Propagation of ultrasonic beams in paratellurite crystal

Ultrasonics 37 (1999) 377–383 www.elsevier.nl/locate/ultras

Propagation of ultrasonic beams in paratellurite crystal V.N. Belyi *, N.S. Kazak, V.K. Pavlenko, E.G. Katranji, S.N. Kurilkina Stepanov Institute of Physics, National Academy of Sciences of Belarus, 70 Fr. Skaryna Avenue, Minsk 220072, Belarus Received 1 March 1998; received in revised form 1 March 1999

Abstract Analytical expressions have been obtained of the main curvatures of the concavity of the wave vector surface, corresponding to the quasi-transverse wave, versus anisotropy parameter C=C −C −2C of the paratellurite crystal. Changes in the form of 11 12 66 the sound beam on scanning in the direction of its propagation in the (001) and (11: 0) planes have been investigated. It has been found that near the acoustic axis lying in (001) considerable transformation of the beam takes place; initially extended, it becomes squeezed along the [001] axis. The conditions for focusing and channelling of ultrasound radiation have been determined. The propagation directions at which anisotropy of the paratellurite crystal does not influence the beam divergence have been found. It is shown that, for a wide range of deviation angles h of the wave normal from the [001] axis in the symmetry plane (11: 0) (h≤50°), the presence of anisotropy leads to weaker diffraction broadening. The character of variation in the direction and divergence of the sound beam on scanning in the vicinity of the [110] direction has been studied experimentally. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Beam divergence; Focusing; Non-diffraction propagation; Ultrasonic beams

1. Introduction The acoustic interaction in a tetragonal paratellurite crystal classed with 422 symmetry is used in various devices of information processing: light beam deflectors and modulators, filters and spectral analysers [1]. Its wide application in acousto-optics is due to a small value of sound velocity in the [110] direction for acoustic waves polarized along the [11: 0] axis and a quite high acousto-optic figure of merit among the known crystals [2,3]. This permits one to obtain a diffraction efficiency close to 100% with a reduction of the control sound field power to 0.5–1 W. However, in the direction mentioned, a considerable anisotropy of the TeO crystal 2 elastic properties is observed, which leads to noticeable distortions of the shape of the ultrasound beam and, consequently, to a change in the diffraction pattern that should be taken into account in calculating correctly and designing acousto-optical devices. In order to describe the peculiarities of elastic wave propagation in anisotropic media one may use the wave vector surface (e.g. see Ref. [4]). Its fundamental role is due to the fact that the direction of the normal vector * Corresponding author. E-mail address: [email protected] ( V.N. Belyi)

at any point on this surface determines the energy flux orientation in space. In addition, the curvatures W of a the wave vector surface determine the divergence of the beam propagation of the corresponding direction: at W <0 the beam is focused; at W >0 it is defocused. a a In W =0 directions the ray surface possesses sharp a faces. In these directions caustics of the acoustic field take place [5,6 ]. Mainly computer simulation is used to analyse the wave vector surface. In Ref. [7] three-dimensional images of the concavities of the paratellurite surface obtained by the Monte Carlo method are given and the existence of directions of ultrasound beam focusing has been established. In Ref. [8], sections of the ray surface, which is a pedal curve of the wave vector surface, by planes orthogonal to (001) and containing a [001] axis have been obtained. Numerical calculation confirmed the existence of caustics. Analysis of variation of the group velocity of elastic waves in the vicinity of the [110] crystallographic direction was also performed in Refs. [9,10]. However, the results obtained require further substantiation. Note that the knowledge of only the wave vector surface and the ray surface received from computer modelling does not enable one to analyse the peculiarities of distortions in the shape of a sound beam when

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scanning. But this question is of great practical importance and can only be resolved on the basis of analysis of the curvature variations of the wave vector surface. The authors of Ref. [11] have obtained analytical expressions for the curvature by characterizing the beam broadening in a single cross-section corresponding to the (001) plane at wave propagation in the symmetrical directions [100], [010], [110]. However, the behaviour of beam transformation is determined by both wave vector surfaces in the scanning plane and in the plane orthogonal to it. In the present paper analytical expressions have been obtained for the two curvatures of the wave vector surface for the direction of ultrasound beam scanning in symmetry planes (001) and (11: 0) of the paratellurite crystal. Theoretical and experimental studies of changes in the energy flux direction and distortions in the form of a beam of quasi-transverse waves of its deviation from the [110] direction have been made.

2. Peculiarities of ultrasonic beam propagation in the (001) and (11: 0) planes of the TeO crystal 2 It is known [12–14] that the vector of elastic displacement for the monochromatic ultrasonic beam in the crystal can be given in the form of the wave A(r, t)=(2p)−2 −

Z 2V

P

2

CA

dq A(q)a exp i

−2

D

qWq exp i(k r−v t), 0 0

B

Z r− U q V (1)

for which the space–time dependence of the amplitude is determined by the group velocity U and its derivative with respect to the wave vector k: W=∂U/∂k. Here q=k−k ; k , v are, respectively, the ‘central’ wave 0 0 0 vector of the beam and the frequency corresponding to it; a is the unit polarization vector; A(q) are the amplitudes of the monochromatic plane waves forming the beam; Z is the coordinate axis, which is collinear to the ‘central’ wave vector of the beam (Zdk ); V is the 0 phase velocity. The beam divergence is determined by the eigenvalues of the planar symmetrical tensor W, which is related to the principal curvatures of the wave vector surface [12]. To calculate the influence of the medium anisotropy on the beam divergence, the quadratic coefficients of anisotropy can be used:

A B A B

Here W is the eigenvalue of the tensor W; Wn , Wn , 1 1 2 are, respectively, the ratios of the beam diffraction divergence in the crystal along the directions (which are collinear to the eigenvectors b and b of the W tensor) 1 2 to the corresponding beam divergence in the isotropic medium. If |Wn |>1, the presence of anisotropy leads 1,2 to an increase in the beam divergence; in the case of |Wn |<0 the diffraction decreases compared with the 1,2 isotropic medium. At Wn =0 non-diffraction beam 1,2 propagation takes place. The beam parameters [Eq. (1)] are determined from the characteristic equation [15] (3) L=1 Sp(LL 9 )=0, 3 where L =V2d −L , L9 =1 d d L L , SpM= il il il il 2 mnl kji mk nj M ; d and d are Levi–Civita and Kronecker symbols ii mnl il respectively. L =C∞ n n ; C∞ are elastic coefficients il ijkl j l ijkl of the crystal divided by its density r. For a given direction of the wave normal n, Eq. (3) determines the phase velocities V of three isonormal waves with polari ization vectors a a =L9 /SpL (4) 9 . i k ik ik The group velocity of isonormal waves U and its derivative v can be obtained by differentiating Eq. (3): it 1 ∂L il a a , U= r 2V ∂n i l r 1 ∂L ∂L ∂(a a ) il a a + il i l −2U U . W = i l r s rs 2v ∂n ∂n ∂n ∂n r s r s (5)

A B GA B A BC

D

H

Let us now consider the peculiarities of the quasitransverse beam propagation in the (001) plane of the paratellurite crystal ( Fig. 1) which has a 422 class symmetry. In the crystallographic coordinate system the X -axis coincides with the axis of fourfold symmetry; 3

k 1 ∂U Wn =W 0 = b b ; 1 1 V V 1 ∂n 1 k 1 ∂U W2 =W 0 = b b . 2 2 V V 2 ∂n 2

(2)

Fig. 1. Cross-section of the slowness surface of the paratellurite crystal. Solid curve corresponds to the quasi-transverse (qT ) wave.

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X and X coincide with the axis of twofold symmetry. 1 2 We now use a new coordinate system XYZ for which the X-axis coincides with the wave normal of the ‘central’ component of the beam. In this coordinate system, the ˜ are related to C∞ by the following elastic coefficients C mg mg relations: C C ˜ =C∞ − sin2 2Q; C ˜ =C∞ − sin2 2Q; C 11 11 2 22 11 2 C ˜ =C∞ ; ˜ =C∞ + sin2 2Q; C C 13 13 12 12 2 C ˜ =− sin2 4Q; C 16 4

˜ =−C ˜ ; C 26 16

˜ =C ˜ =C∞ ; C 44 55 44

C ˜ =C ˜ ; C ˜ =C∞ + sin2 2Q, C 23 13 66 66 2

(6)

where C=C∞ −C∞ −2C∞ is the parameter of elastic 11 12 66 anisotropy; Q is the angle between the wave normal and the [100] axis. At such a transformation of coordinates for the components of the L tensor we can obtain the ik following expressions: ˜ n n ; ˜ n2 +2C ˜ n2 +C ˜ n2 +C L =C 16 1 2 44 3 66 2 11 1 11 ˜ n2 +C ˜ n2 +C ˜ 4n2 −2C ˜ n n ; L =C 22 66 1 11 2 4 3 16 1 2 ˜ n2 ; ˜ (n2 +n2 )+C L =C 33 3 44 1 2 33 ˜ +C ˜ )n n ; L =(C 23 13 44 2 3 ˜ )n n ; ˜ +C L =(C 44 1 3 13 13 ˜ (n2 +n2 )+(C ˜ +2C ˜ )n n . L =C (7) 12 16 1 2 12 66 1 2 Taking into account Eq. (6), it follows from Eq. (4) that the change in the polarizations of quasi-transverse waves propagating in the (001) plane is determined by ˜ −C ˜ +앀A ˜ C −2C 66 11 16 , (8) ˜ ˜ −C ˜ +앀A 2앀A −2C C 16 66 11 ˜ +C ˜ + ˜ )2+4C ˜ 2 , and where V=[1 (C with A=(C 11 66 16 2 11 ˜ 앀 C − A)]0.5 is the phase velocity. Using the expressions 66 in Eqs. (5), (6) and (8), one can show that at wave normal n scanning in the (001) plane the angle y between the direction of beam propagation and the group velocity vector is determined by the relation a · a=

1

C

˜ ) ˜ +C ˜ (C C U 12 . tan y= 2 =− 16 11 V V2앀A

D

(9)

The dependence y(a) calculated on the basis of Eq. (9), where a=Q−(p/4), is given in Fig. 2. In our calculation we used the following values of elastic constants [16 ]: C =55.7 GPa; C =51.2 GPa; C =21.8 GPa; 11 12 13 C =105.8 GPa; C =26.5 GPa; C =65.9 GPa, and 33 44 66 crystal density r=5990 kg/m3. It is seen that the maximum change in the direction of the energy flow of the

Fig. 2. The dependence of angle y between the group velocity U vector and the normal n on the angle a between the normal n and the [110] axis for a slow quasi-transverse wave propagating in the (001) plane of paratellurite.

quasi-transverse wave takes place in the vicinity of the longitudinal normal [110]. Such a property of the wave vector surface of the TeO was used [17,18] for con2 trolling the acoustic beam in a wide range of angles (~70°). Let us consider now how the curvatures of the wave vector surface for quasi-transverse waves propagating in the (001) plane and, consequently, the quadratic coefficients of anisotropy change. Using Eq. (5) we obtain 1 Wn = 1 V2

C

× V2−U2 + 2

D

˜ −2C ˜ −C ˜ ) (C ˜ +C ˜ ) 4V2U2 +(C 2 11 66 12 11 12 , 앀A

˜ −C ˜ +앀A 1 C 11 ˜ +C ˜ )2 . (10) ˜ − 66 Wn = (C 2C 2 2V2 13 44 44 ˜ −V2) 앀A(C 44 The parameter Wn characterises the beam divergence in 1 the direction orthogonal to the wave normal in the (001) plane, and Wn determines the beam divergence 2 along the [001] axis. As can be seen from the curves in Fig. 3 calculated with the help of the expressions in Eq. (10), in the vicinity of the longitudinal normal [110], where the change in the dependence y(a) is most dramatic, there exists a region of elliptical points with positive curvatures of the wave vector surface (25.7°
C

D

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be given in the form: ˜ ∞ =C∞ +1 C; C ˜ ∞ =C∞ −1 C; ˜ ∞ =C∞ −1 C; C C 12 12 2 22 11 2 11 11 2 ˜ ∞ =C∞ +1 C; C ˜ ∞ =C∞ ; C ˜ ∞ =C∞ ; C 66 66 2 13 13 23 13 ˜ ∞ =C∞ ; ˜ ∞ =C ˜ ∞ =C∞ ; C ˜ ∞ =0; C ˜ ∞ =−C C 55 44 44 33 33 16 26 ˜ ∞ =C ˜ ∞ =C ˜ ∞ =C ˜ ∞ =C ˜ ∞ =C ˜ ∞ =0. C (11) 14 24 25 34 35 36 Using Eqs. (3)–(5) and (11) and taking into account that the beam propagates at angle h to the [001] axis, i.e. n∞ =sin h, n∞ =cos h, the following expression for 1 2 the group velocity vector can be obtained: U=

Fig. 3. The dependence of quadratic coefficients of anisotropy Wn 1,2 on angle Q in the (001) plane of paratellurite.

the vicinity of the directions with a zero curvature, planar regions of the wave vector surface appear. This leads to the absence of diffraction beam divergence. From Fig. 3 it follows also that for directions of acoustic ˜ =V2, i.e. at Q$17°, the curvature axes for which C 44 W calculated with the aid of Eq. (5) is not determined. 2 However, for directions near the acoustic axis, the modulus of W characterising the beam divergence 2 along the [001] axis sharply increases, and the sign of W is changed. This means that the ultrasonic beam is 2 defocused along the [001] axis provided that Q<17° and the beam is focused at Q>17°. It is seen that there are two regions of hyperbolic points of the wave vector surface where the principal curvatures have different signs. In the first region of angles 0°
1 V∞

˜ ∞ n∞ e∞ ), ˜ ∞ n∞ e∞ +C (C 44 3 3 66 1 1

(12)

˜ ∞ n∞ 2+C ˜ n∞ 2 is the phase velocity of the where V∞=앀C 66 1 44 3 slow transverse wave propagating in the (11: 0) plane and polarized along the [11: 0] direction. The vector e∞ 1 is collinear with the [110] axis. According to Eq. (12), the angle of deflection b of the energy flow from the [001] axis is determined by ˜∞ C tan b= 66 tan h. ˜∞ C 44 For the principal curvatures we obtain 1 Wn∞ = 1 V∞2

(13)

G

1 ˜∞ − C 11 (L∞ −V∞2) (L∞ −V∞2)−L∞ 2 11 33 13 ˜ ∞ +C ˜ ∞ )2n∞ 2+(L∞ −V∞2) ×[(L∞ −V∞2)(C 11 13 44 3 33

H

˜ ∞ )2n∞ 2−2L∞ (C ˜ ∞ +C ˜ ∞ ) (C ˜ ∞ +C ˜ ∞ )n∞ n∞ ] ; ˜ ∞ +C ×(C 12 66 1 13 13 44 12 66 1 3 1 Wn∞ = 2 V∞2

A

B

˜ ∞ 2n∞ 2+C ˜ ∞ 2n∞ 2 C 44 3 , ˜ ∞ +C ˜ ∞ − 66 1 C 66 44 V∞2

(14)

where ˜ ∞ n∞ 2+C ˜ ∞ n∞ 2; L∞ =C 11 11 1 44 3 ˜ ˜ L∞ =C∞ n∞ 2+C∞ n∞ 2; 33 3 44 1 33 ˜ ∞ )n∞ n∞ . ˜ ∞ +C L∞ =(C 44 1 3 13 13 The dependencies Wn∞ (h) calculated with the help 1,2 of the expressions in Eq. (14) are given in Fig. 4. It can be seen that for all the directions of transverse beam propagation 0°
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Fig. 4. The dependence of quadratic coefficients of anisotropy Wn∞ 1,2 on angle h in the (11: 0) plane; h is the angle between the wave normal n∞ for the quasi-transverse beam and the [001] axis.

3. Experimental arrangement In the present paper an experimental study of the change in the spatial configuration of the quasitransverse wave ultrasound field in the (001) plane depending on the value of wave vector deviation from the direction of the longitudinal normal [110] in the TeO crystal has been carried out. The section by the 2 (001) plane is of primary interest because of the strong energy walk-off at a relatively small deviation of the wave normal from the [110] direction. The spatial distribution of the ultrasound beam was measured by the shadow ( Toepler) method. The method is based on the light wave refraction in a medium with local changes in the refractive index induced by the passing ultrasound beam. The light wave passes through an acousto-optical cell and is focused on the diaphragm. The light scattered from refractive index inhomogeneities passing through the diaphragm reaches the measurement plane where it induces a pattern definitively reflecting the ultrasound field distribution in the medium. The transversely polarized in the direction [11: 0] beam with a 10 MHz frequency was excited by a lithium niobate piezoelectric transducer of +163yz cut. The transducer size was 2 mm along the [11: 0] axis and 8 mm along [001] axis. The ultrasonic wave was launched into the crystal through a glass buffer. In order to optimize the process of ultrasonic beam excitation, the electrical impedance of the piezoconverter was matched with that of the high-frequency generator at the excitation frequency by the autotransformer circuit. At continuous excitation of the acoustic wave the acoustic field distribution represents a standing wave of complex configuration owing to a superposition of the

381

incident and the reflected beams from the crystal faces ultrasonic waves. The pattern obtained in this case by the shadow method corresponds to the Sheffer–Bergman diffraction and gives no idea about the geometrical parameters and the character of ultrasonic beam propagation in a crystal. To observe the propagation of a beam of travelling acoustic waves in the bulk of a crystal the ultrasound was generated by pulses (trains) with a duration of about 5 ms. The piezoelectric transducer was excited by a resonance generator with a matched load of 50 V, which permits us to obtain a high-frequency power up to 50 W at continuous generation and higher by one order of magnitude at amplitude modulation. When the piezoconverter was excited by 5 ms pulses, the peak power density of the ultrasonic beam was ~50 W/cm2. The ultrasonic frequency generator and the recording system were initiated in synchronism to enable us to detect the photometric response of the first ultrasonic train before a standing wave becomes steady in the crystal bulk. The amplitude intensity distribution of the light scattered by the ultrasonic wave in the (001) plane of the crystal was measured in our experiments. Measurements were made using an automated program-controlled system of two-coordinate scanning of the bulk of the crystal by radiation from an He–Ne laser (l=0.63 mm). The scanning and recording system was realized on the basis of equipment of CAMAC standard. Scanning was carried out by displacing the crystal relative to the laser beam by two step-motors that operate in the start–stop regime. The space resolution of the recording system was determined by the probing laser beam diameter. In our case the resolution was 0.5 mm (40×40 points at a 20×20 mm2 area being scanned ). The scanning time for one distribution was determined by the velocity of crystal displacement relative to the probing laser beam and was about 20 min. During measurements of each individual distribution the acoustic beam power was being kept constant. The scattered radiation was registered by a photodiode whose pulsed signal was amplified and detected by a peak detector. The signal level obtained was digitized by a ten-digit analog-to-digital converter and entered into the computer for further processing and graphical representation of the amplitude distribution. In the process of measurement the result was averaged over 100 pulses for each distribution point. Measurements of the ultrasound field distribution were made in the (001) plane for different directions of the acoustic wave vector relative to the [110] axis. The direction of the wave vector of the ultrasonic wave was changed by abrasion, at a certain angle, of the face of the crystal through which the ultrasound came. The angle accuracy of abrasion was 0.03°.

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4. Experimental results The experimental results are presented in the form of sections of measured amplitude distributions of the ultrasound field by surfaces of equal intensities. For each distribution the ultrasound intensity is normalized to the maximum value. Fig. 5(a)–(d) shows the most characteristic configurations of the acoustic field in the plane (001). The direction [110] there corresponds to the vertical axis Y. The pattern of the acoustic field thus obtained permits one to judge the direction of propagation and divergence of the beam. The direction of the acoustic wave energy flux was determined from the condition of maximum intensity of light scattered by the ultrasound, which corresponds to the signal amplitude maxima in the experimentally obtained distributions. In accordance with this, for each fixed value of Y in amplitude distributions ( Fig. 5), the coordinate X corresponding to the maximum value of the amplitude measured at two-coordinate scanning was found. Consequently, a set of points with coordinates X, Y in the crystal with maximum ultrasound intensity observed was separated for each pattern of the field. Using the least-squares technique for each distribution, a straight line was plotted through these points (see dotted arrows

in Fig. 5); this was taken as the direction of the energy flux. Because the ultrasound beam has a complex structure the accuracy of measurement of the energy walkoff angle y ( Fig. 2) was 8°. It is seen that the direction of ultrasound beam propagation depends on the value of its wave vector deviation from the [110] axis. Fig. 5(a)–(d ) shows the amplitude distributions for deviation angles of the sound wave vector from the direction [110]: 1.3°, 0.9°, 0.55°, and 0.27° respectively. The angular shifts of the acoustic field from the [110] axis corresponding to these values are 40°, 33°, 24° and 8° respectively. Fig. 2 gives the experimental measurement data for angle y of ultrasound beam energy walk-off at wave vector deviation from the [110] direction by angle a. A similar change in the ultrasound field also takes place when its wave vector departs from the [110] axis in the opposite direction. From the behaviour of ultrasound field distributions it is seen that for all the sections investigated the deviation of the beam energy flux from this axis is enhanced with increasing angle between the wave vector and the [110] axis. It should be noted that the general structure of the beam is preserved. As can be seen from Figs. 2 and 5, there is a considerable change in the direction of the energy flux at small angles of wave

Fig. 5. Sections of three-dimensional amplitude distribution of ultrasonic field by surfaces having equal intensities in the (001) plane for angles of deflection a of the acoustic wave vector from the [110] axis of: (a) 1.3°; (b) 0.9°; (c) 0.55°; (d) 0.25°.

V.N. Belyi et al. / Ultrasonics 37 (1999) 377–383

vector deviation from [110], but the form of the beam changes slightly, which is in agreement with theoretical data (Figs. 2 and 3).

383

beam energy flow when scanning in the vicinity of the [110] direction. It is shown that the theoretical data are in good agreement with experiment.

Acknowledgements 5. Conclusions The explicit dependence of the principal curvatures of the surface of quasi-transverse wave refraction on elastic constants, the parameter of acoustic anisotropy C=C −C −2C and the direction of beam propaga11 12 66 tion in the coordinate planes (001) and (11: 0) of the TeO crystal has been established. It is shown that for 2 all directions of the wave normal in the (11: 0) plane there are elliptical points with positive curvatures of the wave vector surface. From the expressions obtained in Eq. (10) and numerical values of elastic constants [16 ] it follows that in the (001) plane there are two parabolic points at which one of the curvatures vanishes: Q=12.7° and Q=25.7°. In the [110] direction, normalized curvatures Wn reach their maximum values: Wn (45°)$ 1,2 1 52.7 and Wn (45°)$11.8. At Q=24.8° the principal 2 curvatures Wn =1, meaning that in this direction the 1 beam divergence in the (001) plane is equal to the diffraction limit. It has been found that for angles 0°
The authors wish to thank A.G. Khatkevich for useful comments. This work is supported by the International Science and Technology Centre in the frame of the project ISTC-B-078-97.

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