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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.
A.V. Shubnikov Institute of Crystallography, Federal Research Center “Crystallography and Photonics,” Russian Academy of Sciences, Moscow, Russia
Dmitrii A. Bessonov
A.V. Shubnikov Institute of Crystallography, Federal Research Center “Crystallography and Photonics,” Russian Academy of Sciences, Moscow, Russia
Vasilii N. Lyubimov
A.V. Shubnikov Institute of Crystallography, Federal Research Center “Crystallography and Photonics,” Russian Academy of Sciences, Moscow, Russia
*Address all correspondence to: valshits@mail.ru
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.
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 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 c_{ijkl} 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 wave (α = 3 = l) from the internal sheet:
urti/kσ̂rt=∑α=14CαAαLαexpikα⋅r‒ωt.E1
Here, u is the wave field displacement vector, σ̂ is the stress tensor, C_{α} are the partial waves amplitudes, ω is the frequency, and kα=km+pαn 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_{α} and L_{α} characterize the partial field polarizations. Being not independent (as well as u and σ̂), these vectors are normalized by the condition: Aα2=1.
In terms of Eq. (1), the boundary condition of free surface, σijnjy=0=0, takes the form:
Cr1L1+Cr2L2+ClL3+CiL4=0.E2
The unknown vectors A_{α} and L_{α} are found from the so-called Stroh’s formalism based on the fact that the combined six-vectors ξ_{α} = {A_{α}, L_{α}}^{T} (the superscript T means transposition) together with parameters p_{α} (α = 1,…,6) are eigenvectors and eigenvalues of the 6 × 6 Stroh matrix N̂ [12],
N̂=−nn−1nmnn−1mnnn−1nm−mm−ρv2Îmnnn−1.E3
Here Î is the unit 3 × 3 matrix and the matrices (ab) are defined by the convolutions (ab)_{jk} = a_{i}c_{ijkl}b_{l} of the moduli tensor c_{ijkl} with the vectors a and b. The six eigenvectors ξ_{α} are complete and orthogonal to each other everywhere apart from points of degeneracy. The orthogonality property may be expressed in the form:
Aα⋅Lβ+Aβ⋅Lα=0,α≠β.E4
Depending on v, the vectors ξ_{α} and the parameters p_{α} 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 p_{α}, 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 p_{α}, one localized and the other nonphysical (increasing into the depth of the medium), related to the internal sheet. The second incident wave and the nonphysical inhomogeneous component were naturally excluded (C_{5} = C_{6} = 0) from the sums in Eqs. (1) and (2).
The amplitude C_{i} of the incident wave is assumed to be known, while the remaining amplitudes may be expressed in terms of C_{i} through scalar multiplication of Eq. (2) by the vector products L_{2} × L_{3}, L_{1} × L_{3}, or L_{1} × L_{2}. As a result, we arrive at the following reflection coefficients in the form of the ratios of mixed products:
3. Singular reflection geometry in the vicinity of EBW
The deduced equations (5) represent just exact relations which describe the discussed resonance reflection only in some definite narrow region of orientations of surface and sagittal plane. Let us demonstrate that such singular region really arises close to the geometry of EBW propagation related to the pair {n_{0}, m_{0}}. Indeed, wave superposition (1) in this exceptional geometry is decomposed into two independent solutions—EBW and three-partial reflection—satisfying the boundary conditions, which are fragments of Eq. (2):
L02=0,E6
Cr1L01+ClL03+CiL04=0,E7
where the subscript 0 indicates the initial unperturbed configuration. Of course, near this geometry, the vector L_{2} should be small and the other vectors L_{1}, L_{3}, and L_{4} nearly coplanar. As a result, both the numerator and denominator of the expression for R_{2} in (5) should independently approach zero; i.e., R_{2} is singular. This singularity is responsible for the resonance character of reflection and, therefore, for the discussed effect. The coefficient R_{1} in (5) is regular because the small vector L_{2} appears in both the denominator and the numerator of the expression for this coefficient.
Let us introduce practically important characteristics of the investigated resonance, specifically, the gain of excited wave intensity (K_{2}) and energy loss (K_{1}) in a parasitic wave:
K2=P2P4=R22s2s4,K1=P1P4=R12s1s4.E8
Here, P_{α} are the Poynting vector lengths, which are products of the energy density in the corresponding partial wave (∝Cα2) and its ray speed s_{α}. Another useful characteristic of the resonance is the excitation efficiency η = 1−K_{1}, equal to the fraction of energy transferred from the incident to excited wave.
Thus, a small perturbation of the crystal orientation in the vicinity of the geometry of the EBW propagation may provide a resonance intensification of the reflected wave r2. The control parameters of resonance are the optimized characteristics of geometry of reflection {n, m} (Figure 1a), i.e., the orientations of the surface and sagittal plane, as well as the angle of incidence α of the pump wave i related to these characteristics. The perturbation {n_{0}, m_{0}} → {n, m} is shown in Figure 2a. The new plane boundary P of the crystal is specified by the unit normal n = n(ψ, χ) rotated by a small angle ψ = ∠(n, n_{0}) with respect to the normal n_{0} to the initial boundary P_{0} along which the EBW can propagate. The axis of rotation (the intersection line between P and P_{0} planes) is specified by the angle χ, counting from reference line 1||m_{0} and lying in the range of [0, π]. The direction m in the P plane is specified by the angle φ measured from line 2 along the projection of the unperturbed vector m_{0} on the boundary P.
Below, to compact equations, we will also use the alternative notation for the introduced small angles:
ϕ1=φ,ϕ2=ψ.E9
In fact, as we shall see, the squares of these angles (in units of radians) rather than the angles themselves are small parameters in the theory developed below.
As is seen in Figure 2b, where the scheme of resonance reflection is shown, the angle α of incidence must be chosen near the threshold angle of total internal reflection α̂: α=α̂+δα. This configuration is characterized by the tracing speed v = ω/k of the wave field close to the limiting velocity v̂. The adjusting angle δα controlling reflection resonance is directly related to the difference δv=v−v̂≈δp2/2κ [8]:
δα=δvv̂A=A2κv̂δp2,A=v̂1+pi2∂pi∂vv=v̂,E10
where κ is the radius of curvature of the cross section of the middle sheet of the slowness surface by the sagittal plane at the limiting point corresponding to v=v̂.
4. Analytical description of resonance characteristics
In this section, we shall obtain approximate expressions for the coefficients K_{1} and K_{2} defined by Eq. (8) in the general case of arbitrary anisotropy. Reflectivities R_{1} and R_{2} are expressed in (5) in terms of vectors L_{α}, which will be considered as functions of the parameters δp, φ, ψ, and χ (see Figure 2). The last (not small) angle χ, which determines the axis for n rotation, is considered fixed at this stage. Its influence on the effect will be studied numerically. Expressions (5) can be expanded in the other small parameters.
Given (7), the numerator in Eq. (5) for R_{2} can be approximately represented as:
L4L1L3≈ϕi∂∂ϕiL4L1L30,E11
where the subscript 0 means that after differentiation, one should put ϕ = ψ = 0, δp = 0.
To calculate the denominator in the expressions for reflection coefficients R_{1} and R_{2}(5), let us expand the two cofactors separately:
L3×L1≈L03×L01+ϕi∂∂ϕiL3×L10,E12
L2≈ϕi∂L2∂ϕi0+12ϕiϕj∂2L2∂ϕi∂ϕj0+δp∂L2∂p20.E13
In writing (13), we used the fact that L_{02} = 0 (6). Substituting Eqs. (12) and (13) into the denominators of ratios (5), it is easy to verify that the term linear in φ and ψ drops out of the result. This term is proportional to the mixed product, which is zero,
L03L01∂L2∂ϕi0=0,E14
because, as we will see, it is composed of coplanar vectors. Let us prove that they are all perpendicular to the same real vector A_{02}. From orthogonality condition (4), one has
A02⋅L0α+L02⋅A0α=A02⋅L0α=0.E15
for all α ≠ 2 where the fact that L02=0 was again used.
In order to prove that the derivative ∂L2/∂ϕi0 is also orthogonal to A02, one can use identity A2v̂⋅L2v̂=0 valid for any transonic states of arbitrary geometry {m, n} [13]. Let us differentiate this identity and then set ϕ = ψ = 0:
A2⋅∂L2∂ϕi+L2⋅∂A2∂ϕi0=A02⋅∂L2∂ϕi0=0.E16
Thus, Eqs. (15) and (16) prove vanishing in (14), which means that indeed the denominator of both reflection coefficients (5) does not contain terms linear in φ and ψ. The numerator of R_{1} is found from the same relations (12) and (13) after replacing in them indices 1 → 4, and the numerator of R_{2} is given by Eq. (11). After some straightforward calculations, one obtains
As could be expected, loss coefficient K_{1}(21) is regular in the control parameters φ, ψ, and δp, whereas gain K_{2}(22) is singularly dependent on them: at ϕ_{i}ϕ_{j} ≪ δp ≪ 1, K_{2} tends to zero, and at δp ≪ ϕ_{i}ϕ_{j} ≪ 1, it diverges.
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_{1} 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_{1} = 0 to the system of equations:
λ″ij4ϕiϕj=0,δp=−λ′ij4ϕiϕj,E23
where the first equation determines the relation between the angles of rotation of the sagittal plane ϕ1≡φ and the normal to the surface ϕ2≡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_{44} > c_{66}. 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:
ψ=γ±φ,δp=λ±φ2.E24
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_{2} given by Eq. (22) should obviously be large because it is inversely proportional to the square of the small parameter:
K2con∝φ−2∝ψ−2.E25
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 P_{i}D_{i} = P_{r2}d_{r2} (Figure 1a). This balance gives
K2con≡Pr2/Pi=Di/dr2,E26
i.е., the reflected beam turns out to be narrower than the incident one by a factor of K_{2}. On the other hand, the related diffraction divergence angles, δi∼cs/νDi and δr2∼cs/νdr2 (where c_{s} ∼ 10^{5} cm/s is the sound speed), are in similar proportion:
δr2≈Didr2δi=K2conδi.E27
Thus, the possible increase in the coefficient K_{2} 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 D ∼ 1 cm, one can obtain a coefficient K_{2} ∼ 5–10 at d_{r2} ≈ 1–2 mm, δ_{i} ∼ 10^{−3}, and δ_{r2} ∼ 10^{−2} rad.
For a fixed direction of the normal n(ψ, χ) to the crystal boundary, the surfaces K2φδα and ηφδα 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^{4}. And along the ridges, one obtains
K2max=const/δα,ηmax=const.E28
Of course, with variations of n, the constants in (28) also change. For some definite direction of n, the mode conversion occurs when ηmax=ηcon=1 and K2max=K2con.
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, ηcon=1 and K2con, whereas on the other, an increased extremal gain K2max>K2con and a decreased efficiency η^{max} < 1 occur. It is even more important that on the second trajectory, the same value of coefficient K_{2} 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.
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,ψm,χmandδαm are found for the gains K2max=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 δαm with efficiency ηm retained rather high level. The magnitudes of δαm 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 BaTiO_{3} crystals, the angle δαm 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%.
Systems
Crystal
m_{0,} n_{0}
Normal n
φm, rad
δαm, rad
η, %
χ, rad
ψ, rad
Cubic
LiF
m0‖x′,n0‖y′
0.21
0.29
0.13
0.008
88.7
Silicon
0.29
0.29
0.13
0.007
74.0
Hexagonal
Graphite
m0‖x,n0‖y
0
0.02
0.14
0.034
94.9
ZnS
0.03
0.48
0.32
0.008
93.4
CdCe
0.03
0.69
0.14
0.008
95.2
Tetragonal
Paratellurite
m0‖x′,n0‖y′
0.09
0.03
0.07
0.013
99.8
BaTiO_{3}
m0‖x,n0‖z
0.05
0.28
0.37
0.031
80.0
Trigonal
Quartz
m0‖x1,n0‖y
0.01
0.60
0.48
0.043
75.0
LiNbO_{3}
m0‖x2,n0‖y
0.30
0.78
0.09
0.007
79.0
Orthorhombic
Rochelle salt
m0‖z,n0‖x
0.12
0.44
0.55
0.009
94.6
m0‖x,n0‖z
0.18
0.47
−0.08
0.010
98.4
Table 1.
Characteristics of extremal resonances K_{2} = 5 for acoustic crystals of various symmetry systems.
The unperturbed orientations of vectors m_{0} and n_{0} are determined by the axes x, y, and z of crystallographic coordinates, the bisector axes x′ and y′ in the basal plane xy and by the directions xα‖cosθα0sinθα, where θ_{1} = −0.76 and θ_{2} = 0.46 rad.
Figure 4 shows the optimization of the parameters δαm and ηm for the lithium niobate crystal that corresponds to the value K_{2} = 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 K_{2}(φ, δα) similar to that shown in Figure 3 was plotted from which the φm and δαm values corresponding to the extremal point at the “crest” with the amplitude K_{2} = 5 were determined. Consequently, each point on the δαmψχ and ηmψχ 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 δαm and ηm. 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 K2maxδα versus δα for eight analyzed acoustic crystals shows these dependences as very slowly changing functions close to constants.
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 calculations for some of hexagonal crystals.
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_{0} to be parallel to the axis 6 so that the normal n_{0} to this boundary is directed along the y axis. In this geometry, an EBW with the polarization A_{02}||z can propagate along the crystal surface in the direction m_{0}||х with a speed:
v̂0=c44/ρ.E29
For transverse isotropy, we may change the initial crystal surface orientation P_{0} → P, rotating its normal vector n around m_{0} (i.e., choosing in Figure 2a the angle χ = π/2) by a small angle ψ
n0→n=0cosψsinψ.E30
In addition, as before, we introduce a perturbed propagation direction m rotated relative to the vector m_{0} by a small angle ϕ in the new surface plane P:
m0→m=m0cosφ+n×m0sinφ,n×m0=0sinψ−cosψ.E31
Based on the standard equations of crystal acoustics [14, 15], one can determine the wave parameters entering superposition (1)
We assume the parameters q and p to be real, which holds true at c11>c44>c66. Note that the inequality c11>c66 is always satisfied (this is the crystal stability condition [14]) and the inequality c11>c44 is almost always satisfied (we do not know exclusions). Meanwhile, the condition c44>c66, which indicates that EBW (29) belongs to the middle sheet of the slowness surface (Figure 2b), is valid in far from all crystals (say, in a half of them).
As before, the angle of incidence is chosen near the angle α̂ of total internal reflection (Figure 2b). This angle corresponds to the limiting speed, which now can be directly related to the EBW speed v̂0(29):
v̂2=v̂021−β⌢φ2,β⌢=d2−Δ14Δ34c14Δ14.E38
In turn, the small tuning angle δα corresponds to the interval δv=v−v̂ and, consequently, to the parameter δp:
δv=v̂0pδα=12v̂0δp2,δα=δp2/2p.E39
Substituting expressions (32)–(34) for the vectors L_{α} into Eq. (5), we obtain reflectances R_{1} and R_{2} as functions of the moduli of elasticity and perturbation parameters ϕ, ψ, and δp:
Taking into account Eq. (40), we can reduce the conversion condition R_{1} = 0 to the system of equations:
λ″φ2−aλ′ψ2=0,φ2−λ′δp+φψβ⌢=0.E43
The first equation yields two versions of the mode conversion relationship between the rotation angles of the boundary (ψ) and sagittal plane (ϕ):
ψcon=±γφcon,γ=λ″aλ′.E44
Substituting the parameter values from Eq. (42) into the radicand of Eq. (44), we can easily see that for с_{44} > с_{66} in the case under consideration, the γ value is always real:
In other words, at с_{44} > с_{66}, there are always two crystal cuts, i.e., two versions of coupled orientations of the boundary and sagittal planes, which ensure mode conversion near angle α̂ of total internal reflection. Certainly, each set of chosen angles ψ = ±γφ corresponds to its own definite angle of incidence: δα=δp2/2p(39). The second equation in (43) yields:
δpcon±=φcon21λ′∓γβ⌢,δαcon±=φcon42p1λ′∓γβ⌢2.E46
Thus, the resonance width with respect to the incidence angle is indeed very small: δα∝φ4. As we have seen, it is preferable to choose the sign corresponding to the maximum value δαcon± (the upper sign at β⌢<0 and the lower sign at β⌢>0):
ψcon=−γφconsgnβ⌢,δαcon=φcon42p1λ′+γβ⌢2.E47
Given the found relations for reflectances (40) and (41), we can obtain the gain (K_{2}) and loss (K_{1}) coefficients (8) where one can put
s1≈s4,s2/s4≈c44/c66.E48
Here we took into account that for the studied reflection geometry with the sagittal plane close to transverse isotropy, the ray speeds s_{α} are approximately equal to the phase speeds of the bulk waves involved in the reflection. Under the total conversion conditions (44)–(47), the coefficient K_{1} is 0; therefore, the excitation efficiency of beam r2 is maximum: η = 1 = 100%. In this case, the gain K_{2} can be written as:
K2con=Gconφcon2,Gcon=μ2+μ˜2γ2s2/s441+a2γ4.E49
The above analysis based on expansion of the equations in small parameters φ^{2}, ψ^{2}, δp, and δα is approximate. Therefore, the range of applicability of the results obtained may differ, depending on the degree of crystal anisotropy and other factors. In Figure 6, the analytical linear relation between the conversion angles φ 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 BeCu) hexagonal crystals [16, 17].
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 theoretical ones. In case of Ti, we have β⌢<0 and coincident signs of angles ϕ and ψ, while for BeCu, β⌢>0 and the angle signs are different.
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 mode conversion and the extremal gain are almost identical.
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_{1}(40) acquires a minimal value K1min at φ^{2} = λ′δp which may be estimated as:
K1min=R12=λ″2λ′2∼c44−2c66c444.E50
Thus, any hexagonal crystal with the modulus c_{44} close to 2c_{66} must be very efficient for our effect (K_{1} ≪ 1). Due to the “4” power in the latter estimate, this remains true even when the moduli are not so close to each other. For instance, at |c_{44}−2c_{66}|/c_{44} ∼ 1/3, Eq. (50) gives the estimate K_{1} ∼ 10^{−2} and efficiency η = 1−K_{1} becomes ∼ 99%. In accordance with reference book [16], there are quite a number of hexagonal crystals where K_{1} ≪ 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:
K1=φ2−λ′δp2+λ″δp2φ2+λ′δp2+λ″δp2,E51
K2=μφ2s2/s4φ2+λ′δp2+λ″δp2.E52
And the efficiency of the resonance η = 1−K_{1} is given by:
η=4λ′δpφ2φ2+λ′δp2+λ″δp2.E53
Thus, for a fixed δp, the coefficients K_{2} and η are determined by the same function F(φ):
K2φ=μs2s4Fφ,ηφ=4λ′δpFφ,Fφ=φ2φ2+λ′δp2+λ″δp2.E54
Accordingly, their maximum magnitudes are determined by the same extremum condition:
φ2=λδp=δpλ′2+λ″2≈λ′δp1+12λ″λ′2.E55
The last estimate in (55) is valid when λ″/λ′2≪1. In this approximation, one obtains
K1minφδα≈λ″2λ′2,K2maxφδα≈Gφ2,G≈μs2s4.E56
Thus, one can conclude that the consistent variation of the sagittal plane orientation φ and the tuning incidence angle δα∝δp2(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 K2maxφδα and ηmaxφδα. 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 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 K_{2}(δα) and η(δα) in such extremal trajectories related to ridges on the surfaces K2φδα and ηφδα (of the type shown in Figure 3) conformably to a monoclinic stilbene crystal. The upper curves correspond to the choice of φ = f_{K,η}(δα) in the “proper” trajectory corresponding to the maximum of the shown characteristic, while the lower curves are plotted for φ = f_{η,K}(δα) from the “foreign” trajectory. In this case, of the two possible optimization variants, the trajectory in which K2max 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_{m} for the optimal angle of incidence corresponding to an amplitude of K2max=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_{m} ≥ 0.01 rad for K_{2} = 5). According to such criteria, we found 14 crystals of monoclinic, trigonal, orthorhombic, and hexagonal systems. They are characterized by a rather high efficiency η_{m}, 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].
Here, we limit ourselves to presenting only data for several monoclinic crystals with the best parameters (see Table 2). The values of δα_{m}, φ_{m}, and η_{m} presented in the table for all crystals correspond to the same gain K_{2} = 5 and relate to the choice of optimal trajectory φ_{K} as for stilbene (Figure 7). This results in the maximum width of resonance for a small decrease in efficiency η_{m}. By the way, stilbene is the absolute leader in the table in the value of δα_{m} and one of the leaders in the efficiency η_{m}.
Crystals
φ_{0}, rad
φ_{m}, rad
δα_{m}, rad
η_{m}, %
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
Table 2.
Key parameters of 1D resonance for a series of monoclinic crystals.
The directions of m_{0} are specified by the azimuth angles φ_{0} counted from the x axis of the crystallographic coordinates and the angles φ_{m}—From m_{0}.
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) 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 high-symmetry 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 (5) and (8).
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, λ/D ∼ 10^{−3} rad. Here, we also had in mind the possible manifestations of the effect in phonon physics, where the language of plane waves is more relevant.
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, stringent requirements for a weak divergence of the incident beam, ∼ 10^{−3} rad, which can be realized only at high ultrasonic frequencies ∼100 MHz, arise. For the same reason, the amplitude of the excitation coefficient is also limited to K_{2} ≈ 5–10. However, in the case of retransformation of the emergent beam through its narrowing in the perpendicular dimension as well, the intensification efficiency increases many fold, to ∼ 10^{2}. In the hypersonic frequency range, the amplification amplitudes can be increased significantly. In this case, however, one might expect additional restrictions due to an increase in the absorption of acoustic waves.
The acoustic resonance considered here can also manifest itself in phonon physics as the channeling of high-density energy near the surfaces of crystals with specially chosen orientations and in accompanying nonlinear phenomena. The strategy for an experimental search of such effects can be based on existing techniques for studying the manifestations of phonon focusing [3] and the propagation of ballistic phonons [18].
This work was supported in part by the Ministry of Education and Science of the Russian Federation (state assignment for Federal Research Center “Crystallography and Photonics”).
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Written By
Vladimir I. Alshits, Dmitrii A. Bessonov and Vasilii N. Lyubimov
Submitted: June 27th, 2018Reviewed: November 2nd, 2018Published: December 14th, 2018