ISSN 10637761, Journal of Experimental and Theoretical Physics, 2011, Vol. 113, No. 4, pp. 659–672. © Pleiades Publishing, Inc., 2011. Original Russian Text © V.I. Alshits, V.N. Lyubimov, 2011, published in Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2011, Vol. 140, No. 4, pp. 755–770.
SOLIDS AND LIQUIDS
Conical Refraction of Elastic Waves in Absorbing Crystals V. I. Alshits and V. N. Lyubimov Shubnikov Institute of Crystallography, Russian Academy of Sciences, Moscow, 119333 Russia *email:
[email protected] Received March 2, 2011
Abstract—The absorptioninduced acousticaxis splitting in a viscoelastic crystal with an arbitrary anisot ropy is considered. It is shown that after “switching on” absorption, the linear vector polarization field in the vicinity of the initial degeneracy point having an orientation singularity with the Poincaré index n = ±1/2, transforms to a planar distribution of ellipses with two singularities n = ±1/4 corresponding to new axes. The local geometry of the slowness surface of elastic waves is studied in the vicinity of new degeneracy points and a selfintersection line connecting them. The absorptioninduced transformation of the classical picture of conical refraction is studied. The ellipticity of waves at the edge of the selfintersection wedge in a narrow interval of propagation directions drastically changes from circular at the wedge ends to linear in the middle of the wedge. For the wave normal directed to an arbitrary point of this wedge, during movement of the dis placement vector over the corresponding polarization ellipse, the wave ray velocity s runs over the same cone describing refraction in a crystal without absorption. In this case, the end of the vector moves along a universal ellipse whose plane is orthogonal to the acoustic axis for zero absorption. The areal velocity of this movement differs from the angular velocity of the displacement vector on the polarization ellipse only by a constant fac tor, being delayed by π/2 in phase. When the wave normal is localized at the edge of the wedge in its central region, the movement of vector s along the universal ellipse becomes drastically nonuniform and the refrac tion transforms from conical to wedgelike. DOI: 10.1134/S1063776111100013
1. INTRODUCTION The internal conical refraction of elastic waves in crystals, like a similar effect in crystal optics, is a good example of the nontrivial role of anisotropy, which can create new phenomena in crystals that are basically impossible in isotropic media. The energy flux P of a wave in a crystal, as a rule, is not parallel to its wavevector k = km. For each m, the direction of the Poynting vector P is determined by the orientation of the normal n with respect to the slow ness (refraction) surface v–1(m), where v is the wave phase velocity. Unlike isotropic media, where due to the degeneracy of transverse bulk eigenwaves, both linearly and circularly polarized waves can propagate in any direction, in crystals this can occur only along acoustic axes, which are usually isolated. In particular, for direction m selected along the acoustic axis, when a contact between the sheets of the slowness surface is conical, each direction of the wave polarization relates to its own normal to the slowness surface at the contact point. Therefore, Poynting vector P for elliptic or cir cular polarization should precess along a certain cone corresponding to a complete set of normals at the con sidered contact point of sheets of the slowness surface. This phenomenon is called internal conical refraction. Conical refraction in acoustics was theoretically predicted and experimentally observed for the first time in [1]. The authors of this paper directed an ultra sonic beam at a cubic Ni crystal along the threefold
symmetry axis, to which the conical acoustic axis always corresponds, and found that the wave energy flux traced a circular cone when the transverse polar ization direction was varied in the degeneracy plane. Later, more general cases of elliptic refraction cones for less symmetrical directions of acoustic axes were described theoretically [2–4] and observed experimen tally [5]. The most complete theory of this effect for arbitrary anisotropy is presented in monographs [6, 7]. In this paper, we generalize this theory to the case of the absorption of a wave in a crystal, which always takes place to a particular. Absorption in the acoustics of anisotropic media leads not only to attenuation of waves during their propagation but also to a number of qualitatively new properties and phenomena. In par ticular, according to [8–12], after “switching on” absorption, any conical acoustic axis not coinciding with the direction of the threefold symmetry axis should split into two singular axes (the acoustic axis directed along the threefold axis appears due to sym metry and is not split). Below, we will theoretically describe this splitting and accompanying transforma tions in the local geometry of the slowness geometry and topology of polarization fields in the vicinity of singular axes. Based on this consideration, we will generalize theory [6, 7] to the distribution of elliptic polarizations and ray velocities of elastic waves for propagation directions m near the split axes. As follows from our solution of a similar problem [13] in crystal
659
660
ALSHITS, LYUBIMOV
optics, consideration of absorption leads to rather nontrivial changes in the fundamental properties of the phenomenon. 2. FORMULATION OF THE PROBLEM AND BASIC EQUATIONS Consider a viscoelastic medium with an arbitrary anisotropy characterized by the elastic moduli tensor ˆ = {ηijkl}, and density ρ. ˆc = {cijkl}, viscosity tensor η The elastic displacement field u(r, t) in such a medium is described by the known equation of motion [14] (1) ρu·· i = c ijkl u l, kj – η ijkl u· l, kj , where dots over u signify derivatives with respect to time t, while derivatives with respect to spatial coordi nates are denoted as ∂/∂xk… ≡ …, k. By substituting the displacement field of a bulk elastic wave with fre quency ω = kv, amplitude C, and normalized polar ization vector A, (2) u ( r, t ) = CA exp [ ik ( m ⋅ r – vt ) ] into (1), we obtain the generalized Christoffel equa tion ˆ ' – iQ ˆ '' )A = ρv 2 A, (Q (3) ˆ ' and Q ˆ '' are real symmetric matrices where Q (4) Q ''jk = ωm i η ijkl m l . ˆ '' to the usual Note that the imaginary addition –i Q ˆ ' , unlike the latter, is dispersive; acoustical tensor Q i.e., it depends on frequency. Equation (3) for nondegenerate directions m spec 2 ifies three eigenvectors Aα and three eigenvalues ρ v α ˆ ' – iQ ˆ '' . It is nat of the generalized acoustical tensor Q Q 'jk = m i c ijkl m l ,
ural that polarization vectors Aα and phase velocities vα in this case are complex, A α = A 'α + iA ''α , v α = v 'α – iv ''α , α = 1, 2, 3. (5) We assume below that the frequency ω is real. There fore, it follows from (5) that the quantity kα has a small imaginary addition determining the attenuation of the propagating wave: v ''α⎞ ω ω ⎛ 1 + i (6) k α ≡ k 'α + ik ''α = ≈ . ⎝ v 'α⎠ v 'α – iv ''α v 'α The complex phase velocities of eigenwaves are found from Eq. (3): ˆ ' – iQ ˆ '' )A Aα ⋅ ( Q 2 α (7) ρv α = . Aα ⋅ Aα The polarization vectors Aα, being the eigenvectors of a symmetric matrix, are mutually orthogonal for nondegenerate directions m: (8) A α ⋅ A β = δ αβ , α ≠ β.
As for the normalization of these vectors, we are com 2 pelled to discard the usual condition A α = 1, bearing in mind that the polarization of a wave propagating in the vicinity of the singular acoustic axis can approach 2 the circular one when A α 0. Below, we will use the normalization Aα
2
2
2
= A 'α + A ''α = 1.
(9)
It is easy to verify that at a large distance from the acoustic axes, the imaginary additions to v α' and A α' remain small due to weak absorption. In this case, per turbation theory gives ˆ ''A ' A 'α ⋅ Q 2 ˆ 'A ' , α (10) v ''α = , ρv 'α = A 'α ⋅ Q α ' 2ρv α ˆ ''A ' ˆ ''A ' A 'α ⋅ Q A 'α ⋅ Q β γ 'β + A A 'γ . A ''α = 2 2 2 2 ' ' ' ' ρ ( vβ – vα ) ρ ( vγ – vα )
(11)
At the same time, it follows from (11) that, as the wave of a degenerate branch is approaching the acoustic axis, when the intrinsic velocity v 'α approaches v 'β or v 'γ , the quantity A ''α increases, losing its smallness, and the wave ellipticity also increases. This means that simple perturbation theory cannot be used near the acoustic axis and expressions (10) and (11) should be replaced by others. 3. FORMALISM FOR THE VICINITY OF DEGENERACY DIRECTIONS For the nondegenerate branch, expressions (10) and (11) are valid even along directions where two other branches are degenerate. Below, we will use the number α = 3 for the characteristics of the nondegen erate branch. In this case, according to (11), vector A ''3 for any directions should be a small addition to A '3 : A ''3 A '3 .
(12)
With allowance for orthogonality condition (8), ine quality (12) allows us to calculate in the first approxi mation the complex polarization vectors A1, 2 by neglecting their components along A 3' . This remains valid even near the acoustic axes where imaginary components of vectors A1, 2 are not small. Let m0 be the acoustic axis direction in a crystal with switchedoff absorption. In such a crystal, the phase velocities are degenerate along m0, v1 = v2 ≡ v0, and along with the nondegenerate wave with the veloc ity v3 and polarization A03, an infinite number of waves with the velocity v0 and any linear polarization in the degenerate plane D orthogonal to the vector A03 (Fig. 1) are allowed in this direction. We will choose in the D plane an arbitrary orthogonal pair of unit vectors
JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS
Vol. 113
No. 4
2011
CONICAL REFRACTION OF ELASTIC WAVES IN ABSORBING CRYSTALS
A01 and A02 that with A03 form the righthand triple. We will construct our formalism in this basis. Now we return to an absorbing crystal and consider initial equation (3) for a direction m = m0 + δm close to m0: ˆ ) ( m 0 + δm ) }A α { ( m 0 + δm ) ( ˆc – iωη (13) 2 = ρ ( v 0α + δv α ) A α . In the linear approximation, this equation takes the form ˆ + δQ ˆ )A = ρ ( v 2 + 2v δv )A , (14) (Q 0 α 0α 0α α α
m0
661
A03
A02 A01
D
Fig. 1. Allowed polarization plane of a wave for the degen erate propagation direction m0 in a nonabsorbing crystal. Orthogonal basis {A01, A02, A03}.
where ˆ = m ˆc δm + δmcˆ m – iQ '' , δQ 0 0 0 ˆ ˆ ˆ ˆ '' Q = Q ' ( m ), Q = Q '' ( m ). 0
0
0
Here, (15) R =
0
(16)
where α, β = 1, 2, 3 and the repeated subscripts β assume summation. By substituting expansion (16) for α = 1 and 2 into (14), we obtain ˆ A = 2ρv δv A + ρ ( v 2 – v 2 )a A , δQ 1 0 1 1 0 03 13 03 (17) 2 2 ˆ δQ A 2 = 2ρv 0 δv 2 A 2 + ρ ( v 0 – v 03 )a 23 A 03 . We can see from these equations that coefficients a13 and a23 are indeed small compared to other coeffi cients aαβ in expansion (16). Therefore, in the linear approximation the vectors A1 and A2 can be replaced by their projections on degeneracy plane D: A 1 ≈ a 11 A 01 + a 12 A 02 ,
A 2 ≈ a 21 A 01 + a 22 A 02 .
(18)
By multiplying Eq. (17) scalar by A1 and A2 (18), we obtain two systems determining coefficients aαβ in expansions (18): ( δQ 11 – 2ρv 0 δv 1 )a 11 + δQ 12 a 12 = 0, δQ 12 a 11 + ( δQ 22 – 2ρv 0 δv 1 )a 12 = 0, ( δQ 11 – 2ρv 0 δv 2 )a 21 + δQ 12 a 22 = 0, δQ 12 a 21 + ( δQ 22 – 2ρv 0 δv 2 )a 22 = 0, i, j = 1, 2.
(24)
(25)
1 ( A ˆc A + A ˆc A )m , q = 01 02 02 01 0 2ρv 0 s'' ⎫ Q ''11 ± Q ''22 ⎬ = , 4ρv 0 p'' ⎭
Q ''12 q'' = , 2ρv 0
(26)
ˆ '' A . (27) Q ''ij = A 0i ⋅ Q 0 0j The vectors s0, p, and q have the following properties: (28) s 0 ⋅ m 0 = v 0 , p ⋅ m 0 = q ⋅ m 0 = 0. Taking (23) into account, systems (19) and (20) can be easily solved to obtain the directions of polarization vectors A 1, 2 || – ( q ⋅ δm – iq'' )A 01 + ( p ⋅ δm – ip'' ± R )A 02 (29) of degenerate branches in the region under study. It is easy to verify that vectors (29) are mutually orthogo nal in accordance with relation (8), A1 ⋅ A2 = 0.
(19)
(20)
where ˆA , δQ ij = A 0i ⋅ δQ 0j
2
s0 ⎫ 1 ⎬ = ( A 01 ˆc A 01 ± A 02 ˆc A 02 )m 0 , 2ρv 0 p ⎭
We expand the polarization vectors Aα in (14) into components of the basis {A01, A02, A03}: A α = a αβ A 0β ,
2
( p ⋅ δm – ip'' ) + ( q ⋅ δm – iq'' ) ,
(21)
The conditions for existing nontrivial solutions of homogeneous systems (19) and (20) give the same quadratic equation, 2
( δQ 11 – 2ρv 0 δv ) ( δQ 22 – 2ρv 0 δv ) – δQ 12 = 0, (22) whose roots determine the unknown additions δv1, 2 to the degenerate velocity v0: δv 1, 2 = s 0 ⋅ δm – is'' − (23) + R.
4. SPLITTING OF THE CONICAL AXIS AFTER SWITCHING ON ABSORPTION If we switch off absorption, Eq. (23) transforms to the known expression [8] describing the local geome try of the sheets of the phase velocity surface, v 1, 2 ( m ) = v 0 + δv 1, 2 ( m 0 + δm ) near the degeneracy point v1(m0) = v2(m0) = v0: 2 2 (30) δv 1, 2 = s 0 ⋅ δm − + ( p ⋅ δm ) + ( q ⋅ δm ) . If vectors p and q are nonzero and not parallel to each other (p × q ≠ 0), expression (30) describes the vicinity of the conical contact of sheets v1, 2(m) of the phase velocity surface and, hence, of sheets 1/v1, 2(m) of the slowness surface. This corresponds to conical degen eracy of the general type.
JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS
Vol. 113
No. 4
2011
662
ALSHITS, LYUBIMOV (a)
εt
(b)
θ
Γ
δm+
q
q δm+ |p''|/p
M
M p
p δm−
|p''|/q δm−
Fig. 2. Schematic images of ellipse (34) in the general case (a) and for the conical acoustic axis m0 lying in the symmetry plane of a crystal (b).
As mentioned above, switching on absorption gives rise to a small imaginary addition to the phase velocity: v = v' – iv''. Thus, along with the slowness surface, the absorption surface v''(m) appears. In this case, real component v' of the phase velocity also considerably changes. We will see below that these changes leads to topological transformations in the slowness surface 1/v'(m). All this is described by Eq. (23). The degeneracy condition δv1 = δv2 in (23) is reduced to the requirement that the complex radicand in (24) is equal to zero, i.e., to the system of equations 2
2
2
2
(31) ( p ⋅ δm ) + ( q ⋅ δm ) – p'' – q'' = 0, (32) p'' ( p ⋅ δm ) + q'' ( q ⋅ δm ) = 0. This system has two solutions for δm corresponding to the intersection points of the ellipse (Fig. 2), 2 2 δm ⋅ p⎞ 2 ⎛ δm ⋅ q⎞ 2 ⎛ + = 1, r = p'' + q'' (33) ⎝ r ⎠ ⎝ r ⎠ with the straight line δm ⋅ M = 0 passing through the end of vector m0 perpendicular to the vector M = p''p + q''q = r ( p sin α + q cos α ). (34) Here, angle α is introduced which is defined by the expressions sin α = p''/r, cos α = q''/r. (34') Thus, switching on absorption leads to the splitting of the conical axis m0 into two axes passing along the wave normals m± = m0 + δm±, where m 0 × ( p''p + q''q ) (35) δm ± = ± . m0 ⋅ [ p × q ] It follows from (35) that the projections of vectors δm± on p and q are equal to (Fig. 2b) δm ± ⋅ p = − (36) + q'', δm ± ⋅ q = ± p''. Consider, for example, the splitting of the conical axis m0 lying in the symmetry plane S of the crystal. It
this case, the polarization vector A03 of the nondegen erate branch obviously also lies in plane S. Then, it is reasonable to choose vectors A01 and A02 so that the vector A01 is directed along the normal to plane S, while vector A02 lies in plane S together with vectors m0 and A03 (Fig. 3a). It is easy to verify that in this case, because of the symmetry of tensors cijkl and ηijkl of the crystal under study belonging to a syngony no lower than monoclinic, the relations (37) q'' = 0, q || A 01 , p || [A 01 × m 0 ] should take place. With allowance for (36) and (37), the splitting in this case is determined by the vectors p'' (38) δm ± = ± A 01 . 2 Note that in the case of the mutual orthogonality of vectors p and q found above, ellipse (33) looks espe cially simple (see Fig. 2b). Thus, the splitting of axes in the particular case under study occurs in a plane orthogonal to the symmetry plane, while the splitting angle (Fig. 3b) (39) δψ ≈ 2 δm ± = 2 p'' /q is proportional to absorption. 5. LOCAL GEOMETRY OF THE SLOWNESS SURFACE NEAR SINGULAR AXES Note that the radicand in (24) on the entire straight line δm ⋅ M = 0 is purely real, negative inside ellipse (33) between degeneracy points (35) (see Fig. 2), and positive outside the ellipse. This means that in the part of the line lying inside the ellipse, the value of R in (23) is purely imaginary. Therefore, on the line connecting the degeneracy points, the real components v '1 (m) and v '2 (m) of the phase velocity should coincide. This defines the selfintersection lines both of the phase
JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS
Vol. 113
No. 4
2011
CONICAL REFRACTION OF ELASTIC WAVES IN ABSORBING CRYSTALS (a)
A01
(b)
q || A01
S
S A03
A02
663
p
A02
δψ
m0
A03 m+ δm+
m0 m−
δm−
Fig. 3. Acoustic axis m0 in the symmetry plane S of a crystal (a) and its absorptioninduced splitting (b).
velocity surface v '1, 2 (m) and slowness surface 1/ v '1, 2 (m). Quite similarly, we can conclude that the corresponding absorption surface sheets v ''1, 2 (m) should intersect on the continuation of the line δm ⋅ M = 0 outside the ellipse. Figure 4 shows sche matically the slowness surface 1/ v '1, 2 (m) with the self intersection line and split acoustic axes passing through its ends. Consider now in detail the features of the geometry of this surface in the near vicinity of singular axes. According to expressions (23) and (24), in the general case of split ting the conical acoustic axis m0 in two, m± = m0 ± δm+, the phase velocities along new axes are different: v0
v '0± = v 0 ± s 0 ⋅ δm + .
However, when the initial direction m0 is in the sym metry plane, the vector s0 should obviously also lie in this plane, and therefore s0 ⋅ δm+ = 0 and the differ ence disappears.
phase velocities on the circle Γ, δv '1, 2 ( ε, θ ) = v '1, 2 – v '0+ = − + εRef ( θ ), where f (θ) =
2t ⋅ ( N – iM ) ,
N = – q''p + p''q.
(42) (43)
It follows from (42) that the dependence δ v '1, 2 (ε) ∝ ε for ε 0 has the infinite derivative with respect to ε in each cross section θ ≠ θ0, where the angle θ0 corresponds to passage of vector δmε through the self intersection line where Ref(θ0) = 0. Such a singularity of the dependence δ v '1, 2 (ε) at the end of selfintersec tion wedge corresponds to the sharpened “nose” of surfaces v '1, 2 (m) and 1/ v '1, 2 (m) and the flat “fan” of normals to these surfaces on the circle m = m+ + δmε(θ) for ε 0 (Fig. 5).
We return now to the general case of an arbitrary anisotropy and consider a small vicinity of one of the singular axes, say, the axis m+ = m0 + δm+. Let us find the velocities of isonormal waves on the circle Γ (see Fig. 2): m = m+ + δmε(θ), which has a small radius ε = δm ε δm + and is located in a plane orthogonal to the vector m0. We set δm ε ( θ ) = εt ( θ ),
(40)
where the orientation of the unit vector t is specified by azimuth θ measured with respect to vector p t ( θ ) = ( p cos θ + [ m 0 × p ] sin θ )/p.
m+
δψ
m−
(41)
Thus, the vector εt runs over the entire circle Γ when the azimuth θ changes from 0 to 2π (see Fig. 2). With allowance for (40) and (41), Eq. (23) and (24) in the principal order in ε give increments of the real
Fig. 4. Selfintersection of the slowness surface 1/ v '1, 2 (m) and split acoustic axes.
JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS
Vol. 113
No. 4
2011
664
ALSHITS, LYUBIMOV
Γ m+
Fig. 5. Fragment of the internal degenerate sheet 1/ v 2' (m) of the slowness surface near a singularity at the end of self intersection wedge and a plane fan of normals to the sur face at this point.
6. FEATURES OF POLARIZATION FIELDS NEAR SINGULAR AXES Polarization vectors (29) on the same circle Γ (40) can be written up to a constant factor in the form εf ( θ )⎞ . (44) A 1, 2 = A 01 + i ⎛ 1 − A ⎝ + q'' + ip''⎠ 02 Here, it is appropriate to specify function f(θ) (43) for further analysis in the form 2
f ( θ ) = A p ⋅ q cos θ + g sin θ – ip cos θ , where
(45)
(46) A = 2 ( p'' – iq'' )/p, g = m 0 ⋅ [ p × q ]. It is easy to verify that after going around the circle Γ, i.e., when θ changes from 0 to 2π, function f(θ) (45) changes its sign. Note that the phase Ψ of the complex function (47) f ( θ )/A = R ( θ ) exp [ iΨ ( θ ) ] should be equal to half the phase of the square of this function 2
2
R exp ( 2iΨ ) = p ⋅ q cos θ + g sin θ – ip cos θ. On the other hand, it follows from (48) that
(48)
2
p Ψ ( θ ) = – 1 ⎛ Arctan ⎞ , ⎝ 2 p ⋅ q + g tan θ⎠ 2
(49)
2
∂Ψ gp cos 2Ψ = . 2 ∂θ ( p ⋅ q cos θ + g sin θ )
(50)
Fig. 6. Polarization feature of the singular acoustic axis for g > 0: polarizations rotate through π/2 after a full rotation of the wave normal around this axis. The difference in the velocities of isonormal waves is symbolically represented by the difference in the thickness of the contour of ellipses. The ellipse color (grey or white) symbolizes the difference in the absorption intensity. The equalvelocity line is shown by the solid line, and the equalabsorption line, by the dashed line.
This gives the relation Ψ(2π) – Ψ(0) = π sgn g . Thus, (51) f ( 2π ) = – f ( 0 ), and after a full bypass over a small circle Γ around the singularity axis m+, the identical transformation of polarization field (44) to itself is realized in the form A 1 ( 2π ) = A 2 ( 0 ),
A 2 ( 2π ) = A 1 ( 0 ).
(52)
In other words, each of the two orthogonal polariza tion ellipses rotates through π/2 after going around the singular axis, transforming to the polarization of the other isonormal wave (Fig. 6). It is important that in this case the complex velocities v1, 2 = v '1, 2 – i v ''1, 2 of the same branches also simultaneously interchange their sheets on the wave surfaces, as shown in Fig. 7. The singularity of the polarization field found in the vicinity of directions m+ (Fig. 6) can be character ized by the Poincare index n defined as the total polar ization rotation (in 2π units) after going around the singularity over the circle Γ. The turn of polarization ellipses found above is π/2, while the rotation direc tion, according to (50), is completely determined by the sign of parameter g (46). Therefore, we obtain 1 (53) n = sgn ( m 0 ⋅ [ p × q ] ) 4 (see also [10]). We can show that the same expression is also valid for another singularity corresponding to the singular axis along m–.
JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS
Vol. 113
No. 4
2011
CONICAL REFRACTION OF ELASTIC WAVES IN ABSORBING CRYSTALS
1/v1'(m)
665
m0
v1' = v2'
1/v2'(m) m+
v1'' (m)
v1'' = v2''
v2'' (m)
Fig. 7. Configuration of surfaces 1/ v '1, 2 (m) and v ''1, 2 (m) in the vicinity of the singular axis m+. The heavy lines show continuous passage between the external and internal regions of the slowness and absorption surfaces during full going around the axis m+.
Thus, the physical identity of two pictures for θ = 0 and θ = 2π is provided not by the coincidence of wave characteristics inside each of the branches, as in the case of zero absorption, but by the identity of their combination. This becomes topologically possible owing to such new feature of the wave surfaces of absorbing crystals as their selfintersection. In the absence of absorption, when the degenerate sheets of the slowness surface have the only contact point, one of the two isonormal waves on any path around this point is always faster than the other. Therefore, linear undirected polarization vectors corresponding to them are not equivalent, as if they had different thicknesses. Thus, to make a “cross” formed by a set of such vec tors coincident with itself, it should be rotated mini mally through π rather than π/2, as in the case consid ered above (see Fig. 6). Rotation through π/2 will
Fig. 8. Singular behavior of polarization fields in the vicin ity of the acoustic axis of a nonabsorbing crystal (the Poincare index is n = 1/2). The undirected segments rep resent the polarization configurations of the total wave field of two branches.
superimpose linear crosses only by neglecting their thickness, while a change in their thickness is topolog ically impossible (Fig. 8). It is for this reason, according to [8], that in the absence of absorption the conical acoustic axis along m0 corresponds to a singularity in the polarization field with the Poincaré index n = (1/2)sgng. This is the minimal possible index in the real polarization field. The decomposition of this singularity into two singu larities (53) after switching on absorption satisfies the index conservation law. On the other hand, the same total index ±1/2 is obtained after going around a pair of split points (Fig. 9). 7. CONICAL REFRACTION IN ABSORBING CRYSTALS 7.1. Conical Refraction Neglecting Absorption As mentioned above, in the absence of absorption, apart from the nondegenerate wave with polarization vector A03, an infinite number of waves polarized in degeneracy plane D orthogonal to vector A03 can prop agate along the acoustic axis m0 (see Fig. 1). Thus, in the basis {A01, A02} belonging to this plane, the vector A ( β ) = A 01 cos β + A 02 sin β (54) for any angle β in the interval 0 ≤ β ≤ 2π specifies the polarization of an eigenwave propagating along direc tion m0 with phase velocity v0. Of course, a wave with circular polarization A = A01 + iA02 can also propagate in the same direction. Consider a monochromatic plane wave with the wave normal m0 along the acoustic axis, polarization A, and velocity v0, u ( r, t ) = CA exp ( iΦ 0 ), Φ 0 = k ( m 0 ⋅ r – v 0 t ). (55) The Poynting vector of such a wave is [6] P = ( Reu· )cˆ ( Reu· )m 0 /v 0 .
JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS
Vol. 113
No. 4
2011
(56)
666
ALSHITS, LYUBIMOV Q
q Δs p
s0
v0m0
s
m0 A03
D Fig. 9. Field of elliptical polarizations of degenerate branches in the vicinity of a pair of singular acoustic axes in an absorbing crystal for the case g > 0. The Poincaré indices on small contours around split degeneracy points are n = 1/4. The total index n = 1/2 on the external con tour coincides with the index of the unperturbed linear polarization field around the initial acoustic axis of a non absorbing crystal. Crosses at the vertical dashed straight line correspond to linear polarization.
For linear and circular polarizations, we have Reu lin = CA ( β ) cos Φ 0 , (57) and Reu cir = C [ A 01 cos Φ 0 – A 02 sin Φ 0 ], (58) respectively. In these two cases, the Poynting vector is described by different expressions: 2
2
2
P lin = C ρω ( s 0 + p cos 2β + q sin 2β ) sin Φ 0 , 2
(59)
2
P cir = C ρω ( s 0 – p cos 2Φ 0 + q sin 2Φ 0 ).
(60) Similarly, the elastic energy densities W = ρ(Re u· )2 can be found as 2
2
2
2
2
W lin = C ρω sin Φ 0 , W cir = C ρω . (61) Taking (59)–(61) into account, the ray velocities of these waves are (62) s lin = P lin /W lin = s 0 + p cos 2β + q sin 2β, s cir = P cir /W cir = s 0 – p cos 2Φ 0 + q sin 2Φ 0 . (63) During the period of a circularly polarized wave, after full rotation of the polarization vector in the degener acy plane D, the ray velocity vector scir (63) twice cir cumscribes a cone (Fig. 10). In this case, the end of the vector scir runs twice over the ellipse, (64) Δs = s – s 0 = – p cos 2Φ 0 + p sin 2Φ 0 , in which vectors p and q are conjugated semidiameters [6]. Taking (28) into account, the ellipse plane Q is orthogonal to m0, while the direction of going around
A02 A(β)
β A01
Fig. 10. Internal conical refraction cone.
ellipse (64) and optical rotation (58) coincide for m0 ⋅ [p × q] > 0 and are opposite for m0 ⋅ [p × q] < 0. For a linearly polarized wave, the same refraction cone is circumscribed by vector slin (62) upon varying the orientation of vector A(β) (54), when the angle β changes in the interval 0 < β < π (Fig. 10). It is this scheme that was implemented in experiments [1]. 7.2. Polarization Ellipses A1, 2 on the Edge of SelfIntersection Wedge Consider now the wave characteristics of an absorb ing crystal on the edge of selfintersection wedge of the slowness surface. To describe the movement of the wave normal along the line connecting the split degen eracy points δm±, we introduce the vector δmξ = δm+sinξ. As the angle ξ changes from –π/2 to π/2, this vector passes over the entire edge from one singularity to another. The substitution δm = δmξ into (29) gives the polarization vectors of isonormal waves on the edge of wedge. Taking relations (36) into account, we obtain A 1, 2 = { ( sin α sin ξ – i cos α )A 01 + [ cos α sin ξ − cos ξ ) ]A 02 } + i ( sin α + × [2(1 − + sin α cos ξ ) ]
– 1/2
(65)
,
where normalization (9) is fulfilled and the notation introduced in (34') is used. Recall that the real components of phase velocities coincide at the edge of the wedge, v '1 = v '2 = v 'ξ , but depend, however, on the “observation” point position at the edge. The imaginary components v ''1, 2 coincide only at the end points ξ = ±π/2. Taking (6) into
JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS
Vol. 113
No. 4
2011
CONICAL REFRACTION OF ELASTIC WAVES IN ABSORBING CRYSTALS
account, the real components of displacement vectors u1, 2 can be written in the form
π α ⎫ + A 02 sin ⎛ – ⎞ ⎬ sin Φ 0 , ⎝ 4 2⎠ ⎭
Reu 1, 2 ( r, t ) = C 1, 2 exp ( – k ''1, 2 m ξ ⋅ r ) × Re [ A 1, 2 exp ( iΦ ξ ) ]
(66)
U2
≡ C 1, 2 exp ( – k ''1, 2 m ξ ⋅ r )U 1, 2 , where the wave normal mξ and real phase Φξ on the edge of wedge m ξ = m 0 + δm ξ ,
Φ ξ = k 'ξ m ξ ⋅ r – ωt
(67)
and dimensionless displacements vectors U 1, 2 = Re [ A 1, 2 exp ( iΦ ξ ) ]
(68)
are introduced. It is significant that the trivial decay of the wave in (66), which is proportional to exp(⎯ k ''1, 2 mξr), is separated from vectors U1, 2 describing absorption effects, which are more impor tant for us. The choice of the zero time reference in the sta tionary problem under study, of course, has no effect on measurements and can be different for isonormal waves independent of each other. Therefore, vectors U1, 2 and polarization vectors A1, 2 are defined to the sign. Below, this sign will be selected to make expres sions compact. Note that upon scanning of the edge of the wedge between singular axes with the wave normal mξ, i.e., when ξ changed in the interval –π/2 ≤ ξ ≤ π/2, the elliptic polarization specified by expressions (65) and (68) rapidly changes. According to [6], it is easy to ver ify that the wave ellipticity determined by these expressions corresponds to rotation with sign inde pendent of angle α and corresponding to the right screw along the propagation direction if sinξ > 0 and to the left screw if sinξ < 0. At the ends of the wedge for ξ = π/2 or ξ = –π/2, where singular degeneracy is achieved, isonormal waves coincide: U1 = U2 ≡ U0. In both cases, circular polarization takes place, but with different rotation “signs”: U0 U0
ξ = π/2
ξ = – π/2
1 = ( A 01 cos Φ ξ – A 02 sin Φ ξ ), 2 1 = ( A 01 cos Φ ξ + A 02 sin Φ ξ ). 2
(69)
Here, angle α is excluded from arguments by proper choice of the zero time reference. At all other points of the wedge the polarization ellipses of isonormal waves are different. At the center of the wedge for ξ = 0, the isonormal waves have dif ferent mutually orthogonal linear polarizations: U1
ξ=0
⎧ π α = ⎨ A 01 cos ⎛ – ⎞ ⎝ 4 2⎠ ⎩
ξ=0
667
(70)
⎧ π α = ⎨ A 01 sin ⎛ – ⎞ ⎝ 4 2⎠ ⎩
π α ⎫ – A 02 cos ⎛ – ⎞ ⎬ sin Φ 0 . ⎝ 4 2⎠ ⎭ It can be verified that linear polarization (70) is pre served on the entire line δm || M (34) directed perpen dicular to the wedge edge (see Fig. 9). Expressions for the polarization ellipses of isonor mal waves on the wedge edge are considerably simpli fied in the particular case when the initial conical axis m0 belongs to the symmetry plane of the crystal, which gives q'' = 0. Assuming for definiteness that p'' > 0, we can set α = π/2. Then, instead of (65), we have for the elliptical polarization of isonormal waves on the edge of wedge A 1 = A 01 cos ( ξ/2 ) + iA 02 sin ( ξ/2 ), (71) A 2 = A 01 sin ( ξ/2 ) + iA 02 cos ( ξ/2 ). The rotation of displacement vectors U1, 2 (68) over ellipses is now described by the expressions U 1 = A 01 cos ( ξ/2 ) cos Φ – A 02 sin ( ξ/2 ) sin Φ, (72) U 2 = A 01 sin ( ξ/2 ) cos Φ – A 02 cos ( ξ/2 ) sin Φ. Expressions (72) define polarization ellipses in parametric form. The lengths of the horizontal and vertical semiaxes of the first ellipse are cos ( ξ/2 ) and sin ( ξ/2 ) , respectively. The same lengths for the sec ond ellipse correspond to the vertical and horizontal semiaxes. At the ends of the edge of the wedge for ξ = ±π/2, the lengths of the semiaxes of ellipses are equal, while polarization is circular. During displacement from the ends of the edge to its center, the major semi axes of the ellipses increase, while the minor axes decrease to zero. Thus, general expressions (69), (70), and particular example (72) lead to the same polarization distribu tion pattern at the edge of wedge. As the wave normal moves over the edge from one singular axis to another, the isonormal waves starting from degeneracy and cir cular polarization of a certain sign monotonically decrease their ellipticity which vanishes at the center of the edge where polarization ellipses elongate to vec tors. After passage to the second half of the wedge, the wave ellipticity changes its sign and monotonically increases, transforming to circular polarization at the degeneracy point at the other end of the edge. Figure 11 illustrates the behavior of polarization of isonormal waves at the edge of the wedge. Consider now the kinematics of motion of the dis placement vectors for isonormal waves over polariza tion ellipses. We express the coordinates of the ends of
JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS
Vol. 113
No. 4
2011
668
ALSHITS, LYUBIMOV
ξ = −π/2 ξ = −π/4
ξ=0
ξ = π/4
ξ = π/2
Fig. 11. Polarization distribution of isonormal waves on the edge of selfintersection wedge of the slowness surface.
radius vectors Uα (α = 1, 2) on the ellipse in polar coordinates Uα, ϕα U α = U α ( A 01 cos ϕ α + A 02 sin ϕ α ).
(73)
Of course, the lengths Uα of these radius vectors for different azimuths ϕα are different. By comparing expressions (65), (68), and (73), we obtain 2 U 1, 2 = 1 1 ± cos ξ 2
( sin α − + cos ξ ) cos 2Φ ξ + sin ξ cos α sin 2Φ ξ × , 1− + sin α cos ξ cos α sin ξ – ( sin α − + cos ξ ) tan Φ tan ϕ 1, 2 = ξ . sin α sin ξ + cos α tan Φ ξ
(74)
(75)
By differentiating the latter equation with respect to time, we can easily find the angular velocities ϕ· 1, 2 of the motion of radiusvector ends U1, 2 over the polarization ellipse ω sin ξ ϕ· 1, 2 = , 2 2U 1, 2
(76)
where we took into account that Φ· ξ = –ω. As follows from (76), angular velocities behave differently in time at different “observation” points on the wedge. Along singularity axes (ξ = ±π/2), when isonormal polariza tion ellipses merge in one circle (69), the denominator in (76) becomes unity, and circular motion has a con stant angular velocity ϕ· 1 = ϕ· 2 = ±ω. Here, the upper and lower signs correspond to the righthand and left hand screw rotation, respectively, for ξ = π/2 and ξ = –π/2. As the value of ξ decreases, the nonuniformity of motion increases and acquires the singular character for ξ 1, when velocities ϕ· 1, 2 are small during a considerable part of the period, while the correspond ing azimuthal angles ϕ1, 2 are almost fixed, so that the larger part of the ellipse is passed for a short time with a very high velocity. This is clearly illustrated by ana lytic expressions for the abovediscussed particular
case of splitting of the acoustic axis from the symmetry plane of the crystal (Fig. 12), when ξ ξ tan ϕ 1 = – tan tan Φ ξ , tan ϕ 2 = – cot tan Φ ξ , (77) 2 2 ω sin ξ (78) ϕ· 1, 2 = . 1 ± cos ξ cos 2Φ ξ We can see from expressions (74)–(78) and Fig. 12 that functions ϕ1, 2(Φξ) and ϕ· 1, 2 (Φξ) have periods equal to half the wave period. This means that already half the rotation of the displacement vector over the polarization ellipse exhausts all of its physically differ ent orientations. It is also of interest to find the sector velocities of motion, which, by definition, are equal to the area covered by radius vectors U1, 2 per unit time: sec 1 2 1 v 1, 2 = U 1, 2 ϕ· 1, 2 = ω sin ξ. (79) 2 4 This expression for the sector velocity is valid for arbi trary anisotropy. It is the same for both isonormal waves and is independent of time, which is consistent with the conclusion on nonabsorbing crystals [6]. In this case, velocity (79) strongly depends on the posi tion of the wave normal on the edge of wedge, decreas ing to zero at the wedge center (ξ = 0), where polariza tion becomes linear. 7.3. Universal Refraction Cone on the Edge of Wedge and Kinematics of the Ray Velocity Precession over This Cone Now we can generalize the classical theory of con ical diffraction [6] presented briefly in Section 7.1. We calculate the ray velocities of isonormal waves (66) at the edge of wedge. By substituting (66) into (56), we obtain energy fluxes 2 P 1, 2 = C 1, 2 exp ( – k ''1, 2 m ξ ⋅ r ) (80) 2 2 2 2 2 × ρω [ s 0 ( F + G +− ) – p ( G +− – F ) + 2qFG +− ], where (81) F = sin α sin ξ sin Φ ξ – cos α cos Φ ξ , − G +− = cos α sin ξ sin Φ ξ + ( sin α + cos ξ ) cos Φ ξ . (82) The energy density in isonormal waves under study is 2 W 1, 2 = ρ ( Reu· 1, 2 ) (83) 2 2 2 2 = C 1, 2 exp ( – k ''1, 2 m ξ ⋅ r ) ρω ( F + G −+ ). Correspondingly, the ray velocities of these waves on the edge of wedge are s 1, 2 = P 1, 2 /W 1, 2 = s 0 – p cos 2Θ 1, 2 + q sin 2Θ 1, 2 , (84) where Θ 1, 2 = Θ 1, 2 ( α, ξ, Φ ξ ), tan Θ 1, 2 = F/G −+ . (85) Relation (84) can be obtained from classical expression (63) for nonabsorbing crystals after the substitution Φ0 Θ1, 2 in the latter. This means that
JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS
Vol. 113
No. 4
2011
CONICAL REFRACTION OF ELASTIC WAVES IN ABSORBING CRYSTALS ϕ1 π/2 2
(a)
(b)
3 5
4
2 1 −π/2
π/2 Φξ
0
−π/2
ϕ· 1 /ω ≡
sec v 2 /gω
sec ϕ· 2 /ω ≡ v 1 /gω
(c)
3
(d)
3
2
2
5
5
1
1 4
4
3
3
−π/2
π/2 Φξ
0 −π/2
−π/2
1
ϕ2 π/2
4
5 3
1
669
2
2
1 0
π/2 Φξ
−π/2
π/2 Φξ
0
sec Fig. 12. Dependences of azimuths ϕ1 (a) and ϕ2 (b) (77) and coinciding normalized velocities ϕ· 1 /ω ≡ v 2 /gω (c) and ϕ· 2 /ω ≡ sec
v 1 /gω (d) [(78), (79), (99), respectively] on the wave phase Φξ for a series of “observation” points on the edge of wedge (by the example of the conicalaxis splitting from the symmetry axis). Curves 1–5 correspond to angles ξ = 5°, 15°, 45°, 75°, and 90°, respectively.
the ends of ray velocity vectors at any point on the edge of wedge in an absorbing crystal move along the same paths described by the same ellipse Δs 1, 2 = – p cos 2Θ 1, 2 + q sin 2Θ 1, 2 .
(86)
The shape of this ellipse is completely determined by vectors p and q and is independent of parameters Θ1, 2. In other words, it is sensitive neither to the wave phase Φξ nor to angles α and ξ related to the absorption parameters and position of the observation point on the edge of wedge. The major semiaxes of universal
ellipse (86) coinciding with ellipse (64) for a nonab sorbing medium are [6] 2 2 2 2 2 2 λ 1, 2 = 1 ( p + q ± ( p + q ) – 4 ( p × q ) ). (87) 2 Although vectors p and q (25) depend on the choice of basis {A01, A02}, it is easy to verify that combinations p2 + q2 and p × q are invariant with respect to the rota tions of this basis. Although the mechanical trajectories of the ends of vectors s1, 2 are identical, the kinematics of their
JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS
Vol. 113
No. 4
2011
670
ALSHITS, LYUBIMOV
motion along the universal ellipse is very sensitive to the passage of the “observation” point on the edge of wedge from one singular axis to another. It is easy to verify that parameters Θ1, 2 at both ends of the edge in Eq. (86) coincide (Θ1 = Θ2 = Θ0) and, hence, the instant ray velocities s1, 2 of isonormal waves propagat ing along each of the singular axes (ξ = ±π/2) also coincide. For the properly chosen zero time reference, the quantities Θ0(π/2) and Θ0(–π/2) differ only in sign: (88) Θ 0 ( ± π/2 ) = ± Φ ξ . In this case, s ( π/2 ) = s 0 – p cos 2Φ ξ + q sin 2Φ ξ , (89) s ( – π/2 ) = s 0 – p cos 2Φ ξ – q sin 2Φ ξ . In other words, the precession of the vector of the ray velocity propagating along the singular axis is identical to the analogous process along the conical axis for zero absorption for circularly polarized wave (63). The directions of rotation of ray velocities ±s(π/2) (89) around split singular axes have different signs. It is easy to verify that for m0 ⋅ [p × q] > 0 they coincide with the corresponding directions of circular polarization (69), and for m0 ⋅ [p × q] < 0 they are opposite to them. Despite the discovered identity of cones (63) and (89), there is an important difference between conical refraction patterns corresponding to them. The gener atrices of cone (63) in a nonabsorbing crystal are directed along corresponding normals to the slowness surface at the conical contact point of its sheets, whereas normals to a similar surface around the singu lar axis, as we see from Fig. 5, form a flat fan that strongly differs from ray velocity cone (89) (see Fig. 10). Analysis shows that, as the wave normal moves over the edge of the wedge from the singular axis to the cen ter, the motion of the end of the ray velocity vector over universal ellipse (86) changes its character. The motion becomes less uniform with approaching the wedge center and the vector s longer resides in certain parts of the trajectory. At the central point ξ = 0, the motion ceases at all: the isonormal vectors s1, 2 are “frozen” in certain positions. Indeed, for ξ = 0, expressions (81), (82), and (85) give values of Θ1, 2 independent of time: (90) Θ 1, 2 = α ± π . 2 4 In this case, the corresponding fixed ray velocity vectors are s 1, 2 ( 0 ) = s 0 ± ( p sin α + q cos α ) = s 0 ± M/r. (91) As would be expected, this result is related to expres sions (70) for the linear polarization of isonormal waves in the same way as expression (62) is related to expression (57) from the refraction theory for nonab sorbing crystal. We can show that two normals to the slowness surface at the point ξ = 0 on the edge of wedge are parallel not to vectors (91) but to their com
ponents lying in the plane {m0, M} orthogonal to the edge (see Fig. 2). Note that the positions of ray velocity vectors (91) fixed in time should “follow” a change in the orienta tion of the selfintersection edge (δm± || m0 × M(α)) with changing absorption parameters (angle α (34')) by moving along ellipse (86), which itself, as we know, is insensitive to such changes. This means that points (91), generally speaking, do not coincide with the ends of the major semiaxes of the ellipse. Of course, coinci dences are also possible in more symmetric situations. For example, this occurs upon splitting of the conical axis m0 lying in the symmetry plane of the crystal. Obviously, for any small deviation ξ from zero, constant vectors (91) will acquire small additions, which, being dependent on the phase Φξ, will restore the motion of ray velocities s1, 2 over the cone. How ever, if the deviation from the center of the edge of wedge is not large, vectors s1, 2 will be in the vicinity of orientations (91) for a large part of the period. In this case, of course, the directions of timeaveraged vectors s 1, 2 also will not strongly deviate from the direction of vectors (91). This means that refraction in the middle part of the selfintersection line of sheets of the slow ness surface will have the wedge character rather than the conical one. In the particular case of conical axis splitting from the symmetry plane of the crystal considered above, we can set α = π/2, which considerably simplifies expres sions (81), (82), and expressions (85) for angular parameters Θ1, 2: tan Θ 1 = cot ( ξ/2 ) tan Φ ξ ,
(92) tan Θ 2 = tan ( ξ/2 ) tan Φ ξ . Now we quantitatively describe the kinematics of precession of ray velocities at the edge of the wedge in the general case of arbitrary anisotropy. We introduce polar coordinates (S1, 2, ϕ 1, 2 ) of the ends of radius vec tors on ellipses (86), m0 × p (93) Δs 1, 2 = S 1, 2 ⎛ p cos ϕ 1, 2 + sin ϕ 1, 2⎞ . ⎝p ⎠ p By comparing expressions (93) and (86), we can easily obtain 2
S 1, 2 = ( q sin Θ 1, 2 – p cos Θ 1, 2 ) 2
2
2
2
(94)
2
q F – 2p ⋅ qFG +− + p G +− , = 2 2 F + G −+ 2
p ⋅ q – p cot Θ 1, 2 (95) cot ϕ 1, 2 = . g Differentiation of Eq. (95) gives the angular velocities · g · (96) ϕ 1, 2 = – Θ 1, 2 . 2 S 1, 2
JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS
Vol. 113
No. 4
2011
CONICAL REFRACTION OF ELASTIC WAVES IN ABSORBING CRYSTALS
Here, the derivative Θ· 1, 2 is found by using (85): F· G −+ – FG· +− ω sin ξ = – Θ· 1, 2 = 2 2 2 (97) F + G +− U 1, 2 ( Φ ξ + π/2 ) = – 2ϕ· 1, 2 ( Φ ξ + π/2 ), 2 where U 1, 2 and ϕ· 1, 2 are given by functions (74) and (76) with the phase shift Φξ Φξ + π/2. sec
The sector velocities v 1, 2 of the motion of vector ends Δs1, 2 along universal ellipse (86) are determined similarly to (79): sec 1 2 · 1 v 1, 2 = S 1, 2 ϕ 1, 2 = – gΘ· 1, 2 = gϕ· 1, 2 ( Φ ξ + π/2 ). (98) 2 2 Thus, this velocity differs from the angular velocity of polarization (76) only by a dimensional factor g and the phase shift by π/2. By inserting angular velocity ϕ· 1, 2 (78) into (98) for the symmetrical example con sidered above, we obtain the sector velocity in a more compact form: sec gω sin ξ (99) v 1, 2 ( Φ ξ ) = ≡ gϕ· 2, 1 ( Φ ξ ). 1− ξ cos 2Φ ξ cos + Here we took into account that the phase shift of the velocity ϕ· 1, 2 in simplified variant (78) is equivalent to passage to the conjugate branch: ϕ· 1, 2 ( Φ ξ + π/2 ) = ϕ· 2, 1 ( Φ ξ ). The relation of functions (99) allowed us to character ize the angular polarization velocities and sector ray velocities by the same curves in Fig. 12. These depen dences adequately characterize the precession of the ray velocity upon conical refraction at the edge of wedge discussed above. Angular velocities (96) behave similarly, especially in the region of small ξ. As singu lar axes are approached (ξ = ±π/2), the time variations of angular velocities are “smoothed,” but do not dis sec appear, like for v 1, 2 , and are preserved depending on the difference between p and q. 8. DISCUSSION Thus, the role of absorption in crystals is not simply reduced to the trivial decay of elastic waves during their propagation. We have seen that the switching on of absorption drastically transforms the geometry of wave surfaces and topology of vector polarization fields by splitting the conical acoustic axes of the gen eral position. In this case, selfintersection lines appear on the slowness surface and on the newly formed absorption surface (see Fig. 7). The projec tions of these lines on the unit sphere m2 = 1 of all directions continue each other after passing through split degeneracy points. The plane field of linear polar ization vectors having a singularity with the Poincaré index n = (1/2)sgng at the conical degeneracy point is transformed after the axis splitting to the plane distri
671
bution of polarization ellipses which has two singular points m± with equal Poincaré indices n = (1/4)sgng (see Fig. 9) and circular polarization at these points (see Fig. 11). Topological changes in the wave surfaces and polarization fields give rise to qualitatively new fea tures of the internal conical refraction. Nevertheless, the theory can be still generalized in terms of a classi cal refraction cone cut by an ellipse along which the end of the ray velocity vector of the wave runs twice during a period. As we have seen, the classical conical refraction picture in a nonabsorbing crystal automati cally appears for a circularly polarized wave propagat ing along the conical acoustic axis. In an absorbing crystal, the same cone and the same ellipse are pre served as the precession trajectory of ray velocities, over the entire line of the selfintersection wedge of wave surfaces between the split degeneracy points. The refraction pattern along singular axes them selves does not differ from classical : isonormal waves degenerate to one circularly polarized wave corre sponding to the ray precession with a constant sector sec
velocity over the ellipse v = gω. This will be mani fested in a screen “illumination” picture as a com pletely drawn ellipse (Fig. 13a). The increase in illu mination near the ends of the major semiaxes is explained by the slower movement of vector s0 in these sec
regions (the linear velocity on the ellipse is 2 v /S0). As the observation point moves over the edge of the wedge to the center, both the precession of vectors s1, 2 and the illumination pattern become less uniform (Figs. 13b, 13c). At the center of the wedge (ξ = 0), only two points corresponding to isonormal linearly polarized waves will be illuminated (Fig. 13d): the refraction becomes purely wedge. Thus, when the wave normal scans the edge of the wedge, the refrac tion type gradually changes from purely conical to wedgelike. Consider now the conditions for observing the pre dicted effects. In principle, no threshold absorption is required for axis splitting. Simply, the weaker the absorption, the smaller the solid angle within which all the abovedescribed processes proceed. If this angle is smaller than the divergence angle of an acoustic beam, we will see neither the splitting of acoustic axes nor the effects related to splitting. Therefore, to observe these phenomena, angle δψ (39) should exceed the diffrac tion divergence of the beam of the order of λ/d, where λ is the acoustic wavelength and d is the beam diame ter. Taking into account that angle δψ (39) increases with increasing the wave frequency, while the diver gence angle, on the contrary, decreases, we have the frequency threshold from below. Analysis gives an esti
JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS
Vol. 113
No. 4
2011
672 (a)
ALSHITS, LYUBIMOV (b)
(c)
authors (V.I.A.) acknowledges the support of the Pol ish–Japanese Institute of Information Technologies (Warsaw) and the Kielce University of Technology (Kielce, Poland).
(d) p q
REFERENCES
Fig. 13. Illumination regions of an elliptic contour of internal conical refraction in an absorbing crystal for the wavenormal directions corresponding to four different points on the line of the edge of selfintersection wedge (from the singular axis (a) to the center of the edge (d)). Example of splitting of the conical acoustic axis lying in the symmetry plane.
mate for the choice of the frequency ν = ω/2π in the experiments: cs μ ν > ν th ∼ (100) , 2πηd where cs is the speed of sound, μ is the shear modulus, and η is the material viscosity. Because we are dealing with sufficiently high frequencies and room tempera tures T ≈ 300 K, it is reasonable to estimate the viscos ity η as the socalled phonon viscosity, 3k B T (101) η ph ∼ τ ph , 3 a where τph is the phonon relaxation time, kB is the Bolt zmann constant, and a is the lattice parameter. By inserting cs ≈ 3 × 105 cm/s, μ ~ 1011 dyn/cm2, d ≈ 0.5 cm, τph ~ 10–10 s, and a ≈ 3 × 10–8 cm into (101), we obtain the estimate of the threshold frequency νth ~ 100 MHz. Although this estimate is rather rough, it shows that subtle effects discussed in this paper can be observed. ACKNOWLEDGMENTS The work was supported by the Polish Foundation MNiSW (grant no. NN501252334). One of the
1. J. De Klerk and M. J. P. Musgrave, Proc. Phys. Soc., London, Sect. B 68, 81 (1955). 2. M. J. P. Musgrave, Acta Crystallogr. 10, 316 (1957). 3. A. G. Khatkevich, Sov. Phys. Crystallogr. 7 (6), 742 (1962). 4. P. A. Barry and M. J. P. Musgrave, Q. J. Mech. Appl. Math. 32, 205 (1979). 5. K. S. Aleksandrov and T. V. Ryzhova, Sov. Phys. Crys tallogr. 9 (3), 298 (1964). 6. F. I. Fedorov, Theory of Elastic Waves in Crystals (Nauka, Moscow, 1965; Plenum, New York, 1968). 7. Yu. I. Sirotin and M. P. Shaskol’skaya, Principles of Crystal Physics (Nauka, Moscow, 1979) [in Russian]; (Mir, Moscow, 1983) [in English]. 8. V. I. Al’shits, A. V. Sarychev, and A. L. Shuvalov, Sov. Phys. JETP 62 (3), 531 (1985). 9. A. L. Shuvalov and P. Chadwick, Philos. Trans. R. Soc., A 355, 156 (1997). 10. V. I. Alshits and V. N. Lyubimov, in Proceedings of the 2nd Workshop on Dissipation in Physical Systems, Borkow, Poland, September 1–3, 1997, Ed. by A. Rad owicz (Politechnika Swietokrzyska, Kielce, Poland, 1998), p. 15. 11. A. L. Shuvalov and N. H. Scott, Q. J. Mech. Appl. Math. 52, 405 (1999). 12. A. L. Shuvalov and N. H. Scott, Acta Mech. 140, 1 (2000). 13. V. I. Alshits and V. N. Lyubimov, JETP 98 (5), 870 (2004). 14. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 7: Theory of Elasticity (Nauka, Moscow, 1987; Butterworth–Heinemann, Oxford, 1995).
Translated by M. Sapozhnikov
JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS
Vol. 113
No. 4
2011
