earlier. As in the caseof the evolution at the pulse center, they look exactly the same. Small differences, twoorders of magnitude smaller than the peak height, become apparent when the normalizedpulses are subtracted, as shown in Figure 3.14. The waveforms in this figure result fromsubtracting the normalized MPS pulse from the normalized FWM pulse at v1 2r,Figure 3.14a, and at v1 = 4ir, Figure 3.14b. At both distances, the difference betweenthe two pulses is small, however it increases at the greater distance.It is not surprising that the reconstructions of the two different pulses are so similar.Close to the array (for v1 <ai), they should both have the same behavior, as evidencedby Equations (3.25) and (3.26). It is only when the pulse has propagated a distance awayfrom the array, so that the term which causes the decay in the MPS pulse (1/(1 +v/a))has some effect, that any differences are expected. The comparisons between the twoChapter 3. Localized Wave Performance 34U.-0.4-0.8—1.2 I0 10I-(a)1.2 . .60Figure 3.12: Evolution of reconstructed pulse at the pulse center as a function of v1 for anarray of size 1.6 with 41 x 4lelements: a) the FWM; b) the MPS pulse; c) the differencebetween the two.1 .20.80.41.......60V(b)0.03-0.02-0.01-U)—0.01--0.02-L -0.03f\-/0 10 20 30 40 50V(c)60Chapter 3. Localized Wave Performance 35Figure 3.13: Reconstructions of the FWM at a) v1 = 2K and b) v1 = 4K, and the MPSpulse at c) v1 = 2K and d) v1 = 4K(0) (b)(c) (d)Chapter 3. Localized Wave Performance 36Figure 3.14: The difference between the normalized FWM and MPS pulse at a) v1 = 27rand b) v1 = 47rshow that the slight differences between the two reconstructed pulses increase as thepropagation distance, which is consistent with the expected behavior.3.2.2 Pulse Behavior as it DecaysNow, the effects of the premature decay brought on by the array launching will beexamined. A FWM pulse, reconstructed from an array of size 1.6 x 1.6 with 41 x 41elements will be examined at distances of v1 = 2ir, 4ir, 6ir, and 8ir. According to the graphshowing the evolution at the pulse center as a function of v1, Figure 3.12a, significant(a) (b)Chapter 3. Localized Wave Performance 37decay occurs through this region. Figure 3.15 depicts the pulse shape as a function of u1at Pi = 0 at these values of v1, while Figure 3.16 shows the pulse as a function of P1 atui = 0. In all cases, the pulses have been normalized to their maximum values for easeof comparison.The figures show that, although the array launching causes the pulse to decay beforeit is expected to, the decay is not immediately accompanied by spreading. The reconstruction error increases slightly, but the pulse is still easily distinguishable. Since thefocused nature of the pulse is retained, the decrease in intensity of the pulse should causeno resolution problems.3.3 Current ArraysAn important consideration when deciding upon an array to use is the dimensions whichcan realistically be attained.Development of two-dimensional arrays has become an important part of ultrasoundresearch. A 16 x 16 element transducer array with 0.6mm element spacing, expandedto a 16 x 16 array of connector pins at a standard spacing of 2.5mm has been reported(Smith and Light, 1992; Goldberg et al., 1992). This shows that very dense spacing ispossible, and the value of 0.6mm will be used as a lower bound on element spacing. Sincethe presence of a transducer at the origin is important, in order to maintain a symmetricarray 17 >< 17 will be used as an upper bound on the number of array elements.To summarize the results of this chapter, the size of the array determines the distanceto which the pulse can be reconstructed, and the element spacing is responsible for thesmoothness of the reconstruction. For course element spacing, there is reconstructionerror which manifests itself in a roughness in the trailing wake. Although a larger arrayis required to generate a pulse at greater distances from the source, the required elementChapter 3. Localized Wave Performance 38/H-0.4-0-0.2 .—0.01 —0.006—0,002z—ct(a)0.8-0.6-0.4--0.2-—0.11.——0.01 —0.006—0.002 0.002 0.006 0.01z—ct(c)I....sj...\.........Figure 3.15: FWM normalized pulse as a function of u1 at p = 0 for pulse centers a)= 2ir, b) v1 = 47r, c) v1 = 67r, and d) v1 = 8Kno‘J.u0.6-I.......:noU,U0.60.4-n nV.L1\______ ______ ______ ______Vnfl_U.L—0.01 —0,006—0.002 0.002 0.006 0.01z — Ct0.002 0.006 0.01( b)ci)0.80.6-0.4-0.2-0—0.2-—0.4-—0.01 —0.006—0.002 0.002 0.006z — ct0.01(d)Chapter 3. Localized Wave Performance 39Figure 3.16: FWM normalized pulse as a function of p at u = 0 for pulse centers a)v1 = 27r, b) v1 = 47r, c) v1 = 67r, and d) v1 = 8irr--\•\\__— -1-- -0.10.8-0.6-0.4--a0—0,200.8-0.6-0.40nfl—u,L0.2 0.3p(a)0 0.1noU.Li0.6-a0—0.20.80.6-0.4- 0.2-—0.2-—0.40.2p0.3(b)/\.J..p(c)0 0.1 0.2 0.3 0 0.1 0.2 0.3p(d)Chapier 3. Localized Wave Performance 40spacing does not increase. This means that the array must be made larger by increasingthe number of elements rather than increasing the distance between the elements.The folded array configuration will cause the pulse to propagate farther than thenon-folded, but not as far as anticipated while maintaining a smooth reconstruction.The effective course element spacing makes the pulse indistinguishable from reconstruction error after an unexpectedly short distance. The source functions are much morecomplicated, making the folded array configuration of little practical importance, sincethe gains are not worth the cost.The array determines the propagation distance, overriding the expected propagationcharacteristics of either the MPS pulse or FWM. For this reason, the FWM can bereconstructed as effectively as the MPS pulse. When normalized reconstructions of eachpulse are compared, the difference is two orders of magnitude smaller than the peakheight. For computer simulations, the FWM is better to work with because it has asimpler form, and fewer calculations are required. For this reason, the FWM will be usedin the next chapter for comparisons to other waves.Chapter 4Comparison With Other WavesHaving established the possibility of launching localized waves from a finite array, thenext step is to compare these waves to others. The FWM will be compared to continuouswaves as well as to X waves. The FWM pulse should propagate without diffracting fartherthan the continuous waves, which would give them increased resolution when used forultrasound imaging. The question is how far a FWM will propagate when generatedfrom a realistic array, since the propagation distance is dependent on array size. Thepropagation characteristics of the FWM should be similar those of the X waves, as theyare both localized waves, or limited diffraction beams.To make the comparisons, the variables and parameters for the FWM must havedimensions reintroduced. In the interest of making a fair comparison, comparable continuous waves and X waves must be chosen.4.1 Continuous Waves4.1.1 Making a Fair ComparisonContinuous waves are defined by their wavelength, so in order to compare localized wavesto continuous waves, a comparable value for the wavelength of the continuous wave mustbe chosen. This is somewhat nontrivial, as the localized waves are not sharply definedby a single wavelength. Rather, they have a broad spectrum, which must be measuredsomehow and converted to a wavelength.41Chapter 4. Comparison With Other Waves 42Figure 4.17: FWM spectrum for parameters = 9 x iO and k 2.5m’.Focus Wave Mode SpectrumThe temporal Fourier transform of the focus wave mode at the origin is given by:1 !D(wk_—e kck w>kc= c0C)CoC.)r-J0.10.010.00 1:::: e.,:::::::0.01 0.1 1 10 100c (MHz)=(4.30)Figure 4.17 shows the spectrum for the FWM with parameters z = 9 x i0 andk = 2.5m’. The speed of sound in water, c, is 1.5 x 103m/s.The spectrum of the FWM can be used to select a comparable continuous wave, aswell as the parameter k to dimensionalize the pulse and array discussed previously. Ifa cut-off frequency, c., is defined as the frequency before which the fraction p of theChapter 4. Comparison With Other Waves=1000 0.5 1 1.5 2 25k (1/rn)spectrum is contained, that is:then w, is calculated as:Figure 4.18: w, vs k for p 0.95i: ?= O,w)dw1: (r= O,w)dw43= ke (i— ln(1—(4.32)Figure 4.18 shows the variation of cut-off frequency, w,, as a function of k. The valuesof z and c are as given previously.Conversely, if the cut-off frequency and the fraction of the integral of the spectrumdefining that frequency are determined, then the parameter k can be found and used to1 &1—-0.:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::I —3 3.5 4 4.5 5=p (4.31)Chapter 4. Comparison With Other Waves 44give dimensions to the array and the FWM pulse as follows:4 (433)C \zo—ln(1—p)JHaving developed a method to select a comparable wavelength for the continuouswave, as well as a way to dimensionalize the FWM, the two can be compared.4.1.2 ResultsThe behavior of a Gaussian beam propagating in free space in the z direction is as follows:if w0 is the initial beam waist (at the 1/e point), and ) is the wavelength, then along theline of propagation, the amplitude varies as 1/[1 + (\z/rw)2]hu1’2. The beam half-widthis also a function of z, and is expressed as (Arnaud, 1976):2’/w(z) = WO [1+ (;)] (4.34)The distance to the boundary between the near and far fields, that is the Rayleighlength, where the beam begins to decay as 1/z is reached when z irw/,\. At thatpoint, the amplitude has fallen to i// times it’s original value, and its radius hasincreased to /w0. in the region z > 7rwg/,\, the beam diverges at an angle which canbe approximated by 0 /(irwo). The radius of the beam, therefore, increases as Oz.If the frequency of the Gaussian beam is chosen so that f = w/2ir, then, insertingEquation (4.32) into the relation ). = c/f gives the following expression for the wavelength:2ir4435k(4-ln(l-p))For the FWM, the value at the pulse center (u = 0) is given by:-Z eIklJ= 0,v = 2z) = e ZO (4.36)zoChapter 4. Comparison With Other Waves 45which defines the waist as w0 (z/k2)1/.Using this value for the initial waist of theGaussian beam gives a Rayleigh length of:Lirwgz—ln(1—p)2kInserting this relation into the equation describing the waist of a Gaussian beam gives:2 1/2= + ( , 2kz (4.38)k \zo—lfl(l—p)JBoth w and z can easily be nondimensionalized in the same way as the coordinatesused in the localized wave investigation through multiplication by k, making the the newcoordinates w1 = kw and z1 = kz. With these coordinates, the beam waist is describedby:2 1/2(4.39)Also, with these coordinates, the Rayleigh length is LR1 = kLR [z — ln(1—p)]/2. For= 9 x iO and p = 0.95, the dimensionless Rayleigh length evaluates to 1.5. Thiscorresponds to a value for v1 of 2LR1 = 3.0. For the FWM created from the array studiedin Section 3.2.2, there is no sign of diffractive spreading well past this point. In fact, outto v1 8ir there is no spreading. For a distance of z1 vi/2 = 47r, the waist of theGaussian beam would have increased to over eight times its original size.Results Using Realistic Arrays Using arrays of the size discussed previously, it iscertainly possible to produce a localized wave that will outperform a Gaussian beam.What must now be considered are the results using smaller, more realistic arrays. Todo this, dimensions must be reintroduced. Consider a Gaussian beam with a frequencyof f = 2 MHz, which is within the ultrasound range of 1 — —15MHz. For this cut-offfrequency and a value for p of 0.95, k is calculated to be 2.5m. As before, z = 9 x iO,making the initial waist w0 = 12mm.Chapter 4. Comparison With Other Waves 46003*Gaussian beam88mm arrayo 66mm a r ray ** 44mm array o0.02 —___.r*0* ±0I . -----*010.01*0•0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1z(m)Figure 4.1.9: Half-width as a function of propagation distance for a Gaussian beam (solidline), and FWM beams generated from a 44mm x 44mm array with 9 x 9 elements (*),66mm x 66mm array with 13 x 13 elements (o), and 88mm x 88mm array with 17 x 17elements (x).Three arrays of varying sizes are chosen to generate FWM pulses to compare with aGaussian beam. In all three arrays, the element spacing is 5.5mm, which is well abovethe lower bound for spacing set out in Section 3.3. The number of elements for eacharray is 9 x 9, 13 x 13, and 17 x 17, making the physical dimensions 44mm x 44mm,66mm x 66mm, and 88mm x 88mm.Figure 4.19 shows the theoretical half-width (at the 1/c point) as a function of propagation distance for a Gaussian beam as well as for q produced by the three arrays. Thevalue q was chosen rather than simply charting the real part of the FWM in order tocounter effects of the oscillatory nature of beam. As is expected, the larger array givesChapter 4. Comparison With Other Waves 470.03- . - - p -.0Gaussian beamfolded array oo non—folded array0.02- .-- .- .---—--— ._——- x > )( x x—— xx x x x x- x 00.01- -.......o 00-0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1z(m)Figure 4.20: Half-width as a function of propagation distance for a Gaussian beam (solidline), and FWM beams generated from a 44mm x 44mm array with 9 x 9 elements with(x) and without (o) the folded termthe best results, showing much less diffraction than the Gaussian beam. For all arrays,the FWM beams show limited diffraction up to a distance of approximately ten timesthe array side dimension. For ultrasound use, the waves likely will not have to travelfarther than 0.4m, and all the FWM beams showed limited diffraction to that distance.For the smallest array (44mm x 44mm), the half-width of as produced using sourcefunctions including the folded term was found as a function of propagation distance.Figure 4.20 shows that evolution compared with the half-width for the Gaussian beamand the FWM from the same array without the folded term. The beam generated bythe folded array appears to propagate farther before spreading, but closer inspection ofthe reconstruction shows that the folded array also seems to cause a lot of reconstructionChapter 4. Comparison With Other Waves 48error. Figure 4.21 shows the lateral and axial plots of the reconstructed beam at O.2m and0.3m for both the folded and non-folded arrays. These are distances that an ultrasoundbeam will likely have to travel, and this plot shows the beam from the non-folded arrayto have much less reconstruction error. The folded array has a great deal of interferencewhich shows up especially well in the lateral beam plot. Through the important range forultrasound imaging, up to 0.4m, the reconstruction from the non-folded array actuallyhas a narrower beam half-width. This coupled with the smoother reconstruction makesthe non-folded source functions a better choice than the folded configuration.At z = 0.3m, the half-width of the Gaussian beam has expanded to 13mm, as calculated using Equation (4.34). For the two smallest arrays considered, the beam half-widthof the FWM, taken from Figure 4.19, is 10mm. This is a reduction of 23%, which shouldtranslate to a significant increase in resolution.4.2 X WavesLu and Greenleaf have investigated the possibility of launching X waves using a finiteaperture radiator (Lu and Greenleaf, 1992a). Like the focus wave mode solutions, the Xwaves will travel to infinite distance without spreading if they are produced by an infiniteaperture. They found that even when produced by a finite aperture, the X waves willpropagate large distances without spreading. They used the depth of field, the distancein the direction of propagation where the beam fell to half of its original value, to describehow far the beam produced from a finite aperture would travel without spreading. Theyfound that the depth of field was a function of the size of the radiator, as is the case withthe FWM results. They developed an expression for the depth of field, Zmax, as follows(Lu and Greenleaf, 1992a):Zmax = cot( (4.40)Chapter 4. Comparison With Other Waves 49Figure 4.21: Lateral and axial plots of the reconstructed FWM at O.2m (left) and O.3m(right) from a 9 x 9 element 44mm x 44mm array with (solid line) and without (dashedline) the folded term.Q1— foldednon—folded , I AO.8•DQEjO.4.0.20-0 0,02 0.04 0.06 0.08 0.1p (m)-QDC0E-D1)0E0Ca)-oD0’0Ea,)0E0C‘J.u.._l0.6::____0 0.02 0.04 0.06 0.08 0.1p (m)‘0.6-::0.8-a)0D-4-’C00E0-4/10—0.004 —0.002 0 0.002 0,004 —0.004 —0.002 0 0.002 0.004z—ct (m) z—ct (m)Chapter 4. Comparison With Other Waves-o=3C-CcLCC50Figure 4.22: Evolution of for FWM beams generated from a 44mm x 44mm arraywith 9 x 9 elements (dotted line), 66mm x 66mm array with 13 x 13 elements (dashedline), and 88mm x 88mm array with 17 x 17 elements (solid line).where D is the diameter of the radiator, and is the Axicon angle, which is a parameteraffecting the lateral resolution. The X waves differ from the FWM in that the X wavepeak value does not oscillate, but rather remains constant.To evaluate the FWM pulses using the same criterion as the X waves, the value of Icas a function of the z was charted for the arrays of the previous section (44mm x 44mmwith 9 x 9 elements, 66mm x 66mm with 13 x 13 elements, and 88mm x 88mm with17 x 17 elements) in Figure 4.22. For the 44mm x 44mm array, the depth of field, asdefined in the previous paragraph, was found to be 405mm, for the 66mm x 66mm arraya value of 573mm was found, and 744mm for the 88mm x 88mm array.For a full description of the X waves and their behavior when launched from a finiteaperture, the reader is referred to (Lu and Greenleaf, 1992a). Their results for a 50mm1010.10.0188mm array- 66mm array44mm array-0.1z(m)Chapter 4. Comparison With Other Waves 51diameter radiator, and an Axicon angle of 4° will be quoted here.For a 50mm diameter radiator, Lu and Greenleaf found a depth of field of 358mm,which is very close to the theoretical value of 357.6mm calculated from Equation (4.40).A 50mm diameter round radiator has the same area as the square 44mm x 44mm array,making the two radiators comparable. The FWM has a depth of field of 405mm, whichis slightly better than that for the X-wave, but it is of the same order of magnitude,making the FWM is comparable behavior to the X wave.If there were no cost or technological limits placed on the size of the generating array,a FWM beam could be generated that would handily outperform continuous waves ofa comparable wavelength. Even with an array of currently realizable size and elementspacing, the FWM generated will propagate through the important range for medicalultrasound imaging (up to about 0.4m) with less diffractive spreading than continuouswaves. This reduction in spreading is important in terms of image resolution. Thepropagation characteristics of the FWM are similar to another limited diffraction beam,X waves, in terms of propagation distance before the pulse falls to half its maximumvalue.Chapter 5Conclusions and Future DirectionsIn order to launch an exact replication of a localized wave, theory dictates that an infiniteaperture is required. It is possible to generate a reasonable approximation using a finitearray of discrete sources. The waves launched from a finite array do not exhibit thesame propagation characteristics as those from an infinite aperture in that they will notpropagate to infinity without experiencing decay or diffraction. They do, however, travelan extended distance before diffraction occurs.The size of the generating array determines the length of this diffraction-limited period. The element spacing affects the smoothness in the area around the pulse centerof the reconstructed wave, especially the trailing wake. The distance to which the wavewill propagate varies linearly with the size of the array, so a square array who’s sidedimension is increased ten times (increasing the area one hundred times) will cause thewave to propagate ten times farther. The required element spacing does not change withreconstruction distance. This means that in order to generate beams which propagategreater distances, the array must be made larger by adding elements rather than byincreasing the element spacing.Since it is the array size that limits propagation distance, it is desirable to have alarge array. A ‘folding’ scheme has been proposed to make a small array act like a largerarray (Ziolkowski, 1989). Localized waves generated using this folded array do travelfarther than their non-folded counterparts, but as the distance from the array increases,the quality of the wave decreases. There is greater reconstruction error in the area52Chapter 5. Conclusions and Future Directions 53surrounding the pulse center of the beam. Also, the source functions required for thefolded term are much more complicated and three orders of magnitude larger than forthe non-folded term. This, coupled with the fact that ultrasound waves are not requiredto travel great distances, diminishes the importance of the folded array configuration forthis application.It was previously thought that the infinite energy of the original FWM made it impossible to launch, and that finite energy variations on this wave must be found. Sincethe size of the generating array determines propagation limits rather than the waveformapplied, the FWM can be launched from an array with the same success as its finiteenergy derivatives such as the MPS pulse. The difference between the normalized FWMand MPS pulse is two orders of magnitude smaller than the peak height. The FWM hasa simpler formulation, which makes it preferable to work with over the MPS pulse, atleast for simulations.In a comparison with a traditional Gaussian beam using beam half-width at the l/epoint as a figure of merit, FWM beams generated by realistic arrays showed superiorpropagation characteristics through the area of interest (up to 0.4m). This is an important measure of the wave in that it relates to the resolution of the resulting image. At adistance of 0.3m from the source array, the FWM has a beam half-width of 10mm, whilethat of a comparable Gaussian beam is 13mm. This is a reduction of 23% for the FWM.Compared to X waves, another limited diffraction beam, the FWM showed similarpropagation characteristics. The criterion used in this comparison was the distance thebeam traveled before falling to half its original magnitude. When generated from arraysof the same area, the FWM traveled slightly farther than the X wave, 405mm as opposedto 358mm.Chapter 5. Conclusions and Future Directions 54Future Directions Real transducer behavior and how it might limit the source functions was not considered. An investigation into this could yield important information.The simulations showed that there was little difference between the reconstructed FWMand MPS pulse, however an analysis of the source functions for both could help determinewhich waveform should be used. The folded source functions could also be analyzed.The application of windowing techniques could be studied. The effect of applyinga Hanning or Hamming window the size of the array was examined, but this type ofwindow just made the array act like a smaller one. The use of temporal windows, orspatial windows the size of the individual elements could be investigated.All of the results obtained here were through computer simulations. The next stepwould be to test these findings using a real transducer array. An entire array does nothave to be built to accomplish this. Rather, a single transducer could be used, movingit to each location in the array, measuring the resulting signal, and adding the signals.BibliographyArnaud, J. A. (1976). Beam and Fiber Optics. Academic Press, New York.Brittingham, J. N. (1983). Focus waves modes in homogeneous Maxwell’s equations:Transverse electric mode. J. Appi. Phys., 54(3):1179—1189.Durnin, J. (1987). Exact solutions for nondifFracting beams. I. the scalar theory. J. Opt.Soc. Am. A, 4(4):651—654.Durnin, J., Miceli, Jr., J. J., and Eberly, J. H. (1987). Diffraction-free beams. Phys. Rev.Lett., 58(15):1499—1501.Fatemi, M. and Arad, M. A. (1992). A novel imaging system based on nondiffractingX waves. In McAvoy, B. R., editor, IEEE Ultrasonics Symposium, pages 609—612.Ultrasonics, Ferroelectric, and Frequency Control Society.Goldberg, R. L., Smith, S. W., Ladew, R. A., and Brent, J. C. (1992). Multi-layerPZT transducer arrays for improved sensitivity. In McAvoy, B. R., editor, IEEEUltrasonics Symposium, pages 551—554. Ultrasonics, Ferroelectric, and FrequencyControl Society.Greenleaf, J. F. (1983). Computerized tomography with ultrasound. Proceedings of theIEEE, 71(3):330—337.Hillion, P. (1986). More on focus wave modes in Maxwell equations. J. Appi. Phys.,60(8): 2981—2982.55BIBLIOGRAPHY 56Hykes, D. L., Hedrick, W. R., and Starchman, D. E. (1992). Ultrasound Physics andInstrumentation. Churchill Livingstone Inc., New York, New York, second edition.Jones, D. S. (1964). The Theory of Electromagnetism. Pergamon, New York.Lu, J.-y. and Greenleaf, J. F. (1992a). Nondiffracting X waves—exact solutions to free-space scalar wave equation and their finite aperture realizations. IEEE Trans. Uitrason. Ferroelec., and Freq. Contr., 39(1):19—31.Lu, J.-y. and Greenleaf, J. F. (1992b). Steering of limited diffraction beams with a two-dimensional array transducer. In McAvoy, B. R., editor, IEEE Ultrasonics Symposium, pages 603—607. Ultrasonics, Ferroelectric, and Frequency Control Society.Lu, J.-y., Song, T. K., Kinnick, R. R., and Greenleaf, J. F. (1993). In vitro and invivo real-time imaging with ultrasonic limited diffraction beams. IEEE Trans. Med.Imaging, 12(4):819—829.Macovski, A. (1983). Medical Imaging Systems. Prentice-Hall Inc., Englewood Cliffs,New Jersey.Palmer, M. R. and Donnelly, R. (1993). Focused waves and the scalar wave equation. J.Math. Phys., 34(9):4007—4013.Smith, S. W. and Light, E. D. (1992). Two-dimensional array transducers using hybridconnection technology. In McAvoy, B. R., editor, IEEE Ultrasonics Symposium,pages 555—558. Ultrasonics, Ferroelectric, and Frequency Control Society.Wu, T. T. (1985). Electromagnetic missiles. J. Appl. Phys., 57(7):2370—2373.Wu, T. T., King, R. W. P., and Shen, H.-M. (1987). Spherical lens as a launcher ofelectromagnetic missiles. J. Appl. Phys., 62(10):4036—4040.BIBLIOGRAPHY 57Ziolkowski, R. W. (1985). Exact solutions of the wave equation with complex sourcelocations. J. of Math. Phys., 26(4):861—863.Ziolkowski, R. W. (1989). Localized transmission of electromagnetic energy. Phys. Rev.A, 39(4):2005—2033.Ziolkowski, R. W. (1991). Localized wave physics and engineering. Phys. Rev. A,44(6):3960—3984.Ziolkowski, R. W. and Lewis, D. K. (1990). Verification of the localized-wave transmissioneffect. J. Appi. Phys., 68(12):6083—6086.Ziolkowski, R. W., Lewis, D. K., and Cook, B. D. (1989). Evidence of localized wavetransmission. Phys. Rev. LetI., 62(2): 147—150.

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