245

LEEE TRANSACPIONS ON ULTRASONICS. FwnOELECTRlCS, AND FREQUENCY MWIXOL, VOL 39. NO I , JANVAAY 1992 I Y

Nondiffracting X Waves-Exact Solutions to Free-Space Scalar Wave Equation and

Their Finite Aperture Realizations Jian-yu Lu, Member, LEEE, and James F. Greenleaf, Fellow, E E E

Abstract-Novel families of generalized nandiffracting waves the h t Ju Bessel nondifFracting annular array transducer [21] have k e n discovered. The7 are exact n o n d i c t i n g solutions of withPZT ceramic/polymer composite and applied it to medical the isotropic/homogenous scalar wave equation and are a gener- acoustic imaging and tissue [22]-[25], camp- alization of some of the previoosly known nondiffracting waves such the plane wave, Durnin,s beams, and the bell era/ . had a similar idea to use an annular array to realize

portion of the M~~~ beam equation in to an infini@ a Jo Bessel nondiffracting beam and compared the JO Bessel of new beams. One subset of the new nondihcting waves have beam to the Axicon 1261. X-like shapes that are termed "X waves.'' These nondiffracting [n this paper, we repon families of generalized nondiffract- X waves can be almost exactly realized over a finite depth of ing solutions of the free-space scalar wave equation, and field with finite apertures and by either broad band or band- limited radiators. With a 25 mm diameter planar radiator, a specifically, a subset of these nond~ffracting solutions, which

zeroth-order broadband x wave will have about 2 5 mm lateral we term X waves (so-called X waves because of ole appear- and 0.17 mm axial 6 d B beam widths with a -6-dB depth ance of the field distribution in a plane throu)fh the axis of of field of aboot 171 mm. The phase of the X waves changes smoothly with time across the aperture of the radiator, therefore, X waves can be realized with physical devices. A zeroth-order hand-limited X wave was produced and measured in water by our 10 element, 50 mm diameter, 2.5 MHz PZT ceramicipolymer composite J o Bessel nondirfracting annular array transducer with -&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.

I. INTRODUCTION

I N 1983, J. N. Brittingham [I] discovered the first localized wave solution of the free-space scalar wave equahon and

called it a self focus wave mode. In 1985, another localized wave solution was discovered by R. W. Zmlkowski [2] who developed a procedure to c o n m c t new solutions [3] through the Laplace transform. These localized solutions were Further studied by several investigators [4]-191. An acoustic device was conshucted to realize Z~olkowski's Modified Power Spec- trum pulse that is one of the solutions of the wave equation through the Laplace transform [lo].

In 1987 J. Dumin discovered independently a new exact nondifkacting solution of the free-space scalar wave equation [XI]. Thissolution was expressed in continuous wave form and was realized by optical experiments [12]. Dumin's beams were further studied in optics in a number of papers [13]-1191. Hsu et al. [20] realized a Jo Bessel beam with a narrow band PZT ceramic ultrasonic transducer of nonumform poling. We made

the beam). The X waves are a frequency weighted Laplace transform of the nondiffracliug portion of the Axicon beam equations 1271-[33] (in addition to the nondiffracting portion, the Axicon beam equation contains a factor that is proportional to fi that will go to infinity as z + XJ [31]) and travel in free-space (or isotropic/bomogeneous media) to infinite distance without spreading provided that they are produced by an in6nite aperture.

Even a d h Enite apertures, X waves have a large depth of field. (For example, with 25 mm diameter radiator, a zeroth- order broadband X wave will keep 2.5 mm and 0.17 mm -6-dB lateral and axial beam widths, respectively, and have a 171 mm depth of field that is about 68 times the -6-dB lateral beam width and 7 times the radiating diameter.) We have experhnentally produced a zeroth-order band-limited X wave [MI. We used our 10 element, 50 mm diameter, 2.5 MHz PZT ceramicipolymer composite JO -el nondiffracting annular array transducer [XI, [22]. The X wave had -&dB lateral and axial beam widths of about 4.7 mm and 0.65 mm, respectively, over a 6 - d B depth of hid of aboul358 mm.

In comparison with Ziolkowski's localized wave mode [Z], [3], the peak pulse amplitude of the X wave is constant as it propagates to infinity because of its nondiffracling nature. Durnin's beams are nondiffracting at a single frequency but will diffract (or spread) in the temporal direction for multiple frequencies [22]. X waves are multiple freqwncy wares bur they nre nondiffracfing in both frajrrverse and ariol dzrec- lions.

Like the Jo Bessel nondifEracling beam [21)-[25], X waves Manurnipt iecelved April 10, 1991; revised and accepted June 17. 1991.

This work was Npponed ih by grant CA 43920 from the National could be applied to acoustic imaging and tissue characteriza- ~ ~ ~ t i t ~ t ~ . . nF H ~ S I ~ ~ . tion. In addition, the soace and time localization vrooerties of .. ~. . .

The authors are wilh the Biadynamics Research Unit. Department of x allow pafiicle-like energy transmission through Physiology aad Biophysics, Mnyo Clinic and Foundation, Rochester: MN 55905

large distances, as is the case with Ziolkowski's localized wave IEEE log Number 91203584. mode [7].

LO IEEE TRANSACnONS ON ULTRASOWCS. FERROELECIXICS. AND FREQUENCY CONTROL. VOL 39. NO. 1. JANUARY lq'X2

In Section 11, we first introduce the families of generalized The exact solution of the free space scalar wave equation,@t, nondiffracting waves and the nondiffracting X waves. Then, represents a family of nondifhacting waves because if one we prov~de two specific X wave examples. The production travels with the wave at the speed, c, i.e., z - rt =constant, of X waves with realizable finite apenure radiators and finite both the lateral and axial complex field patterns, Ql(r, 4) and bandwidths are reported in Section IE. Finally, in Sections N @z(z - rt). will be the same for all time, t, and distance, a. and V, we discuss further impl~cations of these novel beams. I f r l (k . C) in (6) is real, "=" in (5) will represent backward

and forward propagating waves, respectively. (Ln the following 11. THEORY OF X WAVES AND TWO EXAMPLES analysis, we consider only the forward propagating waves. All

results will he the same for the backward propagating waves.) We will show hclow three families of generalized non- diffracting solutions of the free space (or isotropic/bomogeneous) Furthermore, QC(s) and ari(s) will also represent families

scalar wave equation. of nondiffracting waves if rl(k.C) is independent of k and C,

The free space scalar wave equation in cylindrical coordl- respectively. These waves will travel to infinity at the speed of

rates is given by cl without diffracting or spreading in either the lateral or axial directions. This is different from Brittingbam's focus wave

I F l ~ = I

mode [I], and Ziolkowski's localized wave [2] and modified [ ( ) + + - n . "-0. (1) power spectrum pulse [3] that contain both z - d and z - dt ,

or both z - ct and z + c+ terns in their variables, where c and where = represents radial coordinate, d, is ,, are different azimuthal angle, z is axial axis. whtch is perpendicular to the we find mC(s) is 'he most interesting solution. In the plane defined by r and d,, f is timime and represents acoustic following, we will discuss @<(s) only. QC(s) is a generalized pressure or Hertz potential that is a function of r, (1, 2, and t.

function that contains fiome of the nondiffracting solutions of The three functions below are families of exact solutions of

the free space scalar wave equation known previously, such (1) (see the Appendix):

as, the plane wave. Durnin's nondlffracting beams, and the nondiffracting portion of the Axicon beam in addition to an

$(s) = [T(k) [$ J4v)f(s)&] dk (2) infinity of new beams.

-C If T(k) = li(k - k'), where S(k - k') is the Diiac-Delta function and k' =w/e > 0 is a constant, f ( s ) = e', rro(k,C)

vr I = - r @ . b(k. C) = = iw/cl, from (2) and (5), one obtains

a ) = j ) [ / (3) Du'"in9s ~o~d i fhc t i ng beam ,111

-77 -li

QD(s) = - [L i A(@)p-%" 'CO ' (6 - f l )n ( ) p ' ( f l*- l ) I (8) QL(T,$. z - r t ) = @1(r. $)@?(z - c Y ) (4) -=

where where

s = ( Y ~ ( ~ . < ) T ~ o s (4 - 8) + b(k, <)[a & cl(k. C)t] (5) / j = ,,&GF (9)

and where where n is a constant and w is angular frequency. If A(@) = i"+"', we obtain the nth-order nondiffracting Bessel beams:

cl(k.C) = cJl+ [na(k, <)/b(k.~)]' ( 6) Q 7, (.F) = J,, ( ( ~ T ) P ' ( ' ~ - ~ ~ + ~ ~ ) . (n = 0. 1. 2, . . .). (10)

where T ( k ) is any complex function (well behaved) of k and could include the temporal frequency transfer function of a If n = 0, one obtains the Jo Bessel nondiffracting beam. From radiator system, A(@) is any complex function (well behaved) (S), it is seen that cl(kl, C) of the Dumin beams is equal to of 0 and represents a weighting function of the integration with w//i and is dependent upon k'. Therefore, if T(k) in (2) is respect to 8, f (s) is any complex lunction (well behaved) a complex function containing multiple kequeucies, a<(.?) of s, D(C) is any complex function (well behaved) of C will represent a dispersive wave and will dBract in the axial and represents a weighting function of the integration with direction, a. If both 7~ and n are zero, (10) represents the respect to 5, which could be the Axicon angle ((I I)), rxo(k,t) plane wave [ll]. is any oomplex function of k and (. b(k.0 is any complex If in (2), we let T(k) = 6(k - k'). ,f(s) = e', A(@) = i"ee, function of k and 5. The term c: is wndant in (I), k and cun(k,i)=-iksinc, b(k,C)= zkcosC, where k' =w/c > 0 is a C are variables that are independent of the spatial position, free parameter, and C represents the Axicon angle (we confine r'= (rcos$, rs in$, a), and time, t, and Qz(a - ct) is any 0 < C < ~ 1 2 ) ~ we obtain the nth-order nondiffracting portion complex function of z - rt. Ql(r,$) is any solution of the of the Axicon beam [31]: following transverse Laplace equation (an example of Ql(r, dl) is given in the Appendix): (PAn(s) = ~ , ( k ' r sin ~)e ' (" ' "~"~-" '~+"* ) , (n = 0, 1, 2, ...).

1 8' (11) [:$(';) +;iw]@l(r , , i=~. Q From (1 l), it is seen that c1(k1,C) = clcosC is independent of k', which means that the waves constructed horn QA,,(B) by (2)

LU ANU GREEM.EAF: NONDIPFRACITNG X WAVES

will have a constant speed ci for all frequency components and will be nondispersive. The X waves derived in the following are such waves that are nondiffracting or nonspreading in both lateral and axial directions.

We obtain the new X wave if in (2), we let T(k) = B ( k ) r - ' ~ ~ , A(@) = znean8, ao(k,() = -iksinC, b(k,C) = ika)s<, f (s) = ee, obtaining

This is an integration of the nth-order nondiffracting portion ((11)) of the Axicon beam equation multiplied by B(k)e-nok, where B(k) is any complex function (well behaved) of k and represents a transfer function of a practical radiator system, k =wlc, a0 > 0 is a constant, and < is the Axi- con angle [31]. From the previous discussion, it is seen that ex, is an exact nondfiacting solution of the free- space scalar-wave equation (1). Equation (12) shows that as,, is represented by a Laplace transform of the function, B(k)J,(krsinC), and an azimuthal phase term, ein*. Because the waves represented hy @x, have an X-like shape in a plane through the axial axis of the waves, we call them nth-order X waves.

We finish this section with two new nondiffracting X wave examples. The first represents an X wave with broad bandwidth, white the second LF a band-limited X wave.

Example I: First, we describe an expression for an X wave produced by infinite aperture and broad bandwidth.

Lf, in (12), B(1r) -- ao, we obtain from the Laplace transform tables [35]

giving an nth-order broadband X wave solution (exact solution of (1). see the Appendix):

ao( r sin <)"em@ @xsn, = , (n = 0, 1, 2, ...) (14) m(7 + mln

\ / where the subscript BE means "broadband," and

M = ( r s i n ~ ) ' + 7' and where

(15)

r = [ao - i(z cos < - ct)]. (16)

For simplicity, we let n = 0, obtaining axially symmetric

zeroth-order nondfiacting X wave:

a0 @SBB~ = (17)

2/(T sin + [a0 - i(zcosC - rt)12

The behavior of the analytic envelope of the real part of (17). R e [ a . s ~ ~ , ] is shown in the upper left panel of Fig. 1 or Fig. 2. The modulus of @XBB" in (17) can be obtained as shown in (18) (see bottom of page).

Now we discuss the lateral, axial, and X branch behaviors of this X wave. It is seen from (18) that for 6xed C and no, at the plane z = dlcosC we have

which represents the lateral pressure distribution of the X wave through the pulse center and has an asymptotic hehavlor proportional to l/r when r is very large.

On the line r = 0, from (18), we obtain

which is the pressure distribution of the X wave along the axial axis, a, and has an asymptotic behavior proportional to 1/ I z - &t I when I z - +t I is very large.

The pressure distribution of the X wave along its two branches (see Figs. 1 and 2) can be obtained when

sin C cos C

where

cos C c - I z - - t I > l Qo ca? <

and from (18) we obtain

which has an asymptotic behavior that is proportional to 114- when I z - +t I isvery large. This is the direction of slowest descent of the X wave ampl~tude from its center.

From (19). (ZO), and (Z), it is seen that the smaller a,, is, the faster the function I @.yns, I diminishes as r or I z - -t I increases. If sin< is small, the diminution of I (P.~:) will he slow with respect to r and fast with respect to I e- &f. ( . This means that if the X waves are applied to

IEEE lXA?\SACTlONS Oh ULTi7ASONICS ~RROLLECIRICF. N FREQUENCY CONTROI.. VOL 39. NO. I. JANUARY 19%

Fig. I. Pmel (a) ruprcsenls !he carul ren, l i~-~rdci snndillincliily X u.llvc solution of Ihc free-spilcc rcnlal. wave cqualion. Panels (h), (c), and (d) are lhc ilppnlpriale rcmlh-order nundilfracling X waver cal~uiuted from !he Raylcigh-SnmmerfcId dilfraclion inlegral wiih a 25-rnm diamelsi planar radValilr a1 dirwncra : =YI mm. 9Ll mm. and !SO mm, re.ipeclively. The dirncnsion of each panel is 12.5 mm x 6.25 mm. U ( f i ) = <!u. <= 4" bmd rr,, = 11.05 nmm.

Fig. 2. Same fumlal :LS Fiz. 1. crcrpl ihiil r 50 rnm diamcmt piansr radiator is used

imaging, smaller ncj will give better resolutions in both lateral example. If B(k) is a Blackman window function [36]: and axial directions, while smaller sin ( will reduce the Lateral resolution while increasing the axial resolution.

{ R" b.32 - 0.Srori 2 + 0.OXrori v]. 0 5 12 < 2L"

At thecenter of the zeroth-order X wave (r = 0. r = rf /cos B ( k ) = C), Qsas, -- 1 for all distance, z , and time, t (see (17)). 0, otherwise

(24) which is unlike Z~olkowski's localized wave [2] and moditied we can approximate the bandwidt~, of a real transducer. power s p c m m pulses (11 that recover their pulse peak values Although we cannot find an analytic expression for Q.s., periodically or aperiodically with distance, z. from using (24) in (12) by directly looking up a Laplace

E.x~mple2: Because B(k1 in (12) is free. it may be chosen transform [able, we will describe in thesection El the resulting to achieve realizable bandwidth as shown in the following nondiffracting beam calculated for some 6nite apertures.

LU AND GREENLEAF: NONUFIRACTING X WAVES 3

111. REALIZATION OF 14 WAVFS WITI~ and F-' represents the inverse Fourier transform. The first and RNTE APERTURE RADLATORS second terms in (30) represent the contribution from high and

The extension and therefore the apenure requirement of X low frequency

waves in space is infinite (see (12)). But we show that nearly Because B ( w / r ) in @8) $ independent of r' and #. (30) can be rewritten in another form I

exact X waves can be realized with b i t e aperture radiators over large axial distance. To calculate X waves kom the b i t e aperture radiators. a specmm of X waves is used. Equalion ( k ) = B ( ) ) ($7 = 0 1 2 . ) (32)

(12) can be rewritten as (25) (see bottom of page). which is where

an inverse Fourier transform of the function (spectrum of X waves)

pik.r.c~l

'"JB (:) J , (>sin C) dl'. kITz

(I 0 "(11

. ( r e = 0.1.2.. . .) 2~ D(2 piiir.ol,

(26) +' ,id1 r ' ~ l r ' ~ ~ . ( l ' . 4'. alix 2rr

i ) I) "01

where (r l =0. 1. 2. ...) (33)

is the Heaviside step function P7J. an,, (7.t) = F-' B At the plane z = 0. (26) becomes

[ (;)c,, (.?. ;)I Lu' I".+, (T, $. ,) = B (%) En, (r . ~[j. z) , (11 = 0.1.2.. . .) (28)

= F B ( 1 * [ ( ) I (34) r c

where

The spectrum m (28) is used to cal~vlate X waves Gom radiators with finite aperture. If we assume that the radiator is circular and has a diameter of D, rhe Raylcigh-Sommerfcld formlilation of diffraction by a plane screen [38] can he written as

and

mB, (F. t ) = F-I pa, (<:)I , ( , = 0. 1. 2, ... ) (31)

where A is wavelength, roi is the distance between the observation point, F, and a point on the source plane (r'. 4').

where * denotes convolution on time, 1, and F '[B(w/r.)] can be taken as an impulse response of the radiator.

We now deqcrihe an example of an X wave produced hy a finite aperture. Let B ( k ) = f ro , C = 8". (sin4' r= 0.0598, cos4" r= 0.9981, nu = 0 05 nim, n = 0, and c = 1.5 mmib~r (speed of sound in water). The analytic envelope of the real part of an,,, RI'[@R~]. calculated fronl (33) and (34) is given in Fig. I and Fig. 2, when the d~ameter of the radiator, D, 15

25 mrn and 50 mm, respectively. The X waves are shown at ,- = 30mm, 90 mm. and 150 mm, respectively. These X waves compared well to the analytic envelope of the real part of the exact nondiffracting X wave solution, RP[@.YBB,,], where @yos,, is given by (17) (see upper-left panel of Fig. 1 or 2).

Fig. 3 shows beam plots of thc X waves in Figs. 1 and 2. Figs. 3(al), 3(a2), and 3(a3) represent the lateral beam plots of the X waves produced at Lhe axial distances z =30 mm, 90 mm, and 150 mm, req)ectively, through the center of the pulses. The full, dotted, and dashed lines represent the beam plots of the X waves produced by a 25 mm diameter radiator, 50 mm diameter radiator, and the exact nondiffracting X wave solution of the free-space scalar wave equation (real pan of (17)), respectively. Fig. 3(a4) represents the peak of the X waves along the axial distance, z, from 6 mm to 400 mm. The 4 - d B depths of field of the X waves are about 171 mm and 348 mm when thc radiator has diameters of 25 mm and 50 mm, respectively. The -6dB lateral heam width throughout the depth of field i6 about 2.5 mm.

24 TRANSACIIONS ON ULTRASONICS. FERROELELTRICS, AND R(EOUENCY CONTROL VOL 39

0'1 U'O 9 0 PO 20 00 epmlufiew p z p u u o ~

0-1 8'0 SO C'o 1'0 0'0 BPnllUSew psz,puuoN

LU AND G B E e N W . NONDlFFRACnNG X WAVES

Fig. 4. RayleigMommerieid aiculalion 01' [he zemlh-ndcr nundill'rilcling X w;tves pmduucd hy a 25 nuu diameter. hand-limiled planar radiator at distances ; =30 mm. 90 mm, and 150 mm. Thc ritdi;dor lrilnslcr rundiun Lllkl is a Blackman window Function peaked at 3.5 MHz with a --(,-dB hilndwidlh about 2.88 MHz. The dirncnsian ol silch psnrl is 12.5 mm x 6.25 mm. C = 4' and a" =0.05 mm.

Figs. 3(bl) to 3(b3) are axial beam plots of the X waves with respect to c l t - z and correspond to Fgs. 3(al) to 3(a3), respectively. ( e l= c/cos<. For ( = 4", cl ;= 1.002c.) The edge waves produced by the sharp mtncation of the pressure at the edge of the radiator are clearly seen. These edge waves can be overcome by using proper aperture apodizalion techniques [39] and will be discussed in the next section. The -6-dB axial beam width obtained from Fig. 3@) is about 0.17 mm.

The depth of field and the lateral and axial beam widths of the zeroth-order broadband X wave produced by a radiator of diameter D can be determined analytically in the follow- ing. For any given angular frequency, wo, (30) represents a continuous field of an aperture shaded by a .lo Bessel function

where

and

The depth of field of this JO Bessel beam is determined by

L111

Substituting (37) into (38). we obtain

which is independent of frequency * a ! If C= -1" and D = 25mm, we obtain, from (39). z,,=178.8 mm, which is very close to what we obtained from Fig. 3(a)(4) (171 mm).

The 6 - d B lateral beam w~dth of the X waves can be calculated from (19). whlch is

and the -&dB axid beam width is obtained from (20)

I . 2dnn 2 1 ; - t ( = - . 1-03 < cos <

When no = 0.05 mm and i = 4*, the calculated -6-dB lateral and axial beam widths are about 2.5 mm and 0.17 mm, respectively, which are the same as we measured from Fig. 3.

Figs. 4 and 5 are the analytic envelope of a zeroth-order hand-limited nondiffracting X wave produced by a radiatar of diameter of 25 mm and 50 mm, respectively, when B(L) is chosen as a Blackman w~ndow funcuon (see (24)). The parameter ice in (24) for these figures is given by

where r. =1.5 mrnll~b, fo =3.5 MHz. and the -6-dB band- width of ihe Blackman window function is about 2.88 MHz. The X waves in Figs. 4 and 5 are a band-limited version of those in Figs. 1 and 2. They are the finite aperture realization of the convolution of function F-l[B~~ic)]/nlr with (14) when ri = 0, that is

D Z,,, = - rot 5

2 (39)

IEEE TRANSACTIONS ON ULTRASONICS. FERROELECIWCS. ANII FREQUENCY CONTROL VOL 39. NO. I. JANUARY 1992

Fig. 5. Samr formal as Fig. 4. cxcepl lhal a 511 mm d ime le r planar radiator ir used.

with n = 0, where B(lo/c) (w 2 Oj can be any complex Rayleigh-Sommerfeld formulation of diffraction 138) can be function that makes the integral in (25) converge, "*" denotes used to simulate the resulting beams. the convolution with respect to time, t, and the subscript BL Besides the X waves. there are many other families of represents "band-limited." nondiffracting waves that are also exact solutions of the free-

The three panels in Figs. 4 and 5 represent band-limited space scalar wave equation (see (3) and (4) and the Appendix). X waves at dislance z =30 mm, 90 rum, and 150 mm, respectively. The parameters <and ao are the same as those in B. Resnluhon, Depth of Field, and Sidelobes o f X Waves Figs. 1 and 2, which are 4' and 0.05 mm, respectively. From Figs. 4 and 5 , it is seen that the band-limited X waves have lower field amplitude in the X branches than the widehand X waves in Figs. 1 and 2.

Fig. 6 is the same as Fig. 3, except that it represents the lateral and axial beam plots of the band-limited X waves in Figs. 4 and 5. The 6 - d B lateral and axial beam widths of the band-limited X waves obtained from Fig. 6 are about 3.2 mm and 0.5 mm, respectively, at aU three distances ( z =30 mm, 90 mm, and 150 mm). Therefore, these band-limited X waves are also nondifhacting waves. Theu -6-dB maximum nondiffracting distances derived from Fig. 6 are 173 mm aod 351 mm, when the diameters of the radiators are 25 mm and 50 mm, respectively. The depths of field of the band-limited X waves are almost the same as those of the broad band nondiffracting X waves obtamed lrom Fig. 3 and are very close to those calculated from (39).

JV. DISCUSSION

A. Other Nondiffmding Solutions

From (43). it is seen that an infinity of nondiffracting X wave solutions with lower field amplitude in their X branches can be obtained by choosing different complex functions R(k) . In practice, B(k) can be a h-ansfer function of physical de- vices, such as acoustical transducers, electromagnetic antennas or other wave sources and their associated electronics. The

. .

From (IS), it is seen that as n, decreases, the X waves diminiqh faster with both r and I r - (el eos Ct) I, and hence they have higher lateral and axial resolution. This requires ihat the radiator system has a broader bandwidth because the lerm expi-aowjc) in (26) will diminish slower for smaller 0,). For larger values of sin 5, the lateral resolution will he increased while the axial resolution is decreased. The -6-dB lateral and axial beam widths of the zeroth-order broadband nondiffrscung X waves ((17)) are determined by (40) and (41), respectively.

For finite aperture, the X waves are nondiffracting only within the depth of fieId The depth of field of finite aperture nondiffracting X waves is related only to the sue of the radlator and the Axicon angle C ((39)). Bigger values of sinC will increase the lateral resolution ((40)) but reduce the depth of field. This trade-off is similar to that of conventional focused beams. For band-limited nondiffracting X waves, the - laleral and axial beam widths are increased from those of the broadband X waves, but the depth of field is about the same.

The X waves have no or lower sidelobes in the (i - elt) plane than the .& Bessel nondtact ing beam [21], see Figs. 3 and 6. They do have energy along the branches of the X that becomes smaller as the tadius r increases (see Figs. 1, 2, 4, and 5). It is fortuitous that the realizable band-limited X waves piresent lower field amplitude in their X hmches than the broadband X waves (Figs. 4 and 5). In acoustical imaging, the effect of energy in the X branches could be suppressed dramatically by combining X wave transmit with dynamic

28 I E E TRANSACCIONS ON ULTRASONICS. F ~OELFCTRICS, AND FREQUENCY CONTROL. VOL 39. NO 1. JANUARY 1992

spherical focused receiving as was proposcd fa1 the J" Bessel nondiffracting beam [2.5].

C. Edge Waves and Propagation Speed ofthe X Waves

It is seen from Figs. 3@) and 6(b) that there is some energy in the waves produced by the edge of the aperture because of sharp truncation of the pressure at the edge of the radiators. These edge waves can be reduced by apodization techniques [39] that use window functions to apodize the edge of the aperture. (The results in Figs. 1 to 6 are obtained with a rectangular window aperture weighting.)

Fig. 7 shows reduction of the edge waves with aperture apodization. Figs. 7(a) (except panel (4)) and 7(b) show the lateral and axial beam plots, respectively, of band-limited X waves produced by a 25-mm-diameter radiator. The full and dotted lines represent the beam plots before and after the aperture apodization, respechvely. The apodkation function is given by (44) (see bottom of page). where v1 = 3D/8, D is the diameter of the radiator, and r is the lateral distance from the center of Ule radiator. Panels (I), (2), and (3) in both Figs. 7(a) and 7(b) represent the beam plots at distance 2 =30 mm, 90 mm, and 150 rnm, respectively. Fig. 7(a4) plots the peak value of the X wave along the axial axis from z =6 mm to 400 mm. The edge waves are reduced around 10 dB within the depth of field by the aperture apodization. Thc depth of field of the X wave is reduced from 173 mm to about 147 mm after the aperture apodization because of the reduced effective size of the aperture.

From f17). it is seen that the X waves travel with a speed faster than sound or light speed, c (superluminal). For example, If C = J", the X waves will travel about 0.2% faster lhan r. This is also the case of Durnin's Jo Bessel beam. From (10). for 71 = (1, we obtain

where cl = w//3 and ll = < k. Theretore. c =w/k < PI.

This is explained in the following: For large radius, r, on the surface of a radiator, the wavefront of the X wave is advanced. It is these wave fronts that construct the X wave at large distances as the lneuence of the wave fronts from the center of the radiator dies out. The wave energy radiated from each side of the radiator travels at the speed of sound. r.. but the intersection, or peak. of these waves travels greater than the speed of sound.

D. Total Energy, Energy Dettsir?. U J I ~ Causality

Like the plane wave and Durnin's beam, the total energy of an X wave is infinite. But, the energy densi~y of all these waves is finite. Approximate X waves are realizable with finite apertures and with finite energy over a deep depth of field.

From (17). it is seen that at a given distance z, the X waves extend from t> - co t o t < m. Therefore, they are not causal. but the X waves produced by fininte aperture diminish very last as I t I -%. lf we ignore the X waves for I t I > t o , where I e.~,, (r'. to) I<< 1, these modified X waves can be treated as causal waves. Similarly. Ziolkowski's localized wave 121, and modified power spectrum wave 131 suffer from the problem of causality, but they are closely approximated with finite aperture by using this same causal approximation [10].

We do not wony about the superluminal solutions, non- causal solutions. infinite total energy or infinite apertures as long as the pressure distributions over an aperture are slrFficienLly well behaved that they can be physically realized and truncated in time and space and still produce a wave of practical usefulness. Theoreucal X waves are superluminal, ooncausal, and have infinite total energy and apertures, but rhey can be truncated to produce practical waves. One example waq given in reference [34], where a zeroth-order X wave was physically realized over a deep depth of field in an experiment using an acoustic annular array transducer [21], [22].

E. Application of X Waves to Acoustic Imaging and EIecrromagtie~ic Energy Tra~ismi.~sion

The real part of (17), R C [ ~ ~ ~ ~ , ] . has a smooth phase change aver time, t , across the transverse direction, r . This makes it possible to realize X waves with physical devices. Therefore, a broadband acoustic transducer could be used to produce X waves in Figs. 1 to 3, while a band-limited transducc~ could generate those in Figs. 4 to 6. The electrodes of an acoustic broadband PZT ceramictpolymer composite transducer can be cut into annular elements [21] and each elcmcnt driven with a proper waveform depending upon its radial position. Annular array transducers can be used for both X wave transmitling and conventional dynamically spherical focused receiving [34]. The low sidelobes of a Gaussian shaded receiving would suppress the off-center X wave energy and produce high resolution, high frame rale, and large depth of field acoustic images.

The X wave solution to the free-space scalar wave equation csn also be applied lo electromagnetic energy transmission for private communication and military applications [q. (Non- diffracting electromagnetic X waves are discussed in detail in r401.1

We have developed a new family of nondiffracting waves that provides a novel way to produce focused beams without lenses. They are realizable with finite apertures and with broadband and band-limited planar radiators w ~ t h deep depth of field. The energy on the X branches of the X waves could be suppressed dramatically in maging by combining the X

LU AND GREENLMF. NONDIFFRACrING X WAVES

I -- -

3II UCB TRANSACIIONS ON ULTRASOMCS. FWIROELECTRLCS. AND FREQuoVCi CDWROL VOL 39. NO I. JANUARY 1992

waves with a conventional dynamically spherical focused receiving system to produce high contrast images 1221. Char- acterizahon of tissue properties could be simplified because of the absence of diffraction in X waves. In addition, the space and time localization properties of X waves will allow particle- ltke energy transmission through large distances, as is the case of Ziolkowski's localized wave mode [7].

APPENDIX

Theorem: Functions @f(s), P,K(.v), and mL(s) are exact solutions of the free-space scalar-wave equatton (1).

Proofl The functions QC(s) and are linear sum- mations of function f (B) over free parameters k and C, respectively, see (2) and (3). Therefore, if we can prove f (s) is an exacl solution of (I), @C(s) and ah-(s) will also be exact solutions. (For example, one can directly prove that (14) is an exact solution of the free-space scalar wave (I). Using the expression of s in (51, we obtain

Using (6), the right-hand side of (53) is zero. Therefore, a c t s ) and Q K ( s ) are exact solutions of the free-space scalar wave equation (1).

Now we prove eL( s ) in (4) is also an exact solution of (1):

Because @l(r,d) is an exact solution of the transverse Laplam equation ((7)). the right-hand side of (54) is zero. ThLs proves that @L is an exact solution of the free-space scalar wave equation (I).

For example, if we choose Q ~ ( z - ct) as a Gaussian pulse

*2(t - d) = e-(:-ct12/"l (55)

and Ql (r.4) as Poisson's formula 141, p. 3731

where

and .f($) is any function (well-behaved) of 4. and n and a are constants, the Gaussian pulse with a lransverse field pattern @,(T,&) will travel to infinity at the speed of sound or light without any change of pulse shape.

The banweme field ((56)) has the following properties:

This means that if yL%@l(~.r/l) = 0, then @1(0,31) = 0.

The authors appreciated the secretarial assistance of Elnine C. Qnarve and the graphic assistance of Christine A. Welch.

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Jian-yu Lu (M'88) was ham in Fwhou. Fujian Province. Pcople's Republic of Chins. an Augs t 20. 1959. He received the B.S. degree in elcclri- cal engineering in lY8Z Erom Fudan University. Shanghni, Chin*, Lhe M.S. degree in Ill85 fmm Tnngji University, Shanghai. China, and the Ph.D. degree in 1988 frnm Sootheast Univenily. Nanjing, China.

He is an Assistant Prnfessnr olBiophgsicr. Maye Mcdical School. and a Research Assnciule. at the Biodynamics Rcsearch Unil. Dcpanment ofPhysiol-

ogy and Biophy~ic.;. Mayo Clinic and Foundalion. Rnchesler, MN. Prom 1988 lo lYq(1. he was a postdnctoral Research FeUow Ihcre. Prior to that, he was a kcacully mcmher of the Dcp;lrlrnenl of Biomedical Engineering, Sdulhcasl Univmily, :nrd worked with Prof. Xu Wei. His research interesls are in acoustical imaging and librue characlerhttion, medical ullrasanic Iransdurrrs. itnd nondifliubing wiwc transmission.

Dr La is u member of the I E E UFFC Society. the American institute of Ultrasuund in Medicine. and Sigma Xi.

James F. GreenleaF(M'7.?-SM'8GF'88) was horn in Salt Lake Ciy. UT. on Fcbnrilr/ 10, 1942. HI. received the B.S. degee in electrical engineering in 1964 born the University of Utah. Salt Lake City. the M.S. degrcc in engineering science in 1968 from PurdueUnivcnity, Lafayette, IN. and the Ph.D. degrce in engineering science from Lhe Mayo GTrduate School or Medicine. Rwhesrer. MN, and Purduc University in 197f1.

He is a Professor o i Biophysics and Medicine, Mavo Medical School. and Consultant. Biodvnam-

ics Rcscuch Unit. Depnrtrnenl of Physiology, Biophysics, and Cnrdiavuscuiar Dise.~xe and Medicine. Mayo Foundation.

Dr. Greenled- has served on Lhe IEE Technical Commiltcc of thc Ultra- s o n i s Sympusium since 1%. He served on Lhe IEEE-UFFC Suhcommiltou lor the Ultrasonics in Medicine/lE!Z Measurement G~lidc Ediloa, and on lhe IEEE Mcdiusi Ultrasound Committee. He is Vice Preridcnl of the UFFC Sociay, Hc holds Ave patents and is the recipienl of the 1986 1. Holmes Pioneer award form the American rnstirutt of Ultrasound in Medicine, and is u Fellow of the IEEE and ihc ANM. He is thc Distinguished Lecturer for the E E W C Society for 1991!-1991. 13s spacial lield of iaerest is in ullraonic hic~medical imaging science and hss published more than 150 articlcs and edited four hooks.