Nondiffracting X Waves-Exact Solutions to Free

245
LEEE TRANSACPIONS ON ULTRASONICS. FwnOELECTRlCS, AND FREQUENCY MWIXOL, VOL 39. NO I , JANVAAY 1992
IY
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
have k e n discovered. The7 are exact n o n d i c t i n g solutions of
the isotropic/homogenous scalar wave equation and are a generalization of some of the previoosly known nondiffracting waves
the plane wave, Durnin,s beams, and the
such
to an infini@
portion of the M~~~
beam equation in
of new beams. One subset of the new nondihcting waves have
X-like shapes that are termed "X waves.'' These nondiffracting
X waves can be almost exactly realized over a finite depth of
field with finite apertures and by either broad band or bandlimited radiators. With a 25 mm diameter planar radiator, a
zeroth-order broadband x wave will have about 2 5 mm lateral
and 0.17 mm axial 6 d B beam widths with a -6-dB depth
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 M H z 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
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'sModified Power Spectrum 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
I
Manurnipt iecelved April 10, 1991; revised and accepted June 17. 1991.
by grant CA 43920 from the National
This work was Npponed ih
~ ~ ~ t i tnF
I ~ ~ .
..~ tH~~. .S~.
The authors are wilh the Biadynamics Research Unit. Department of
Physiology aad Biophysics, Mnyo Clinic and Foundation, Rochester: MN
55905
IEEE log Number 91203584.
the h t Ju Bessel nondifFracting annular array transducer [21]
withPZT ceramic/polymer composite and applied it to medical
acoustic imaging and tissue
[22]-[25], campbell era/. had a similar idea to use an annular array to realize
a Jo Bessel nondiffracting beam and compared the JO Bessel
beam to the Axicon 1261.
[n this paper, we repon families of generalized nondiffracting solutions of the free-space scalar wave equation, and
specifically, a subset of these nond~ffractingsolutions, which
we term X waves (so-called X waves because of ole appearance of the field distribution in a plane throu)fh the axis of
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 zerothorder 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 dzreclions.
Like the Jo Bessel nondifEracling beam [21)-[25], X waves
could be applied to acoustic imaging and tissue characterization. In addition, the soace and time localization vrooerties
of
. .
x
allow pafiicle-like energy transmission through
large distances, as is the case with Ziolkowski's localized wave
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
nondiffracting waves and the nondiffracting X waves. Then,
we prov~detwo specific X wave examples. The production
of X waves with realizable finite apenure radiators and finite
bandwidths are reported in Section IE.Finally, in Sections N
and V, we discuss further impl~cationsof these novel beams.
The exact solution of the free space scalar wave equation,@t,
represents a family of nondifhacting waves because if one
travels with the wave at the speed, c, i.e., z - rt =constant,
both the lateral and axial complex field patterns, Ql(r, 4) and
@z(z rt). will be the same for all time, t, and distance, a.
I f r l ( k . C) in (6) is real, "=" in (5) will represent backward
and forward propagating waves, respectively. (Ln the following
analysis, we consider only the forward propagating waves. All
11. THEORYOF X WAVES AND TWO EXAMPLES
results will he the same for the backward propagating waves.)
We will show hclow three families of generalized nondiffracting solutions of the free space (or isotropic/bomogeneous) Furthermore, QC(s) and ari(s) will also represent families
of nondiffracting waves if rl(k.C) is independent of k and C,
scalar wave equation.
The free space scalar wave equation in cylindrical coordl- respectively. These waves will travel to infinity at the speed of
cl without diffracting or spreading in either the lateral or axial
rates is given by
directions. This is different from Brittingbam's focus wave
mode [I], and Ziolkowski's localized wave [2] and modified
"-0.
(1) power spectrum pulse [3] that contain both z - d and z - d t ,
I F
l ~ = I
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
(2) infinity of new beams.
$(s) = [T(k)
J4v)f(s)&]
dk
If T(k) = li(k - k'), where S(k - k') is the Diiac-Delta
-C
function and k' =w/e > 0 is a constant, f ( s ) = e', rro(k,C)
vr
= -r@. b(k. C) =
= iw/cl, from (2) and (5), one obtains
(3) Du'"in9s ~ o ~ d i f h c t i n gbeam ,111
a
)=
-
[ ( ) + + - n .
[$
j )[ /
-li
-77
I
QL(T,$. z - r t ) = @1(r.$)@?(z - cY)
[Li
QD(s) = -
-=
(4)
where
A(@)p-%"'CO'(6-fl)n()
I
p'(fl*-l)
(8)
where
s = ( Y ~ ( ~ . < ) T(4
~o s8)
+ b(k, <)[a & cl(k. C)t]
/j=
(5)
,,&GF
(9)
where n is a constant and w is angular frequency. If A(@)=
i"+"', we obtain the nth-order nondiffracting Bessel beams:
and where
cl(k.C) = cJl+
[na(k, <)/b(k.~)]'
(6)
where T ( k ) is any complex function (well behaved) of k and
could include the temporal frequency transfer function of a
radiator system, A(@)is any complex function (well behaved)
of 0 and represents a weighting function of the integration with
respect to 8, f (s) is any complex lunction (well behaved)
of s, D(C) is any complex function (well behaved) of C
and represents a weighting function of the integration with
respect to 5, which could be the Axicon angle ((I I)), rxo(k,t)
is any oomplex function of k and (. b(k.0 is any complex
function of k and 5. The term c: is wndant in (I), k and
C are variables that are independent of the spatial position,
r'= (rcos$, rsin$, a), and time, t, and Qz(a - ct) is any
complex function of z rt. Ql(r,$) is any solution of the
following transverse Laplace equation (an example of Ql(r, dl)
is given in the Appendix):
-
[:$(';)
1 8'
+;iw]@l(r,,i=~.
Q
Q 7,
(.F)
= J,, ( ( ~ T ) P ' ( ' ~ - ~ ~ + (n
~ ~=
) .0. 1. 2,
. . .).
(10)
If n = 0, one obtains the Jo Bessel nondiffracting beam. From
(S), it is seen that cl(kl, C) of the Dumin beams is equal to
w//i and is dependent upon k'. Therefore, if T(k) in (2) is
a complex function containing multiple kequeucies, a<(.?)
will represent a dispersive wave and will dBract in the axial
direction, a. If both 7~ and n are zero, (10) represents the
plane wave [ll].
If in (2), we let T(k) = 6(k - k'). ,f(s) = e', A(@)= i"ee,
cun(k,i)=-iksinc, b(k,C)= zkcosC, where k' =w/c > 0 is a
free parameter, and C represents the Axicon angle (we confine
0 < C < ~ 1 2we
) ~obtain the nth-order nondiffracting portion
of the Axicon beam [31]:
(PAn(s)
= ~ , ( k ' r sin ~ ) e ' ( " ' " ~ " ~ - " ' ~ + " * ) , (n= 0, 1, 2, ...).
(11)
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
zeroth-order nondfiacting X wave:
a0
@SBB~ =
+ [a0 - i(zcosC - rt)12
2/(T sin
(17)
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
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 Axicon angle [31]. From the previous discussion, it is seen
that ex, is an exact nondfiacting solution of the freespace 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]
<
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
- IQo
z - - t I > l ca?
<
and from (18) we obtain
giving an nth-order broadband X wave solution (exact solution
of (1). see the Appendix):
@xsn, =
a o ( r sin <)"em@
m(7+ mln
\
, (n = 0, 1, 2, ...)
(14)
/
where the subscript BE means "broadband," and
M = ( r s i n ~ ) '+ 7'
(15)
and where
r = [ao- i(z cos
< - ct)].
(16)
For simplicity, we let n = 0, obtaining axially symmetric
which has an asymptotic behavior that is proportional to
114when z - +t I isvery large. This is the
direction of slowest descent of the X wave ampl~tudefrom
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 t o r and fast with respect
( . This means that if the X waves are applied to
to I e- &f.
I
-
IEEE lXA?\SACTlONS O h ULTi7ASONICS
~RROLLECIRICF.N
FREQUENCY CONTROI.. VOL 39. NO. I. JANUARY 19%
snndillincliily X u.llvc solution of Ihc free-spilcc rcnlal. wave cqualion. Panels (h),
Fig. I. Pmel (a) ruprcsenls !he carul ren,li~-~rdci
(c), and (d) are lhc ilppnlpriale rcmlh-order nundilfracling X waver cal~uiutedfrom !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
imaging, smaller ncj will give better resolutions in both lateral
and axial directions, while smaller sin ( will reduce the Lateral
resolution while increasing the axial resolution.
At thecenter of the zeroth-order X wave ( r = 0. r = rf/cos
C), Q s a s , 1 for all distance, z , and time, t (see (17)).
which is unlike Z~olkowski'slocalized wave [2] and moditied
power s p c m m pulses (11 that recover their pulse peak values
periodically or aperiodically with distance, z.
E.x~mple2:Because B(k1 in (12) is free. it may be chosen
to achieve realizable bandwidth as shown in the following
--
50 rnm diamcmt piansr radiator is used
example. If B ( k ) is a Blackman window function [36]:
B ( k )=
{
R"
b.32 - 0.Srori
0,
2 + 0.OXrori v].
0 5 12 < 2L"
otherwise
(24)
can approximate
the bandwidt~,of a real transducer.
Although we cannot find an analytic expression for Q.s.,
from using (24) in (12) by directly looking up a Laplace
transform [able, we will describe in thesection El the resulting
nondiffracting beam calculated for some 6nite apertures.
we
LU AND
GREENLEAF: NONUFIRACTING X WAVES
3
111. REALIZATION OF 14 WAVFSWITI~
RNTE APERTURE RADLATORS
The extension and therefore the apenure requirement of X
waves in space is infinite (see (12)). But we show that nearly
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
(12) can be rewritten as (25) (see bottom of page). which is
an inverse Fourier transform of the function (spectrum of X
waves)
):(
'"JB
and F-' represents the inverse Fourier transform. The first and
second terms in (30) represent the contribution from high and
low frequency
Because B ( w / r ) in @8) $ independent of r' and #. (30)
can be rewritten in another form
(
k
)= B
(
C)
J , (>sin
(I
is the Heaviside step function P7J.
At the plane z = 0. (26) becomes
Lu'
=B
(11
= 0.1.2.. . .) (28)
where
The spectrum m (28) is used to cal~vlateX 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
I):<(
mB, (F. t )= F-I pa,
,
)
(32)
+'2rr
"(11
D(2
2~
kITz
piiir.ol,
r ' ~ l r ' ~ ~ . ( l4''
..alix
,id1
"01
i)
I)
= 0 . 1. 2. ...)
(33)
I);
( )I
an,,(7.t) = F-' [B (;)c,,(.?.
(%)
En,( r . z)
,
r
c
~[j.
.
0
(rl
),
=0 1 2
pik.r.c~l
where
$.
($7
dl'.
(26)
(T,
)
where
. ( r e = 0.1.2.. . .)
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').
=
F
B
( 1*
[
(34)
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 ) = fro, 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
R I ' [ @ Rcalculated
~].
fronl (33) and (34) is given
part of an,,,
in Fig. I and Fig. 2, when the d~ameterof 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
where
the exact nondiffracting X wave solution, RP[@.YBB,,],
@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.
I
24
TRANSACIIONS ON ULTRASONICS. FERROELELTRICS, AND R(EOUENCY CONTROL VOL 39
0'1
U'O
90
PO
20
00
epmlufiew p z p u u o ~
0-1
8'0
SO
C'o
BPnllUSew psz,puuoN
1'0
0'0
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 following. For any given angular frequency, wo,(30) represents a
continuous field of an aperture shaded by a .lo Bessel function
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~dthof the X waves can be
calculated from (19). whlch is
and the -&dB axid beam width is obtained from (20)
I.
<
2 1 ; - t1-03
(=-.
2dnn
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~ndowfuncuon (see (24)). The
parameter ice in (24) for these figures is given by
where
and
The depth of field of this JO Bessel beam is determined by
L111
Substituting (37) into (38). we obtain
Z,
D
= - rot 5
2
(39)
where r. =1.5 mrnll~b,fo =3.5 MHz. and the -6-dB bandwidth 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. 1and 2. They are the finite aperture realization of
the convolution of function F-l[B~~ic)]/nlr
with (14) when
ri = 0, that is
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 m m d i m e l e r planar radiator ir used.
with n = 0, where B(lo/c) (w 2 Oj can be any complex
function that makes the integral in (25) converge, "*" denotes
the convolution with respect to time, t, and the subscript BL
represents "band-limited."
The three panels in Figs. 4 and 5 represent band-limited
X waves at dislance z =30 mm, 90 rum, and 150 mm,
respectively. The parameters <and ao are the same as those in
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).
Rayleigh-Sommerfeld formulation of diffraction 138) can be
used to simulate the resulting beams.
Besides the X waves. there are many other families of
nondiffracting waves that are also exact solutions of the freespace scalar wave equation (see (3) and (4) and the Appendix).
B. Resnluhon, Depth
. of
. Field, and Sidelobes o f X Waves
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 s u e 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.
JV. DISCUSSION
The X waves have no or lower sidelobes in the (i - elt)
plane than the .& Bessel nondtacting beam [21], see Figs.
A. Other Nondiffmding Solutions
3 and 6. They do have energy along the branches of the X
From (43). it is seen that an infinity of nondiffracting X that becomes smaller as the tadius r increases (see Figs. 1,
wave solutions with lower field amplitude in their X branches 2, 4, and 5). It is fortuitous that the realizable band-limited X
can be obtained by choosing different complex functions R ( k ) . waves piresent lower field amplitude in their X hmches than
In practice, B ( k ) can be a h-ansfer function of physical de- the broadband X waves (Figs. 4 and 5). In acoustical imaging,
vices, such as acoustical transducers, electromagnetic antennas the effect of energy in the X branches could be suppressed
or other wave sources and their associated electronics. The 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
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, noncausal 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].
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 E. Application of X Waves to Acoustic Imaging
value of the X wave along the axial axis from z =6 mm to and EIecrromagtie~icEnergy Tra~ismi.~sion
400 mm. The edge waves are reduced around 10 dB within
The real part of (17), R C [ ~ ~ ~has
~ ,a ]smooth
.
phase
the depth of field by the aperture apodization. Thc depth of
t,
across
the
transverse
direction,
r . This
change
aver
time,
field of the X wave is reduced from 173 mm to about 147 mm
after the aperture apodization because of the reduced effective makes it possible to realize X waves with physical devices.
Therefore, a broadband acoustic transducer could be used
size of the aperture.
to
produce X waves in Figs. 1 to 3, while a band-limited
From f17). it is seen that the X waves travel with a speed
transducc~
could generate those in Figs. 4 to 6. The electrodes
faster than sound or light speed, c (superluminal). For example,
of
an
acoustic
broadband PZT ceramictpolymer composite
If C = J", the X waves will travel about 0.2% faster lhan r.
transducer
can
be
cut into annular elements [21] and each
This is also the case of Durnin's Jo Bessel beam. From (10).
elcmcnt
driven
with
a proper waveform depending upon its
for 71 = (1, we obtain
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
< k. Theretore. c =w/k < shaded receiving would suppress the off-center X wave energy
where cl = w//3 and ll =
PI.
and produce high resolution, high frame rale, and large depth
This is explained in the following: For large radius, r, on of field acoustic images.
the surface of a radiator, the wavefront of the X wave is
The X wave solution to the free-space scalar wave equation
advanced. It is these wave fronts that construct the X wave csn also be applied lo electromagnetic energy transmission for
at large distances as the lneuence of the wave fronts from the private communication and military applications [q.(Noncenter of the radiator dies out. The wave energy radiated from diffracting electromagnetic X waves are discussed in detail in
each side of the radiator travels at the speed of sound. r.. but
r401.1
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~yof all these
waves is finite. Approximate X waves are realizable with finite
apertures and with finite energy over a deep depth of field.
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 hdeep 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
waves with a conventional dynamically spherical focused
receiving system to produce high contrast images 1221. Characterizahon 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 particleltke energy transmission through large distances, as is the case
of Ziolkowski's localized wave mode [7].
APPENDIX
CDWROL VOL 39. NO I. JANUARY
1992
Now we prove e L ( 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
Theorem: Functions @f(s), P,K(.v), and mL(s) are exact
solutions of the free-space scalar-wave equatton (1).
are linear sumProofl The functions QC(s) and
*2(t - d )= e-(:-ct12/"l
mations of function f (B) over free parameters k and C,
respectively, see (2) and (3). Therefore, if we can prove f (s) and Ql(r.4) as Poisson's formula 141, p. 3731
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
where
(55)
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 A u g s t
20. 1959. He received the B.S. degree in elcclrical 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 ofPhysiology 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 o f 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. Biodvnamics Rcscuch Unit. Depnrtrnenl of Physiology, Biophysics, and Cnrdiavuscuiar
Dise.~xe and Medicine. Mayo Foundation.
Dr. Greenled- has served on Lhe IEETechnical Commiltcc of thc Ultras o n i s Sympusium since 1%. He served on Lhe IEEE-UFFC Suhcommiltou
lor the Ultrasonics in Medicine/lE!Z Measurement G~lidcEdiloa, 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~medicalimaging science and hss published more than 150
articlcs and edited four hooks.