Abstract

The resonant excitation of an intense elastic wave through nonspecular reflection of a special pump wave in a crystal is described. Geometric criteria are found under which mode conversion, when the incident and reflected beams belong to different acoustic branches, coexists with total internal reflection of an acoustic beam. In this case, the entire energy of an incident pump wave is spent on the excitation of a narrow intense reflected beam close in structure to an eigenmode. A consistent choice of orientations of the sagittal plane and crystal surface that excludes the reflection of a parasitic wave of leakage is found. The resonance parameters have been found for a medium with an arbitrary anisotropy. General relations are concretized for monoclinic, orthorhombic, trigonal, tetragonal, cubic, and hexagonal systems. Estimates and illustrations are given for a series of such crystals. The intensity of the reflected beam increases with its narrowing, but its diffraction divergence also increases with this narrowing. Nevertheless, the intensity of the beam can be increased by a factor of 5–10 at sufficiently high frequencies while keeping its divergence at an acceptable level. Amplification by two orders of magnitude can be achieved by compressing the beam in two dimensions through its double reflection.

Keywords: crystals, elastic waves, acoustic beams, total internal reflection, mode conversion, efficiency of transformation, diffraction divergence

## 1. Introduction

Modern crystal acoustics is an important base for numerous instruments and devices using concentrated ultra- and hypersonic beams, delay lines, surface and bulk waves, etc. Many of acoustic effects in crystals arise exclusively due to their anisotropy. In particular, piezoelectricity exists only in crystals and is widely used in acoustic devices [1, 2]. Another spectacular example of a nontrivial role of anisotropy is phonon focusing [3], the concentration of energy in a crystal along special directions for which the acoustic beam in Poynting vectors is much narrower than that in wave vectors. In this chapter, we will consider another principle of energy concentration in acoustic waves that is also entirely related to crystal anisotropy. Intense ultrasonic beams are widely used in engineering, medicine, scientific instrument making, etc. [4]. The reflection and refraction of such beams at the interfaces between layered isotropic structures are commonly used for their transformation. Crystals open up new opportunities for beam transformation.

A method for producing intense beams in crystals based on the features of their elastic anisotropy was proposed in our paper [5]. "Compression" of an acoustic beam is achieved by choosing the geometry of the beam incidence on a surface close to the angle of total internal reflection, when one of the reflected beams (r2) propagates at a small angle to the surface (Figure 1a) and, its width can be made arbitrarily small. However, we need the compression not of the beam width, but of its energy density. For instance, in isotropic medium, a compression of such beam is accompanied by a decreasing amount of energy entering it, without any growth of its intensity. The same occurs in the crystal without a special choice of reflection geometry. And still in anisotropic media, there are specific orientations which admit the beam intensification.

checking the established relations between the basic parameters determining the unusual resonance phenomenon with features quite promising for applications.

Consider a semi-infinite elastic medium of unrestricted anisotropy with a free boundary. It will be characterized by the tensor of moduli of elasticity cijkl and the density ρ. The sagittal plane is specified by two unit vectors: the propagation direction m along the surface and the normal n to the surface. Reflection shown schematically in Figure 1b is the superposition of four partial waves: the incident (α =4= i) and reflected (α =1= r1) waves from the outer sheet of the slowness surface, the reflected wave (α =2= r2) from the middle sheet, and the localized

> A<sup>α</sup> Lα

The unknown vectors A<sup>α</sup> and L<sup>α</sup> are found from the so-called Stroh's formalism

!

means transposition) together with parameters p<sup>α</sup> (α = 1,…,6) are eigenvectors and

Here ^<sup>I</sup> is the unit 3 � 3 matrix and the matrices (ab) are defined by the convolutions (ab)jk = aicijklbl of the moduli tensor cijkl with the vectors a and b. The six eigenvectors ξ<sup>α</sup> are complete and orthogonal to each other everywhere apart from points of

Depending on v, the vectors ξ<sup>α</sup> and the parameters p<sup>α</sup> may be real or form complex conjugated pairs. The reflection considered in this chapter (Figure 1b) belongs to the second supersonic region of the slowness surface. In the above terms, here the wave superposition formally may include four bulk partial waves with real parameters pα, two incident and two reflected, from the external and middle sheets. In addition, at our disposal, there are two inhomogeneous partial waves with complex conjugated parameters pα, one localized and the other nonphysical (increasing into the depth of the medium), related to the internal sheet. The second incident wave and

degeneracy. The orthogonality property may be expressed in the form:

� � exp f g <sup>i</sup>ð Þ <sup>k</sup><sup>α</sup> � <sup>r</sup>‒ω<sup>t</sup> : (1)

<sup>α</sup> ¼ 1.

Cr1L<sup>1</sup> þ Cr2L<sup>2</sup> þ ClL<sup>3</sup> þ CiL<sup>4</sup> ¼ 0: (2)

ð Þ nm ð Þ nn �<sup>1</sup>

A<sup>α</sup> � L<sup>β</sup> þ A<sup>β</sup> � L<sup>α</sup> ¼ 0, α 6¼ β: (4)

ð Þ� nm ð Þ� mm <sup>ρ</sup>v<sup>2</sup>^I mn ð Þð Þ nn �<sup>1</sup>

� �

<sup>T</sup> (the superscript T

<sup>y</sup>¼<sup>0</sup> <sup>¼</sup> 0, takes

: (3)

2. Formulation of the problem and basic relations

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

wave (α =3= l) from the internal sheet:

the form:

161

u rð Þ ; t ð Þ i=k σ^ð Þ r; t � �

and σ^), these vectors are normalized by the condition: A<sup>2</sup>

based on the fact that the combined six-vectors ξ<sup>α</sup> = {Aα, Lα}

eigenvalues of the 6 � 6 Stroh matrix <sup>N</sup>^ [12],

<sup>N</sup>^ ¼ � ð Þ nn �<sup>1</sup>

ð Þ mn ð Þ nn �<sup>1</sup>

¼ ∑ 4 α¼1 C<sup>α</sup>

In terms of Eq. (1), the boundary condition of free surface, σijnj

Here, u is the wave field displacement vector, σ^ is the stress tensor, C<sup>α</sup> are the partial waves amplitudes, <sup>ω</sup> is the frequency, and <sup>k</sup><sup>α</sup> <sup>¼</sup> <sup>k</sup> <sup>m</sup> <sup>þ</sup> <sup>p</sup>α<sup>n</sup> � � are the wave vectors of partial components with a common projection k onto the direction of propagation m (Figure 1b). The k value determines the tracing speed v = ω/k of stationary motion of the wave field (1) along the boundary. Vectors A<sup>α</sup> and L<sup>α</sup> characterize the partial field polarizations. Being not independent (as well as u

As is shown in [5], this happens when the wave field of the beam r2 is close to the eigenmode—an exceptional bulk wave (EBW) satisfying the free surface boundary condition [6, 7]. The perturbation of the selected EBW propagation geometry transforms this one-partial solution to the resonance reflection component. To obtain such a special resonance reflection near the eigenmode, the EBW should exist on the middle sheet of the slowness surface, while the incident "pump" wave should belong to the external sheet (Figure 1b). The proper cuts can be found almost in any crystal. However, one should bear in mind that in this case, apart from the reflected wave r2 excited from the middle sheet of the slowness surface, another reflected wave r1 belonging to the external sheet inevitably exists. Energy losses related to this parasitic wave can be minimized by choosing crystals or geometries with parameters corresponding to the closeness of reflection to mode conversion, when such a parasitic wave does not appear.

In [5, 8], we considered perturbed geometries, where the surface of the crystal remained unchanged and the plane of reflection (sagittal plane) was rotated by a small angle φ about the normal n to the surface (Figure 1a). In this case, an increase in the intensity of excited beam was controlled by the angle φ, whereas the energy loss to the parasitic beam was completely determined by the relation between the moduli of elasticity and could be reduced only by an appropriate choice of the crystal.

In [9], we analyzed another variant of the theory where a similar resonance in a hexagonal crystal was governed by the angle of rotation of the surface about the direction of propagation of the unperturbed exceptional bulk wave. In this case, conversion also occurs only under a certain relation between moduli of elasticity.

In [10, 11], the more general analysis was accomplished which allowed us to demonstrate that the mode conversion of resonance near total internal reflection (i.e., the scheme in Figure 1a without the parasitic beam r1) can be implemented in almost any acoustic crystal by a consistent variation of orientations of both the boundary and the sagittal planes.

In this chapter, we shall summarize the results of mentioned and some other studies of ours and present the combined theoretical consideration of the problem with both analytical approximate calculations and numerical exact computations

#### Figure 1.

Scheme of excitation of a narrow beam near total internal reflection in the r space (a) and k space (b) with Pi, ki; P<sup>r</sup>1, k<sup>r</sup>1; and P<sup>r</sup>2, k<sup>r</sup><sup>2</sup> to be mean Poynting and wave vectors of the incident, parasitic, and excited beams, respectively.

A method for producing intense beams in crystals based on the features of their elastic anisotropy was proposed in our paper [5]. "Compression" of an acoustic beam is achieved by choosing the geometry of the beam incidence on a surface close to the angle of total internal reflection, when one of the reflected beams (r2) propagates at a small angle to the surface (Figure 1a) and, its width can be made arbitrarily small. However, we need the compression not of the beam width, but of its energy density. For instance, in isotropic medium, a compression of such beam is accompanied by a decreasing amount of energy entering it, without any growth of its intensity. The same occurs in the crystal without a special choice of reflection geometry. And still in anisotropic media,

As is shown in [5], this happens when the wave field of the beam r2 is close to

In [5, 8], we considered perturbed geometries, where the surface of the crystal remained unchanged and the plane of reflection (sagittal plane) was rotated by a small angle φ about the normal n to the surface (Figure 1a). In this case, an increase in the intensity of excited beam was controlled by the angle φ, whereas the energy loss to the parasitic beam was completely determined by the relation between the moduli of elasticity and could be reduced only by an appropriate choice of the crystal. In [9], we analyzed another variant of the theory where a similar resonance in a hexagonal crystal was governed by the angle of rotation of the surface about the direction of propagation of the unperturbed exceptional bulk wave. In this case, conversion also occurs only under a certain relation between moduli of elasticity. In [10, 11], the more general analysis was accomplished which allowed us to demonstrate that the mode conversion of resonance near total internal reflection (i.e., the scheme in Figure 1a without the parasitic beam r1) can be implemented in almost any acoustic crystal by a consistent variation of orientations of both the

In this chapter, we shall summarize the results of mentioned and some other studies of ours and present the combined theoretical consideration of the problem with both analytical approximate calculations and numerical exact computations

Scheme of excitation of a narrow beam near total internal reflection in the r space (a) and k space (b) with Pi, ki; P<sup>r</sup>1, k<sup>r</sup>1; and P<sup>r</sup>2, k<sup>r</sup><sup>2</sup> to be mean Poynting and wave vectors of the incident, parasitic, and excited beams,

the eigenmode—an exceptional bulk wave (EBW) satisfying the free surface boundary condition [6, 7]. The perturbation of the selected EBW propagation geometry transforms this one-partial solution to the resonance reflection component. To obtain such a special resonance reflection near the eigenmode, the EBW should exist on the middle sheet of the slowness surface, while the incident "pump" wave should belong to the external sheet (Figure 1b). The proper cuts can be found almost in any crystal. However, one should bear in mind that in this case, apart from the reflected wave r2 excited from the middle sheet of the slowness surface, another reflected wave r1 belonging to the external sheet inevitably exists. Energy losses related to this parasitic wave can be minimized by choosing crystals or geometries with parameters corresponding to the closeness of reflection to mode

there are specific orientations which admit the beam intensification.

conversion, when such a parasitic wave does not appear.

boundary and the sagittal planes.

Acoustics of Materials

Figure 1.

respectively.

160

checking the established relations between the basic parameters determining the unusual resonance phenomenon with features quite promising for applications.
