1 Introduction

Acoustic anisotropy of crystals, which is usually associated with the bulk acoustic wave (BAW) slowness surface as its geometrical image, plays an important role in solution of various problems related to design and optimization of devices based on propagation of acoustic waves (bulk waves, surface acoustic waves, Lamb waves, etc.) in single crystals, layered, and composite structures. The basics of acoustic wave propagation in homogeneous anisotropic media were previously well developed theoretically [1,2,3,4,5,6,7,8,9,10,11,12] and include the number and type of BAW degeneracy directions (acoustic axes) in crystals of different symmetry, the behavior of polarization fields and their transformation under perturbation of material constants, propagation of acoustic beams in anisotropic media, and other important aspects.

New impulse in investigation of acoustic anisotropy has been recently given by fast development of novel material fabrication technologies, which allow combining crystals with different acoustic properties in one structure. In this case, analysis and visualization of acoustic anisotropy is very important for efficient selection of combined materials and crystal orientations and helps to achieve the required set of acoustic wave characteristics in the final structure. Deep understanding of BAW anisotropy is also crucial for design of acoustic metamaterials.

Among other crystal orientations, the areas characterized by strong or even extreme anisotropy are of special interest, because they allow variation of acoustic wave velocities and other characteristics in a wide range and can be potentially useful for design of devices with diversity of required parameters. Strong anisotropy of BAW properties leads to such phenomena as oblique beam propagation with extremely large walk-off angles and anomalous backward reflection [13], important for acousto-optic and some other applications. Several following examples illustrate intentional use of strong BAW anisotropy in ultrasonic devices.

Anisotropic propagation of bulk acoustic waves in solids is used in the design of various acousto-optic devices [14,15,16,17,18,19]. Those examples include oblique BAW beam propagation enabling quasi-collinear type of diffraction and BAW autocollimation used in multichannel modulators. During the last decade, the effect of BAW beam structure in strongly anisotropic crystals on performance of acousto-optic devices has been studied [20,21,22]. More recently, attention has been focused on crystals of lower symmetry systems, namely orthorhombic and monoclinic ones [23,24,25,26,27,28]. Acousto-optic interaction at GHz-range frequencies in anisotropic microstructures and resonators is also at the edge of applied research [29,30,31].

Acoustic anisotropy plays an important role in investigation of surface acoustic waves (SAW) and optimization of single-crystal substrates for SAW devices: filters, delay lines, sensors, etc. [32, 33]. The symmetry of a substrate material determines orientations, in which the generalized SAW degenerates into the shear-horizontally (SH) polarized quasi-bulk SAW or sagittally polarized Rayleigh wave solutions required for application in certain types of SAW filters and sensors. More detailed analysis of acoustic anisotropy enables finding leaky waves suitable for application in high-frequency SAW devices. Though a leaky SAW propagates faster than the slowest BAW and attenuates because of BAW radiation into the substrate, the existence of “non-leaky” waves with negligible attenuation and their location on a leaky wave branch [34, 35] can be also predicted from analysis of acoustic anisotropy, for example via calculation of “exceptional wave” (EW) lines [36]. These lines comprise one-partial homogeneous solutions of SAW problem and give rise to the branches of quasi-bulk low-attenuated leaky SAWs [37]. In some crystals characterized by strong acoustic anisotropy (quartz, \(\hbox {TeO}_2\), \(\hbox {Li}_2\hbox {B}_4\hbox {O}_7\)) such leaky SAWs show quasi-longitudinal structure [35, 38, 39]. They are 1.5–2 times faster than typical SAWs and facilitate manufacturing of high-frequency SAW devices.

BAW properties are known to change rapidly around conical acoustic axes [8, 10]. These singular directions are characterized by at least two BAWs with different polarizations propagating with equal velocities, and exist in all known acoustic crystals. Therefore, the search and classification of BAW degeneracy directions is an important part of analysis of acoustic anisotropy. Degeneracy directions can be found as solutions of a set of algebraic equations. The possible number of acoustic axes in a crystal of given symmetry and additional constraints, e.g., position of acoustic axes relative to symmetry elements of the point group, can be predicted theoretically. This approach was characterized as “analyze rather than solve” [10]. It predicts the directions of extreme acoustic anisotropy around singular directions, but it is not able to reveal such areas if they exist in non-singular directions, because they do not satisfy formal degeneracy conditions.

A different approach to the search of orientations with extreme acoustic anisotropy is based on computational analysis of BAW properties in a crystal. In this work, we propose to use normalized curvature of the slowness surface as a quantitative measure of anisotropy. In any point except singular directions, each sheet of the BAW slowness surface is smooth and has two principal curvature values. In normalized form, the principal curvatures are the eigenvalues of the beam diffraction tensor. The choice in favor of this quantity has several reasons. First, it is a clear physical interpretation of the slowness surface curvature since its magnitude is proportional to far field divergence of the beam. Second, it does not just allow searching for acoustic axes, but also highlights non-singular directions of extreme BAW anisotropy. Finally, computation is performed by the same procedure for any crystal system, either with or without piezoelectric effect, and requires only the knowledge of elastic moduli and piezoelectric constants, as well as the dielectric tensor and material density.

As a contribution to the field, we analyze the usage of BAW slowness surface curvature tensor to find directions with strong acoustic anisotropy. Thus, a robust numerical procedure for complete visualization BAW beam anisotropy is proposed. This procedure does not impose any limitations on symmetry of the crystal. The proposed method helps to solve some problems related to propagation of acoustic waves in crystals and optimization of crystal orientations for different applications. As the examples, we observe acoustic crystals widely used in acousto-optic and acousto-electronic technology: paratellurite (TeO\(_2\)), lithium niobate (LNO, LiNbO\(_3\)) and potassium gadolinium tungstate (KGW, KGd(WO\(_4\))\(_2\)). The examined crystals belong to different crystal systems: tetragonal for paratellurite, trigonal for LNO and monoclinic for KGW. Our calculation results fully comply with theoretical predictions on properties of conical BAW axes in crystal and give insight on applications of BAW anisotropy in device physics.

2 Methodology

2.1 Acoustic axes in anisotropic solids

In this section, we briefly observe the state of art in theoretical analysis of acoustic axes in crystals.

An acoustic axis is one of specific directions in a crystal, in addition to the longitudinal and transverse normals [1]. It is characterized by equal phase velocities of at least two BAWs propagating along the same direction. The theory of acoustic wave degeneracy was previously developed by Khatkevich [3], Alshits et al. [4, 5, 12], Shuvalov and Every [6, 7], Boulanger and Hayes [9], Vavryčuk [11]. The most comprehensive surveys on the topic were made by Shuvalov [8] and Alshits and Lothe [10]. Here, there are some basic statements about acoustic axes in crystals obtained theoretically, which are important for correct understanding and interpretation of numerical results presented in Sect. 3.

  1. 1.

    The local geometry of the velocity or slowness surface sheets in the vicinity of acoustic axis depends on the type of degeneracy of the acoustic tensor. Tangent degeneracy occurs along a fourfold symmetry axis in tetragonal or cubic crystals. Conical degeneracy is more typical and exists along a threefold symmetry axis, in the planes of crystal symmetry or outside such planes (“off-plane” acoustic axes). In transversely isotropic media, acoustic axes build the cones or the lines on the stereographic projection of propagation directions.

  2. 2.

    The maximum number of acoustic axes in monoclinic, triclinic and orthorhombic crystals is sixteen. One, five or nine acoustic axes can exist in a tetragonal crystal. In a trigonal crystal, the total number of acoustic axes is four, ten or sixteen.

  3. 3.

    Acoustic axes were found in all known crystals. However, Alshits and Lothe proved that a model crystal without acoustic axes can exist [10].

  4. 4.

    The maximum number and types of acoustic axes do not change if piezoelectric effect is present in a crystal but the actual number and locations of acoustic axes generally change. In piezoelectric crystals the “stiffened” acoustical tensor, with added piezoelectric contributions, is used instead of pure elastic tensor.

  5. 5.

    Conical acoustic axes are stable to perturbations of elastic or piezoelectric properties. They shift or split rather than disappear when these properties change because of variable growth conditions or after-growth treatment (for example, in chemically reduced LNO wafers with suppressed pyroelectric effect for easier fabrication of SAW devices).

  6. 6.

    Due to degeneracy of acoustical tensor, an infinite number of BAW modes with different polarization vectors can propagate parallel to the acoustic axis. For conical axes, the Poynting vectors form a cone while propagation direction rotates around the axis. Hence, the phenomena of conical refraction can be observed [2, 40, 41].

  7. 7.

    Acoustic axes are associated with singularities in BAW polarization field. Tangent axes have integer topological charge (Poincaré index) \(N=-1,0,1\). Conical and wedge-point axes have fractional topological charge \(N=\pm 1/2\) (or possibly \(N=0\) for wedge-point axes). Disappearance and splitting of axes under phase transitions and small perturbations of material constants is constrained by conservation of the topological charge.

2.2 BAW diffraction tensor

The solution of Christoffel equations determines three scalar functions of phase velocities V for any wave normal \({\mathbf{n}}\). They are commonly represented as the slowness surface \(V^{-1}({\mathbf{n}})\). The normal vector to the slowness surface is the group velocity \(\mathbf{S}\). The second derivatives of the slowness surface characterize its curvature.

For calculation of the BAW diffraction tensor, the following procedure is used [42, 43]. The normalized transverse component of the group velocity vector \(\mathbf{S}\) is defined as

$$\begin{aligned} \mathbf{g} = \mathbf{S} /V - \mathbf{n} \end{aligned}$$
(1)

A fragment of a slowness surface with the group velocity vector is illustrated in Fig. 1. Hereinafter, the wave normal direction is characterized by two Euler angles \(\theta \) (between \(\mathbf{n}\) and \(x_3\) axis) and \(\varphi \) (between projection of \(\mathbf{n}\) on \(x_1 x_2\) plane and \(x_1\) axis). In a general case, principal curvature planes can be arbitrarily rotated with respect to global coordinate axes \(x_1 x_2 x_3\). The local frame of reference \(x'_1 x'_2 x'_3\) is associated with the curvature of the slowness surface in the direction of the wave normal \(\mathbf{n}\).

The components of the diffraction tensor are defined as

$$\begin{aligned} W_{ij} = \delta _{ij} - n_i n_j + g_i g_j + \frac{\partial g_i}{\partial n_j}, \end{aligned}$$
(2)

where \(\delta _{ij}\) is the Kronecker delta. The diffraction tensor \({\widehat{W}}\) is symmetrical and planar, i.e., \({\mathbf{n}} {\widehat{W}} {\mathbf{n}}=0\). The tensor can be diagonalized to the form

$$\begin{aligned} {\widehat{W}}' = \begin{pmatrix} w_1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad w_2 &{}\quad 0 \\ 0 &{}\quad 0 &{} 0\quad \\ \end{pmatrix}, \end{aligned}$$
(3)

where \(w_1, w_2\in {\mathbb {R}}\) are the eigenvalues. Local axes \(x'_1\) and \(x'_2\) are selected along the eigenvectors of the tensor \({\widehat{W}}\) corresponding to the eigenvalues \(w_1\) and \(w_2\), so that \(w_1>w_2\) for each of the BAW modes. Wave normal \(\mathbf{n}\) is the eigenvector of \({\widehat{W}}\) with zero eigenvalue; therefore, \(x'_3\) axis is collinear with \(\mathbf{n}\). Thus, any direction in a crystal can be characterized with six \(w_{\beta }^{(\alpha )}\) coefficients, where \(\alpha =1,2,3\) is the index of the BAW mode and \(\beta =1,2\) is the index of the \({\widehat{W}}\) tensor eigenvalue.

As defined by (2), the diffraction tensor is a normalized and dimensionless curvature tensor. The eigenvalues \(w_1\) and \(w_2\) are the scaling factors of the Green’s function in the convolution integral along axes \(x_1'\) and \(x_2'\) [44, 45]. As the result, the far-field beam divergence is \(|w_\beta |\) times larger than in an isotropic medium. The beam area in the far field is proportional to \(|w_1 w_2|\), which therefore characterizes total diffraction loss.

Fig. 1
figure 1

Illustration to the definition of BAW diffraction tensor and its relation to the slowness surface curvature

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The procedure for computation of the BAW diffraction tensor is the following. First, the phase velocities \(V^{(\alpha )}({\mathbf {n}})\) are calculated from the Christoffel equation. The modes are sorted by their phase velocity, \(V^{(1)}\geqslant V^{(2)}\geqslant V^{(3)}\) (fast, medium and slow). Second, the components of the group velocity vector \({\mathbf {S}}\) are calculated as partial derivatives of \(V({\mathbf {n}})\) and the transverse component (1) is found. Third, the derivatives of \(\mathbf{g}\) are calculated and the diffraction tensor \({\widehat{W}}\) is composed according to Eq. (2). Fourth, the eigenvalues of \({\widehat{W}}\) are found and sorted.

Six diffraction coefficients are calculated as functions of propagation direction. The calculation procedure takes into account piezoelectric effect as an additional term in the acoustic tensor. To visualize the simulation results we use contour plots of each coefficient \(w_{\beta }^{(\alpha )}\) presented in a stereographic projection on a Wolff net. The numerical results for three crystals of different symmetry are discussed in Sect. 3.

2.3 Singular points of the diffraction tensor

A conical axis is an isolated singularity direction on the slowness surface where degeneracy takes place [10]. The geometry of the slowness surface in a conical axis point is shown in Fig. 2. Curvature of the slowness surface is infinite in the direction of a conical axis. Near the axis, the first eigenvalue for the faster mode is increasing and unbounded while the second eigenvalue for the slower mode is simultaneously decreasing and unbounded:

$$\begin{aligned} w_{2}^{(s)} \rightarrow -\infty ,\qquad w_{1}^{(f)} \rightarrow +\infty , \end{aligned}$$
(4)

where \(f=1,2\) is the index of faster degenerate mode and \(s=f+1\) is the index of slower degenerate mode. Any plane section of the slowness surface through a conical singularity point is a union of smooth intersecting curves. Thus, two other eigenvalues are continuous functions and the equality holds in the degeneracy point [6]:

$$\begin{aligned} w_{1}^{(s)} = w_{2}^{(f)} . \end{aligned}$$
(5)
Fig. 2
figure 2

General features of the BAW slowness surface at the vicinity of a conical acoustic axis

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Unbounded growth and decreasing of the diffraction coefficients is visualized as a region with increased density of isolines in contour plots. According to Eq. (4), unbounded values of the \({\widehat{W}}\) tensor eigenvalues simultaneously exist for both of the degenerate modes. Thus, the condition for acoustic degeneracy of two faster modes is large positive \(w_{1}^{(1)}\) and large in magnitude and negative \(w_{2}^{(2)}\). This case is rare in crystal acoustics, but is explicitly illustrated for paratellurite crystals in Sect. 3.1. The common case is degeneracy of the slower modes. The condition for acoustic degeneracy for this case is large positive \(w_{1}^{(2)}\) and large in magnitude and negative \(w_{2}^{(3)}\).

Contour plots of the diffraction tensor eigenvalues \(w_{\beta }^{(\alpha )}\) are very helpful for visualization regions in crystals with strong acoustic anisotropy. In those directions, BAW diffraction coefficients are large in magnitude but finite. Such regions are often observed between conical acoustic axes as demonstrated in Sect. 3. Moreover, such regions may be not neighboring any of the real acoustic axes in the case when degeneracy condition is not fulfilled, but the difference of phase velocities of two modes is small. Large diffraction coefficients are also associated with regions where BAW velocity rapidly changes without mode degeneracy. This case is discussed in detail below for paratellurite crystal.

Fig. 3
figure 3

Acoustic diffraction tensor in paratellurite crystal: a symmetry elements; b BAW axes and regions with high anisotropy \(w_{1}^{(1)}w_{2}^{(2)}<-40\) (hatch) and \(w_{1}^{(2)}w_{2}^{(3)}<-40\) (fill); c–h contour plots of BAW diffraction coefficients in stereographic projection (contour spacing is 0.2, for \(w_{1}^{(3)}>10\) only major contours with spacing 10 are shown)

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3 Analysis of selected crystals

3.1 Paratellurite

Paratellurite (\(\alpha \)-TeO\(_2\)) is a tetragonal crystal of 422 point group. The elastic constants of paratellurite and their temperature coefficients were measured by Silvestrova et al. [46]. Paratellurite is the crystal with the strongest anisotropy of BAW properties [13, 47, 48]. The ratio of the maximum to the minimum BAW velocity is 7.2, and the energy flow angle reaches \(74^\circ \). Piezoelectric effect in paratellurite is weak and does not significantly affect acoustic anisotropy of the crystal, including the number of acoustic axes and their directions.

Stereographic projections of BAW diffraction coefficients in paratellurite are shown in Fig. 3. Symmetry of the BAW slowness surface and its derivatives is 4/mmm. The high density of contour lines and the specific behavior of diffraction coefficients in the vicinity of acoustic axes highlight the locations of such axes in \(\hbox {TeO}_2\). The positions of acoustic axes are marked with filled circles. Moreover, the areas characterized by fast variation of diffraction coefficients are also clearly seen. These areas connect acoustic axes and may include orientations with zero coefficients \(w_2^{(\alpha )}\). Such orientations mean autocollimation of acoustic beam along one direction and can be very useful for application in acoustic or acousto-optic devices.

Acoustic axes in tetragonal crystals are restricted to symmetry planes [3]. One of the axes is always associated with the fourfold symmetry axis Z and belongs to tangential type of degeneracy, label “A” in Fig. 3. Therefore, all six coefficients \(w_{\beta }^{(\alpha )}\) in this direction are bounded. In XY plane the slowness surface is decomposed to a circle and a quartic. The conical acoustic axes are the intersection points of the quartic outer sheet with the circle, label “B” (4 axes). Hereinafter, the axes obtained by multiplication with the symmetry elements of a crystal are labeled with the same letter and different subscripts.

One of specific features of acoustic anisotropy in TeO\(_2\) is \(c_{11} < c_{66}\) that was first mentioned by Uchida and Ohmachi [47]. This leads to anomalous BAW degeneracy, which involves the two fastest modes, in XZ and YZ planes. A special type of BAW degeneracy in tetragonal crystals takes place when \(c_{11}=c_{66}\). In this particular case, the quartic of slowness in XY plane is a union of two ellipses with axes along directions \(X+45^\circ \) and \(X-45^\circ \). Elastic constants \(c_{11}=56.12\) GPa and \(c_{66}=66.14\) GPa in paratellurite are rather close. Correspondent velocities of two fastest BAW modes are 3058 and 3320 m/s. A single acoustic axis that exists along the X axis in the case \(c_{11}=c_{66}\) is split into two symmetrical axes in XZ plane, label “C” in Fig. 3 (4 axes). The diffraction coefficients \(w_{1}^{(1)}\) and \(|w_{2}^{(2)}|\) are high in the whole sector of XZ plane between the anomalous axes.

Thus, paratellurite has 9 acoustic axes, the theoretical maximum for tetragonal crystals, that agrees with theoretically predicted. Due to strong acoustic anisotropy, EW lines in \(\hbox {TeO}_2\) were found for all three bulk modes, including the fastest one [49]. Another peculiarity of BAW propagation in paratellurite is extremely low BAW velocity of 617 m/s for a pure shear BAW propagating along \(X+45^\circ \) symmetry axis. The velocity quickly increases with the angle between \(\mathbf{n}\) and the slowest direction leading to high but finite diffraction coefficients \(w_{1}^{(3)}=50.9\) and \(w_{2}^{(3)}=11.7\). Unlike acoustic axes, there is no discontinuity in \(w_{\beta }^{(3)}\) values.

Fig. 4
figure 4

Acoustic diffraction tensor in lithium niobate crystal: a symmetry elements; b BAW axes and regions with high anisotropy \(w_{1}^{(2)}w_{2}^{(3)}<-10\); c–h contour plots of BAW diffraction coefficients in stereographic projection (contour spacing is 0.1)

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Fig. 5
figure 5

Acoustic diffraction tensor in KGW crystal: a symmetry elements; b BAW axes and regions with high anisotropy \(w_{1}^{(2)}w_{2}^{(3)}<-10\); hatched regions indicate the directions \(w_{1}^{(3)},w_{2}^{(3)}<0\), in which plane wavefront becomes concave; c–h contour plots of BAW diffraction coefficients in stereographic projection (contour spacing is 0.2)

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3.2 Lithium niobate

Lithium niobate (LNO, LiNbO\(_3\)) is a trigonal crystal of 3m point group. We use the material constants of LNO measured by Kushibiki et al. [50]. The results of simulations are shown in Fig. 4. Symmetry of the BAW slowness surface and its derivatives is \({\overline{3}}\)m. Though acoustic anisotropy in LNO is not as strong as in paratellurite, zero contour lines (green lines in the figures) still can be found on the contour plots of diffraction coefficients \(w_{2}^{(2)}\) and \(w_{2}^{(3)}\).

The total number of acoustic axes is 10, with three axes in each crystal symmetry plane (YZ and equivalent ones). Between the oblique axes there are no continuous regions of strong anisotropy. Moreover, there are autocollimation points, for the slow quasi-shear mode at the Euler angles \(\theta = 44^\circ \) (between axes “B” and “C”) and \(\theta = 54^\circ \) (between axes “C” and “D”).

LNO is a piezoelectric crystal, which is most widely used due to very strong piezoelectric effect. It is a common material for high-frequency piezoelectric transducers for excitation of BAW waves in solids [14]. In the last years, it found the main application in low-loss SAW filters and wireless SAW sensors. New application fields for this crystal appear due to increasing requirements to the new generation of SAW devices for communication systems and rapidly developing direct bonding technologies [51, 52]. That allows combining LNO with quartz [53], langasite [54], sapphire [55], and other materials in multilayered structures and requires optimization of crystal orientations based on good knowledge of their acoustic anisotropy. The multiple EW lines arising from acoustic axes give rise to the branch of low-attenuated leaky waves (see Sect. 4.1.1). Some of these leaky waves (e.g., propagating in \(41^\circ \)–\(64^\circ \) YX-cuts) found wide application in low-loss SAW filters for mobile communication systems, due to combining of high velocity with high electromechanical coupling coefficient and low propagation losses.

3.3 Potassium gadolinium tungstate (KGW)

Potassium gadolinium tungstate (KGW, KGd(WO\(_4\))\(_2\)) is a monoclinic crystal of 2/m point group. We choose the coordinate axes XYZ in the following setting: \(Z||{\mathbf {b}}\) is twofold symmetry axis, \(Y||{\mathbf {c}}\). Elastic constants of KGW were measured by Mazur et al. [23].

KGW is a laser host and nonlinear optical crystal. The family of potassium rare-earth tungstates has been recently rediscovered as a group of promising acousto-optic materials owing to their high laser damage threshold, good mechanical and thermophysical properties, established growth technology and processability [23, 24, 28].

Table 1 Acoustic axes in KGW crystal
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Calculation results for KGW are shown in Fig. 5. There are 8 isolated acoustic axes. The axes positions are listed in Table 1. Two axes labeled “A” and “B” are confined to XY plane. The others are 6 off-plane axes, labels “C,” “D,” and “E.” The plots for \(w_{1}^{(2)}\) and \(w_{2}^{(3)}\) show that there are continuous regions of high BAW divergence between axes “A” and “C” and between axes “B” and “D.” Those regions are the “ridges” and “valleys” on the slowness surface, though the surface itself remains smooth and does not contain other singularities between mentioned degeneracy points.

Another peculiarity of BAW in KGW is existence of regions where \(w_{1}^{(3)}<0\). This case is discussed in Sect. 4.1.2.

Fig. 6
figure 6

Exceptional wave lines in LNO crystal

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4 Discussion

4.1 Applications

4.1.1 Exceptional waves

Visualization of acoustic anisotropy helps solving different problems referred to design of BAW and SAW devices, for example, optimization of substrate orientations for high-frequency SAW devices. Fast variation of BAW velocities with propagation direction observed in the neighborhoods of acoustic axes is accompanied by fast rotation of polarization vectors. As a result, some BAWs can satisfy the stress-free mechanical boundary condition on selected crystal surfaces. Such “exceptional” BAWs can be slow shear, fast shear, or even longitudinal in some crystals. Propagation directions permitting existence of EWs form continuous lines on a stereographic projection of unit wave vectors (“EW lines”). In SAW devices, electrical boundary conditions and mass load of electrode structure modify the bulk wave into the quasi-bulk SAW or leaky SAW, but it usually stays low-attenuated and provides low propagation losses. Moreover, the branches of quasi-bulk low-attenuated leaky SAWs can arise from the fast shear and longitudinal EWs. Higher velocities of these waves, compared to common SAWs, facilitate fabrication of high-frequency SAW devices.

Figure 6 shows a fragment of the stereographic projection of the BAW diffraction coefficient \(w_{1}^{(2)}\) with added EW lines in LNO. Solid and dashed lines refer to the fast and slow quasi-shear BAWs, respectively. Longitudinal EWs do not exist in this material. In addition to SH-polarized BAWs propagating in the symmetry plane YZ and switching between the fast and slow BAW modes in directions of four acoustic axes, three off-plane EW lines can be observed in the neighborhood of acoustic axes “B” and “D.” Two lines arising in vicinity of the axis “D” cross XY plane. The corresponding orientations are \(38^{\circ }YX\) and \(128^{\circ }YX\) cuts. \(128^{\circ }YX\) cut supports propagation of the Rayleigh SAW and is widely used in SAW filters and sensors. \(38^{\circ }YX\) cut gives rise to a branch of low-attenuated SH-type leaky SAWs. They propagate up to 1.25 faster than the Rayleigh SAW. The corresponding orientations are often applied in low-loss resonator SAW filters for communication systems [56].

Calculation and analysis of EW lines together with visualization of acoustic anisotropy can be an efficient tool in optimization of multilayered structures for SAW devices. Today such structures often replace regular SAW substrates in resonator SAW filters with improved Q-factors. For example, a thin LNO plate bonded to quartz substrate provides a unique combination of high velocity, low attenuation, and high electromechanical coupling required for high-performance SAW devices if orientations of both crystals are properly optimized based on analysis and visualization of their acoustic anisotropy [53]. Non-attenuated quasi-longitudinal SAWs existing in these structures were found due to application of the method described here to investigation of acoustic anisotropy of quartz and calculations of EW lines in this crystal [57].

4.1.2 Autocollimation directions

Among special directions of BAW propagation, there are autocollimation directions where \(w_{\beta }^{(\alpha )}=0\). In such directions, self-diffraction of BAW beams is suppressed by anisotropy. Low beam divergency is used to reduce cross-talk in multichannel devices [17, 18]. Since any conical axis is associated with negative diffraction tensor eigenvalues for the slower degenerate mode, autocollimation lines for this mode may exist close to the axis. The behavior of isolines near conical axes is illustrated in Fig. 7 where the fragments of contour plots around conical axes are magnified.

In a general case, autocollimation directions are not necessary related to acoustic axes. For example, Fig. 5 demonstrates large BAW regions with \(w_{\beta }^{(\alpha )}<0\) in KGW crystal, including those for pure shear mode in XY plane. For this mode \(w_{2}^{(2)}<0\) between \(X+40^\circ \) and \(X+130^\circ \) directions, including the symmetry axis of the dielectric permittivity tensor \(N_g\) (\(X+101.5^\circ \)), along which pure collinear Brillouin scattering takes place. Another configuration useful for practical applications is around the direction \(X+60^\circ \) in XY plane where both diffraction coefficients for slow quasi-shear wave polarized in XY plane, \(w_{1}^{(3)}\) and \(w_{2}^{(3)}\), are negative. Those directions are specially marked in Fig. 5b. In this region, we predict propagation of ultrasonic BAW beams with reduced spreading and concave wavefronts because the diffraction coefficients are small, \(|w_{1}^{(3)}|<0.45\) and \(|w_{2}^{(3)}|<0.62\). An autocollimation point of the same kind for the slow quasi-shear wave in XY plane exists in potassium yttrium tungstate (KY(WO\(_4\))\(_2\)) crystals having similar properties to KGW [28].

4.2 Diffraction tensor and other measures of anisotropy

A close view on isolines of BAW diffraction tensor in a neighborhood of a conical acoustic axis (Fig. 7) provides an insight of geometry of the slowness surface, which depends on crystal symmetry and position of the acoustic axis. The isolines in an infinitesimal neighborhood of a conical axis have 2 orthogonal symmetry planes. That complies with asymptotic representation of the slowness surface as an elliptical cone [7, 8]. In the case of the threefold symmetry axis in LNO (Fig. 7a), the isolines become circles.

Eigenvalues of the diffraction tensor can be compared to “anisotropy coefficients” for BAW beams defined by Kastelik et al. [17] and by Balakshy and Mantsevich [20]. Those coefficients were defined as the ratio of the beam ray angular spectrum width to the width of the plane wave angular spectrum. This phenomenological definition is necessary when the parabolic approximation is not valid, but it makes anisotropy coefficients dependent not only on the properties of the material, but also on configuration of the piezotransducer and acoustic frequency. Moreover, calculation of anisotropy coefficients is a highly resource-demanding numerical method based on computation and analysis of 3D distributions of BAW fields for each direction in the crystal and each eigenwave. Calculations of the BAW diffraction tensor eigenvalues for paratellurite numerically coincide with anisotropy coefficients for far-field and small beam divergency [17, 20].

Fig. 7
figure 7

Isolines of BAW diffraction tensor eigenvalues (labeled solid lines) and slowness surface (dashed lines) in the neighborhood of conical acoustic axes: a \(V^{(3)}\) and \(w_{2}^{(3)}\) for threefold symmetry axis A axis in LNO; b \(V^{(2)}\) and \(w_{2}^{(2)}\) for anomalous \(\hbox {C}_1\) axis in \(\hbox {TeO}_2\); c, d \(V^{(3)}\) and \(w_{2}^{(3)}\) for off-plane \(\hbox {E}_1\) and \(\hbox {C}_1\) axes in KGW

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Finally, it should be noted that the eigenvalues of the diffraction tensor are a generalization of parabolic diffraction coefficient (also called “anisotropy parameter”) widely used in simulation and design of SAW devices. In SAW delay lines, this parameter is responsible for diffraction losses. In resonator SAW filters, the anisotropy parameter estimated in direction parallel to electrodes of an interdigital transducer (IDT) determines insertion losses and parasitic modes caused by coupling between SAW and transverse modes guided by IDT [57]. Similarly, diffraction losses of BAWs should be taken into account in design of other ultrasonic devices. One of important practical cases is a slow shear wave in paratellurite \(X+45^\circ \) symmetry plane. This BAW mode is widely used in acousto-optic devices owing to its high coupling to optical waves.

5 Conclusions

Contour plots of eigenvalues of BAW diffraction tensor provide fast and robust method for survey and visualization of acoustic anisotropy of crystals, including a search for singular directions (conical acoustic axes), areas of fast or slow variation of BAW velocities, and directions where autocollimation of acoustic beams is expected. The number and position of BAW axes obtained using the diffraction tensor fully complies with theoretical predictions. Besides that, the proposed visualization method provides a comprehensive picture of BAW anisotropy in a crystal not only highlighting the degeneracy directions. We demonstrated that analysis of crystal acoustic anisotropy is closely related to applications since this method helps to locate crystallographic areas with characteristics required for a wide range of ultrasonic, acousto-electronic and acousto-optic devices. This facilitates further detailed optimization of crystal orientation for these devices.