ISSN 1063-7850, Technical Physics Letters, 2008, Vol. 34, No. 1, pp. 11–13. © Pleiades Publishing, Ltd., 2008.Original Russian Text © I.A. Borodina, B.D. Za

œ

tsev, I.E. Kuznetsova, A.A. Teplykh, 2008, published in Pis’ma v Zhurnal Tekhnichesko

œ

Fiziki, 2008, Vol. 34, No. 1, pp. 26–32.

11

It is well known that hybrid (coupled) waves canexist in magnetic materials [1], dielectric electromag-netic waveguides, and optical fibers [2]. Previously, wepredicted [3, 4] the existence of antisymmetric andsymmetric hybrid acoustic Lamb waves and the zero-and high-order transverse shear (SH) waves in piezo-electric plates. Recently, the phenomenon of hybridiza-tion was theoretically studied in much detail for the for-ward acoustic waves in piezoelectric plates of lithiumniobate [5].

This paper presents the results of theoretical investi-gation of the hybridization of backward waves in piezo-electric plates, in particular, of potassium niobate(KNbO

3

).

We have formulated and solved the problem ofacoustic wave propagation in a piezoelectric crystalplate. The corresponding system of equations includedthe equations of motion of the elastic medium, theLaplace equation for the medium and vacuum, theequations of state for the piezoelectric crystal and vac-uum, and the corresponding mechanical and electricalboundary conditions [6]. The general solution of thissystem of equations was found in the form of a systemof plane inhomogeneous partial waves. Then, the wavephase velocity and the distributions of amplitudes of allmechanical and electrical variables in depth of the platewere determined using the method proposed by Farnell[7]. We considered the main crystallographic cuts (

X

,

Y

,

Z

) of the crystal and various directions of wave propa-gation in the plates of piezioelectric materials (lithiumniobate, lithium tantalate, langasite, langanite, and

potassium niobate). The material constants for thesecrystals were taken from [8–10].

Using the obtained solution, we constructed plots ofthe phase velocities versus parameter

hf

(where

h

is theplate thickness and

f

is the wave frequency) for theacoustic Lamb waves and the high-order SH wavespropagating in the materials indicated above. An analy-sis of these dependences allowed us to select frequencyintervals in which the coupled backward waves couldexist. A specific feature of these waves is the growth intheir phase velocity with increasing value of the

hf

product. The results of this analysis showed that thebackward acoustic waves could exist in all materialslisted above, but the hybridization of these waves ispossible only in potassium niobate. This fact is proba-bly related to an extremely pronounced anisotropy inthe properties of this material in comparison to theother piezoelectric crystals studied. For example,potassium niobate is characterized by strongly differentdielectric permittivity components

ε

11

and

ε

22

, amount-ing to 37 and 780, respectively.

Figure 1 shows plots of the phase velocity

V

ph

versusparameter

hf

for acoustic waves propagating in the

Y

+30

°

direction of an

X

cut potassium niobate crystal. Ascan be seen, most dispersion curves consist of twobranches (solid lines correspond to forward waves,while dashed lines represent backward waves). Thesedependences show that the phase velocity grows withthe

hf

product on the backward branches and there aretwo cutoff frequencies. In addition, the branches ofbackward waves have characteristic regions of spa-

Hybridization of Backward Acoustic Wavesin Piezoelectric Plates

I. A. Borodina*, B. D. Za

œ

tsev, I. E. Kuznetsova, and A. A. Teplykh

Institute of Radio Engineering and Electronics (Saratov Branch), Russian Academy of Sciences, Saratov, Russia*e-mail: borodinaia@yandex.ru

Received May 10, 2007

Abstract

—The phenomenon of hybridization of the backward acoustic waves propagating in a piezoelectriccrystal plate has been studied. In an electrically free plate (in particular, of potassium niobate) with a crystalorientation for which a sagittal plane is the symmetry plane, the dispersion curves of backward acoustic wavesexhibit points of intersection and hybridization is absent. However, for a small change in the direction of wavepropagation, the dispersion curves exhibit “repulsion” and the waves become coupled. The degree of hybrid-ization is quantitatively evaluated in terms of the hybridization coefficient, which is defined as the ratio of thetotal mutual energy density and the total energy density of the interacting waves. It is demonstrated that theextent of repulsion of the dispersion curves for the interacting waves is determined by the dependence of thehybridization coefficient on the product of the plate thickness and the wave frequency.

PACS numbers: 43.20.+g

DOI:

10.1134/S1063785008010045

12

TECHNICAL PHYSICS LETTERS

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No. 1

2008

BORODINA et al.

tiotemporal synchronism, where these curves exhibitrepulsion and the waves become coupled.

Detailed analysis showed that the phenomenon ofhybridization of the backward waves has generally thesame character as that of the forward waves [5]. It wasestablished that, on the passage through the region ofspatiotemporal synchronism, the interacting wavessmoothly change their polarization and type dependingon the

hf

product. In addition, it was found that, if a sag-ittal plane coincided with the symmetry plane, the dis-persion curves had points of intersection where hybrid-ization was absent. This behavior implies that, despitethe equality of phase velocities and frequencies, thewaves do not interact and only exhibit degeneracy at theindicated points.

Figure 2 shows the dispersion curves plotted as

V

ph

versus

hf

product for the SH

4

and S

4

backward waves in

X–Y

cut potassium niobate crystal. In this crystallo-graphic situation, a sagittal plane is the symmetry planeand the dispersion curves of backward waves merelyintersect (Fig. 2a). However, this degeneracy isremoved when the wave propagation direction

δ

is onlyslightly changed. This is illustrated in Fig. 2b, whichshows that the dispersion curves of the given pair ofwaves already begin to repulse for

δ

= 0.1

°

. The furtherincrease in the detuning

δ

within small limits leads toan increase in the extent of repulsion of the dispersioncurves and in the corresponding difference between thephase velocities of the SH

4

and S

4

waves (see Fig. 2c for

δ

= 2

°

).The degree of hybridization of the backward acous-

tic waves was quantitatively characterized in terms of

the hybridization coefficient

M

defined as the ratio ofthe total mutual energy density and the total energydensity of the interacting waves [5]. Figure 3 shows theplots of

M

versus

hf

for various detuning

δ

of the S

4

andSH

4

waves propagating in the electrically open

X

–

Y

+

δ

cut KNbO

3

crystal plate. As can be seen, the hybrid-ization coefficient vanishes for

δ

= 0 (curve

1

). For

δ

≠

0, the coefficient

M

exhibits a resonance behavior(curves

2

and

3

) and the width of this resonance growswith the extent of detuning. Thus, the degree of cou-pling of the hybrid waves is characterized by the widthof the dependence of

M

on the

hf

product [5].We have also studied the effect of the electrical

shorting of the piezoelectric plate on the hybridizationof backward acoustic waves. It was found that the cou-pling of waves is retained upon the shortage of one or

70

10

02 14

hf

, km/s

V

ph

, km/s

60

50

40

30

20

4 6 8 10 12

Fig. 1.

Plots of the phase velocity

V

ph

versus

hf

product forhigh-order acoustic waves propagating in the

X

–

Y

+ 30

°

cutof a potassium niobate crystal (solid and dashed branchescorrespond to the forward and backward waves, respec-tively).

hf

, km/s

V

ph

, km/s50

40

30

SH

4

S

4

SH

4

S

4

(a)

50

40

30

SH

4

S

4

SH

4

S

4

(b)

50

40

30

SH

4

S

4

SH

4

S

4

(c)

8.92 9.00 9.02

Fig. 2.

Plots of the phase velocity

V

ph

versus

hf

product forthe S

4

and SH

4

waves propagating in the

X

–

Y

+

δ

cutKNbO

3

crystal for

δ

= 0 (a), 0.1

°

(b) and 2

°

(c).

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HYBRIDIZATION OF BACKWARD ACOUSTIC WAVES IN PIEZOELECTRIC PLATES 13

both sides of the plate, but the region of spatiotemporalsynchronism appears at significantly smaller values ofthe

hf

product.In conclusion, our investigation showed that back-

ward acoustic waves can exist in all of the piezoelectricmaterials under consideration (lithium niobate, lithiumtantalate, langasite, langanite, and potassium niobate),but the hybridization of these waves is only possible inpotassium niobate. This fact can be explained by a

much more pronounced anisotropy in the properties ofthis material in comparison to the other piezoelectriccrystals studied. From this we infer that strong anisot-ropy is a necessary condition for the hybridization ofbackward acoustic waves.

Acknowledgments.

This study was supported bythe Russian Foundation for Basic Research (projectno. 05-02-16947a).

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Translated by P. Pozdeev

1.0

–0.58.98 9.02

hf, km/s

M

0.5

0

8.99 9.00 9.01

3

2

1

Fig. 3. Plots of the hybridization coefficient M versushf product for the S4 and SH4 waves propagating inthe electrically open X – Y + δ cut KNbO3 crystal plate forδ = 0 (1), 0.1° (2) and 2° (3).