Acoustic Bessel-like beam formation by an axisymmetric grating

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by  N. Jiménez

Acoustic Bessel-like beam formation by an axisymmetric grating
N. Jim´enez,1 V. Romero-Garc´ıa,2 R. Pic´o,1 A. Cebrecos,1 V.J.
S´anchez-Morcillo,1 L.M. Garcia-Raffi,3 J.V. S´anchez-P´erez,4 and K. Staliunas5
1Instituto de Investigacio´n para la Gestio´n Integrada de Zonas Costeras,
Universitat Polit`ecnica de Val`encia, Paranimf 1, 46730 Grao de Gandia, Spain
2LUNAM Universit´e, Universit´e du Maine, CNRS,
LAUM UMR 6613, Av. O. Messiaen, 72085 Le Mans, France
3IUMPA, Universitat Polit`ecnica de Val`encia, Cam´ı de Vera s/n, 46022 Valencia, Spain
4Centro de Tecnolog´ıas F´ısicas: Acu´stica, Materiales y Astrof´ısica,
Universitat Polit`ecnica de Val`encia, Cam´ı de Vera s/n, 46022 Valencia, Spain
5ICREA, Departament de F´ısica i Enginyeria Nuclear,
4 Universitat Polit`ecnica de Catalunya, Colom, 11, E-08222 Terrassa, Barcelona, Spain
1
0 We report Bessel-like beam formation of acoustic waves by means of an axisymmetric grating of
2 rigid tori. The results show that the generated beam pattern is similar to that of Bessel beams,
characterizedbyelongatednon-diffractingfocalspots. Amultiplefocistructureisobserved,dueto
n
the finite size of the lens. The dependence of the focal distance on the frequency is also discussed,
a
J on the basis of an extended grating theory. Experimental validation of acoustic Bessel-like beam
formationisalsoreportedforsoundwaves. Theresultscanbegeneralizedtowavebeamsofdifferent
7
nature, as optical or matter waves.
2

PACSnumbers: 43.20.Mv,43.20.Hq,43.20.Fn
]
i
c
s
- Besselbeams,originallyproposedinoptics1,2,arenow (a)
l at the basis of many applications due to their unusual
r
t propagationproperties3–8. Themostcelebratedproperty
m
ofaBesselbeamisthat, intheidealcase, thefieldprop-
.
t agates invariantly, i.e. without any diffracting broaden-
a
ing,incontrasttotheothercanonicalcase,theGaussian
m
beam, where the beam experiences diffractive broaden-
- inginfreespacepropagation. Asaconsequence,thefield
d
n patterninaBesselbeampossessesaninfinitelyextended
o focal line. (b) M
c
Strictlyspeaking,Besselbeamisasolutionofthewave
[ m=3
equation in the form of a monochromatic wave with a
m=2
1 transverse profile given by a Bessel function of the first
m=1
v kind, which by definition presents an infinite spatial ex-
9
tension. Thisidealcasecannotberealizedinpractice(in
6
the same way as ideal, infinitely extended plane waves
7
cannotexist). However,approximateorimperfectBessel
6
. beams of finite transverse extent can be excited by dif- a
1
ferentmeans,displayingnotaninfinitebutanextremely
0 n=2 n=1
elongated focal line.
4
f(r)| f(r)|
1 In optics, Bessel-like beams are usually formed by fo- n=2 n=1
: cusingaGaussianbeambyanaxicon9, atransparentre-
v
i fractive element of conical shape, as shown in Fig. 1(a). FIG. 1. (Color online) (a) Illustration of the formation of
X
The beam in propagation through the axicon acquires BesselbeambyanaxiconresultinginimperfectBesselbeam
r linearly tilted (conical) wave-fronts, which results in an showing a focus-line of finite extent; (b) Illustration of the
a
elongated focus behind the axicon. As the axicon is formationofBessel-likebeamsbyaplaneofconcentricrings,
not infinitely extended in transverse space, the result- whereconvergingdiffractedwavesresultintwoelongatedfoci.
ing Bessel beam is not perfect, and displays a focal line
of finite extent. Optical Bessel beams have been also
obtained by acoustic gradient index lenses10. In electro- entmaterials,orforetchingofdeepnarrowholesinlaser
magnetism, Bessel-like beams have been generated from manufacturing of opaque materials, among others3,4,6.
a subwavelength aperture by adding a metallic circular In acoustics, Bessel beams of sound waves were also
grating structure in front of the aperture11. Such im- reported7,12, however are still not so broadly applied
perfect Bessel beams find multiple applications, e.g. in as in optics, which is perhaps related with the lack
opticsforlaserinscriptionofpatternsdeepintotranspar- of convenient techniques of formation of such kind of
2
4 4
(a) (b)
a/λncy ()3.35 a/λncy ()3.35
e e
qu2.5 qu2.5
e e
d fr 2 d fr 2
e e
aliz1.5 aliz1.5
m m
or (c) or
n 1 n 1
0.5 0.5
0 10 20 30 40 −1 −0.5 0 0.5 1
axial position (z/λ) sin (α)
15
(c) 1 (i) Rings 1 (d)
0.8 Bessel
r/λon () 105 (i) p/pmax000...246 y 0.8 B
siti 0 0−4−3−2−1 0 1 2 3 4 nsit 0.6
o axial position (r/λ) e
al p −5 Int 0.4
di
ra−10 0.2
−15 0
0 5 10 15 20 25 30 35 40 45 0 10 20 30 40 50
axial position (z/λ) z / λ
FIG. 2. (Color online) (a) Map of acoustic pressure along the symmetry axis (horizontal axis) for varying frequency (vertical
axis). Analytical estimations from Eq. (1) of focal line extent from f to f are shown by solid white lines. (b) Map of
n1 nM
far field in angular direction (horizontal axis) for varying frequency (vertical axis), where analytical estimations are shown in
dashedblacklines. (c)2Dpressuredistributionforcaseofsinglefocus(a=1.03λ)onacrosssectionalongthesymmetryaxis.
(i) Inset shows axial cross-section of the pressure field at a distance z = 5λ (solid black) compared with Bessel beam (solid
gray). (d) On-axis intensity for the cases a/λ=1.1717 (red line) and a/λ=1.6313 (blue line). Dashed line is an eye-guide to
show the linear dependence.
acoustic waves. Acoustic Bessel beams have been ex- culations (using finite difference time domain (FDTD)
cited using acoustical axicons8, in analogy to the op- techniques18)ofacousticwavespropagatingthroughsuch
tical case. However the most convenient way to form axisymmetricgratingswereusedtoobservethecomplete
acoustic Bessel beams is by using annular transducer acoustic field. Finally, the experimental verification of
arrays13. Related theoretical studies include the scatter- Bessel-like beam formation by an axisymmetric grating
ing of Bessel beams by spheres14, nondiffracting bulk- is reported.
acoustic X waves15 or non linearly generated Bessel
beams of higher harmonics16,17. Each element of the concentric ring structure is char-
acterized by two parameters, (i) the toroidal radius, r
m
The present work proposes and demonstrates a tech- and (ii) the radius of the tube (circular section), R .
m
nique for acoustic Bessel-like beam formation using a The rings in the axisymmetric grating have increasing
planar structure made of of concentric tori of circular toroidal radii as r =ma, where a is the separation be-
m
section, called here rings for simplicity. We show that, tween rings and m is an element index, as shown in Fig.
under specific conditions, part of a diffracted wave col- 1(b). Thecontinuityofthetransversalcomponentofthe
limates, producing an elongated focus. Moreover, dif- wavevectorattheinterfacebetweenthefreepropagation
ferent diffraction orders can result in different elongated medium and a linear diffraction grating with periodicity
foci, as illustrated in Fig. 1(b). In the present work a, results in diffraction of normal incident plane waves
we demonstrate the feasibility of this idea by analytical at diffraction angles given by sinα = nλ/a where λ is
n
estimations, numerical simulations and experiments. A thewavelengthandnisthediffractionorder. Itisworth
simpleanalyticalmodelbasedonanapproachofaxisym- notingthatapproximatelyhalfpartofthediffractedradi-
metric diffraction gratings is used to estimate the focal ationconvergestowardsthesymmetryaxisandtheother
positions and the extent of the focal line. Numerical cal- half diverge. Resulting from the converging radiation, as
3
it follows from simple trigonometry considerations, each
ring with major radius r is mapped to a particular dis- (a)
m
tance along the symmetry axis, given by
(cid:115)
r a (cid:18)nλ(cid:19)2
f (r )=f = m 1− . (1)
n m nm nλ a
Ifthesystemofconcentricringsextendsfromr (toroidal
1
radius of inner ring) to r (radius of outer ring) in the
M
transverse plane, the focal line for the n-th diffraction
order will extend approximately from f to f . In a
n1 nM
limitingcaseofinfinitelyextendedringstructure(r =0,
1
r = ∞) Eq. (1) predicts an infinitely extended focus,
M
similarly to that of an ideal Bessel beam. 2
(b)
First,weperformnumericalsimulationsinordertoex-
plorethecharacteroftheelongatedfocalline(alsodiffer- a/λ) 1.8
ent foci) due to the axisymmetric diffraction. For that, n ( 1.6
o
we use a structure composed by a set of 50 concentric ati
rings with constant minor radius R = a/3, irradiated ar 1.4
p
by a plane wave. We notice that this structure is much se 1.2
larger than that used in experiments (detailed below), ng
therefore, the Bessel-like features of focal line are more ri 1
pronounced. For the numerical simulations we calculate
0.8
the wave propagation using the FDTD technique. Fig. 0 2 4 6 8 10
2(a) and 2(b) represent the frequency dependence of the axial position (z/a)
on-axis amplitude and radial far field amplitude respec-
tively. Color map represents the amplitude of the acous- 2
(c)
tic field, |p|, and continuous lines in Fig. 2(a) show the
λ) 1.8
predictionsfromEq. (1)forthecasesm=1andm=50 a/
for the first three diffraction orders, n = 1,2,3. The n ( 1.6
o
focal spots appear at normalized frequencies a/λ = n, ati
so thediffraction anglesat these frequenciescorresponds ar 1.4
p
e
to αn =π/2 (sinαn =1). With increasing frequency, as g s 1.2
showninFig. 2(a),focielongatecorrespondingtoanalyt- n
ri 1
icalestimationsbyEq. (1)(whitelinesinFig. 2(a)). The
diffractionangledecreaseswiththeincreasingfrequency. 0.8
Concerning the angular far field distribution shown in 0 2 4 6 8 10
Fig. 2(b), excellent agreement between theory and nu- axial position (z/a)
merical results are observed. Different maxima appear
duetodifferentdiffractionorder,asaconsequenceofthe FIG. 3. (Color online) (a) Experimental set-up. (b) Numeri-
callycalculatedand(c)experimentallymeasuredmapsofon-
focusing effect at near field due to the finite size of the
axis amplitude dependence for varying frequency (vertical).
structure, in agreement with Fig. 2(a).
Solid white lines indicate the extend of the focal line from
Next,weanalyzetheacousticfieldbehindtheaxisym-
analytical estimations using Eq. (1) for f (r ) and f (r ).
metric grating for a particular frequency corresponding 1 1 1 7
Dashed white lines represent the frequencies shown in Fig. 4
to a = 1.033λ (see horizontal dashed line in Fig. 2(a)).
We study here the case when only one focal line ap-
pears. Fig. 2(c) shows the field distribution on axial
cross-section. The focus is substantially long, which is a with a linear trend overlaid of stray oscillations from
signatureofaBessel-likebeam. Thelatterisprovenand fn1 to fnM, as shown in Fig. 2(d). These oscillations
illustrated in the inset of Fig. 2(c), where the transverse are mainly due to the finite size of the structure: the
field profiles at the indicated distance behind the ring edges of the diffraction grating result in fringes. We also
structure is plotted and compared with Bessel function. note, thatalthoughthefocallinesassociatedwithdiffer-
Finally, Fig. 2(d) shows the intensity distribution entdiffractionorderspartiallyoverlap, theradiationdue
alongthesymmetryaxisfortwodifferentfrequencies: im- tohigherorderdiffractionclearlydominatesintheinter-
portant to note that the amplitude of the field increases ferencepicture. Thelineardependenceofacousticinten-
with distance until a maximum focusing distance, where sity along the focus is however modified if the thickness
the amplitude drops. The scattered energy is propor- oftheringsdependsontheirradius(seetheexperiments
tional to the area between neighbouring rings. Hence- below).
forth, the acoustic intensity along the focus increases For the experimental validation we designed a system
4
−5 −5
10 kHz 16 kHz
m) m)
c 0 c 0
y ( y (
(a) (b)
5 5
5 10 kHz 5 16 kHz
m) m)
c 0 c 0
y ( y (
(c) (d)
−5 −5
10 20 30 40 50 10 20 30 40 50
z (cm) z (cm)
5 z=9 cm z=14 cm z=29 cm 5 z=14 cm z=29 cm z=44 cm
m) m)
c 0 c 0
y ( y (
−5 (e) (f) (g) −5 (h) (i) (j)
−5 0 5 −5 0 5 −5 0 5 −5 0 5 −5 0 5 −5 0 5
x (cm) x (cm) x (cm) x (cm) x (cm) x (cm)
FIG.4. (Coloronline)(a-d)Amplitudedistributionsinplanesalongthesymmetryaxisr(z,x)obtainedbynumericalsimulations
(a,b)andexperimental(c,d)measurementsfortwodifferentfrequencies,showingtheformationofelongatedfoci. Experimental
transversalamplitudecross-sections(x,y)atdifferentdistancesbehindtheringstructure,showingtheformationoftheBessel-
likebeam. (e-g)correspondtotheexperimentaltransversalplanesat10kHzmarkedwithdashedlinesin(c). (h-j)correspond
to the experimental transversal planes at 16 kHz marked with dashed lines in (d).
composed by a set of 7 concentric rigid rings embedded We have also measured field cross-sections: an hori-
inair. Theringsaremadeofmethylmethacrylate(plex- zontal plane containing the symmetry axis (see Figs. 4
iglass), which acoustic impedance is much larger than (a-d)) as well as several axial cross-sections (see Figs. 4
that of the air (Z /Z ∼ 6000); therefore, they (e-j)). We focused on two particular frequencies, 10 kHz
plexiglass air
can be considered acoustically rigid. The variation of (a/λ = 1.17) and 16 kHz (a/λ = 1.88) (indicated by
the toroidal radius in the structure is r = ma, where white dashed-dotted horizontal lines in Figs. 3(b) and
m
a = 4 cm. The minor radius now is a function of the 3(c)). For the analyzed structure the frequency of 10
major radius, R(r ), following a hyperbolic secant pro- kHz is close to the condition a/λ = 1, consequently, the
m
filelikeintheaxisymmetricgradientindexlensanalyzed first diffraction spot appears just behind the structure
in Ref. [19]. This profile modifies the linear dependence (see white dashed line in Fig. 3(c) as a reference). How-
of the intensity along the focus, however also eliminates ever, the frequency of 16 kHz produces the elongated fo-
the sharp drop of intensity at the end of the focal line, cus or Bessel-like beam. These two phenomena are both
which results in a more smooth and better reproducible numerically and experimentally shown in Figs. 4(a)-(c)
diffraction pattern. All measurements were performed and Figs. 4(b)-(d) with good agreement.
in anechoic chamber to avoid unwanted reflections. The Inordertoseethesymmetryqualityofthebeamspro-
source was a loudspeaker radiating a sound wave with a duced experimentally, we have also measured the axial
whitenoisespectrumplacedat1.5minfrontoftheplane cross-sectionsofthepressurefieldatdifferentz-positions
ofconcentricrings,asufficientdistanceinordertoensure forabovediscussedcasesof10kHzand16kHz(Fig. 4(e)-
thatanearlyplanewaveradiatesthestructureatthefre- (j)). In both cases (i) the diffracted pattern is highly
quencies of interest. A movable microphone located be- axisymmetric and (ii) the diffracting broadening of the
hind the structure recorded the transmission spectrum. central beam, along the extended focus, is almost negli-
The experimental set-up is shown in Fig. 3(a). gible. The transversal profiles in Figs. 4(e)-(j) also illus-
ThequantitativestudyofBessel-likebeamformationis trate the typical shape of the truncated Bessel function.
summarized in Figs. 3(b) and 3(c). Pressure color maps It is simple to predict the tendencies of the amplitude
of Figs. 3(b) and 3(c) show the numerical and exper- distribution along the focus for such a small number of
imental frequency dependence of the on-axis amplitude rings. The longitudinal shape along the elongated focus
produced by the used structure. In the experiment, we seemstobenotlinearlyincreasing/sharplydropping,but
were able to measure frequencies until 20 kHz (a <2.3), correspondinglysmoother. Thisisduetothesmallnum-
λ
i.e. we could achieve Bessel beam formation by the first berofringsinthestructure; alsoduetothedifferencein
diffraction order only. The expected dependence of the thickness of the rings.
focaldistanceonfrequency(comparewithFig. 2)isalso Concluding, we have demonstrated the principle of
evident in both numerical and experimental plots. Bessel-like focusing in a system of concentric rigid rings.
5
Although the size of experimental system was reduced (makingtheradiiincrementingnotlinearly)whichisan-
(to 7 rings), the main properties of the Bessel beam for- otherparameterallowingtheengineeringofthefocusing,
mation were demonstrated: the elongated foci along the and allowing to optimize the focal structure; finally (iv)
symmetry axis, the Bessel-like distributions of the field by using multiple layers of rings at equidistant separa-
in axial cross-sections, and the expected dependence of tions along the symmetry axis one could not only en-
the focal distance with the frequency. The Bessel beam hancetheeffect,butalsointroduceanotherpossibilityof
formationcanbesubstantiallyimproved, asfollowsfrom tailoringthefocalspot,astheinterferencefromdifferent
numerical calculations with larger number of rings. planes will come into play.
For technical applications, the Bessel beam forma-
tion can be modified and improved, according to specific
needs, by some means as for example: (i) the use of not ACKNOWLEDGMENTS
toroidal rings, but rather the ones with more sophisti-
cated shapes, which would favour the converging part of The work was supported by Spanish Ministry of
the diffracted wave (i.e. to convert into Bessel beams Science and Innovation and European Union FEDER
nearly all radiation (note that the rings can converge as through projects FIS2011-29731-C02-01 and -02, also
maximum the half of initial radiation); (ii) the use of a MAT2009-09438, MTM2012-36740-C02-02 and UPV-
ring structure of adequate thickness which would allow PAID 2012/253.. VRG acknowledges financial support
tailoring the longitudinal profile of the focus, according from the “Pays de la Loire”through the post-doctoral
totherequirements;(iii)modifyingtheradiioftherings programme.
1 J. Durnin, J. Opt. Soc. Am. A 4, 651 (1987). Lett. 102, 143901 (2009).
2 J. Maddox, Nature 327, 183 (1987). 12 J. Lu and F. Greenleaf, IEEE Trans. Ultrason. Ferroelec.
3 M. Duocastella and C. Arnold, Laser Photonics Rev. 6, Freq. Contr. 37, 438 (1990).
607 (2012). 13 H.Masuyama,T.Yokoyama,K.Nagai, andK.Mizutani,
4 F. Fahrbach and A. Rohrbach, Nat. Commun. 3, 632 Jpn. J. Appl. Phys. 38, 3080 (1999).
(2011). 14 P. L. Marston, J. Acous. Soc. Am. 121, 753 (2007).
5 X. Chu, Eur. Phys. J. D 66, 259 (2012). 15 J.Salo,J.Fagerholm,A.T.Friberg, andM.M.Salomaa,
6 Y. Matsupka, Y. Kizuka, and T. Inoue, Appl. Phys. A- Phys. Rev. Lett. 83, 1171 (1999).
Mater. 84, 423 (2006). 16 D.DingandJ.y.Lu,J.Acous.Soc.Am.107,1212(2000).
7 D. Hsu, F. Margetan, and D. O. Thompson, Appl. Phys. 17 K. B. Cunningham and M. F. Hamilton, J. Acous. Soc.
Lett. 55, 2066 (1989). Am. 108, 519 (2000).
8 P. Katchadjian, C. Desimone, and A. Garcia, AIP Conf. 18 A. Taflove and S. Hagness, Computational Electrodynam-
Proc. 1211, 1043 (2010). ics: The Finite-Difference Time-Domain Method (Artech
9 J. H. McLeod, J. Opt. Soc. Am. 44, 592 (1954). House, Boston, 2000).
10 E. McLeod, A. B. Hopkins, and C. B. Arnold, Opt. Lett. 19 V. Romero-Garc´ıa, A. Cebrecos, R. Pico´, V. Sa´nchez-
31, 3155 (2006). Morcillo, L. Garcia-Raffi, and J. Sa´nchez-P´erez., App.
11 Z.Li,K.B.Alici,H.Caglayan, andE.Ozbay,Phys.Rev. Phys. Lett. 103, 264106 (2014).