7. The particular case of 1D perturbation (ψ = 0)

The presented theory of mode conversion in the vicinity of the angle of total internal reflection was based on coupled perturbations of both the sagittal plane and the surface orientations. At the rotations of only one of these planes, the mode conversion can be achieved only for crystals with certain relations between the moduli of elasticity. However, sometimes, these requirements might appear to be not very limiting. This can be explicitly shown on the example of the considered above case of hexagonal crystals. Indeed, for the unperturbed surface at ψ = 0, the loss coefficient K<sup>1</sup> (40) acquires a minimal value Kmin <sup>1</sup> at φ <sup>2</sup> = λ<sup>0</sup> δp which may be estimated as:

$$K\_1^{\min} = \left| R\_1 \right|^2 = \left| \frac{\lambda''}{2\lambda'} \right|^2 \sim \left( \frac{c\_{44} - 2c\_{66}}{c\_{44}} \right)^4. \tag{50}$$

Thus, any hexagonal crystal with the modulus c<sup>44</sup> close to 2c<sup>66</sup> must be very efficient for our effect (K<sup>1</sup> ≪ 1). Due to the "4" power in the latter estimate, this remains

Resonance Compression of Acoustic Beams in Crystals DOI: http://dx.doi.org/10.5772/intechopen.82364

true even when the moduli are not so close to each other. For instance, at |c44�2c66|/ <sup>c</sup><sup>44</sup> � 1/3, Eq. (50) gives the estimate <sup>K</sup><sup>1</sup> � <sup>10</sup>�<sup>2</sup> and efficiency <sup>η</sup> = 1�K<sup>1</sup> becomes � 99%. In accordance with reference book [16], there are quite a number of hexagonal crystals where K<sup>1</sup> ≪ 1. Below, we shall also give examples of crystals of monoclinic symmetry systems having got the same property.

This motivates us to a short consideration of the simplified approach to the reflection resonance with unchanged crystal surface orientation. As was shown in [8], the description becomes especially compact if to choose the crystal boundary parallel to the plane of crystal symmetry. By the way, such planes exist in all crystals, except triclinic [14, 15]. In this case, expressions (21) and (22) acquire the structure:

$$K\_1 = \frac{\left(\rho^2 - \lambda'\delta p\right)^2 + \left(\lambda''\delta p\right)^2}{\left(\rho^2 + \lambda'\delta p\right)^2 + \left(\lambda''\delta p\right)^2},\tag{51}$$

$$K\_2 = \frac{\left(\mu\rho\right)^2 s\_2 / s\_4}{\left(\rho^2 + \lambda'\delta p\right)^2 + \left(\lambda''\delta p\right)^2}. \tag{52}$$

And the efficiency of the resonance η = 1�K<sup>1</sup> is given by:

$$\eta = \frac{4\lambda'\delta p\rho^2}{\left(\rho^2 + \lambda'\delta p\right)^2 + \left(\lambda''\delta p\right)^2}.\tag{53}$$

Thus, for a fixed δp, the coefficients K<sup>2</sup> and η are determined by the same function F(φ):

$$K\_2(\rho) = \frac{\mu \mathfrak{s}\_2}{\mathfrak{s}\_4} F(\rho), \qquad \eta(\rho) = 4\lambda' \delta p F(\rho), \qquad F(\rho) = \frac{\rho^2}{\left(\rho^2 + \lambda' \delta p\right)^2 + \left(\lambda'' \delta p\right)^2}. \tag{54}$$

Accordingly, their maximum magnitudes are determined by the same extremum condition:

$$
\rho^2 = |\lambda| \delta p = \delta p \sqrt{\lambda'^2 + \lambda''^2} \approx \lambda' \delta p \left[ \mathbf{1} + \frac{\mathbf{1}}{2} \left( \frac{\lambda''}{\lambda'} \right)^2 \right].\tag{55}
$$

The last estimate in (55) is valid when <sup>λ</sup>″=λ<sup>0</sup> ð Þ<sup>2</sup> ≪ 1. In this approximation, one obtains

$$K\_1^{\min}(\boldsymbol{\varrho}, \delta \boldsymbol{\alpha}) \approx \left(\frac{\lambda''}{2\lambda'}\right)^2, \quad K\_2^{\max}(\boldsymbol{\varrho}, \delta \boldsymbol{\alpha}) \approx \frac{\boldsymbol{G}}{\boldsymbol{\varrho}^2}, \quad \boldsymbol{G} \approx \frac{\mu \mathfrak{s}\_2}{\mathfrak{s}\_4}.\tag{56}$$

Thus, one can conclude that the consistent variation of the sagittal plane orientation φ and the tuning incidence angle δα ∝ δp<sup>2</sup> (10) along one extremal trajectory (55) provides simultaneous optimization of both the gain and the efficiency of the resonance reflection. However, as was shown above, this coincidence is not an exact result but a consequence of our approximate calculations. The computer analysis based on exact formulas (5) without their expansion in small parameters leads to distinct extremal trajectories for the functions Kmax <sup>2</sup> ð Þ <sup>φ</sup>; δα and <sup>η</sup>maxð Þ <sup>φ</sup>; δα . A difference between them depends on angle φ and crystal anisotropy. On the other hand, as was discussed above, the occurrence of those trajectories might be used for

and ψ (47) is compared with the results of numerical calculations based on formulas (5), without their expansion in small parameters, on the example of two (Ti, and

Correlations between the conversion angles ψcon and φcon (in radians) obtained by numerical calculation (solid

In both cases, our computations not only practically confirm linearity of the relation between ϕ and ψ but also yield slopes of these dependences close to theo-

We can only wonder why the predictions in the first order of the perturbation theory are confirmed so well by the exact numerical calculation in a wide range of angles ϕ, which are far from small. Anyhow, but the Ti crystal reveals one more "loyalty" with respect to our approximate theory: in its case, the geometries of the

The presented theory of mode conversion in the vicinity of the angle of total internal reflection was based on coupled perturbations of both the sagittal plane and the surface orientations. At the rotations of only one of these planes, the mode conversion can be achieved only for crystals with certain relations between the moduli of elasticity. However, sometimes, these requirements might appear to be not very limiting. This can be explicitly shown on the example of the considered above case of hexagonal crystals. Indeed, for the unperturbed surface at ψ = 0, the loss coefficient K<sup>1</sup>

<sup>2</sup> = λ<sup>0</sup>

 

Thus, any hexagonal crystal with the modulus c<sup>44</sup> close to 2c<sup>66</sup> must be very efficient for our effect (K<sup>1</sup> ≪ 1). Due to the "4" power in the latter estimate, this remains

2

� <sup>c</sup><sup>44</sup> � <sup>2</sup>c<sup>66</sup> c<sup>44</sup> <sup>4</sup>

<sup>1</sup> at φ

<sup>2</sup> <sup>¼</sup> <sup>λ</sup>″ 2λ<sup>0</sup> 

, 0 and coincident signs of angles ϕ and ψ,

δp which may be estimated as:

: (50)

\_

lines) and approximate theoretical analysis (dashed lines) for the (1) Ti and (2) BeCu crystals.

mode conversion and the extremal gain are almost identical.

7. The particular case of 1D perturbation (ψ = 0)

. 0 and the angle signs are different.

BeCu) hexagonal crystals [16, 17].

retical ones. In case of Ti, we have β

\_

(40) acquires a minimal value Kmin

172

Kmin <sup>1</sup> ¼ j j R<sup>1</sup>

while for BeCu, β

Figure 6.

Acoustics of Materials

substantial increasing of the width of the resonance in the range of tuning incidence angle δα at the expense of a slight decrease in the efficiency η.

Figure 7 shows dependences K2(δα) and η(δα) in such extremal trajectories related to ridges on the surfaces K2ð Þ φ; δα and η φð Þ ; δα (of the type shown in Figure 3) conformably to a monoclinic stilbene crystal. The upper curves correspond to the choice of φ = fK,η(δα) in the "proper" trajectory corresponding to the maximum of the shown characteristic, while the lower curves are plotted for φ = fη,K(δα) from the "foreign" trajectory. In this case, of the two possible optimization variants, the trajectory in which Kmax <sup>2</sup> is realized is more advantageous. This corresponds to curves 1 in Figure 7. Obviously, for such a choice in the stilbene case, there is approximately only a 2.5%, loss in efficiency, but in the return gain, twice in the resonance width: the value of δα<sup>m</sup> for the optimal angle of incidence corresponding to an amplitude of Kmax <sup>2</sup> ¼ 5 increases from 0.08 to 0.155 rad.

Here, we limit ourselves to presenting only data for several monoclinic crystals with the best parameters (see Table 2). The values of δαm, φm, and η<sup>m</sup> presented in the table for all crystals correspond to the same gain K<sup>2</sup> = 5 and relate to the choice of optimal trajectory φ<sup>K</sup> as for stilbene (Figure 7). This results in the maximum width of resonance for a small decrease in efficiency ηm. By the way, stilbene is the

Crystals φ0, rad φm, rad δαm, rad ηm, % Stilbene 1.54 0.16 0.155 97.2 Triglycine sulfate 1.35 0.12 0.089 91.0 Benzyl 1.03 0.73 0.069 97.6 Tartaric acid 1.13 0.30 0.021 97.4 The directions of m<sup>0</sup> are specified by the azimuth angles φ<sup>0</sup> counted from the x axis of the crystallographic coordinates

absolute leader in the table in the value of δα<sup>m</sup> and one of the leaders in the

Key parameters of 1D resonance for a series of monoclinic crystals.

Resonance Compression of Acoustic Beams in Crystals DOI: http://dx.doi.org/10.5772/intechopen.82364

The analytical theory developed above is constructed within the theory of elasticity and, within the range of its applicability, it is exact to the extent of Eqs. (5)

Based on the same principles, we used the image of acoustic beams in our reasoning only for clarity. Actually, we did not go outside the plane wave approximation in our calculations by assuming it to be sufficient in the short wavelength limit of interest, <sup>λ</sup>/<sup>D</sup> <sup>10</sup><sup>3</sup> rad. Here, we also had in mind the possible manifestations of the effect in phonon physics, where the language of plane waves is more

Based on our analysis, we can probably count on the realization of resonant reflection in crystals, whereby a wide incident acoustic beam converts almost all of its energy into a narrow high-intensity reflected beam. A special choice of crystals with a definite relation between the elastic moduli is required to optimize the resonance. In addition, since the resonance region is narrow in angles of incidence,

and (8), which express the resonant reflection coefficients for an arbitrary anisotropic medium in terms of the eigenvectors of Stroh's matrix (3). Generally, the dependence of these eigenvectors on the geometrical parameters of reflection can easily be found by numerical methods. An analytical alternative is to expand the exact formulas (5) into a series in small angular parameters. Finally, an explicit analytical calculation based on these formulas for a number of geometries in highsymmetry crystals (for example, hexagonal ones) is also possible. Here, we used all three approaches: the numerical calculations based on Eqs. (5) and (8), their expansion into a series, and even an explicit representation of the results via the elastic moduli. In this case, avoiding the cumbersomeness of our calculations and the unmanageability of the analytical formulas, we retained only the first nonvanishing terms in all expansions that conveyed the key dependences and the effect being investigated on physical parameters. On the other hand, all graphical results of our analysis were obtained through computations based on the exact formulas

efficiency ηm.

Table 2.

and the angles φm—From m0.

8. Conclusions

(5) and (8).

relevant.

175

In [8], we accomplished the numerical search of crystal candidates for possible future observations of the discussed effect. In all cases, the surface was supposed to be parallel to the symmetry plane while the sagittal plane orientation varied. The "casting" involved about 350 crystals. The basic criteria for the crystal selection were the closeness of the resonance to mode conversion and not too small resonance widths over the angles of incidence (δα<sup>m</sup> ≥ 0.01 rad for K<sup>2</sup> = 5). According to such criteria, we found 14 crystals of monoclinic, trigonal, orthorhombic, and hexagonal systems. They are characterized by a rather high efficiency ηm, while the width δα of resonances over the angles of incidence satisfies the formulated selection rule. The resulting parameters of the resonance in these crystals are described in [8].

#### Figure 7.

Dependences of the gain coefficient K2 (a) and the efficiency η (b) of the resonance on the incidence angle δα in a stilbene crystal with the surface parallel to the symmetry plane for optimal trajectories corresponding to the "ridges" of surfaces K2(φ, δα) (curves 1) and η(φ, δα) (curves 2).


#### Resonance Compression of Acoustic Beams in Crystals DOI: http://dx.doi.org/10.5772/intechopen.82364

The directions of m<sup>0</sup> are specified by the azimuth angles φ<sup>0</sup> counted from the x axis of the crystallographic coordinates and the angles φm—From m0.

#### Table 2.

substantial increasing of the width of the resonance in the range of tuning incidence

Figure 7 shows dependences K2(δα) and η(δα) in such extremal trajectories related to ridges on the surfaces K2ð Þ φ; δα and η φð Þ ; δα (of the type shown in Figure 3) conformably to a monoclinic stilbene crystal. The upper curves correspond to the choice of φ = fK,η(δα) in the "proper" trajectory corresponding to the maximum of the shown characteristic, while the lower curves are plotted for φ = fη,K(δα) from the "foreign" trajectory. In this case, of the two possible optimization variants, the

curves 1 in Figure 7. Obviously, for such a choice in the stilbene case, there is approximately only a 2.5%, loss in efficiency, but in the return gain, twice in the resonance width: the value of δα<sup>m</sup> for the optimal angle of incidence corresponding to

<sup>2</sup> ¼ 5 increases from 0.08 to 0.155 rad.

In [8], we accomplished the numerical search of crystal candidates for possible future observations of the discussed effect. In all cases, the surface was supposed to be parallel to the symmetry plane while the sagittal plane orientation varied. The "casting" involved about 350 crystals. The basic criteria for the crystal selection were the closeness of the resonance to mode conversion and not too small resonance widths over the angles of incidence (δα<sup>m</sup> ≥ 0.01 rad for K<sup>2</sup> = 5). According to such criteria, we found 14 crystals of monoclinic, trigonal, orthorhombic, and hexagonal systems. They are characterized by a rather high efficiency ηm, while the width δα of resonances over the angles of incidence satisfies the formulated selection rule. The resulting parameters of the resonance in these crystals are described in [8].

Dependences of the gain coefficient K2 (a) and the efficiency η (b) of the resonance on the incidence angle δα in a stilbene crystal with the surface parallel to the symmetry plane for optimal trajectories corresponding to the

"ridges" of surfaces K2(φ, δα) (curves 1) and η(φ, δα) (curves 2).

<sup>2</sup> is realized is more advantageous. This corresponds to

angle δα at the expense of a slight decrease in the efficiency η.

trajectory in which Kmax

Acoustics of Materials

an amplitude of Kmax

Figure 7.

174

Key parameters of 1D resonance for a series of monoclinic crystals.

Here, we limit ourselves to presenting only data for several monoclinic crystals with the best parameters (see Table 2). The values of δαm, φm, and η<sup>m</sup> presented in the table for all crystals correspond to the same gain K<sup>2</sup> = 5 and relate to the choice of optimal trajectory φ<sup>K</sup> as for stilbene (Figure 7). This results in the maximum width of resonance for a small decrease in efficiency ηm. By the way, stilbene is the absolute leader in the table in the value of δα<sup>m</sup> and one of the leaders in the efficiency ηm.
