- Email: [email protected]

S0022-460X(18)30736-3

DOI:

https://doi.org/10.1016/j.jsv.2018.10.056

Reference:

YJSVI 14473

To appear in:

Journal of Sound and Vibration

Received Date: 24 May 2018 Revised Date:

6 October 2018

Accepted Date: 29 October 2018

Please cite this article as: B. Zaitsev, I. Kuznetsova, I. Nedospasov, A. Smirnov, A. Semyonov, New approach to detection of guided waves with negative group velocity: Modeling and experiment, Journal of Sound and Vibration (2018), doi: https://doi.org/10.1016/j.jsv.2018.10.056. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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New approach to detection of guided waves with negative group velocity: Modeling and experiment 1

Boris Zaitsev , Iren Kuznetsova* , Ilya Nedospasov2, Andrey Smirnov2,3, Alexander Semyonov1 1

2

Kotelnikov Institute of Radio Engineering and Electronics of RAS, Saratov branch, Saratov

410019, Russia Kotelnikov Institute of Radio Engineering and Electronics of RAS, Moscow 125009, Russia

3

Chernyshevskii N.G. Saratov State University, Saratov 410012, Russia

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2

Keywords: forward and backward plate acoustic wave, phase and group velocities, dispersion

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dependencies, interdigital transducer, electrical impedance, finite element analysis.

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Abstract

A new method for detecting the forward and backward acoustic waves in the piezoelectric plates by using a system of the acoustically isolated interdigital transducers (IDT) with the different spatial period is developed. This method was tested on the piezoelectric acoustic modes A1 and SH1 propagating in Y-X LiNbO3 plate. Theoretical analysis has shown that the SH1 mode in the entire frequency range represented a forward wave. At that the

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dispersion dependence of the A1 mode near the cutoff frequency has a smooth transition from a forward wave to a backward one with decreasing frequency. In order to experimentally observe this transition 19 IDTs with different periods were deposited on a single lithium niobate plate. The measurement of the frequency dependence of the real part of the electrical impedance of

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these IDTs showed that for SH1 mode the resonance frequency decreased monotonically with a growth in the spatial period. This behavior corresponded to a forward wave. For A1 mode the

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resonance frequency initially decreased with the growth of the IDT period and then began to increase after reaching a value of ZGV frequency. This behavior is explained by the smooth transition from the region of the forward wave to the region of the backward wave. The calculation of the frequency dependences of the real part of the electrical impedance of each transducer for the waves considered, carried out by the finite element method, turned out to be in good agreement with the experiment. 1. Introduction Recently, a great interest of the researchers has attracted by the backward acoustic waves, which are characterized by the oppositely directed phase and group velocities [1-7]. Such waves can exist in the isotropic plates [8, 9], multilayered structures [1, 10, 11], phonon crystals [12, 13], pipes, shells and hollow cylinders [14-19]. In the elastic plates, the backward acoustic waves

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can be polarized both in the sagittal plane (Lamb waves) and in the shear-horizontal direction (SH waves). The possibility of the existence of the backward acoustic waves was first predicted for Lamb waves in the isotropic plates [8]. The analysis of the dispersion relations showed that for a free isotropic plate there are the narrow frequency ranges in which the energy transfer velocity has a negative sign. In the subsequent theoretical works, the characteristics of these waves were

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actively investigated. The values of the material constants of the isotropic plates for which the backward waves of the higher orders could exist were analytically and numerically estimated [20, 21]. It has been also analytically shown that the presence of the negative curvature on the slowness surface can make a decisive contribution to the significant broadening of the frequency

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ranges for the existence of the backward Lamb waves [21]. The possibility of the existence of the waves with negative group velocity for the shear-horizontal waves in the piezoelectric plates was

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predicted theoretically [6, 22-24]. As a result of these works, it has been concluded that the necessary condition for the existence of the backward Lamb waves is the presence of two polarizations that interact at the boundaries of the plate. For the backward shear-horizontal waves, a piezoelectric effect is a prerequisite. The other condition for the existence of such waves is the presence of the concavity of the section of the slowness surface for shear bulk waves in the propagation direction of the backward SH wave.

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Another aspect of the study of the backward acoustic waves is the problem of their excitation and registration. Along with the fundamental interest, this problem has the practical value. As shown in [3] it is possible to create a new type of resonator based on an aluminum nitride film by using an acoustic Lamb wave, part of the dispersion dependence of which

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corresponds to the backward wave. Obviously, in this case, the wave has a zero group velocity (ZGV) at a certain range of the frequency. In this case, when the wave is excited with the help of

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the IDT, its Q-factor could be the greatest due to the fact that the wave energy is not radiated in both directions from the transducer. Otherwise the Q-factor will decrease with increasing the power flow along the film. Since the frequency range of existence of wave with ZGV is rather small, for realizing such device one can provide an accurate relationship between the IDT period and thickness of the film. At present, there are the several methods of exciting the backward acoustic waves. One of these methods is the excitation of the backward Lamb waves in the isotropic plates by means of a laser. As a result of heating the surface layer, the non-stationary thermo-elastic stresses arise, leading to the excitation of both forward and backward acoustic waves [2, 25-27]. The possibility of obtaining the negative reflection and focusing of the backward Lamb waves is described in [27]. It opens the prospect of the development of the acoustic super lens.

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It is known that acoustic waves in the plates could be investigated by means spatial light modulation (SLM) method [28]. In this case acoustic waves are generated by laser pulses depicted on the plate surface as regular lines. This method allows excite forward and backward acoustic waves corresponding to fixed points on the dispersion curves including ZGV Lamb modes [29, 30]. It is necessary to say that non-contact methods are favorable for generate ZGV resonances

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and also the associated Q-factor which could be high, because as soon as there is contact, the ZGV resonance is highly damped and becomes difficult to detect.

The acoustic waves in the non-piezoelectric plates can also be excited by the wedge method. In this case, the piezoelectric transducer is placed in the upper part of the wedge placed

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on the plate surface. When an alternating voltage is applied to the transducer, a bulk acoustic wave is excited, passing through the wedge into the plate. As a result, the elastic stresses arise on

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the plate surface, which lead to the appearance of an acoustic wave propagating in the plate. By varying the angles of the wedge, one can change the projection of the wave vector onto the sample surface, which makes it possible to efficiently excite both the forward and backward acoustic waves in the non-piezoelectric plates [4, 31].

It should be noted that when using the above-described method of the wedge excitation of the waves, it is important to ensure a good acoustic contact between the wedge and the plate. In

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this connection, in many works, backward waves in plates immersed in a liquid are studied both theoretically and experimentally [32-40]. As is known, the backward acoustic waves can exist near the so-called "thick resonances", at which the acoustic wave simply rereflects between the sides of the plate [21]. In this connection, the phase velocity of these waves propagating along

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the plate can reach the very large values. In the case of the contact with a liquid, this results in the emission of the wave energy into the liquid. It is known that, in contrast to the forward

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acoustic waves, the backward acoustic waves, including the leaky waves are characterized by the complex wave numbers. This leads, in the general case, to the inequality of the group and the energy transfer velocities. In addition, unlike the forward leakage waves, the amplitude of which increases with the depth, the backward leaky waves attenuate with a depth in the liquid. It follows that the application of the formula for calculating the group velocity of the acoustic leaky waves is correct only for small values of their attenuation. In connection with the foregoing, it is proposed to use other methods to analyze the group velocity of the leaky waves. These include the analysis of the energy fluxes of the leaky waves in a plate and liquid [41, 42], as well as an analysis of the phase shift of the transmitted wave through a plate immersed in a liquid [34]. Obviously, that for the excitation of the backward acoustic waves in the piezoelectric plates it is possible to use the interdigital transducers (IDTs) [3, 7]. The main problem with this

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excitation method is the proof of the excitation of the backward wave. In the general case, the IDT radiates an acoustic wave in both directions and the problem of fixing the oppositely directed phase and group velocities in this case is a nontrivial problem. One way to register exactly the backward acoustic wave is to study its propagation in a piezoelectric plate in the contact with a semiconductor layer. This layer forms a longitudinal stream of the charges and, accordingly, an acoustic electromotive force. Knowing the sign of the acoustic electromotive

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force and the type of the charge carriers, one can uniquely determine the direction of the group velocity [42].

It should be noted that the most rapt attention of the researchers is caused by the studies related to the excitation of the waves with zero group velocity in the backward mode spectrum

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[3, 7, 43, 44]. The existence of such waves is due to the presence of the sections in the spectrum of the plate backward mode, both with the negative and positive group velocity. Thus, there exist

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a point at which the value of the energy transfer velocity is equal to zero, and the phase velocity of the wave has a finite value. The possibility of exciting a Lamb wave with the ZGV in an AlN membrane 2.5 µm thick at a frequency of 2 GHz was demonstrated [3]. Thus, the problem of the excitation and registration of the backward acoustic waves in the piezoelectric plates have the fundamental and practical interest. In the presented work, a new approach to the solution of this problem is realized, based on the numerical simulation and

(wavelength).

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experimental realization of the several IDTs with the different values of the spatial period

The presented paper has the following structure. In the second section, the theoretical analysis of the dispersion dependences of the acoustic waves is made and the corresponding

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wavelengths are chosen for the experimental excitation of the backward waves. In the 3rd section we present the results of an experimental study. In the 4th section, the results of the experiment

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were compared with the theoretical data obtained with the finite element method and the Comsol package. The Section 5 presents the conclusions and here the applicability of the results obtained is discussed.

2. Theoretical analysis To find the dispersion dependences of the forward and backward acoustic waves in a

piezoelectric plate, we consider the following problem (Fig.1). The wave propagates along the axis x1 of the plate bounded by the planes x3=0 and x3=h. The regions x3<0 and x3>h corresponds to the vacuum. All mechanical and electrical variables are assumed to be constant in the direction of the x2 axis, i.e. the problem is two-dimensional. The equation of the motion, the

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Laplace’s equation, and the material equations for the piezoelectric medium have the following form [45, 46]:

ρ ∂ 2U i ∂t 2 = ∂Tij ∂x j ,

∂D j ∂x j = 0

(1)

Tij = Cijkl ∂U l ∂xk + ekij ∂Φ ∂xk , D j = −ε jk ∂Φ ∂xk + e jlk ∂U l ∂xk

(2)

Here Ui is the component of the mechanical displacement of the particles, t is time, Tij is the

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component of the mechanical stress, xj is the coordinate, Dj is the component of the electrical displacement, Φ is electrical potential, ρ, Cijkl , eikl , and ε jk are the density, elastic, piezoelectric, and dielectric constants, respectively.

In the regions x3<0 and x3>h the electrical displacement must satisfy to Laplace’s equation: (3)

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∂DVj ∂x j = 0,

where DVj = −ε 0∂ΦV ∂x j . Here index v denotes the values referred to the vacuum and ε0 is the

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vacuum permittivity.

Fig.1. Geometry of the problem

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We also should consider the mechanical and electrical boundary conditions. On the planes

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x3=0 and x3=h they have the following form: T3j =0 , Φ V = Φ ; D3V = D3 .

(4)

The solution of the boundary problem described above can be represented as a set of the

planar inhomogeneous waves [47-49]:

[ (

)]

Yi ( x1, x3 , t ) = Yi ( x3 ) exp jω t − x1 V ph ,

(5)

where i=1, 2, …, 8 for the piezoelectric plate and i=1, 2 for the vacuum; Vph is the phase velocity, ω is the angular wave frequency. Here the following normalized values are introduced [49]: * * Yi = ωC11 U i V ph , Y4 = T13 , Y5 = T23 , Y6 = T33 , Y7 = ωe*Φ V ph , Y8 = e*D3 ε11 ,

(6)

where i=1,2,3; C11* , ε11* are the normalizing material constants of the piezoelectric medium in the crystal-physical coordinate system; e* =1 and has the dimension of the piezoelectric constant.

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The substitution of (5) in the equations (1) - (4) yields the systems of 8 and 2 ordinary differential equations for the piezoelectric medium and vacuum, respectively. Each of these systems can be written in the following matrix form [49]: [A][dY/dx3] = [B][Y]

(7)

Here [dY/dx3] and [Y] are the eight-dimensional and two-dimensional vectors for piezoelectric medium and vacuum, respectively. Their components are determined in accordance with the

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expressions (6). The matrices [А] and [В] are square with the dimension of 8×8 for piezoelectric medium and one of 2×2 for vacuum. Because the matrix [А] is non-singular (det[А]≠0) one can write for each medium [49]

[dY/dx3]= [А-1][В][Y]= [C][Y].

(8)

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Further, to solve the system of the equations (7), it is necessary to find the eigenvalues β(i) of the matrix [С] and corresponding eigenvectors [Y(i)], which determine the parameters of the

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partial waves for each medium. In this case the eigenvalues β(i) and corresponding eigenvectors [Y(i)] depend not only on the material constants, but also on unknown phase velocity Vph. The general solution is a linear combination of all partial waves for each medium [49]: N

(i )

(

) ( [

Yk = ∑ Ai Yk exp β (i ) x3 exp jω t − x1 V ph i =1

]) ,

(9)

where the numbers of the eigenvalues N=8 for the piezoelectric medium and N=2 for the

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vacuum, Аi are the unknown values. For the determination of the values Аi and velocity Vph we use the mechanical and electrical boundary conditions (4), which were presented in the normalized form with considering (6). For a vacuum located in the regions x3 <0 and x3>h, the

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eigenvalues with a negative and positive real part, respectively, are excluded from the consideration, since all variables in a vacuum must have a decreasing amplitude when removed from the piezoelectric plate.

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Thus, the unknown quantities Ai and velocity Vph for each type of wave can be determined

from a system of 10 homogeneous algebraic linear equations (4). The study of secular relationships for lithium niobate plates with various crystallographic

orientations is presented in [50, 51]. As a result of the calculations, the dispersion dependences of the phase velocities of the acoustic waves in YX lithium niobate plate were constructed. As it is known LiNbO3 is an anisotropic material, i.e. it has a trigonal structure in class 3m [52]. The material constants for lithium niobate taken from [53] are presented in Table 1. For calculation of the properties of acoustic waves in YX LiNbO3 plate one should recalculate all material constants in a new coordinate system [54] by using Euler’s angles [55]. Obtained values are presented in Table 1 in corresponding cell after slash.

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Table 1. The initial/rotated material constants of LiNbO3 crystal [53] Elastic modulii, CEij,1010 [N/m2]

СE11

СE12

СE13

СE14

СE33

СE44

СE66

5.73/7.52 7.52/5.73 0.85 24.24/20.3 5.95 7.28/5.95 2 S Piezoconstants, eij [C/m ] Dielectric permittivity, ε ij /ε0 Density,[kg/m3] e15 e22 e31 e33 εS11 εS33 ρ 3.83/-2.37 2.37/1.3 0.23/-2.37 1.3/2.37 44.3 27.9/44.3 4650 It is known that for a given crystallographic orientation in lithium niobate in the range hf =

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20.3

2.5 – 3.4 km/s there are only two piezoelectric waves of the first order: shear – horizontal SH1 wave and antisymmetric A1 Lamb wave [56]. In this range of hf there exists also a nonpiezoelectric mode S1. Figure 2 shows the detailed dispersion curves for these waves. The

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velocities of two quasishear QS1, QS2 and quasi-longitudinal QL bulk acoustic waves (BAW) in

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Y-direction of LiNbO3 are equal 3958 m/s, 4523 m/s, and 6864 m/s, respectively.

Fig.2. The dispersion curves of SH1, A1, and S1 acoustic waves in YX LiNbO3 plate. It has been found that the dispersion dependence for the wave A1 has both the forward and

backward branches.

To determine the type of the aforementioned waves, we constructed the distribution of the

amplitude of the mechanical components U1, U2, and U3, normalized to the value of the displacement U surf =

(U ) + (U ) + (U ) surf 2 1

surf 2 2

surf 2 on the surface, along the thickness of the 3

plate near the cut-off frequency. These distributions are shown in Fig. 3 for these waves. The values of the parameter hf and corresponding phase velocity used in the calculation of the structure of S1, A1, and SH1 waves are shown in Table 2.

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Fig.3. The distribution of the amplitudes of mechanical displacement components U1, U2, and U3

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of (a) S1, (b) A1, and (c) SH1 acoustic waves versus normalized thickness of the plate of Y−X LiNbO3 for the parameter hf close to the cut off: (a) 2107 m/s; (b) 3286.044 m/s; (c) 2264 m/s.

One can see from Fig. 3 that for all three wave types of the first-order the instantaneous profile of the plate relatively to the central plane is always symmetrical. In addition, for the S1, A1, and SH1 waves, the displacement components U1, U3, and U2, respectively, are maximum.

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They also show that these waves are the ones of the first order. The title of the wave SH1 does not cause any questions. Now we clarify the definition of the waves S1 and A1. In order to determine their wave type (A or S), we used the traditional criterion of the maximality of the corresponding component of the displacement [9]. In accordance with this criterion, waves with

respectively.

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maximum displacement components U1 and U3 should be defined as S1 and A1 modes,

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Table 2. The parameters used in the calculation of the structure of S1, A1 and SH1 waves. S1

A1

SH1

hf [km/s]

2.107

3.286044

2.264

Vph [km/s]

76.096

10.666

90.523

It is obvious that for the excitation of the backward waves, it is possible to use the IDT. In general, this transducer excites the acoustic waves propagating in both directions. In order to record the appearance of the backward wave, it was decided to create the several IDTs on a single lithium niobate plate, differing in the spatial period, and, correspondingly, the wavelength of the excited waves. In order to determine the needed values of the spatial period λ of the IDTs

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we have used the auxiliary lines V ph = λf for different values of λ, which are presented in Fig. 2. By using these lines the pointed values of the spatial period of the IDTs which allowed us to observe the transition from the forward wave region of the dependence to the backward one were determined. It is seen from Fig. 2 that at a fixed plate thickness (h = 0.37 mm) with the growth of the period of the IDT, the resonant frequency of SH1 wave should decrease monotonically. For A1 wave in the region of the existence of a forward wave, the frequency should also decrease,

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and when the dispersion dependence passes to the backward wave region, the resonant frequency should increase.

3. Experimental part

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Based on the calculations performed, a photomask with the 19 IDTs, which differed from each other in a spatial period in the range from 0.98 mm to 1.47 mm was created on the celluloid

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carrier.

Fig. 4. The photomask for IDTs with different spatial dimensions. The creation of a matrix consisting of 19 IDTs with the different spatial period on a plate

of lithium niobate of the Y-cut with the thickness of 0.37 mm and diameter of 3.5" was carried out by using the photolithography. For this we used the photomask which is shown in Fig. 4. After photolitography, the gold wires with a diameter of 25 µm and a length of 30 mm were glued to the lags with the help of the conducting "Silver Print" glue. All the space around each transducer was covered with a quick-drying absorbing varnish for the acoustic isolation of the transducers. Obviously, without the acoustic isolation of the IDTs, the acoustic Lamb wave emitted by the selected transducer will be multiply reflected, both from neighboring IDTs and

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from the edges of the piezoelectric plate. This will lead to the appearance of a huge number of parasitic resonant peaks, against which the useful informative resonances will simply be invisible. Therefore, for the acoustic isolation of each IDT we used the pointed varnish, which was applied by a thin brush under a microscope. The layer thickness was equal to ~0.2 mm. This was quite enough to suppress the re-reflected Lamb waves. Such a method, proposed by us earlier [57, 58], was applied to the piezoelectric resonators with a lateral electric field. It has

waves and, as a result, a good resonance quality is obtained.

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been shown that the applied varnish around the electrodes effectively suppresses parasitic Lamb

The piezoelectric plate was fixed in a specially made holder having two pairs of contact legs for the connection to the LCR meter 4285A (Agilent). Figure 5 shows a photograph of the

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experimental device obtained.

Fig.5. Photo of the experimental device By using a soldering micro-iron, the gold wires of the selected transducer were soldered to

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the back side of the contact legs. The device was connected to 4285A LCR meter, and the frequency dependences of the real and imaginary parts of the electrical impedance of the IDT were measured. Then the wires of the other transducer were soldered to the contact legs and the measurements were repeated. Figure 6 shows the frequency dependences of the real part of the electrical impedance of the IDT with the spatial periods of 0.98 mm (a), 1.113 mm (b), 1.33 mm (c), and 1.47 mm (d). Here the frequency is normalized by the plate thickness h = 0.37 mm. It can be seen that, as predicted in theory, the resonant frequency of the direct SH1 wave decreases with increasing period of the IDT.

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Fig.6. Frequency dependencies the real part of the impedance of the IDTs with the spatial period (a) 0.98 mm, (b) 1.113 mm, (c) 1.33 mm and (d) 1.47 mm. Left column is experiment, right column is modeling.

As for the A1 wave with increasing the spatial period of the IDT at first its resonance

frequency decreases in the forward wave region of the dispersion dependence, and then in the region of the backward wave the resonant frequency increases. It should be noted that a small change in the resonant frequencies of the wave A1 is due to the strong steepness of the dispersion dependence in the region of the transition from the forward wave to the backward one. Figure 7 shows the experimental dependences of the resonant frequency of the SH1 (a) and A1 (b) waves on the period of the IDTs under study.

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It should be noted that the resonant frequencies of the excited waves turned out to be very sensitive to the thickness of the plate. In this connection, the dependences in Fig. 7 were constructed for the parameter hf (h is the real thickness of the plate under the chosen IDT, and f is the wave frequency). The theoretical analysis carried out in Section 2 did not take into account the problems of the excitation of the acoustic waves. For the mathematical simulation of the experimental

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situation, the finite element method and the programming package Comsol Multiphysics 5.2

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were used.

Fig.7. Dependencies of the resonant frequencies of (a) SH1 and (b) A1 waves vs IDT period.

4. Mathematical modeling by using finite element simulation The geometry of the investigated IDT and the net used in the simulation are shown in Fig.

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8. On both faces of the plate, a perfectly matched layers (PMLs) are located to prevent the reflections of the excited waves. It was assumed that these absorbing layers were characterized

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by a quadratic frequency dependence of the attenuation.

Fig.8. Topology of (a) FEM model and (b) corresponding net. It was also assumed that the plate regions outside of the IDT were mechanically free, i. e. the mechanical stresses were equal to zero. In the area of the contact of the IDT fingers with the plate, the mechanical continuities of the mechanical displacements and stresses between the finger and the plate were used as the mechanical boundary conditions. The excitation of an

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acoustic wave was modeled by applying a variable electrical potential difference to the IDT fingers (Fig.8a). In the model the IDT was represented by a set of equipotential rectangles, and the region under the electrodes was divided into the smaller elements. The linear size of these elements was equal to λ / 50 (Fig.8b). As already mentioned above, due to the use of the PML on the faces of the plate, the wave re-reflection was fully eliminated, which led to the disappearance of the parasitic peaks on the

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simulated resonance curves. As a result of the simulation, the frequency dependences of the real part of the electrical impedance of the IDT for different values of their spatial period were obtained. Figure 7 (circles) shows the theoretically obtained values of the resonance frequencies of SH1 (a) and A1 (b) waves as the functions of the period of the investigated IDT. One can see

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their good quantitative and qualitative agreement.

Figure 9 shows 2D modes shape for A1 wave at cutoff and ZGV frequencies in agreement

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with Fig.6.

Fig.9. 2D modes shape for A1 wave at cutoff and ZGV frequencies in agreement with Fig.5. The distribution of the displacements U3 and U1 are presented in left and right columns, respectively. The spatial period of IDT and hf were (a) 0.98 mm and 3.2987 km/s, (b) 1.113 mm and 3.2912 km/s, (c) 1.33 mm and 3.2668 km/s, (d) 1.47 mm and 3.28263 km/s.

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Thus, as a result of the theoretical and experimental studies carried out, it has been shown for the first time that it is possible to detect the backward acoustic waves by means of the IDTs with the different values of the spatial period.

5. Conclusion A new method for detecting the forward and backward acoustic waves in the piezoelectric plates by using a system of the acoustically isolated interdigital transducers (IDT)

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with the different spatial period is developed. This method was tested on the piezoelectric acoustic modes A1 and SH1 propagating in Y-X lithium niobate plate. Theoretical analysis has shown that the SH1 mode in the entire frequency range represented a forward wave. At that the dispersion dependence of the A1 mode near the cutoff frequency has a smooth transition from a

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forward wave to a backward one with decreasing frequency. In order to experimentally observe this transition 19 IDTs with different periods were deposited on a single Y-cut lithium niobate

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plate. The measurement of the frequency dependence of the real part of the electrical impedance of these IDTs showed that for SH1 mode the resonance frequency decreased monotonically with a growth in the spatial period. This behavior corresponded to a forward wave. For A1 mode the resonance frequency initially decreased with the growth of the IDT period and then began to increase after reaching a certain value corresponding to ZGV. This behavior is explained by the smooth transition from the region of the forward wave to the region of the backward wave. The

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calculation of the frequency dependences of the real part of each transducer for the waves considered, carried out by the finite element method, turned out to be in good agreement with the experiment.

Acknowledgement

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In frame of development and production of the photomasks for Lamb waves in LiNbO3 the work has been partially supported by Russian Science Foundation grant # 18-49-08005. In frame of

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theoretical investigation and detection of backward acoustic waves the work has been partially supported by Russian Foundation of Basic Research (grant #17-07-00608). A. Smirnov thanks Russian Foundation of Basic Research (grant #17-307-50007) for the personal financial support.

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A new method to detect backward acoustic waves in piezoelectric plates is suggested Thereto a system of acoustically isolated IDTs with various spatial periods is used Dependence of IDT resonant frequency for A1 mode on IDT period has a local minimum This minimum corresponds to transition of forward type of wave into backward one IDT resonant frequency for forward SH1 mode decreased with increase in IDT period

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