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1 arc Solal, Olli Holmren, and Kimmo Kokkonen. Desin, simulation and visualization of R SUDT devices with transverse mode suppression. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, accepted for publication IEEE reprinted with permission. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of Helsinki University of Technoloy's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertisin or promotional purposes or for creatin new collective works for resale or redistribution must be obtained from the IEEE by writin to pubs permissions@ieee.or. By choosin to view this document, you aree to all provisions of the copyriht laws protectin it.

2 Accepted for publication in IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control Desin, Simulation and Visualization of R-SUDT Devices with Transverse ode Suppression arc Solal, ember, IEEE, Olli Holmren and Kimmo Kokkonen Abstract When desinin narrow band resonant SUDT devices, the excitation of undesired transverse modes may result both in extra ripple in the passband and in spurious response in the stop band. To avoid these issues, it was proposed to use an approach similar to the one used for bulk-acoustic-wave devices. The principle is to add a low velocity reion at the ede of the transducer. If this ede reion is properly desined, the transducer supports a so-called piston mode, i.e., a mode havin a flat transverse amplitude profile across the aperture. A -matrix model is extended to account for transverse modes in SUDTs. The model is used to analyze both reular and piston mode devices. Different physical possibilities to implement the low velocity reion are investiated and compared. In particular, it was found important to desin the transducer so that the acoustical sources and reflectors extend into the ede reion to minimize the couplin to hiher order modes. From these considerations, a new implementation for piston mode devices is proposed and demonstrated on a GS base station 99 Hz filter. Electrical measurements as well as acoustical wave fields measured with an optical interferometer are analyzed and compared to simulations. Index Terms SUDT, transverse modes, piston mode, interferometry I. ITRODUCTIO The excitation of transverse modes in SAW filters is known to result in spurious responses in the reection band and-or extra ripple in the passband. The effect on the performance is the larest in the case of narrow band resonant filters on quartz. A usual counter measure is to apodize the transducer in order to match the excitation profile in the aperture to one transverse mode shape (typically the first symmetric mode) in order to excite predominantly this mode []. Other approaches include usin multiple tracks in order to excite predominantly a chosen mode [2,3] or adustin the portion of the aperture to be active to cancel the couplin to spurious modes[3,4] Another approach described in [5] in the case of a anuscript received, S. is with TriQuint Semiconductor, Apopka, Florida, USA marc.solal@ tqs.com. O. H. and K.K. are with Helsinki University of Technoloy Espoo, Finland very wideband filter on Lithium iobate is to use weihted dummy electrodes. A similar issue exists for bulk-acoustic-wave (BAW) resonators in which several modes can be excited simultaneously. An efficient way to suppress the spurious modes in BAW resonators is to implement a border reion havin correct width and acoustical properties. By implementin this rin, it is possible to chane the mode shape to make it almost rectanular. Then, since the mode shape is matched to the excitation, only one mode is excited and spurious modes are suppressed [6]. ore recently, ayer et al. [7] proposed a similar approach for surface-acoustic-wave (SAW) devices. The principle is to reduce the wave velocity at the ede reion of the transducer aperture. Then, it is also possible to obtain a so called piston mode havin an almost rectanular shape. The technique they used to obtain this lower velocity at the ede reion consists of addin short-circuited ratin havin more electrodes per wavelenth than the center track. Even thouh the analoy between desinin piston mode SAW and BAW devices is very stron, some differences exist: Firstly, for BAW devices the excitation reion includes the edes while it does not for SAW when usin the technique described in [7]. Secondly, an important effect when analyzin wave uidin in SAW devices is the varyin reflectivity in the different reions. This does not exist for BAW devices. Besides the method proposed by [7], there are other possibilities to implement the low velocity edes, such as increasin the metal thickness, addin a dielectric overlay or increasin the mark-to-pitch ratio. These techniques allow the extension of the sources and the reflectors in the ede reion. To model transverse modes, a scalar approximation of the wave is enerally sufficient [8-]. A specific problem for SAW devices is the presence of reflectors inside the device. This was addressed in [2] by usin a 2D version of the CO model. Later, matrix based models were developed to account for the non uniform profile of the reflectivity in the transverse direction [3-4]. In this work, the model presented in [4] was extended to handle sinle-phase-unidirectionaltransducer (SUDT) devices. It was used to compare ede desins (with and without sources) for a 99 Hz GS base

3 Accepted for publication in IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 2 station filter. As a result, a new way to realize piston mode SAW devices is proposed. To confirm this piston mode desin, electrical measurements as well as acoustical fields measured with the optical interferometer [5], are analyzed and compared to simulation. II. RICILE OF SAW ISTO ODE DEVICES A. Width of the ede reion Fiure illustrates the simplified principle of piston mode devices. The center reion of the active aperture (reion ) has a lower velocity than the busbars (reion 3) resultin in waveuidin. The ede reion (reion 2) is desined to have a velocity lower than that in the center reion. The end aps between the finers and the busbars are nelected at this stae, but are included later in the numerical simulations. Under these conditions, it is possible for the transverse wave vector k y, to be zero in the reion. The transverse wave vector is then imainary (evanescent wave) in the reion 3 and real in the reion 2. Usin a scalar model for the waves and applyin the continuity of the amplitude and its y derivative as boundary conditions, it is straihtforward to express the conditions to allow a mode to propaate with k y, = 0. Assumin the parabolic approximation [6] is valid for the slowness curves and small velocity differences between the reions, the followin expression is obtained: m arctan Wf + γ. () v x 2π 2 In (), W is the width of the ede reion, γ is the anisotropy coefficient and the velocities are: vm v x ( + m v = v x + where = ) ( ) v m is the metallized velocity, i.e. the velocity in the busbars (reion 3 in Fi. ), reion (reion ) and v v x is the velocity in the center is the velocity inside the ede reion (reion 2). Accordin to (), the optimal ede width is independent from the width of the reion and depends only on the anisotropy and velocities in the different reions. Therefore, the same ede width can be used independently of the transducer aperture. The piston mode can be obtained with a relatively small ede width. For example, for = =0 3 m and γ = 0. 38, the ede width is only 3.2 wavelenths. In the case of our quartz devices, it was found that the effect of the aps at the edes of the electrodes is small. However, the ede width is slihtly adusted later when the ap is included in the model. v v Fi.. Simplified velocity profile and mode shape for the piston mode device. B. SAW specific characteristics for piston mode operation When tryin to apply a piston mode technique to a R- SUDT [7] desin, one difficulty is to determine the ratin velocity y v x v x v m v m velocity used in (). A typical R-SUDT desin includes several reions with different periods. Typically, when usin standard electrode-width-controlled (EWC) or distributedacoustic-reflection-transducer (DART) cells, the device contains some reions havin reflectors only, i.e., with half a wavelenth period, some reions havin sources only, i.e., havin a quarter wavelenth period and reions havin one reflector and one source per wavelenth. All these reions have different velocities. As shown in (), the correct width of the ede reion is very sensitive to the velocity differences between the ede reion, the center reion and the busbars. Therefore, the transversal modes are expected to be different in different reions alon the wave propaation direction of the R-SUDT device. To illustrate this, Fi. 2 depicts the transversal mode shapes simulated for the different reions of a 99 Hz DART SUDT filter on Y+36 derees quartz with a metal thickness of 2500 Å. In this case, the velocities are m/s for the reflector reions and m/s for the source reions. The velocity in the busbars is m/s. The anisotropy coefficient is The ede velocity is m/s in the reflector reions and m/s in the source reions. The ede velocity is computed assumin the ede is implemented by increasin the mark to pitch ratio (see below). The ede width is chosen such that a piston mode is obtained when the velocity of the center reion is equal to the averae of the source velocity and the reflector velocity. Then the calculated mode shape is close to rectanular (solid line in Fi. 2). When computin the mode shapes usin velocities correspondin to reflector and source reions, however, the mode shape is not rectanular. In the source reion (dotted line), the enery of the main mode is concentrated near the edes, whereas in the reflector reion (dashed line), the mode has a cosine-like shape. y mode shape

4 Accepted for publication in IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 3 Fi. 2. Transversal mode shapes simulated for the different reions of a 99 Hz DART SUDT filter with constant ede width. The solid curve corresponds to a center reion havin a velocity equal to the averae of two electrodes per wavelenth and four electrodes per wavelenth while the dashed and dotted curves correspond to the reions with two and four electrodes per wavelenth, respectively. III. SUDT DEVICE AALYSIS A. General presentation of the model In [4], a model of SAW devices includin both a discrete number of uided modes and a continuum of propaatin waves was presented. The model takes into account also the effect of the reflectors limited aperture. The principle is only recalled briefly here. The first step of the method is to derive the uided modes and continuum. Our model allows us to analyze the case of a symmetric structure comprisin 2+ reions of different velocities. The parabolic approximation is used to take into account for the crystal anisotropy. To simplify the theory, a scalar approximation is used for the waves. The wave vector component in the propaation direction is the same in all the reions while the wave vector component in the transverse direction is, for each reion, derived from the slowness curve. At each interface between two reions, the wave amplitude and its transverse derivative are assumed continuous. By applyin these boundary conditions and assumin an evanescent wave (imainary ) in the external reions, the discrete uided modes velocities and shapes are derived. In the case when k y is real in the external medium, a continuum of waves is found. This continuum is sampled with a fixed increment in k y. k y k y The second step of the method is to derive the multimode matrix for one period of the transducer. [] I [ bl ] [ b ] R = [ ] [ 2] [ 3] [ ] [ 22] [ 23] [ ] [ ] [ ] [ V] [ a ]. L [ ] ar k x () Here electrical ports and 2 acoustical ports are considered, where e m is the number of discrete modes plus the number of continuum samples. The matrix relates the voltaes V, the incomin wave amplitudes at the left e m side a and the incomin wave amplitudes at the riht L m side a to the currents I, the outoin wave R e amplitude on the left side b and the outoin amplitude L on the riht side. The block is the admittance matrix, br the blocks and are the acoustical reflection matrices on the left and riht sides. and are the acoustical transmission matrices. The blocks and 3 are the electroacoustic couplin matrices while the blocks and are the acoustoelectric couplin 2 3 matrices. As described in [4], the reflection occurs only in the portion of the aperture where the reflector is present. For example there is no reflection in the busbars or on the free surface. This means that the amplitude of the reflected wave is proportional to the amplitude of the incomin wave multiplied by a atin function. If (y) is the amplitude of the i th mode normalized such i that ( y) dy = i center of the electrode is : rect( y). ( y) = R R i = i * rect( y) ( y) ( y) dy i m m m 32 23, the reflected wave amplitude at the i ( y) where rect(y) is a function defined to be equal to the unity in the reflective reion and to zero in the non reflective reions. Based on these considerations, the scatterin part of the matrix can be derived. If the phase of the mode (or continuum sample) i under the period is ϕ and the reflection and transmission coefficients on the electrode are cos i sin and, the expressions of the acoustical scatterin blocks are: ϕi + ϕ i i 22 = 33 = sin R i exp 2 (3) ϕi + ϕ i i ( ) ) 23 = 32 = cos R i + δi exp 2 The acoustoelectric and electroacoustic blocks are obtained by computin the mode expansion of the rectanular source of W aperture correspondin to the electrical port.

5 Accepted for publication in IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 4 i 2 i W * i i = = = ( y) dy i 3 i 3 = ϕi + Ga. i exp 2 = * ϕi + Ga. i exp 2 In the previous expression, Ga is the radiation conductance of one electrode. In the simple case of one track devices, the admittance is iven by G + H G ) where H stands for the a ( a Hilbert transform. The next step of the method is to cascade the matrix of the consecutive cells. For the case of a DART or EWC device, two different kinds of cells have to be considered: the reflective cells and the transducer cells. In the transducer cells (four electrodes per wavelenth), there is no reflectivity (. In this case, the expressions for the acoustical = 0) scatterin blocks are reatly simplified. The reflection block matrices are simply zero while the transmission block matrices are diaonal matrices: i i 23 = 32 = δ i exp( ϕi ). (5) The matrices of several consecutive transducer cells can then be cascaded directly by summin the wave amplitudes for the different sources at the boundaries of the four electrodes per wavelenth reion. If the electrode connectivities l in are defined to be zero except if the electrode n is connected to the electrical port i, the expression of the matrix for the cascade of periods is straihtforward:, = = 0 δ i 22 i 23 i 2 i 3 i i i = = = = = = G 33 i 32 G G ( Ga + Ba ) = exp( ϕ ) δ G G a a * i lin * i lin a in m m= k = m n a i in a i in l l l in.exp n ϕ 2.exp n + ϕ 2 m i ik i l.exp n ϕ 2 l.exp n + ϕ 2 * k exp ( nϕ ) k + (6) To reduce the computation time, the four electrodes per wavelenth reions are detected and treated at once usin (6). The reflectors are cascaded period by period. B. ode conversion between the different reions Due to the results shown in Fi. 2, it was assumed it was very important to account for the difference in the mode profiles between the reflective (two electrodes per wavelenth) and transductive (four electrodes per wavelenth) reions. This difference in mode shapes results in conversion at the boundary between the reions. To analyze this conversion, the two sets of mode shapes ( y) ϕ ϕ ( y) and and phases and for two and four electrodes per wavelenth, respectively, are computed. Then, the matrices are derived as described above for the correspondin modes. The two electrodes per wavelenth modes are chosen as the base for the device matrix. So, for each transductive reion, the matrix in the four electrodes per wavelenth base is computed usin (6). The next step is to convert this matrix to a new matrix on the two electrodes per wavelenth modes. =. A =. A = 3. A ~ = A. = = = 0 ~ = A. 23. A = 3. A ~ = A. The matrix A represents the conversion between the two sets of modes (each one assumed to be orthoonal): Ai, = i ( y) ( y). dy (8) * Since the number of modes may be different for the two electrodes per wavelenth reions and the four electrodes per wavelenth reions, A may not be a square matrix and cannot always be inverted. A ~ stands for the pseudo inverse of A. One has to notice that this expression is valid for the uided modes only. A correct expression to represent the conversion for the continuum was not found. The reason is that this would probably be possible only by considerin reflected waves at the boundary between two different reions. The model described above is straihtforward to implement. The matrix of a complete device is computed by cascadin the individual matrices (defined for the two electrodes per wavelenth modes). To compute the wave amplitude fields, an additional step has to be performed. After cascadin the matrices, the output wave for iven load and source impedances are known on the boundary of the structure. Then, the amplitude fields are computed by multiplyin the transfer matrices. (7)

6 Accepted for publication in IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 5 IV. DEVICE ILEETATIO Fi. 3. Estimated relative levels of several spurious modes as a function of the source aperture. The left vertical line corresponds to the center reion width while the riht line corresponds to the total aperture of the center reion and the two ede reions. The points between the two vertical lines correspond to sources extendin inside the ede reions. The levels are normalized to the level of the main mode. The estimated overall spurious level as a function of the source aperture is thus obtained as an envelope of all the lines (marked with ray). The devices described in [7] used short-circuited ratins havin a lare number of electrodes per wavelenth in the ede reions allowin for a reduction in the wave velocity. In their approach, the sources do not extend in the ede reion. To study the effect of extendin the sources to the ede reion, relative levels of spurious modes were estimated with varyin source widths for the confiuration described in section IIB. This estimation is done by computin the interal of the mode shapes on the source aperture relative to the interal of the main mode. Fi. 3 shows an example of estimated relative levels of numerous spurious modes and the overall level as their envelope as a function of the source aperture. The same eometry is used for all points of the curve, and only the source aperture (i.e., the interation bounds) is varied. The lowest spurious levels are obtained when the sources extend far in the ede reion, almost to the boundary between the ede reion and the busbar. For example, if the widths of the center and ede reions are mm and 0.5 mm, the optimum width of the sources is.4 mm. In this case, the expected relative spurious level enerated by the transducer is reduced by ~7 db compared to the case where the sources do not extend to the ede reion. Obviously, this effect may become less important if the velocity difference between the reions increases. Fi. 4. Simulated conductance for different implementations of the piston mode device without impedance matchin: reflectors and source extend in the center reion only (dotted line), reflectors and sources extend both in the center and the ede reion with mass loadin (dashed line) and with hiher mark to pitch ratio (m/p) (solid line). The four main peaks are due to lonitudinal main modes. The spurious modes are seen as additional smaller peaks between the main modes and above the passband. Overall response (top) and zoomed view of the passband (bottom). To confirm this effect, Fi. 4 shows the simulations results obtained for the same filter for three different implementations of the piston mode device. The best results are obtained when the sources and reflectors extend both in the center and the ede reions. This confiuration could be implemented by reducin the velocity in the ede reion by increasin its metal thickness or addin a dielectric overlay, i.e., with mass loadin (dashed line in Fi. 4). However, it was decided to stay with the one metal layer process and so to increase the mark to pitch ratio (m/p) in the ede reion. This approach allows both to lower the velocity of the ede reion and to extend the sources to the ede reion. The split finer width was increased from λ / 8 to 3λ / 6 and the reflector width was increased to et constant aps, see Fi. 5. The simulation of this confiuration is shown in solid line in Fi. 4. The improvement when comparin with the initial piston mode confiuration, in dotted line, is obvious.

7 Accepted for publication in IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 6 Fi. 5. Detailed view showin the ede reion of the implemented piston mode device usin hiher mark-to-pitch-ratio electrodes. The dark reions correspond to the electrodes. The period of the wide electrodes is half a wavelenth while it is a quarter wavelenth for the narrow electrodes. V. EXERIETAL RESULTS A. easured electrical responses Two versions of the same basic filter desin were manufactured: a standard device without the ede reion and a piston mode device with the ede reion. The ede reions with constant width were implemented usin hiher mark-topitch ratio. To choose the width of the ede reion, the averae of the velocities in the two electrodes per wavelenth and in the four electrodes per wavelenth was assumed. The expression () was used as a first trial. Then, the actual mode shapes were computed by usin the numerical method and the ede width was slihtly adusted in order to et a flat transverse profile in the center reion. Fiures 6 and 7 show the measured and simulated transfer functions obtained for the standard and piston mode devices, respectively. Both devices were matched to 50 Ω. Due to the narrow relative passband, the minimum insertion is 4 db for the standard device while it is about 4.8 db for the piston mode device. The device is also stronly resonant, meanin that the reflections inside the transducers as well as the reflections between the two transducers allow to increase the impulse response duration. The standard device has stron ripple in the pass band due to the transverse modes as well as spurious on the hih frequency. Usin the ede reion, this ripple was reduced from.5 db to about 0.2 db. This demonstrates the efficiency of the piston mode operation to reduce the effect of the transverse modes. B. Comparison of the measured and simulated electrical responses Two sets of simulations are shown in Fis. 6 and 7. The first one (labeled without conversion in the fiures) uses the averae velocities of the four and the two electrodes per period reions to compute the modes everywhere. In this case, there is no difference in the mode shapes in the different reions and no conversion has to be accounted for. The second simulation (labeled with conversion ) takes into account the different mode shapes in the different reions. Despite the very lare difference in mode shapes, both simulations ive very similar results. This indicates that the averae velocity in the center reion is a ood approximation when computin the width of the ede reion and the mode shapes for both the reflective and transductive reions. When the averae velocity is used, the continuum can be taken into account. The match between measurement and simulation is very ood especially for the standard device (Fi. 6.). While there are only sliht differences in the passband, the continuum analysis helps to predict more accurately the hih frequency sidelobes. For the piston mode device (Fi. 7.), the areement is also ood. The passband ripples are reatly reduced but they can still been seen on the measurement while the simulation predicts that they are completely suppressed. C. Amplitude profiles To better understand the electrical results, the acoustic amplitude profiles were measured usin a scannin laser interferometer [5]. The interferometer detects mechanical vibrations perpendicular to the sample surface. The spatial resolution of the interferometer is better than µm. It should be noted that amplitudes in areas with different reflectivity, e.., ratin and metal surface, cannot be directly compared since the sensitivity of the interferometer depends on the optical reflectivity of the sample surface. Fi. 6. Comparison between measurement and simulation for the standard device matched to 50 Ω: measurement (solid), simulation without conversion and with continuum (dotted), simulation with conversion and without continuum (dashed), simulation without conversion and without continuum (dash-dotted). o sinificant difference is found between the results of the simulation with and without conversion. The continuum is explainin the hih side reection. Overall response (top) and zoomed view of the passband (bottom).

8 Accepted for publication in IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 7 Fi. 7. Comparison between measurement and simulation for the piston mode device matched to 50 Ω: measurement (solid), simulation with conversion and without continuum (dotted), simulation without conversion and with continuum (dashed). Overall response (top) and zoomed view of the passband (bottom). The interferometer measurements were carried out for 49 frequencies in the frequency rane from Hz to Hz with a frequency step of 0.02 Hz. The dimensions of the measured filters are approximately 0 mm.2 mm, while the finer period in x-direction is relatively small, approximately 4-8 µm. This results in complementary requirements for the scan parameters: the scan step should be small enouh to resolve all the details (finers) while on the contrary the lare size of the device would require a lare scan step to keep the scan time (and data size) reasonable. In practice, it is not feasible to measure the whole device with hih resolution. Therefore, seven smaller scan areas were chosen. Each scan area has 300 x 20 scan points with scan steps of 0.99 µm (x) and 9.90 µm (y), resultin in an area of 297 µm x 88 µm. Fiures 8-2 show measured and simulated acoustic amplitudes at selected frequencies for the standard and piston mode devices. o impedance matchin was used in the measured devices in order to obtain better separation of modes. The simulations assume the averae velocity and include the continuum. The frequencies correspondin to the main modes and spurious modes were selected by locatin peaks from the electrical responses, measured from the unmatched devices. There is a frequency offset of 0. Hz between the measured and simulated frequency responses. The dashed rectanles in the simulated amplitude imaes indicate the locations correspondin to the measured scan areas. In eneral, the simulation predicts well the distribution of the acoustic amplitudes in both devices. Fiure 8 compares the measured and simulated amplitude distributions in the standard device (without the ede desin) for one of the main modes. The mode shape is smooth and round in y direction, as expected. Fiure 9 shows the correspondin mode in the piston mode device (with the ede desin). In this case, the transversal mode shape is close to rectanular. Fiure 0 depicts measured and simulated amplitude distribution at the frequency of a spurious mode existin in the standard device. In the fiure, the transversal mode structure is clearly visible. Fiures and 2 show the amplitude distributions for hiher-order modes above the passband for the standard and piston mode devices, respectively. To illustrate the effect of the ede desin to the transverse amplitude profiles more clearly, Fis. 3 and 4 show the measured and simulated transversal amplitude profiles for the main mode shown in Fis. 8 and 9. The measured amplitude profile was obtained by averain all the lines of the scanned area between x = 2.5 mm and x = 2.8 mm. In addition, the simulated curve was shifted such that the maximum values of the measured and simulated curves coincide. The correspondence is very ood for the standard device in Fi. 3. The discontinuities in the measured profiles are due to the difference of optical reflectivity between the transducer and the busbar. For the piston mode device in Fi. 4, the measured profile is slihtly less rectanular than predicted by the model. This indicates that the velocity in the ede reion is probably slihtly underestimated. This would also explain why some transverse mode effect is seen on the electrical filter response. VI. COCLUSIO A new approach to desin piston mode R-SUDT devices is proposed. Different physical possibilities to implement the low velocity reion enablin piston mode operation were investiated. By extendin the sources in the ede reions, a reduction of spurious mode levels is obtained. For a GS base station 99 Hz filter considered here, the passband ripple due to the transverse spurious modes was reduced from.5 db to about 0.2 db. Experimental results of both electrical responses and acoustical amplitude fields are presented validatin both the simulations and the successful application of the new desin concept.

9 Accepted for publication in IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 8 ACKOWLEDGETS O.H. and K.K. thank the Alfred Kordelin Foundation, the Finnish Cultural Foundation and the okia Foundation for scholarships. Fi. 8. Acoustic wave fields in the standard device without the ede desin for one of the main modes: simulation at Hz (top), measurement at 99.0 Hz (bottom). Fi. 9. Acoustic wave fields in the piston mode device with the ede desin for one of the main modes: simulation at 98.9 Hz (top), measurement at 99.0 Hz (bottom). Fi. 3. Comparison between the measured and simulated transverse profile of the acoustic amplitude for the standard device at x=2.65mm. The discontinuities on the measured curve are due to the difference in optical reflectivity between the busbars and the transducer. Fi. 0. Acoustic wave fields in the standard device for a spurious mode: simulation at Hz (top), measurement at Hz (bottom) Fi.. Acoustic wave fields in the standard device above the passband: simulation at Hz (top), measurement at Hz (bottom) Fi. 2. Acoustic wave fields in the piston mode device above the passband: simulation at Hz (top), measurement at Hz (bottom). Fi. 4. Comparison between the measured and simulated transverse profile of the acoustic amplitude for the piston mode device at x=2.65mm. The discontinuities on the measured curve are due to the difference in optical reflectivity between the busbars and the different reions of the transducer. REFERECES [] S. A. Wilkus, C. S. Hartmann, and R. J. Kansy, "Transverse mode compensation of surface acoustic wave filters", in roc. IEEE Ultrasonics Symposium, 985, pp [2] G. artin, B. Wall, and. Waihnacht, "An alternative method for suppressin undesired transverse modes in lonitudinally coupled SAW resonator filters", IEEE Trans. Ultrason., Ferroelect., Freq. Contr. vol. 42, pp , 995. [3]. ayer, A. Bermann, K. Waner,. Schemies, T. Telmann, and A. Glas, Low resistance quartz resonators for automotive applications without spurious modes, IEEE Ultrason. Symp., 2004, pp [4] Y. Yamamoto and S. Yoshimoto, "SAW transversely uided mode spurious elimination by optimization of conversion efficiency usin W/W0 electrode structure", in roc. IEEE Ultrasonics Symposium, 998, pp [5] T. Omori, K. atsuda,. Yokoyama, K. Hashimoto, and. Yamauchi, "Suppression of transverse mode responses in ultra-

10 Accepted for publication in IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 9 wideband SAW resonators fabricated on a Cu-ratin/5YX-LibO3 structure", IEEE Trans. Ultrason., Ferroelect., Freq. Contr. vol. 54, pp , [6] J. Kaitila,. Ylilammi, J. Ella, and R. Ainer, Spurious resonance free bulk acoustic wave resonators, in roc. IEEE Ultrasonics symp, 2003, pp [7]. ayer, A. Bermann, G. Kovacs, and K. Waner, Low loss recursive filters for basestation applications without spurious modes, in roc. IEEE Ultrasonics symp., 2005, pp [8] J. K Knowles, A note on elastic surface waves, Jour. Geophysical Res. vol. 7, pp , ov. 5,966 [9] J. Schoenwald, "Optical waveuide model for SAW resonators", in roc. IEEE Int. Freq. Contr. Symp., 976, pp [0] C. K. Campbell, "odellin the transverse-mode response of a two-port SAW resonator", IEEE Trans. Ultrason., Ferroelect., Freq. Contr. vol. 38, pp , 99. [] G. Clark, R.-F. ilsom, and J. Schofeld, "3-D modal analysis of SAW filters", in roc. IEEE Ultrasonics Symposium, 985, pp [2] H. A. Haus and K. L. Wan, "odes of ratin waveuide", J. Appl. hys vol. 49, pp , 978. [3] K. Waner,. ayer, A. Bermann, and G. Riha, A 2D -atrix odel for the Simulation of Waveuidin and Diffraction in SAW Components, in roc. IEEE Ultrasonics Symp., 2006, pp [4]. Solal, V. Laude, S. Ballandras, A -atrix Based odel for SAW Gratin Waveuides Takin into Account odes Conversion at the Reflection, in IEEE Trans. on UFFC, vol. 5, no 2, pp , Dec [5] J. V. Knuuttila,. T. Tikka, and.. Salomaa, Scannin ichelson interferometer for imain surface acoustic wave fields, Opt. Lett. vol. 25, pp , [6] T. L. Szabo and A. Slobodnick, The effect of diffraction on the desin of acoustic surface wave devices, IEEE Trans. Sonics Ultrason., vol. 20, pp , Jul [7]. Ventura,. Solal,. Dufilie, J.. Hode, and F.Roux, A ew Concept in SUDT Desin: the RSUDT (Resonant SUDT)" in roc. IEEE Ultrasonics symp., 994, pp -6. [8]. Solal, A -atrix-based odel for the Analysis of SAW Transversely Coupled Resonator Filters, Includin Guided odes and a Continuum of Radiated Waves, in IEEE Trans. on UFFC, vol 50, no 2, pp , Dec