Nondiffracting X Waves-Exact Solutions to Free-Space Scalar

IEEE TRANSACTIONS ON ULTRASONICS. FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 39, NO. I , JANUARY 1YY2
Nondiffracting X Waves-Exact Solutions to
Free-Space Scalar Wave Equation and
Their Finite Aperture Realizations
Jian-yu Lu, Member, IEEE, and James F. Greenleaf, Fellow, IEEE
Abstract- Novel families of generalized nondiffracting waves
have been discovered. They are exact nondiffracting solutions of
the isotropic/homogenous scalar wave equation and are a generalization of some of the previously known nondiffracting waves
such as the plane wave, Durnin’s beams, and the nondiffracting
portion of the Axicon beam equation in addition to an infinity
of new beams. One subset of the new nondiffracting waves have
X-like shapes that are termed “X waves.” These nondiffracting
Xwaves can be almost exactlyrealized over a finite depth of
field with finite apertures and by either broadband or bandlimited radiators. Witha 25 mm diameter planarradiator, a
zeroth-order broadband X wave will have about 2.5 mm lateral
beam widths witha-6-dB
depth
and 0.17 mmaxial-6-dB
of field of about 171 mm. The phase of the X waves changes
smoothly with time across the apertureof the radiator, therefore,
X waves can be realizedwithphysicaldevices.A
zeroth-order
band-limited Xwavewas produced and measured in water
by
our 10 element, 50 mm diameter, 2.5 MHz PZT ceramic/polymer
composite Jo Bessel nondiffracting annulararraytransducer
with -6-dB lateral and axial beam widths of about 4.7 mm and
0.65 mm, respectively, over a -6-dB depth of field of about 358
mm.Possible applications of X wavesin acoustic imaging and
electromagnetic energy transmission are discussed.
I. INTRODUCTION
I
N 1983, J. N.Brittingham [ l ] discovered the first localized
wave solution of the free-spacescalar wave equation and
called it a self focus wave mode. In 1985, anotherlocalized
wave solution was discovered by R. W. Ziolkowski [2] who
developed a procedure to construct new solutions 131 through
the Laplace transform. These localized solutions were further
studied by severalinvestigators[4]-[9].
An acousticdevice
was constructed to realize Ziolkowski’s Modified Power Spectrum pulse that is one of the solutions of the wave equation
through the Laplacetransform [lo].
In 1987 J. Durnindiscoveredindependently
a new exact
nondiffracting solution of the free-space scalar wave equation
[ l l]. This solution was expressed in continuous wave form and
was realized by optical experiments [12]. Durnin’s beamswere
further studied in optics in a number of papers [13]-[19]. Hsu
et al. [20] realized a J O Bessel beam with a narrow band PZT
ceramic ultrasonic transducer of nonuniform poling. We made
Manuscript received April 10, 1991; revisedandacceptedJune
17. 1991.
Thisworkwassupported
in part bygrant CA 43920 from the National
Institutes of Health.
of
Theauthorsarewith
the BiodynamicsResearchUnit,Department
Physiology and Biophysics,MayoClinicandFoundation,Rochester,
MN
55905
IEEE Log Kumber 9103583.
the first .IO Bessel nondiffracting annular array transducer 1211
with PZT ceramiclpolymercomposite and applied it to medical
acoustic imaging and tissue characterization [22]-1251. Campbell et al. had a similar idea to use an annular array to realize
a .Io Bessel nondiffracting beam and compared the .Io Bessel
beam to the Axicon [26].
In this paper, we report families of generalized nondiffracting solutions of the free-spacescalar wave equation, and
specifically. a subset of these nondiffracting solutions, which
we term X waves (so-called X waves because of the appearance of the field distribution in a planethrough the axis of
the beam).The X waves are a frequencyweightedLaplace
transform of the nondiffracting portion of the Axicon beam
equations [27]-1331 (in addition to the nondiffracting portion,
the Axicon beam equation contains a factor that is proportional
to fi that will go to infinity as 2 + x [31]) and travel
in free-space (or isotropiclhomogeneousmedia)
to infinite
distancewithout spreading provided that they areproduced
by an infinite aperture.
Even with finite apertures, X waves have a large depth of
field. (For example, with 25mm 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 h8 times the -6-dB lateral
beamwidthand7
times the radiatingdiameter.) We have
experimentally produced a zeroth-order band-limited X wave
[34]. We used our 10 element, 50 mm diameter, 2.5 MHz PZT
ceramic/polymer composite .J, Besselnondiffractingannular
array transducer [21], [22]. The X wave had -&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.
In comparison with Ziolkowski’s localized wave mode 121.
[3], the peakpulseamplitude
of the X wave is constantas
it propagates to infinity because of its nondiffracting nature.
Durnin’s beamsarenondiffracting at a single frequency but
will diffract (or spread) in the temporal direction for multiple
frequencies [22]. X waves are multiple frequency waves hut
theyarenondiffracting
in both transverseandaxialdirrctions.
Like the .l, Bessel nondiffracting beam 1211-1251, X waves
could be applied to acoustic imaging and tissue characterization. In addition, the space and time localization properties of
X waves will allow particle-like energy transmission through
large distances, as is the case with Ziolkowski’s localized wave
mode[7].
IEEE
TRANSACTIONS
20
ON ULTRASONICS,
FERROELECTRICS,
AND FREQUENCY
CONTROL,
VOL. 39, NO. 1, JANUARY 1992
The exact solution of the free space scalar wave equation,@L,
representsa familyof nondiffractingwavesbecause
if one
travels with the wave at the speed, c, i.e., z - ct =constant,
both the lateral and axial complex field patterns, al(r,4)and
@2(z - ct), will be the same for all time, t , and distance, z .
If c l ( k )C) in (6) is real, ''k" in (5) will represent backward
and forward propagating waves,respectively. (In the following
11. THEORYOF
WAVES AND TWO EXAMPLES
analysis, weconsider only the forwardpropagatingwaves.
All
results will be the same for the backward propagating waves.)
We will show belowthree families of generalized nonFurthermore, Q, S and @ ~ ( . s will
)
also represent families
)
diffracting solutions of the free space(or isotropic/homogeneous)
of nondiffracting waves if cl(k,C) is independent of IC and C,
scalar wave equation.
respectively. These waves will travel to infinity at the speed of
The free 'space scalar wave equation in cylindrical coordic1 without diffracting or spreading in either the lateral or axial
nates is given by
directions. This is different from Brittingham'sfocuswave
mode [l],and Ziolkowski's localized wave [2] and modified
power spectrum pulse [3] that contain both z - ct and z - dt,
or both t - ct and z ct terms in their variables, where c and
where r =
represents
radial
coordinate, 4 is c' are different constants.
azimuthal angle, z is axial axis, which is perpendicular to the
We find
S is the
most
interesting
solution. In the
plane defined by T and
t is time and @ represents acoustic following, we will1 discuss @ (S) only. @<(S)is a generalized
pressure or Hertz potential that is a function of T , 4. z , and t.
function that contains some 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 nondiffracting beams, and the
nondiffractingportion of theAxiconbeam in addition to an
infinity of newbeams.
If T ( k ) = S( k - k ' ) , where S(k - k ' ) is the Dirac-Delta
0
-71
functionand I;' =w/c > 0 is aconstant, f ( s ) = e", ao(k,<)
= - i a . b ( k . < ) = i a = i w / c l , from (2) and (5), one obtains
Durnin's nondiffracting beam [ 1l ]
In Section 11, we first introduce the families of generalized
nondiffracting waves and the nondiffracting X waves. Then,
weprovidetwo specific X wave examples. The production
of X waves with realizable finite aperture radiators and finite
bandwidths are reported in Section 111. Finally, in Sections IV
and V, we discuss further implications of these novel beams.
x
d
+
m
c(
a,
r
c
r
l
where
where
(9)
/j =
where
andwhere
iRe'"H
Q is a constant and
W is angular frequency. If A(8) =
, we obtain the nth-order nondiffracting Bessel beams:
QJ,<(S) = J,(nr)ei(~~L-"'+"O)
. ( n = 0. 1, 2,
where T ( k ) is any complex function (well behaved) of k and
couldinclude the temporalfrequencytransferfunction
of a
radiator system, A ( H ) is any complex function (well behaved)
of H and represents a weighting function of the integration with
respect to 8, f ( s ) is any complex function (well behaved)
of S, D(<) is any complex function(wellbehaved)
of C
and representsa weighting function of the integration with
respect to which could be the Axicon angle ((ll)),ao(k,C)
is any complex function of k and
b(k,<) is any complex
function of IC and
The term c is constant in (l), k and
arevariables that areindependent of thespatialposition,
,F' = (rcos&,r s i n 4 , z ) , and time, t, and @
'(t
- c t ) is any
complex function of t - ct. @ l ( r .4) is any solution of the
following transverse Laplace equation (an example of @ 1 ( r Q)
,
is given in the Appendix):
c,
c
c.
c,
' '
.).
(10)
If n = 0, one obtains theBesselnondiffracting
beam. From
(S), it is seen that cl(X.'. C) of the Durnin beams is equal to
U J / / ~and is dependent upon k'. Therefore, if T ( k ) in (2) is
a complex function containing multiple frequencies, @ < ( S )
will represent a dispersive wave and will diffract in the axial
direction, z . If both 71 and a arezero, (10) representsthe
planewave [ll].
If in (z), we let T ( k ) = d(k - V).f ( s ) = e', 4 8 ) =
aO(k,c)=-iksin(, b(k,<)= ikcosC, where k' =w/c > 0 is a
free parameter, and { represents the Axicon angle (we confine
0 < < 7r/2), we obtain the nth-order nondiffracting portion
of theAxicon beam [31]:
c
@A,(.$)
= .lTJ,,(/c'rsin<)e
l (k ' Z c o s < - d t + n q 5 )
, ( n = 0: 1. 2, ...).
(111
From (1 l), it is seen that (:I(k',<) = c/cos( is independent of k',
which means that the waves constructed from @*"(S) by (2)
21
LU AND GREENLEAF:NONDIFFRACTING X WAVES
will have a constant speedc1 for all frequency componentsand
will be nondispersive. The X waves derived in the following
are such waves that are nondiffracting or nonspreading in both
lateralandaxialdirections.
We obtain the new X wave if in (2), we let T ( k ) =
B ( k ) e P a o k ,A(B) = Pein@,ao(IE,<)= -&sin<, b ( k , C ) =
ikcos<, f ( s ) = e', obtaining
@dy,a
=
ein4
i
B ( ~ ) J , ( ~ Tsin < ) e - k [ a o - z ( z
cosc-c*)ldk,
0
( n = 0, 1, 2, ...).
zeroth-order nondiffracting X wave:
The behavior of the analytic envelope of the real part of
(17), R ~ [ @ ~ Bis B
shown
~ ] intheupperleft
panel of Fig. 1
or Fig. 2. The modulus of @ ' S B B ~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 fixed and ao,
at the plane z = ct/cos<, we have
<
(12)
This is an integration of the nth-order nondiffracting portion
((11))of the Axicon beam equation multiplied by B ( k ) e - a o R ,
where B ( k ) is any complex function(wellbehaved)
of k
andrepresentsatransferfunction
of apracticalradiator
system, k =w/c, a0 > 0 is aconstant,and
C is the Axiconangle
[31]. From the previous discussion, it is seen
that
is an exact
nondiffracting
solution
of the freespacescalar-waveequation
(1). Equation (12) shows that
@,yn is represented by a Laplace transform of thefunction,
B(k)J,(krsin<), and an azimuthal phase term, einm. Because
the waves represented by @*yn 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 sectionwith
two newnondiffractingX
wave examples. The first represents an X wave withbroad
bandwidth, while the second is a band-limited X wave.
Example 1: First, we describe an expression for an X wave
produced by infinite aperture and broad bandwidth.
If, in (12), B ( k ) = ao, we obtain from the Laplace transform
tables [35]
which representsthelateralpressuredistribution
of the X
wave through the pulse center and has an asymptotic behavior
proportional to 1/r when r is very large.
On the line T = 0, from (18), we obtain
which is the pressuredistribution of the X wavealong the
axial axis, z , and has an asymptotic behavior proportional to
l/ I z - s
t I when j z - +t I is very large.
The pressuredistribution
of the X wave along itstwo
branches (see Figs. 1 and 2 ) can be obtained when
sin
J'J,(cuk)e-"'
0
UOCP
dk = @ T 7 ( s
+d m ) "
(13)
cos C
cos
<
where
cos
C
-Iz--tI>l
cos
a0
33
a0
<
andfrom (18) we obtain
giving an nth-order broadband X wave solution (exact solution
of (l), see the Appendix):
QXBB,,
=
ao(r sin <)"ein+
r n ( T
+ m)"
, ( n = 0, 1, 2, ...) (14) whichhas
where the subscript\BB mean's "broadband," and
hl= ( r sin C)' + T 2
(15)
and where
For simplicity, we let n = 0, obtaining axially symmetric
an asymptotic behavior that is proportional to
1/4when I z - s t I is very large. This is the
direction of slowest descent of theX wave amplitude from
its center.
From (19), (20), and (23), it is seen that the smaller a0
is,thefaster
the function I @ a y ~1 ~
diminishes
o
as T or
Iz-s
t I increases. If sin is small, the diminution of
I @ d y ~I will
~ o be slow with respect to r and fast with respect
to I z - s t 1 . This means that if the X waves are applied to
<
IEEETRANSACTIONSONULTRASONICS.FERROELECTRICS.ANDFREQUENCYCONTROL.VOL.
22
3Y, NO. 1, JANUARY 1992
Fig. 1. Panel (a) represents the exact zeroth-order nondiffracting X wave solution of the free-space scalar wave equation. Panels (h),
(c). and (d) are the appropriate zeroth-order nondiffracting
X waves calculated from the Rayleigh-Sommerfeld diffraction integral
at distances : =30 mm, 90 mm, and 150 mm. respectively. The dimension of each panel
with a 25-mmdiameterplanarradiator
is 12.5 mm x 6.25mm. B ( k ) = r d o , C= 4" and i f , , = 0.05 mm.
Fig. 2. Same format as Fig.
1. except that a 50 mmdiameterplanarradiator
imaging, smaller no will give better resolutions in both lateral
and axial directions, while smaller sir1 will reduce the lateral
resolution while increasing the axial resolution.
At the center of the zeroth-order X wave ( T = 0. z = c t / COS
@ , ~ B B ~ l for all distance, z , andtime,
t (see (17)),
which is unlike Ziolkowski's localized wave [2] and modified
power spectrum pulses [3] that recover their pulse peak values
periodically or aperiodically with distance, z .
Example 2: Because B ( k ) in (12) is free, it may be chosen
to achieverealizablebandwidth
asshown in the following
<
c),
=
is used.
example. If B ( k ) is a Blackman window function [ 3 6 ] :
B(k)=
0.42 - 0.5 cos
0.
g + 0.08 cos
k0
1,
0
5 k 5 2ko
otherwise
We canapproximate
the bandwidth of a real transducer.
Although we cannot find an analyticexpression
for @Ay,z
fromusing (24) in (12) by directlylookingup
a Laplace
transform table, we will describe in the Section 111the resulting
nondiffracting beam calculated for some finite apertures.
23
LU AND GREENLEAF NONDIFFRACTING X WAVES
x WAVES WITH
FINITE APERTURE RADIATORS
111. REALIZATION
OF
The extension and therefore the aperture requirement of X
waves in space is infinite (see (12)). But we show that nearly
exact X waves can be realized with finite apertureradiators
over large axial distance. To calculate X waves from the finite
apertureradiators, a spectrum of X waves is used.Equation
(12) can be rewritten as (25) (see bottom of page), which is
an inverse Fourier transform of the function (spectrum of X
waves)
and F ' represents the inverse Fourier transform. The first and
second terms in (30) represent the contribution from high and
low frequency components, respectively.
Because B(w/c) in (28) is independent of r' and q5', (30)
can be rewritten in another form
6R,t(F,k)= D ( k ) G , z ( F . k ) .
(71.
= 0. 1. 2. ...)
(32)
where
0
where
0
(7t
= 0. 1. 2. ...)
(33)
and
is the Heavisidestepfunction[37].
At the plane z = 0, (26) becomes
where * denotes convolution on time, t. and F - ' [ B ( d / r )can
]
be taken as an impulse response of the radiator.
We now describe an example of an X wave produced by
a finite aperture. Let B ( k ) = 0.0, = 4O,(sin4" E 0.0698,
c o d " z 0.998), n o = 0.05 mm, 71 = 0, and = 1.5 mm//ss
(speed of sound in water). The analytic envelope of the real
part of @ R ( , . R P [ @ Rcalculated
~].
from (33) and (34) is given
in Fig. 1 and Fig. 2, when the diameter of the radiator, D , is
The spectrum in (28) is used to calculate X waves from 25 mm and 50 mm, respectively. The X waves are shown at
radiators with finite aperture. I f we assume that the radiator is z = 30mm, 90 mm, and 150 mm, respectively. These X wavcs
circular and has a diameter of D, the Rayleigh-Sommerfeld
compared well to theanalytic envelope of the real part of
formulation of diffraction by a plane screen [38]can be written the exact nondiffracting X wave solution, Rc[@-x-BB,,].
where
as
@ p s ~ ~is "given by (17) (see upper-left panel of Fig. 1 or 2).
?T
Di2
Fig. 3 shows beam plots of the 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 the axialdistances t =30 mm,
90 mm, and 150 mm. respectively, 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
( n = 0 , 1. 2 . ...)
(30) solution of the free-space scalar wave equation(real part of
(17)), respectively.Fig. 3(a4) represents the peak of the X
and
waves along the axial distance, z , from 6 mm to 400 mm. The
-6-dB
depths of field of the X waves are about 171 mm and
@ ~ , , ( ? .=F-'[m,,<
t)
. ( T I , = 0. 1. 2. ...) (31)
<,
348 mm when the radiatorhasdiameters of 25 mm and 50
where X is wavelength,rol
is the distancebetween
the mm, respectively. The -6-dB lateral beam widththroughout
observation point, F, and a point on the source plane ( r ' . 4'). the depth of field is about 2.5 mm.
where
<
( f
(?:)l
I
-....,
0' l
8'0
01
8'0
P'O
9'0
epnw6ew
1'0
0'0
2'0
00
~~ZI~UJJON
\
90
P'O
epnl!u6ew PeZ!plWON
I
0' l
8'0
90
P'O
epnl!u6eyy peZ!pUJJON
Z'O
0'0
L
I
I
1
25
LU AND GREENLEAF: NONDIFFRACTING X WAVES
Fig. 4. RayleighSommerfeld calculation of the zeroth-order nondiffracting X waves produced by a 25 mm diameter, band-limited
planar radiator at distances z =30 mm, 90 mm, and 150 mm. The radiator transfer function D ( k ) is a Blackman window function
2.88 MHz.Thedimension of eachpanel is 12.5 mm x 6.25 mm. C = 4'
peakedat 3.5 MHz witha-6-dBbandwidthabout
and uo =0.05 mm.
Figs. 3(b1) to 3(b3)areaxial beam plots of the X waves
with respect to c l t - z and correspond to Figs. 3(al) to 3(a3),
respectively. (c1 = c/cos<. For C = 4*, c1 z 1 . 0 0 2 ~The
) edge
waves produced by the sharp truncation of the pressure at the
edge of the radiator are clearly seen. These edge waves can
be overcome by using proper aperture apodization techniques
[39] andwillbediscussed
in the next section.The-6-dB
axial beam width obtained from Fig. 3(b) 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 canbedeterminedanalytically
in the following.For any givenangular frequency, W O , (30) representsa
continuous field of an aperture shaded by a Jo Bessel function
S(~O)JO(~O~)
(35)
where
and
and D =
which is independent of frequency W ( , ! If <= io
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-dB
lateralbeamwidth
of the X waves can be
calculatedfrom (19), which is
and the -6-dB axial beam width is obtained from (20)
r
cos {
21z--tI=-
2f l a o
cos
<
When uo = 0.05 mm and
= 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 azeroth-order
band-limitednondiffracting X wave produced by aradiator
of diameter of 25 mm and 50 mm, respectively, when D ( k )
is chosen as a Blackman window function(see (24)).The
parameter h.0 in (24) for these figures is given by
(37)
The depth of field of this Jo Besselbeam is determined by
P11
z,,, = ;/p)
2
- 1.
Substituting (37) into (38), we obtain
z,,,
D
= -cot
2
(39)
where c =1.5 mmIps, f o =3.5 MHz, andthe -6-dB bandwidth of the Blackman window function is about 2.88 MHz.
The X waves in Figs. 4 and 5 are a band-limitedversion of
those in Figs. 1 and 2. They are the finite aperture realization of
the convolution of function .F1[B(w/c)]ino with (14) when
n = 0, that is
(43)
IEEE TRAPiSACTlONS ON ULTRASONICS,FERROELECTRICS.ANDFREQUENCYCONTROL,
Fig. 5. Sameformat as Fig. 4. except that a 50 mm diameter planar radiator
with I I = 0: where B ( u / c ) (UJ 2 0 ) can be any complex
function that makes the integral in (25) converge, “*” denotes
the convolution with respect to time, f , and the subscript BL
represents“band-limited.”
The threepanels in Figs. 4 and 5 representband-limited
X waves at distance .z =30 mm, 90 mm, and 1.50 mm,
respectively. The parameters C and (L(, are the same as those in
Figs. 1 and 2, which are 4” and 0.05 mm, respectively. From
Figs. 4 and S , i t is seen that the band-limited X waves have
lower field amplitude in the X branches than the wideband X
waves in Figs. 1 and 2.
Fig. 6 is the same asFig. 3, except that it represents the
lateral and axial beamplots of the band-limited X waves in
Figs. 4 and 5. The--6-dB 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, atall three distances ( z =30
mm, 90 mm, and 150 mm).Therefore,theseband-limited
X
waves are also nondiffracting waves. Their -6-dB maximum
nondiffracting distances derived from Fig. 6 are 173 mm and
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 arealmost the same asthose of thebroad band
nondiffracting X waves obtainedfromFig.
3 and are very
close to thosecalculatedfrom (39).
VOL 39, NO. 1, JANUARY 1Y92
is used.
Rayleigh-Sommerfeld formulation of diffraction [38] can be
used to simulate the resulting beams.
Besides the X waves,thereare
many otherfamilies of
nondiffracting waves that are also exact solutions of the freespace scalar wave equation (see (3) and (4) and the Appendix).
B. Resolution, Depth of Field, and Sidelohes of X Waves
From (18), it is seen that as (I,
decreases, the X waves
diminish faster with both and 1 2 - (c/ cosCt) 1. and hence
they havehigher lateral and axial resolution. This requires
that the radiator system has a broader bandwidth because the
term exp{ -nou/c} in (26) will diminishslower for smaller
U ( ] . For larger values of sill C. the lateralresolution
will be
increased while the axial resolution is decreased. The -6-dB
lateral and axial beam widths of the zeroth-orderbroadband
nondiffracting X waves (( 17)) are determined by (40) and (41),
respectively.
For finite aperture, the X waves arenondiffracting only
within the depth of field. The depth of field of finite aperture
nondiffracting X waves is relatedonly to the size of the
radiator and the Axicon angle C ((39)).Bigger
values of
sill
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
lateral and axial beam widths are increased from those of the
broadband X waves, but the depth of field is about the same.
IV. DISCUSSION
The X waves have no or lower sidelobes in the ( 2 - q t )
plane than the Jo Bessel nondiffractingbeam [21], see Figs.
A. Other Nondiffructing 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 theradius r increases(seeFigs.
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 B ( k ) . waves present lower field amplitude in their X branches than
In practice, D ( k ) can be a transferfunction 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 branchescouldbesuppressed
or other wave sources and theirassociatedelectronics.
The dramatically by combining X wave transmitwith dynamic
I’
<
LU ANDGREENLEAF:UONIXFFRACTING
E
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AND
FREQUENCYCOKTROL.
28
spherical focused receiving as was proposed for the .l" Bessel
nondiffractingbeam [25].
C . Edge Wuves and Propugation Speed of the X Wu1,e.s
It is seen from Figs. 3(b) 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 windowfunctions to apodize the edge of the
aperture. (The results in Figs. 1 to 6 are obtainedwith
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 representthe beam plotsbefore and after the
apertureapodization,respectively.Theapodizationfunction
is given by (44) (see bottom of page), where 7 ' 1 = 3 0 / 8 , D is
the diameter of the radiator, and 7' is the lateral distance from
the center of the radiator. Panels (I), ( 2 ) , and (3) in both Figs.
7(a) and 7(b) represent the beam plots at distance z =30 mm,
90 mm, and 150 mm, respectively. Fig. 7(a4) plots the peak
value of the X wave along the axial axisfrom 2 =6 mm to
400 mm. The edge waves are
reducedaround 10 dB within
the depth of field by the apertureapodization. The 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 theaperture.
From (17). it is seen that the X waves travel with a speed
faster than sound or light speed, C (superluminal). For example.
if C = il", the X waves will travel about 0.2% faster than r .
This is also the case of Durnin's .To Bessel beam. From (IO),
for 71 =
we obtain
O 1
@J(,(?..Z
where
c1
-
= u,(\jand /j =
q t )=,
d
(45)
~ , ~ ( f ~ 7 . ~ ~ : / ~ ~ ~ - ~
m < k . Therefore. t' = ~ ? : i k
('1.
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 thesewavefronts
that construct the X wave
at large distances as the influence 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, c, but
the intersection, or peak, of these waves travels greater than
thespeed of sound.
D.Total Energy, Energy Density and Causulit):
Like the plane wave and Durnin's beam. the totalenergy
of an X wave is infinite. But, the energy density of all these
waves is finite. Approximate X waves are realizable with finite
apertures and with finite energy over a deep depth of field.
VOL. 39. NO. 1, JANUARY 1992
From (17), it is seen that at a given distance 2 , the X waves
extend from t> - x to f < x.Therefore, they are not causal,
but the X waves produced by fininte aperturediminish very
fast as 1 t j +x.If we ignore the X waves for ' f 1 > to, where
I @-x-, (I:: t o ) I<< 1. these modified X waves can be treated as
causal waves. Similarly: Ziolkowski's localized wave [2], and
modified power spectrumwave (31 suffer from the problem
of causality, but they are closelyapproximatedwithfinite
aperture by using this same causal approximation [lo].
We do not worryabout the superluminal solutions,noncausal solutions, infinite total energy or infinite apertures
as longas the pressuredistributionsover
an apertureare
sufficiently well behaved that they can be physically realized
and truncated in time and space and still produce a wave of
practicalusefulness.Theoretical
X waves are superluminal,
noncausal, and have infinite total energyand apertures, but
they can be truncated to produce practical waves. One example
was given in reference [34], where a zcroth-order X wave was
physically realized over a deep depth of field in an experiment
using an acoustic annular array transducer 1211, [22].
E. Application of X LVuves to Acoustic Imaging
and Electromagnetic Energy Transmission
The real part of (17), R r [ @ . ~ ~ ~has
, , ] a, smooth phase
change over time, t , across the transversedirection, 1 ' . This
makes it possible to realize X waves with physicaldevices.
Therefore, a broadbandacoustictransducercould
be used
to produce X waves in Figs. 1 to 3, while a band-limited
transducer could generate those in Figs. 4 to 6. The electrodes
of an acousticbroadbandPZT
ceramic/polymer composite
transducer can be cut into annular elements 1211 and each
elementdrivenwith
a properwaveform depending upon its
radial position. Annular array transducers can be used for both
.X] wave
~ ~ .transmitting
and conventional dynamically spherical
focusedreceiving
[34]. The low sidelobes of a Gaussian
shaded receiving would suppress the off-center X wave energy
and produce high resolution, high frame rate, and large depth
of field acoustic images.
The X wave solution to the free-space scalar wave equation
can also be applied to electromagnetic energy transmission for
privatecommunication and militaryapplications [7]. (Nondiffracting electromagnetic X waves are discussed in detail in
[ W )
V. CONCLUSION
We have developed a new family of nondiffracting waves
that provides a novel way to produce focused beams without
lenses.Theyarerealizablewith
finite apertures and with
broadband and band-limited planar radiators with deep depth
of field. The energy on the X branches of the X waves could
be suppresseddramatically in imaging by combining the X
LU 4 N D GREENLEAF:NONDIFFRACTING X W.4VES
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ULTRASONICS.
ON
FERROELECTRICS,
31)
waves with a conventionaldynamicallysphericalfocused
receiving system to produce high contrast images [22]. Characterization 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 particlelike energy transmission through large distances, as is the case
of Ziolkowski's localized wave mode [ 7 ] .
APPENDIX
4ND FREQUENCY
CONTROL,
Now we prove
j
( S~) in (4) is also an exact solution of ( 1 ) :
Because Q,l(r.cb) is an exact solution of the transverse Laplace
equation ((7)),the right-hand side of (54) is zero. This proves
that @ L is an exactsolution of the free-space scalar wave
equation (1).
For example. if we choose @ 2 ( ; ; - c+) as a Gaussian pulse
Theorem: Functions @ < ( . S ) , @ K ( s ) and
. @I,(.%)
are exact
solutions of the free-space scalar-wave equation ( 1 ).
Proof: The functions @ < ( . S ) and @ ~ ( s are
) linear summations of function f ( s ) overfreeparameters
L. and <,
respectively, see (2) and (3). Therefore, if we can prove f ( . s ) and
isan exactsolution of ( l ) , Q, - ( ) and Q,K( S ) will also be
exact solutions. (For example. one can directly prove that (14)
is an exact solution of the free-space scalar wave (1). Using
(47)
d
VOL. 39. NO. I . JANUARY 1092
Q(
Q,,(/,,cj)
-
(.t)=
(,-(:-Ct:F/fl?
(5s)
as Poisson's formula (41, p. 3731
and f ( + ) is any function (well-behaved) of {), and (7 and a are
constants, the Gaussian pulsc w i t h a transverse field pattern
Q ~ ( / , , t b )will travel to infinity at thespeed
of sound or light
without any change of pulse shape.
The transverse field ((56)) has the followingpropcrties:
ACKNOWLEDGMENT
The authors appreciated the secretarial assistance of Elaine
(49)
C. Quarve and the graphicassistance of Christine A . Welch.
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+
/,'(h..
1
- ?/,?(h..
<)(,;(h..
c-
,
(53)
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side
of (53) is zero. Therefore,
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cb((s) and
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X WAVES
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their
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[I61 K. Uehara and H.Kikuchi,“Generation
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[l91 S. Y.Cai,A.Bhattacharjee,
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Jian-yu 1.u (h1’88) wa\ born in Fuzhou,Fujian
Province.People’sRepublic
of China, o n August
20. 1959. He received the B.S. degree in electricalengineering
in 1982 fromFudan
Univer\it!.
Shanghai,China, the M.S. degree in 198.5 from
Tongji University,Shanghai,China,and
the Ph.D.
degree i n 1Y88 from Southeast University, h’anjing.
China.
He is a n Assistant Professor o f Biophysics, Mayo
a ResearchAssociate, at the
MedicalSchool.and
Biodynamics Research Unit, Dcpartment of Physlologq and Biophysics. Mayo Clinic and Foundation. Rochc.;ter, MN. From I Y X X
to lY90, he was a po\tdoctoralResearchFellowthere.Prior
t o that, he was
afacultymember o f the Department of Biclmedical Engineering,Southeast
w i t h Prof. Yu Wei. Hisrcscarchinterest\are
in
Univenity,andworked
acoustical imaging and tissue characterization. medical ultrasonic tranduccrs,
and nondiffractingwavetransmission.
Dr. Lu is a member of the IEEE UFFC Society. the American Institute o l
Ultrasound in Medicine, and Sigma Xi.
James F. Greenleaf (21’73-SM’X4-F’88) \\:IS born
In SaltLake City, UT, on February IO, 1942. He
received the B.S. degree in electricalengineering
in 1964 from the University o f Utah. SaltLake
in
City. the M.S.degree in engineeringscience
1968 from Purdue University. Lafayette, IN, and the
Ph.D. degree in engineuing \cience from the Mayo
Graduate School of Medicine. Rochester, MN, and
PurdueUniversity in 1070.
He is aProfessor of Biophysics and Medicine.
Mayo Medical School, and Consultant,Biodynamics Research Unit. Department of Physiology, Biophysics, and Cardiovascular
DiseaseandMedicine,MayoFoundation.
Dr. Greenleaf has served o n the IEEE Technical Committee o f the Ultrasonics Symposium since 1985. He served o n the IEEE-UFFC Suhcommittec
for the Ultrawnics in Medicine!IEEE MeasurementGuidc Editors. and on
the IEEE Medical Ultrasound Committee. He
i i Vice President o f the UFFC
is the recipient of the 1986 J. Holmes
Society.Heholdsfirepatentsand
Pioncerawardform the AmericanInstitute of Ultrasound in Mcdicine. and
isaFellow
of the IEEEand the AIUM. He is the Di\tinguishedLecturcr
for the IEEEUFFCSocietyfor
I99(&199l. Hisspecial field of interest is
i n ultrasonicbiomedicalimagingscience
and has published more than 150
article\and edited four hooks.