The effect of Dirac phase on acoustic vortex in media with screw dislocation
aa r X i v : . [ c ond - m a t . o t h e r] J un The effect of Dirac phase on acoustic vortex in media with screwdislocation
Reza Torabi ∗ and Zahra Rezaei Department of Physics, University of Tafresh,P.O.Box: 39518-79611, Tafresh, Iran
Abstract
We study acoustic vortex in media with screw dislocation using the Katanaev-Volovich theory ofdefects. It is shown that the screw dislocation affects the beam’s orbital angular momentum andchanges the acoustic vortex strength. This change is a manifestation of topological Dirac phaseand is robust against fluctuations in the system.
Keywords: Acoustic vortex, Dirac phase, Screw dislocation, Thermal noise ∗ Corresponding Author.Email: [email protected], Tel:+989125581265, Fax:+988626227814 . INTRODUCTION The linearized equations of elasticity are analogous with Maxwell equations [1]. Thisanalogy enables us to suggest new phenomena for elastic waves by knowing their opticalcounterparts. In particular, vortex phenomena in optics can be mapped onto acoustic vortex.An optical vortex beam focuses on rings rather than points and has helical wavefront, Fig.1, [2]. The difference between this kind of beam and a plane wave is just an overall phase
FIG. 1: The circular trajectory of a vortex beam [2]. factor, e ilϕ . The angle ϕ is the polar angle in cylindrical coordinates for a beam with axisparallel to z and l is the optical vortex strength or the angular momentum that is carriedby the helical beam [3, 4]. When the helical beam interacts with a microscopic particle, theorbital angular momentum can be transferred to the particle and make it spin around thebeam axis. A wide range of applications have been recently found for this orbital angularmomentum transfer. For example we can mention, particle trapping [5] in optical tweezersto manipulate micrometer-sized particles [6] and remote control of particles [7]. Opticalvortex also has application in information encoding [8].Acoustic vortex as the classical counterpart of optical vortex has been studied [9–12] andgenerated [13–15], recently. Since acoustic vortices can transfer orbital angular momentumto particles [15, 16], like their optical counterparts, they can be applied to particle trappingin acoustic tweezers and remote controlling, too. In addition acoustic vortices, potentially,can be used in sonar experiments [13]. Although similar properties to optical vortex isexpected for acoustic vortex, there are limited studies in this area.In this letter, acoustic vortex in media with screw dislocation is studied using theKatanaev-Volovich theory of defects. The motivation for studying defects is that they usu-ally exist in crystalline solids and have strong effect on their physical properties [17–25].2n the presence of defects, we are confronting with complicated boundary conditions. Thisdifficulty persuades physicists to introduce new approaches such as Katanaev-Volovich the-ory of defects in solids [24–30]. Katanaev-Volovich theory is a geometrical approach basedon the isomorphism existing between the theory of defects in solids and three-dimensionalgravity. In this formalism, elastic deformation which is introduced in the medium by defectsis replaced by a non-Euclidean metric. According to this theory, at distances much largerthan the lattice spacing where the continuum limit is valid, the solid can be described bya Riemann-Cartan manifold. Dislocations and disclinations of the medium are respectivelyassociated with torsion and curvature of the manifold. We will show that the screw disloca-tion changes the acoustic vortex strength. This change is due to Dirac phase and is robustagainst fluctuations. Dirac phase belongs in the category of non-integrable phase factorsthat appear in many different areas of Physics [31–33]. Dirac showed that when a particletransports in an external electromagnetic field, its wave function acquires a phase term inaddition to usual dynamic phase factor [34, 35].The letter is organized as follow. In section II, we review the screw dislocation inKatanaev-Volovich formalism. Section III is devoted to the Dirac phase of acoustic wavesin media with screw dislocation. The effect of noise on Dirac phase is discussed in sectionIV. Finally the conclusion is presented in section V. II. SCREW DISLOCATION IN KATANAEV-VOLOVICH FORMALISM
Consider a point in an undeformed medium with coordinates x i with Euclidean metric δ ij . If the deformation due to the defect is described by a displacement vector U i ( x ), thepoint will have coordinates y i = x i + U i ( x ). According to the Katanaev-Volovich approachthe effect of the elastic deformation U i is considered by introducing metric g ij which can beexpressed in terms of the initial metric δ ij as [27–29] g ij := ∂x k ∂y i ∂x l ∂y j δ kl , i, j = 1 , , . One of the defects, which we are interested in, is a screw dislocation. In a screw dislocationthe Burgers vector is parallel to the dislocation line. This kind of defect, which correspondsto a singular torsion along the defect line, is described by the following metric [26, 36] ds = g ij dx i dx j = ( dz + βdφ ) + dρ + ρ dφ , β is related to the Burgers vector, b , by β = b π and the screw dislo-cation line is oriented along the z-axis of the cylindrical coordinates ( ρ, ϕ, z ). To derive theabove metric, we have used the displacement vector U = (0 , , βφ ) associated with a screwdislocation in cylindrical coordinates. The metric tensor g ij is g ij = β + ρ β β (1)and carries no curvature.The torsion two-form associated with this defect is defined as T a = T aij dx i ∧ dx j , ( a ≡{ ρ, φ, z } ), that the only non-vanishing component is given by [25] T z = 2 πβδ ( ρ ) dρ ∧ dφ, where δ ( ρ ) is the two-dimensional delta function in flat space and reveals the singularity intorsion. Also the torsion in tensor notation can be written as T aij = ∂ i e aj − ∂ j e ai , (2)where e ai are triad components. Comparison of equation (2) with the field strength F ij = ∂ i A j − ∂ j A i in the electromagnetism and the singular value of the torsion field, indicates asimilarity between this case and the Aharanov-Bohm effect [37], where the Burgers vectorplays the role of the magnetic flux. III. DIRAC PHASE OF ACOUSTIC WAVES
The dynamic of displacement vector field U ( x , t ) in an elastic medium without defect isgoverned by (see e.g. [38]) ∂ t U i = µρ ∇ U i + ( λ + µ ) ρ ∂ i ∂ j U j , (3)where λ and µ are the Lame coefficients and ρ is the density of the medium. According tothe Katanaev-Volovich approach, media with defects can be treated by nontrivial metric, g ij . Therefore, the covariant generalization of equation (3) gives the displacement vectordynamics in media with defects ∂ t U i = µρ ˜ ∇ U i + ( λ + µ ) ρ ˜ ∇ i ˜ ∇ j U j . (4)4he displacement vector can be decomposed covariantly into transversal, U T i , and longitu-dinal, U Li , parts [28], U i = U T i + U Li , which satisfy the following relations ˜ ∇ i U T i = 0 , ˜ ∇ i U Lj − ˜ ∇ j U Li = 0 . As far as we know from vector analysis, it is always possible to express a vector as the sumof the curl of a vector and the gradient of a scalar. So equation (4) decomposes into twoindependent equations for transverse and longitudinal parts of the displacement vector field,1 v T ∂ t U T i − ˜ ∇ U T i = 0 , v L ∂ t U Li − ˜ ∇ U Li = 0 , (5)where v T = µρ , v L = ( λ + 2 µ ) ρ , are the speeds of transverse and longitudinal parts in the medium. So by decompositionof (4) every longitudinal or transverse component of the displacement vector, U , satisfiesa separate scalar wave equation in the curved space. ˜ ∇ in (5) is the Laplace-Beltramioperator which is given by ˜ ∇ = √ g ∂ i ( g ij √ g∂ j ), where g is the determinant of the metrictensor g ij and g ij = ( g ij ) − is its inverse. This decomposition enables us to study eachmode separately. Here we are interested in the longitudinal mode but similar results can bededuced for the transverse mode, too.Using the metric tensor (1) for screw dislocation, the longitudinal part of the elastic wavein (5) takes the form (cid:26) ρ ∂ ρ ( ρ∂ ρ ) + 1 ρ ( ∂ ϕ − β∂ z ) + ∂ z (cid:27) U Li ( ρ, ϕ, z, t ) = 1 v L ∂ t U Li ( ρ, ϕ, z, t ) . Considering a monochromatic paraxial wave as U Li ( ρ, ϕ, z, t ) = e − iωt e ikz u Li ( ρ, ϕ ) , yields to the following equation for longitudinal elastic wave (cid:26) ρ ∂ ρ ( ρ∂ ρ ) + 1 ρ ( ∂ ϕ − iβk ) − k (cid:27) u Li ( ρ, ϕ ) = − ω v L u Li ( ρ, ϕ ) . (6)5quation (6) implies that ∂ ϕ → ∂ ϕ − ikβ with respect to the defect free case ( β = 0) inwhich the Laplacian operator is given in a flat space. In the other words, the z -componentof angular momentum has changed according to L z → L z − kβ . The angular momentumof the acoustic wave along it’s axis is modified by the presence of the defect that is due tothe torque exerted by the strain field of the dislocation. Introducing a momentum operatoras P = − i ∇ converts (6) into a time-independent Schr¨odinger-like equation with a gaugepotential ( P − A ) u Li ( ρ, ϕ ) = ω v L u Li ( ρ, ϕ ) , (7)where the corresponding vector gauge potential is A = kβρ ˆ e ϕ . (8)Since this gauge is curl free, ∇ × A = 0, the perfect analogy is seen between acoustic wavesin media with screw dislocation and the Aharanov-Bohm effect. Note that, the torsion fieldis invariant under gauge transformations of the potential, A → A + ∇ Λ . According to this correspondence, Dirac phase factor method [34, 39] can be used here (Seethe appendix). Thus, the solution of the wave equation (7) has the following property u Li ( ρ, ϕ ) = exp (cid:26) i Z C A · d r (cid:27) u Li ( ρ, ϕ ) , (9)where u T i ( ρ, ϕ ) is the solution of the defect free case and C is the beam trajectory (See Fig.1). Substituting (8) into (9) yields u Li ( ρ, ϕ ) = e i R ϕ kβdϕ u Li ( ρ, ϕ ) . (10)This means that u Li ( ρ, ϕ ) differs from u Li ( ρ, ϕ ) just in a phase factor e iγ that γ = Z ϕ kβdϕ, is called Dirac phase. In other words, the coupling of torsion with angular momentum leadsto an additional phase factor in the solution when the screw dislocation is present.Hitherto we found that the difference between the solutions to the acoustic wave equationin the presence and in the absence of defects is just manifested in a phase factor, (10). So6e only need to find the solution for the defect free case. In this case, β = 0, the solutionof the wave equation (6) can be easily found as u Li ( ρ, ϕ ) = R ( ρ ) e ilϕ , (11)where R ( ρ ) is the radial solution of the Helmholtz equation and e ilϕ represents acousticvortex carrying the angular momentum l , acoustic vortex strength, along the paraxial axis.According to equations (10) and (11) the solution of the wave equation in the presence ofscrew dislocation is shown as u Li ( ρ, ϕ ) = R ( ρ ) e i ( l + βk ) ϕ . Therefore, the screw dislocation results in the change of the acoustic vortex strength from l to l + βk . This change, due to Dirac phase, is proportional to the magnitude of Burgersvector or in other words the flux of torsion. IV. THE EFFECT OF NOISE ON DIRAC PHASE
The presence of noise is an inevitable subject in physical systems, such as the the ubiq-uitous thermal fluctuation. The effect of noise on the Dirac phase can be treated similar tothe problem of electrons in media with screw dislocations [40]. The same procedure is usedto show that considering a white noise leads to the average zero for the Dirac phase. Thiskind of noise coincides with uncorrelated nature of thermal noise. Indeed, the variance ofthe Dirac phase diminishes with time as h△ γ ( T ) i ∝ T where T is the period of the beam’s rotation on its circular trajectory (Fig. 1). Therefore, γ coincides with its noiseless value in the limit T → ∞ . As a result, in spite of the dynamicphase, the Dirac phase of elastic waves is robust against fluctuations in the system. V. CONCLUSION
In this letter we studied the effect of the screw dislocation on an acoustic wave usingthe Katanaev-Volovich theory of defects. This theory which is a geometrical approach uses7he isomorphism between the theory of solids and three-dimensional gravity to suppress thetechnical complications due to adding defects to the system. It was shown that the screwdislocation changes the acoustic vortex strength by coupling of torsion with orbital angularmomentum. This change is a manifestation of the topological Dirac phase and is robustagainst fluctuations. For a white noise, coincides with the nature of thermal noise, the effectof fluctuations on the Dirac phase diminishes as T where T is the period of beam’s rotation. Appendix: Dirac phase factor method
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