5. Optimization of control parameters of reflection

Until now, we were free with a choice of geometry of the considered resonance reflection. It looks natural to choose the parameters φ, ψ, and δp so that the loss coefficient K<sup>1</sup> would be as small as possible, i.e., the efficiency η would be close to 100%. In terms of Figure 1, this means an exclusion of the parasitic reflected beam r1 which is equivalent to a realization of the mode conversion. Formula (21) allows reducing the criterion of conversion K<sup>1</sup> = 0 to the system of equations:

$$
\lambda''^{(4)}\_{\,\,\,ij} \phi\_i \phi\_j = 0, \,\,\delta p = -\lambda'^{(4)}\_{\,\,ij} \phi\_i \phi\_j. \tag{23}
$$

where the first equation determines the relation between the angles of rotation of the sagittal plane ð Þ ϕ<sup>1</sup> � φ and the normal to the surface ð Þ ϕ<sup>2</sup> � ψ at a fixed position (χ) of the axis of rotation of the vector n (see Figure 2a). The second equation in (23) at the found relation between φ and ψ specifies the dependence δp(φ) and, by (10), the incidence angle δα(φ).

The first requirement in (23) is reduced to a quadratic equation with respect to the ratio ψ/φ. The existence of real roots of this equation (and, therefore, mode conversion) is generally not guaranteed. However, numerical calculations for a number of crystals of different symmetry systems did not give us examples of the absence of such roots. Furthermore, as is shown in the next section, for hexagonal crystals, this equation always has real roots for the case c<sup>44</sup> > c66. Thus, in many crystals, the consistent variation of orientations of the surface and sagittal plane really can provide the mode conversion near the total internal reflection, i.e., the effect which is under consideration.

The general conditions of mode conversion (23) can be represented in the compact form:

$$
\eta = \gamma\_{\pm} \varrho,\ \delta p = \lambda\_{\pm} \varrho^2. \tag{24}
$$

These conditions with real roots γ � specify two variants of the orientations of the surface, sagittal plane, and angle of incidence (for each angle χ, see Figure 2a) that ensure the energy concentrating in the reflected beam r2.

As was expected, the gain K<sup>2</sup> given by Eq. (22) should obviously be large because it is inversely proportional to the square of the small parameter:

$$K\_2^{\rm con} \propto \rho^{-2} \propto \psi^{-2}. \tag{25}$$

But the unlimited increase in (25) with a decrease in the angle φ should not mislead us. Indeed, an increase in the amplitude (25) of the resonance peak (22) is accompanied by its narrowing. However, when this width in angles of incidence δα becomes smaller than the natural diffraction divergence of the beam, the further approach of the incident wave to the total internal reflection angle becomes senseless. Instead of the energy concentrating in the reflected beam r2, the more and more fraction of the incident beam will be out of resonance. Thus, a small divergence of the both beams proves to be an important requirement which, in turn, limits a permissible sound frequency ν from below. Let us estimate these limitations.

In the case of total mode conversion, the condition for the balance of energy fluxes in the incident and reflected beams has the form PiDi = Pr2dr<sup>2</sup> (Figure 1a). This balance gives

$$K\_2^{\rm con} \equiv P\_{r2}/P\_i = D\_i/d\_{r2} \,\tag{26}$$

i.е., the reflected beam turns out to be narrower than the incident one by a factor of K2. On the other hand, the related diffraction divergence angles, δ<sup>i</sup> � cs=νDi and <sup>δ</sup><sup>r</sup><sup>2</sup> � cs=νdr<sup>2</sup> (where cs � <sup>10</sup><sup>5</sup> cm/s is the sound speed), are in similar proportion:

$$
\delta\_{r2} \approx \frac{D\_i}{d\_{r2}} \delta\_i = K\_2^{\rm con} \delta\_i. \tag{27}
$$

Thus, the possible increase in the coefficient K<sup>2</sup> is limited by the diffraction divergence of the r2 beam. To decrease this divergence, the frequency ν must be high. The simple estimation gives the following characteristic values: at ν � 100 MHz and <sup>D</sup> � 1 cm, one can obtain a coefficient <sup>K</sup><sup>2</sup> � <sup>5</sup>–10 at dr<sup>2</sup> <sup>≈</sup> <sup>1</sup>–2 mm, <sup>δ</sup><sup>i</sup> � <sup>10</sup>�<sup>3</sup> , and δr<sup>2</sup> � <sup>10</sup>�<sup>2</sup> rad.

For a fixed direction of the normal n(ψ, χ) to the crystal boundary, the surfaces K2ð Þ φ; δα and η φð Þ ; δα have specific "ridges" (Figure 3). They are determined by the special extremal relations between the angles φ and δα. In framework of our approximate theory [8] (see Section 6), these trajectories coincide and are described by the relation of the type δα = Cφ<sup>4</sup> . And along the ridges, one obtains

$$K\_2^{\text{max}} = \text{const} / \sqrt{\delta a}, \quad \eta^{\text{max}} = \text{const.} \tag{28}$$

Table 1 demonstrates the results of such analysis for a number of acoustic crystals. We present the geometries related to the extremal gains. The angles

Characteristics of extremal resonances K<sup>2</sup> = 5 for acoustic crystals of various symmetry systems.

Numerical plot of surfaces K2ð Þ φ; δα (a) and η φð Þ ; δα (b) for graphite crystal with the fixed boundary parallel

Systems Crystal m0, n0 Normal n φm, rad δαm, rad η, %

Hexagonal Graphite m0kx, n0ky 0 0.02 0.14 0.034 94.9

Trigonal Quartz m0kx1, n0ky 0.01 0.60 0.48 0.043 75.0

Orthorhombic Rochelle salt m0kz, n0kx 0.12 0.44 0.55 0.009 94.6

The unperturbed orientations of vectors m<sup>0</sup> and n<sup>0</sup> are determined by the axes x, y, and z of crystallographic coordinates, the bisector axes x<sup>0</sup> and y<sup>0</sup> in the basal plane xy and by the directions xαkð Þ cos θα; 0; sin θα , where

χ, rad ψ, rad

Silicon 0.29 0.29 0.13 0.007 74.0

ZnS 0.03 0.48 0.32 0.008 93.4 CdCe 0.03 0.69 0.14 0.008 95.2

BaTiO3 m0kx, n0kz 0.05 0.28 0.37 0.031 80.0

LiNbO3 m0kx2, n0ky 0.30 0.78 0.09 0.007 79.0

, n0ky<sup>0</sup> 0.21 0.29 0.13 0.008 88.7

, n0ky<sup>0</sup> 0.09 0.03 0.07 0.013 99.8

m0kx, n0kz 0.18 0.47 �0.08 0.010 98.4

compromise choice of the extremal geometry, instead of mode conversion one, indeed leads to a substantial increase in adjusting angles δα<sup>m</sup> with efficiency η<sup>m</sup> retained rather high level. The magnitudes of δα<sup>m</sup> remain fairly small (�0.01 rad on average) even after the compromise but still look to be acceptable for an experiment. On the other hand, for quartz, graphite, and BaTiO3 crystals, the angle δα<sup>m</sup> is several times larger than the mentioned mean values. In the case of graphite, the mentioned compromise leads to the increase of the tuning angle by a factor of 1.5

<sup>2</sup> ¼ 5. In the presented examples, the

φm, ψ m, χ<sup>m</sup> and δα<sup>m</sup> are found for the gains Kmax

to the hexad axis 6; the φ angle is counted from the vector m0⊥6.

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

Cubic LiF m0kx<sup>0</sup>

Tetragonal Paratellurite m0kx<sup>0</sup>

θ<sup>1</sup> = �0.76 and θ<sup>2</sup> = 0.46 rad.

Table 1.

167

Figure 3.

via reducing the efficiency η by only 5%.

Of course, with variations of n, the constants in (28) also change. For some definite direction of <sup>n</sup>, the mode conversion occurs when <sup>η</sup>max <sup>¼</sup> <sup>η</sup>con <sup>¼</sup> 1 and Kmax <sup>2</sup> <sup>¼</sup> <sup>K</sup>con <sup>2</sup> .

Unfortunately, the obtained identity of the extremal trajectories is just the consequence of our approximations. As numerical analysis shows, usually they are close but not identical. And on one of them, we deal with the mode conversion, <sup>η</sup>con <sup>¼</sup> <sup>1</sup> and Kcon <sup>2</sup> , whereas on the other, an increased extremal gain Kmax <sup>2</sup> . Kcon <sup>2</sup> and a decreased efficiency ηmax < 1 occur. It is even more important that on the second trajectory, the same value of coefficient K<sup>2</sup> is obtained at a larger angle δα. And the numerical analysis shows that one can significantly increase δα in this case at the expense of a relatively small decrease in the efficiency η.

In Figure 3 and in further considerations, we illustrate the performed analysis by numerical calculations for the series of crystals of various symmetry systems. The parameters of the resonance are calculated by exact formulas (5) and (8). We also vary the angle χ and do not limit ourselves to small angles φ and ψ.

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

#### Figure 3.

Kcon

us. Indeed, an increase in the amplitude (25) of the resonance peak (22)

Kcon

<sup>δ</sup><sup>r</sup><sup>2</sup> <sup>≈</sup> Di dr<sup>2</sup>

limitations.

� <sup>10</sup>�<sup>2</sup> rad.

Kmax <sup>2</sup> <sup>¼</sup> <sup>K</sup>con <sup>2</sup> .

and Kcon

166

by the relation of the type δα = Cφ<sup>4</sup>

Kmax

expense of a relatively small decrease in the efficiency η.

<sup>2</sup> <sup>¼</sup> const<sup>=</sup> ffiffiffiffiffi

<sup>2</sup> , whereas on the other, an increased extremal gain Kmax

also vary the angle χ and do not limit ourselves to small angles φ and ψ.

Of course, with variations of n, the constants in (28) also change. For some definite direction of <sup>n</sup>, the mode conversion occurs when <sup>η</sup>max <sup>¼</sup> <sup>η</sup>con <sup>¼</sup> 1 and

This balance gives

Acoustics of Materials

approach of the incident wave to the total internal reflection angle becomes senseless. Instead of the energy concentrating in the reflected beam r2, the more and more fraction of the incident beam will be out of resonance. Thus, a small divergence of the both beams proves to be an important requirement which, in turn, limits a permissible sound frequency ν from below. Let us estimate these

<sup>2</sup> ∝ φ�<sup>2</sup> ∝ ψ�<sup>2</sup>

But the unlimited increase in (25) with a decrease in the angle φ should not mislead

is accompanied by its narrowing. However, when this width in angles of incidence δα becomes smaller than the natural diffraction divergence of the beam, the further

In the case of total mode conversion, the condition for the balance of energy fluxes in the incident and reflected beams has the form PiDi = Pr2dr<sup>2</sup> (Figure 1a).

i.е., the reflected beam turns out to be narrower than the incident one by a factor of K2. On the other hand, the related diffraction divergence angles, δ<sup>i</sup> � cs=νDi and <sup>δ</sup><sup>r</sup><sup>2</sup> � cs=νdr<sup>2</sup> (where cs � <sup>10</sup><sup>5</sup> cm/s is the sound speed), are in similar proportion:

Thus, the possible increase in the coefficient K<sup>2</sup> is limited by the diffraction divergence of the r2 beam. To decrease this divergence, the frequency ν must be high. The simple estimation gives the following characteristic values: at ν � 100 MHz and

For a fixed direction of the normal n(ψ, χ) to the crystal boundary, the surfaces K2ð Þ φ; δα and η φð Þ ; δα have specific "ridges" (Figure 3). They are determined by the special extremal relations between the angles φ and δα. In framework of our approximate theory [8] (see Section 6), these trajectories coincide and are described

δα

Unfortunately, the obtained identity of the extremal trajectories is just the consequence of our approximations. As numerical analysis shows, usually they are close but not identical. And on one of them, we deal with the mode conversion, <sup>η</sup>con <sup>¼</sup> <sup>1</sup>

decreased efficiency ηmax < 1 occur. It is even more important that on the second trajectory, the same value of coefficient K<sup>2</sup> is obtained at a larger angle δα. And the numerical analysis shows that one can significantly increase δα in this case at the

In Figure 3 and in further considerations, we illustrate the performed analysis by numerical calculations for the series of crystals of various symmetry systems. The parameters of the resonance are calculated by exact formulas (5) and (8). We

. And along the ridges, one obtains

<sup>p</sup> , <sup>η</sup>max <sup>¼</sup> const: (28)

<sup>2</sup> . Kcon

<sup>2</sup> and a

<sup>D</sup> � 1 cm, one can obtain a coefficient <sup>K</sup><sup>2</sup> � <sup>5</sup>–10 at dr<sup>2</sup> <sup>≈</sup> <sup>1</sup>–2 mm, <sup>δ</sup><sup>i</sup> � <sup>10</sup>�<sup>3</sup>

<sup>δ</sup><sup>i</sup> <sup>¼</sup> <sup>K</sup>con

<sup>2</sup> � Pr2=Pi ¼ Di=dr2, (26)

<sup>2</sup> δi: (27)

, and δr<sup>2</sup>

: (25)

Numerical plot of surfaces K2ð Þ φ; δα (a) and η φð Þ ; δα (b) for graphite crystal with the fixed boundary parallel to the hexad axis 6; the φ angle is counted from the vector m0⊥6.


The unperturbed orientations of vectors m<sup>0</sup> and n<sup>0</sup> are determined by the axes x, y, and z of crystallographic coordinates, the bisector axes x<sup>0</sup> and y<sup>0</sup> in the basal plane xy and by the directions xαkð Þ cos θα; 0; sin θα , where θ<sup>1</sup> = �0.76 and θ<sup>2</sup> = 0.46 rad.

#### Table 1.

Characteristics of extremal resonances K<sup>2</sup> = 5 for acoustic crystals of various symmetry systems.

Table 1 demonstrates the results of such analysis for a number of acoustic crystals. We present the geometries related to the extremal gains. The angles φm, ψ m, χ<sup>m</sup> and δα<sup>m</sup> are found for the gains Kmax <sup>2</sup> ¼ 5. In the presented examples, the compromise choice of the extremal geometry, instead of mode conversion one, indeed leads to a substantial increase in adjusting angles δα<sup>m</sup> with efficiency η<sup>m</sup> retained rather high level. The magnitudes of δα<sup>m</sup> remain fairly small (�0.01 rad on average) even after the compromise but still look to be acceptable for an experiment. On the other hand, for quartz, graphite, and BaTiO3 crystals, the angle δα<sup>m</sup> is several times larger than the mentioned mean values. In the case of graphite, the mentioned compromise leads to the increase of the tuning angle by a factor of 1.5 via reducing the efficiency η by only 5%.

The positions of maxima on the surfaces do not coincide but are quite close to each

acoustic crystals shows these dependences as very slowly changing functions close

In this section based on [11], we present the explicit analytical theory of the effect under consideration for a hexagonal medium with transversely isotropic elastic properties, which makes it possible to specify the above general relations and express the geometric conditions for mode conversion in terms of the moduli of elasticity of crystal. Analytical considerations will be supplemented with numerical

To describe a hexagonal crystal, we use a standard crystallographic system of coordinates with the z axis oriented parallel to principal symmetry axis 6 and the x and y axes orthogonal to the z axis and lying in the basal plane of transverse isotropy [14, 15]. We choose the crystal boundary P<sup>0</sup> to be parallel to the axis 6 so that the normal n<sup>0</sup> to this boundary is directed along the y axis. In this geometry, an EBW with the polarization A02||z can propagate along the crystal surface in the direction

<sup>v</sup>^<sup>0</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffi

For transverse isotropy, we may change the initial crystal surface orientation P<sup>0</sup> ! P, rotating its normal vector n around m<sup>0</sup> (i.e., choosing in Figure 2a the

In addition, as before, we introduce a perturbed propagation direction m rotated

m<sup>0</sup> ! m ¼ m<sup>0</sup> cos φ þ ½ � n � m<sup>0</sup> sin φ,

c66=c<sup>44</sup> p ,

L1,<sup>4</sup> ¼ f g 2c<sup>66</sup> � c44; �2c66p;ð Þ � pψ � φ c<sup>44</sup>

A<sup>2</sup> ¼ ð Þ φd=Δ14; 0; 1 ,

<sup>k</sup><sup>2</sup> <sup>¼</sup> <sup>k</sup> <sup>1</sup> � <sup>φ</sup><sup>2</sup>

<sup>L</sup><sup>2</sup> <sup>¼</sup> <sup>ψ</sup>; βφ; <sup>δ</sup><sup>p</sup> <sup>þ</sup> βφψ <sup>~</sup> <sup>g</sup>c44, �

<sup>=</sup>2; <sup>δ</sup><sup>p</sup> <sup>þ</sup> φψ; �<sup>φ</sup> � �;

Based on the standard equations of crystal acoustics [14, 15], one can determine

relative to the vector m<sup>0</sup> by a small angle ϕ in the new surface plane P:

Let us now check to what extent the found relations between parameters of our resonance reflection found in the first order of the perturbation theory retain their validity in a more precise numerical description. To be exact, we are checking the analytical dependence (28) of the extremal gain on the tuning angle δα. And indeed,

2

ffiffiffiffiffi

δα <sup>p</sup> versus δα for eight analyzed

<sup>c</sup>44=<sup>ρ</sup> <sup>p</sup> : (29)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi c66=c<sup>44</sup> p , (32)

(33)

n<sup>0</sup> ! n ¼ ð Þ 0; cos ψ; sin ψ : (30)

½ �¼ <sup>n</sup> � <sup>m</sup><sup>0</sup> ð Þ <sup>0</sup>; sin <sup>ψ</sup>; � cos <sup>ψ</sup> : (31)

other, which allows a reasonable compromise at the choice of geometry.

the numerical plot in Figure 5 of the products Kmax

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

6. Explicit theory for hexagonal crystals

calculations for some of hexagonal crystals.

to constants.

m0||х with a speed:

169

angle χ = π/2) by a small angle ψ

the wave parameters entering superposition (1)

<sup>A</sup>1,<sup>4</sup> ¼ �ð Þ <sup>p</sup>; <sup>1</sup>; <sup>0</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k1,<sup>4</sup> ¼ kð Þ 1; � p; �φ ;

Figure 4.

Angle of incidence δα<sup>m</sup> and efficiency η<sup>m</sup> versus the angles ψ and χ at K2 = 5 for lithium niobate (LiNbO3) crystal at m0||x2, n0||y, x<sup>2</sup> = (cosθ2, 0, sinθ2), and θ<sup>2</sup> = 0.46.

Figure 4 shows the optimization of the parameters δα<sup>m</sup> and η<sup>m</sup> for the lithium niobate crystal that corresponds to the value K<sup>2</sup> = 5 at the variation of the orientations of the normal n to the surface, i.e., the angles ψ and χ. For each n direction, the surface K2(φ, δα) similar to that shown in Figure 3 was plotted from which the φ<sup>m</sup> and δα<sup>m</sup> values corresponding to the extremal point at the "crest" with the amplitude K<sup>2</sup> = 5 were determined. Consequently, each point on the δαmð Þ ψ; χ and ηmð Þ ψ; χ surfaces in Figure 4 corresponds to a certain angle φm. As is seen in the figure, the variations of the angles ψ and χ can significantly increase δα<sup>m</sup> and ηm.

#### Figure 5.

Numerical plot of the product Kmax 2 ffiffiffiffiffi δα <sup>p</sup> versus δα for the series of acoustic crystals (1—BaTiO3, 2—paratellurite, 3—graphite, 4—CdCe, 5—ZnS, 6—LiF, 7—LiNbO3, 8—Si).

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

The positions of maxima on the surfaces do not coincide but are quite close to each other, which allows a reasonable compromise at the choice of geometry.

Let us now check to what extent the found relations between parameters of our resonance reflection found in the first order of the perturbation theory retain their validity in a more precise numerical description. To be exact, we are checking the analytical dependence (28) of the extremal gain on the tuning angle δα. And indeed, the numerical plot in Figure 5 of the products Kmax 2 ffiffiffiffiffi δα <sup>p</sup> versus δα for eight analyzed acoustic crystals shows these dependences as very slowly changing functions close to constants.
