We propose and demonstrate experimentally the concept of the annular
interdigital transducer that focuses acoustic waves on the surface of a
piezoelectric material to a single, diffraction-limited, spot. The shape of the
transducing fingers follows the wave surface. Experiments conducted on lithium
niobate substrates evidence that the generated surface waves converge to the
center of the transducer, producing a spot that shows a large concentration of
acoustic energy. This concept is of practical significance to design new
intense microacoustic sources, for instance for enhanced acouto-optical
interactions

Abdelkrim Khelif

Carlos F. Jerez-Hanckes

Davy Gérard

Naima Khelfaoui

Royer D.

Sarah Benchabane

Vincent Laude

English

arXiv

International audienceWe propose and demonstrate experimentally the concept of the annular interdigital transducer that focuses acoustic waves on the surface of a piezoelectric material to a single, diffraction-limited, spot. The shape of the transducing fingers follows the wave surface. Experiments conducted on lithium niobate substrates evidence that the generated surface waves converge to the center of the transducer, producing a spot that shows a large concentration of acoustic energy. This concept is of practical significance to design new intense microacoustic sources, for instance for enhanced acouto-optical interactions

Laude, Vincent

Gérard, Davy

Khelfaoui, Naima

Jerez-Hanckes, Carlos, F.

Benchabane, Sarah

Khelif, Abdelkrim

HAL: Hyper Article en Ligne

Subwavelength focusing of surface acoustic waves generated by an annular interdigital transducer

HAL - Université de Franche-Comté

Subwavelength focusing of surface acoustic wavesgenerated by an annular interdigital transducerVincent Laude, Davy Ge´rard, Naima Khelfaoui, Carlos Jerez-Hanckes, SarahBenchabane, Abdelkrim KhelifTo cite this version:Vincent Laude, Davy Ge´rard, Naima Khelfaoui, Carlos Jerez-Hanckes, Sarah Benchabane,et al.. Subwavelength focusing of surface acoustic waves generated by an annular interdigi-tal transducer. Applied Physics Letters, American Institute of Physics, 2008, 92, pp.094104.<10.1063/1.2891055>. <hal-00201067>HAL Id: hal-00201067https://hal.archives-ouvertes.fr/hal-00201067Submitted on 23 Dec 2007HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.hal-00201067, version 1 - 23 Dec 2007Annular interdigital transducer focuses piezoelectric surface waves to a single pointVincent Laude, Davy Ge´rard∗, Naima Khelfaoui, Carlos F. Jerez-Hanckes†, Sarah Benchabane, and Abdelkrim KhelifInstitut FEMTO-ST, Universite´ de Franche-Comte´, CNRS, F-25044 Besanc¸on, FranceWe propose and demonstrate experimentally the concept of the annular interdigital transducerthat focuses acoustic waves on the surface of a piezoelectric material to a single, diffraction-limited,spot. The shape of the transducing fingers follows the wave surface. Experiments conducted onlithium niobate substrates evidence that the generated surface waves converge to the center of thetransducer, producing a spot that shows a large concentration of acoustic energy. This conceptis of practical significance to design new intense microacoustic sources, for instance for enhancedacouto-optical interactions.A variety of anisotropic propagation phenomena areobserved in solids, that are mainly determined by thesymmetries of the rank-four elastic constant tensor [1].The piezoelectric effect offers a convenient means of gen-erating elastic waves, by transforming electrical to me-chanical energy. The wide majority of elastic wave ex-periments in piezoelectric solids employ planar or inter-digitated transducers (IDT) emitting plane waves withwell-defined wave vectors [2]. In such a situation, a dis-placement field mimicking an ideal plane wave is oftenlooked for and propagation is adequately described byphase quantities, such as the phase velocity. In nonpiezoelectric materials, surface acoustic waves can be ex-cited using a focused laser beam [3]. In many instances,the surface can be considered to be excited at a singlesource point. In this case, the surface supports outgo-ing waves that depart from the source and propagationin the far field depends on the angular dispersion of thespatial group velocity. At a distance exceding a few wave-lengths, the formed ripple pattern has the shape of thewave surface, obtained by plotting the group velocity asa function of the emission angle. This effect was observedvery neatly for surface waves in the experiments of Vineset al. [4] and Sugarawa et al. [5]. Here, we investigate theconverse problem: can we construct an extended sourceusing an appropriate IDT that will focus elastic energyto a single point on the surface of a piezoelectric solid?A circular IDT has already been proposed by Dayand Koerber [6] for substrates with in-plane isotropy (or-thotropy). Since phase and group velocities are equal inthis case, the wave surface reduces to a simple circle. Amore recent alternative, the so-called focused interdigitaltransducer (FIDT), makes use of acoustic emission onlywithin a circular arc [7, 8]. In this case, the acoustic fieldis strongly dependent on the angular extent of the arcand on the focal length of the transducer. Another limitcomes from the anisotropy of the substrate which leads∗D. Ge´rard is now with the Institut Fresnel, Domaine Universitairede Saint Je´roˆme, F-13397 Marseille cedex 20, France.†C. F. Jerez-Hanckes is also with the Centre de Mathe´matiques Ap-plique´es, CNRS, Ecole polytechnique, F-91128 Palaiseau, France,and EPCOS Sophia Design Center, 1300 Route des Creˆtes, F-06560, Sophia-Antipolis, France. 0 3 0  3Slowness (10-4 s/m) 0 0.5 1 1.5 0  0.5  1  1.5plane wave strength (a.u.)(b)(a)FIG. 1: [color online] Dependency of the anisotropic propa-gation of surface acoustic waves on the Y cut (solid red line)and the Z cut (dotted blue line) of lithium niobate (LiNbO3):(a) slowness curve and (b) plane wave strength of the verticalsurface displacement. In both plots, the horizontal axis is thecrystallographic X axis.to aberrations at the focal point. In the following, weconsider using synchronous surface wave emission fromall directions to obtain a diffraction limited focal point.The anisotropic propagation of surface acoustic waveson a planar substrate is governed by two functions of theemission angle ψ. The first one, s(ψ), is the slowness (theinverse of the phase velocity). The second function, a(ψ),is a measure of the anisotropic plane wave strength; it isa number giving the rate of SAW emission in each di-rection [9]. Examples of slowness curves and plane wavestrengths are given for lithium niobate (LiNbO3) in Fig-ure 1, for propagation on both the (010) (Y cut) and the(001) (Z cut) planes. These functions depend acutely onthe material cut considered and are clearly anisotropic.The displacement field of the outgoing wave originatingfrom a point source is given by a Green’s function witha free surface boundary condition. The general form anda practical procedure to compute this Green’s functionwere given in Ref. [9], for a time harmonic dependence ofthe form exp(ıωt). Of all possible acoustic contributions,only surface waves can efficiently transport energy in thefar field. Figures 2a and 2b display the modulus of theSAW contribution to the Green’s function, with evanes-cent waves excluded, for the same orientations of lithiumniobate as in Fig. 1. A system of ripples is seen to orig-inate from the source center. As the wave advances, the2FIG. 2: [color online] Vertical displacement component of thesurface Green’s function for (a) Y cut and (b) Z cut lithiumniobate. x and y are the coordinates of the observation pointwith respect to the source, and f is the frequency. Wavesurfaces for Y cut (solid red line) and Z cut (dotted blue line)lithium niobate are shown in (c). The principle of the annularinterdigital transducer is depicted in (d).ripples become homothetic to the wave surface, as canbe inferred from the following argument. The SAW con-tribution to the far field of the Green’s function can beestimated using the stationary phase principle [10], withthe resultG(R, θ) ≈Aa(ψ) exp(−ıωRh(ψ) − ıpi4sgn{h′′(ψ)})√ωR∣∣h′′(ψ)∣∣(1)with h(ψ) = s(ψ) cos(ψ − θ). In this expression, thebeam steering angle θ − ψ is defined by the conditiondh(ψ)/dψ = h′(ψ) = 0; it is a function of the direc-tion of observation θ. Note that the group velocity isvg(θ) = h−1(ψ). A is a constant independent of the posi-tion of observation. Equation (1) indicates that in the farfield, propagation occurs in the direction θ at the groupvelocity vg(θ) and that the amplitude decreases with thesquare root of the distance. The term√|h′′(ψ)| is pro-portional to the so-called phonon focusing factor, whichgives cuspidal points and the caustics when it vanishes.The wave surface, which is the locus of the group or en-ergy velocity as a function of the emission direction [1],can be obtained directy from the slowness curve by nu- 0 2 4 6 60  65  70  75  80  85  90  95  100conductance (mS)frequency (MHz)-4-2 0 2susceptance (mS)FIG. 3: [color online] Measured admittance of annular inter-digital transducers on Y cut (solid red line) and Z cut (dottedblue line) lithium niobate.merical differentiation [11]. Fig. 2c displays the wavesurface for both the Y and the Z cuts of lithium niobate.We next define a pattern of electrodes with alternat-ing electrical potentials such that the radiated SAW willbe focused to a single point. One possibility is to bina-rize the phase of the Green’s function, thus producinga hologram of a source point. Further considering thatthe source is in the far field of the focus point, we inferthat the electrodes must be placed along a curve homo-thetic to the wave surface. The design principle of suchan annular interdigital transducer (AIDT) is depicted inFig. 2d, with the wave surface for Y cut lithium niobateused as an example. Considering propagation along thedirection given by angle θ, it is well-known that the phasevelocity vector is orthogonal to the wave surface [1]. Byconstruction, the local pitch of the AIDT is then pro-portional to the phase velocity s−1(ψ). Hence, the dis-tance p(θ) between two electrodes can be chosen suchthat fp(θ)s(ψ) = 1/2. This expression guarantees thatthe angular variation of p(θ) is such that the entire elec-trode array is at resonance at frequency f . It also im-plies that the only wave that can be synchronous for alldirections is the one selected when computing the wavesurface.Experiments have been conducted on Y and Z cutlithium niobate (LiNbO3) substrates. AIDT masks weredesigned such that the average acoustic wavelength λ is50 µm and the resonance frequency is 75 MHz. The num-ber of finger pairs is 200 and the metallization ratio is 0.5.Figure 3 shows measurements of the admittance of AIDTdevices for both cuts. The response is roughly the super-position of a capacitance, the SAW contribution around75 MHz, and a broad leaky-SAW contribution extendingfrom 80 to 90 MHz approximately. The SAW contribu-tion has the form of two sharp peaks, one for the entranceand one for the exit of the stop band of the periodic elec-trode array. In contrast, the leaky-SAW contribution isdispersed along the frequency axis since the emission ofthis type of wave is not synchronous.3FIG. 4: [color online] Measured vertical displacements at thecenter of the annular interdigital transducers on (a) Y cutand (b) Z cut lithium niobate. The scans are taken at theresonance frequency and cover an area of 400 by 400 µm2.Cross sections through the focal point are shown above thescans.To directly image and characterize the focal spot, thespatial distribution of the vertical displacement was mea-sured thanks to an heterodyne optical probe [12]. Theresulting maps are shown in Figure 4. Their spatial res-olution, limited by the collection optics, is estimated tobe 5 µm. Waves emitted in all directions interfere in thecentral part of the device and form a concentric fringepattern. An important focusing in the center of the de-vices is clearly obtained at the main resonance frequency.The maximum vertical displacement could be optimizedby better electrical matching to the electrical source andfurther amplification. We note, however, that local strainvariations in the vicinity of the focus can be quite high.Comparison between the experimental maps and thetheoretical calculations of the acoustic field diffracted bya point source (Fig. 2) evidences that they bear a strik-ing resemblance. This shows that the AIDT achieves thetime reversal [13] of the Dirac source considered for theGreen’s functions of Fig. 2. Since the generated angularspectrum misses the evanescent waves required to recon-struct the near field, resolution is limited by diffraction.For isotropic propagation on a surface, the amplitudeat the focus would ideally be given by the Bessel func-tion J0(kR), with k = 2pi/λ, resulting in a full widthat half maximum (FWHM) of 0.48λ [14], or 24 µm forλ = 50 µm. The experimental focus spot is slightlyanisotropic, and its FWHM is approximately 20 µm forboth the Y and the Z cut. This value is slightly smallerthan or comparable to the isotropic limit, though the dif-ference is within the estimated spatial resolution of thescans. This observation however calls for further studiesof the diffraction limit for anisotropic propagation.As a conclusion, we have proposed and demonstratedexperimentally the concept of an annular interdigitaltransducer (AIDT) such that the fingers shape followsthe wave surface. The generated surface acoustic wavesconverge to the center of the transducer, producing an in-tense spot that is limited in resolution by diffraction only.The AIDT can indeed be seen as a time-reversal device,reforming a diffraction-limited image of a point sourcethat would be located at the resonator center. This re-sult is promising for several applications including intensemicroacoustic sources, e.g. for integrated acousto-optics,or to enhance non linear effects.The authors wish to thank B. Wacogne and P. Vairacfor their help with the AIDT masks and in setting up theheterodyne interferometer, respectively, and K. Kokko-nen for insightful comments.[1] D. Royer and E. Dieulesaint, Elastic waves in solids, vol.1 (Wiley, New York, 1999).[2] R. M. White and F. W. Voltmer, Appl. Phys. Lett. 17,314 (1965).[3] R. Von Gutfeld, D. Vigliotti, C. Ih, and W. Scott, Appl.Phys. Lett. 42, 1018 (1983).[4] R. E. Vines, S. I. Tamura, and J. P. Wolfe, Phys. Rev.Lett. 74, 2729 (1995).[5] Y. Sugawara, O. B. Wright, O. Matsuda, M. Takigahira,Y. Tanaka, S. Tamura, and V. E. Gusev, Phys. Rev. Lett.88, 185504 (2002).[6] C. K. Day and G. G. Koerber, IEEE Trans. Sonics andUltrason. SU-18, 461 (1972).[7] T.-T. Wu, H.-T. Tang, Y.-Y. Chen, and P.-L. Liu,IEEE Trans. Ultrason., Ferroelec., Freq. Control 52, 1384(2005).[8] Z. Wang, T. Tang, S. Chen, and B. Chen, J. Phys. D:Appl. Phys. 39, 4902 (2006).[9] V. Laude, C. F. Jerez-Hanckes, and S. Ballandras,IEEE Trans. Ultrason., Ferroelec., Freq. Control 53, 420(2006).[10] J. G. Harris, Linear elastic waves (Cambridge UniversityPress, Cambridge, 2001).[11] V. Laude and S. Ballandras, J. Appl. Phys. 94, 1235(2003).[12] K. Kokkonen, M. Kaivola, S. Benchabane, A. Khelif, andV. Laude, Appl. Phys. Lett. 91, 083517 (2007).[13] G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, Science315, 1120 (2007).[14] S. S. Hong, B. K. P. Horn, D. M. Freeman, and M. S.Mermelstein, Appl. Phys. Lett. 88, 261107 (2006).

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