Generation of optical vorticity from topological defects
aa r X i v : . [ c ond - m a t . s o f t ] A ug Generation of helical modes from a topological defect
S´ebastien Fumeron , Fernando Moraes , and Erms Pereira . Laboratoire d’ ´Energ´etique et de M´ecanique Th´eorique et Appliqu´eeCNRS UMR 7563Nancy Universit´e54506 Vandoeuvre Cedex, France. Departamento de F´ısica, CCEN, Universidade Federal da Para´ıba,Caixa Postal 5008, 58051-970 Jo˜ao Pessoa, PB, Brazil and Instituto de F´ısica, Universidade Federal de Alagoas,57072-900, Macei´o, Alagoas, Brazil.
The propagation of an electromagnetic wave in a medium with a screw dislocation is studied.Adopting the formalism of differential forms, it is shown that torsion is responsible for quantizedmodes. Moreover, it is demonstrated that the modes thus obtained have well defined orbital angularmomentum, opening the possibility to design liquid-crystal-based optical tweezers.
PACS numbers: 42.50.Tx,02.40.-k,61.72.Bb,61.72.Lk
I. INTRODUCTION
During the last decades, the interaction of the orbital angular momentum of light with matter has become avery active research field [1–4] due to its large number of potential applications such as optical tweezers (for themanipulation of living cells and nanoobjects), micromachines (molecular engines) or quantum cryptography devices.In electrodynamics, it is indeed well known that light carries an angular momentum. This latter can be divided intotwo parts: a spin contribution associated to the polarization of the wave and an orbital contribution. A possible wayof controling the orbital angular momentum state of light beams is to use q-plates built out of liquid crystals [5].These q-plates coincide with the cross sections of topological line defects [6] and therefore, understanding how theangular momentum of light interacts with a line defect is of prime interest.From this point of view, the most relevant line defects are probably screw dislocations, because they locally inducetorsion. A screw dislocation is a line defect that may occur in smectic C* liquid crystals [7], in ordinary crystals [8] andeven in spacetime [9]. The generation of the defected topology is achieved through a “cut and glue” Volterra process,based on ideas of the homology theory [10]: basically, the screw dislocation is generated by cutting the medium alonga half-plane, moving the part located over the cut by a vector ~b (named Burgers vector) parallelwise to the edge of thecutting plane, and finally gluing the upper and lower sides. Thus, a screw dislocation is associated with a breakingof translational symmetry and it also exhibits an explicit helicity which, as we show below, has a profound influenceon the angular momentum of a propagating electromagnetic field. Fig. 1 depicts a screw dislocation in a genericcontinuous medium. Assuming cylindrical coordinates and taking the axis of the defect to be the z -axis, it is clearthat the screw dislocation mixes the r and ϕ degrees of freedom. In other words, by going clockwise a complete turnaround the axis, one moves up by one unit of Burgers vector ~b . FIG. 1: Screw dislocation
An elegant way of taking into account the boundary condition ϕ → ϕ + 2 π implies z → z + b (1)is to use an Einstein-Cartan background [11]. This approach has also been used to describe elastic continuous mediain analogy with gravity [12]. In this work, we look for a simple solution for an electromagnetic wave propagatingalong the axis of a screw dislocation. Since the main purpose of this article is to demonstrate the acquisition oforbital angular momentum by the propagating fields, this work may be relevant for applications both in condensedmatter physics (particularly smectic C* liquid crystals) and also in cosmology. For example, searching the cosmicmicrowave background for orbital angular momentum beams could give some clues on the existence of cosmic screwdislocations in the early universe. For simplicity we consider c = ε = µ = 1. Even though the problem at handinvolves non-relativistic systems, for convenience, we work in a four-dimensional spacetime as it provides a frameworkin which Maxwell’s equations are naturally covariant. In cylindrical coordinates, the background geometry inducedby the screw dislocation is given by the line element [9, 11] ds = − dt + dr + r dϕ + ( dz + βdϕ ) , (2)where β = b/ π . Explicitly, the metric tensor is therefore g µν = − r + β β β . (3)It must be emphasized that for solid crystals, this geometrization of matter is actually qualitatively equivalent todeterminations of actual properties of screw-dislocated dielectrics based on usual elasticity theory. An elastic defectis indeed expected to modify the dielectric properties in the vicinity of the dislocation as prescribed in eq. (11) ofreference [13]. This latter has to be compared to the spatial part of the metric (3) rewritten in Cartesian coordinates,that is g ij = δ ij + b πr − y x − y x . (4)up to first order in b πr . Therefore, it is clear that the anisotropy introduced in the dielectric medium by the screwdislocation, to a good approximation, can effectively be described by a background space with unit dielectric constantgiven by metric (4). However, this is done in a qualitative way since the coupling constant P , between the strainfield and the electromagnetic field, does not appear explicitly in our model. This is due to the fact that the startingpoint of the geometric approach is the boundary condition (1) and not the elasto-optic effect.On the other hand, in order to describe electromagnetic waves propagating along the axis of a cosmic screwdislocation, we assume that there is no other source of geometry (gravitational field) in the vicinity of the defect.Also, the dislocation is supposed not to be rotating, which would include a coupling between t and ϕ in the metric,just like the one between z and ϕ due to the dislocation. Moreover, we consider a fixed background geometry, that is,we assume that the electromagnetic wave energy contribution to the local gravitational field is negligible. In eithercase of propagating electromagnetic fields along a screw dislocation, be it in condensed matter or in the cosmos, wehave a geometrical background given by (2). Once the metric tensor is known, the language of General Relativityprovides a powerful tool to determine the equations governing electrodynamics in the distorted background. This isthe object of the next section. II. MAXWELL’S EQUATIONS
In reference [20], Maxwell’s equations were found in the geometry induced by the presence of a cosmic dislocationusing the differential forms formalism [21, 22]. Besides its natural elegance, the main advantage of this formalismis the fact that it provides a coordinate-free formulation of electrodynamics. Coordinates are introduced only whena specification of the field components is required. In what follows, the derivation of Maxwell’s equations in thescrew-dislocated background is presented, following the steps of reference [20]. From the point of view of spacetime,we consider the approximation [23] where the electromagnetic field is taken as a weak perturbation on the spacetimemetric. That way, the contribution of the electromagnetic field to the spacetime geometry is neglected and theEinstein-Maxwell equations are decoupled. From the point of view of solid-state physics, this approximation meansthat the electromagnetic field is supposed not to affect the elastic properties of the material medium.In language of differential forms, Maxwell’s equations can be concisely expressed as [22] dF = 0 (5)and ⋆ d ⋆ F = J . (6)Here, d denotes the exterior derivative, ⋆ is the Hodge star operator (see Appendix), F is the Faraday 2-form definedas: F ≡ B + E ∧ d t. (7)and J is the current density 1-form J = − ρ d t + J r d r + J ϕ d ϕ + J z d z, (8)The electric field 1-form is written as E = E r d r + E ϕ d ϕ + E z d z, (9)and the magnetic field 2-form is given by B = B φz d φ ∧ d z + B zr d z ∧ d r + B rφ d r ∧ d φ. (10)In order to express Maxwell’s equations in terms of the electric and magnetic field components of the usual Euclideanspace, it is necessary to find the transformation laws between the components of a differential form and its componentsin the Euclidean basis. The vector basis of 3-dimensional Euclidean space is the space subset of B ˆ v = { ~e ˆ t , ~e ˆ r , ~e ˆ ϕ , ~e ˆ z , } such that ~e ˆ µ · ~e ˆ ν = η µν , where η µν is the flat Minkowski metric. The above basis is not B v = { ~e t , ~e r , ~e ϕ , ~e z } , the dualbasis of B , which is such that ~e µ · ~e ν = g µν . The relation between the vectors in the basis B v and B ˆ v is therefore: ~e ˆ µ = ~e µ p | g µµ | . (11)As a consequence, the transformation law between the Euclidean components of a vector ~v and its contravariantcomponents are related by [24] v µ = v ˆ µ p | g µµ | . (12)(Notice that, in the last two equations, the sum convention for repeated indices should not be used.) Using the metricto obtain the dual vectors of the 1-form (that is the contravariant vectors) and (12), one finally gets the transformationlaws between the components of a 1-form A and its components in the Euclidean basis: A r = A ˆ r , A ϕ = p α r + β A ˆ ϕ + βA ˆ z e A z = A ˆ z + β p α r + β A ˆ ϕ . (13)After some algebra, one finally obtains Maxwell’s equation in the presence of the screw dislocation as [20]:1 r ∂∂r ( rE ˆ r ) + 1 p r + β ∂E ˆ ϕ ∂ϕ + ∂E ˆ z ∂z = ρ, (14)for Gauss’ law, whereas for the three components of Amp`ere-Maxwell’s law, it comes:1 r " β p r + β ∂∂ϕ − p r + β ∂∂z ! B ˆ ϕ + (cid:18) ∂∂ϕ − β ∂∂z (cid:19) B ˆ z = J ˆ r + ∂E ˆ r ∂t , (15) p r + β r " ∂B ˆ r ∂z − ∂∂r B ˆ z + β p r + β B ˆ ϕ ! = J ˆ ϕ + ∂E ˆ ϕ ∂t , (16)1 r (cid:20) ∂∂r (cid:16)p r + β B ˆ ϕ + βB ˆ z (cid:17) − ∂B ˆ r ∂ϕ (cid:21) = J ˆ z + ∂E ˆ z ∂t . (17)The equations describing the absence of magnetic monopoles and Faraday’s law are both obtained by making ρ = 0 , J ˆ r = J ˆ ϕ = J ˆ z = 0 ,B ˆ i → E ˆ i e E ˆ i → − B ˆ i , where i corresponds to the indices r , ϕ and z . The absence of magnetic monopoles leads to1 r ∂∂r ( rB ˆ r ) + 1 p r + β ∂B ˆ ϕ ∂ϕ + ∂B ˆ z ∂z = 0 , (18)and the three components of Faraday’s law are given by:1 r β p r + β ∂∂ϕ − p r + β ∂∂z ! E ˆ ϕ + 1 r (cid:18) ∂∂ϕ − β ∂∂z (cid:19) E ˆ z = − ∂B ˆ r ∂t , (19) p r + β r " ∂E ˆ r ∂z − ∂∂r E ˆ z + β p r + β E ˆ ϕ ! = − ∂B ˆ ϕ ∂t , (20)1 r (cid:20) ∂∂r (cid:16)p r + β E ˆ ϕ + βE ˆ z (cid:17) − ∂E ˆ r ∂ϕ (cid:21) = − ∂B ˆ z ∂t . (21) III. AN HEURISTIC SOLUTION
Except for the plane wave, most of the propagating solutions of Maxwell’s equations are of quite complicatedform. They are usually described as superpositions of plane waves or, depending on the coordinate system, specialfunctions or polynomial expansions. Since we are interested in orbital angular momentum, it would appear natural tolook for solutions of the Laguerre-Gaussian beam type [18], for example. Nevertheless, the aim of this article, beingpedagogical, is to find a simple propagating solution that contains the essential physics of the system.We consider a wave propagating along the Burgers vector of the defect. Therefore, it is licit to assume that variablescan be separated according: ~E = ~E ( r ) u ( ϕ ) exp [ ikz − iωt ] ,~B = ~B ( r ) u ( ϕ ) exp [ ikz − iωt ] , (22)whith u ( ϕ ) a complex-valued function. That way, after some calculations, (14) writes as the sum of two terms: p r + β E ˆ ϕ (cid:20) r ∂∂r ( rE ˆ r ) + ikE ˆ z (cid:21) + 1 u ( ϕ ) dudϕ = 0 , (23)The first term depends only on r whereas the second depends only on ϕ . As a consequence, it is mandatory that:1 u ( ϕ ) dudϕ = C, (24)with C being a constant complex number. Moreover, for symmetry reasons, it is required that under the transformation φ → φ + 2 π, z → z + b , the field remains unchanged so that finally: u ( ϕ ) = exp [ idϕ ] , (25)with d a real number such that ν = id . As a consequence, the field is expected to have the following form: ~E = ~E ( r ) exp [ ikz + idϕ − iωt ] ,~B = ~B ( r ) exp [ ikz + idϕ − iωt ] , (26)where k is the wavevector along the z direction, d a real number, and ω the angular frequency.By direct substitution of (26) into the Maxwell’s equations (14)-(21), separating the real and imaginary parts ofeach equation, and solving the resulting system of equations, it comes that: ~E ( r ) = ar ~e ˆ r − a p r + β r ~e ˆ ϕ + βar ~e ˆ z (27)and ~B ( r ) = ar ~e ˆ r + a p r + β r ~e ˆ ϕ − βar ~e ˆ z , (28)where the parameter a sets the intensity of the fields. We also get the dispersion relation k = ± ω (29)but more importantly that the integer m is related to the Burgers vector and to the wavevector by m = βk. (30)(Notice that the solution (26) satisfies the boundary condition (1) since equation (30) holds). This implies thatsolutions of the type (26) are quantized, that is only modes with definite wavevector k m = m/β are allowed. Thisbrings about interesting applications such as using the medium endowed with a defect as a filter for specific frequencies.From equations (27) and (28), we obtain the Poynting vector ~S = 12 ~E × ~B ∗ = βa r ~e ˆ ϕ + a p r + β r ~e ˆ z . (31)Moreover, with only ϕ and z components, it appears that the Poynting vector spirals along the direction of propa-gation. To verify this, we identify its components with the components of a tangent vector to a yet unknown spacecurve given in parametric form by r ( t ) , ϕ ( t ) , z ( t ). That is˙ r ( t ) = 0 r ( t ) ˙ ϕ ( t ) = βa r (32)˙ z ( t ) = a p r + β r After straightforward calculations, the solutions of ( ?? ) are obtained as: r ( t ) = r o ϕ ( t ) = βa r o t + ϕ o (33) z ( t ) = a p r o + β r o t + z o , where r o , ϕ o and z o are integration constants. It is clear that the set of equations (33) describes a helix of radius r o and pitch 2 π r o √ r o + β β . Notice that, in the absence of the defect, r = r o , ϕ = ϕ o and z = z o + const · t , whichrepresents a straight line along the z -axis.Now, we turn our attention to vector potential ~A and the scalar potential V . These latter can be obtained from theelectric and magnetic fields given by (26) assorted by an appropriate gauge condition. For convenience, the Coulombgauge is used in all that follows, so that potentials are going to be obtained from: ~B = −→∇ ∧ ~A (34) ~E = − ∂ ~A∂t − −→∇ V (35) −→∇ . ~A = 0 (36) △ V = 0 (37)Using the amplitudes of the fields as prescribed by (27) and (28), the previous set of equations gives for (34)1 r ∂A z ∂φ − ∂A φ ∂z = ar e i ( mφ + kz − ωt ) (38) ∂A r ∂z − ∂A z ∂r = ar p r + β e i ( mφ + kz − ωt ) (39) ∂∂r ( rA φ ) − ∂A r ∂φ = − βar e i ( mφ + kz − ωt ) (40)This system suggests that each of the unknown functions A r , A φ and A z are linear with respect to the factor e i ( mφ + kz − ωt ) . In particular, this implies that: ∂ ~A∂t = − iω ~A (41)Therefore, (35) gives the system: iωA r − ∂V∂r = ar e i ( mφ + kz − ωt ) (42) iωA φ − r ∂V∂φ = − ar p r + β e i ( mφ + kz − ωt ) (43) iωA z + ∂V∂z = βar e i ( mφ + kz − ωt ) (44)This in turn implies that the scalar potential V is linear with respect to the factor e i ( mφ + kz − ωt ) . Bearing in mindthere is a similar property for the vector potential, this strongly suggests that for both potentials, variables can beseparated in the following way: A r = a r ( r ) e i ( mφ + kz − ωt ) (45) A φ = a φ ( r ) e i ( mφ + kz − ωt ) (46) A z = a z ( r ) e i ( mφ + kz − ωt ) (47) V = v ( r ) e i ( mφ + kz − ωt ) (48)Therefore, expressing the Coulomb gauge equations (36)-(37), it comes straightforwardly that:1 r ∂∂r ( rA r ) + 1 r ∂A φ ∂φ + ∂A z ∂z = 0 (49)1 r ∂∂r (cid:18) r ∂V∂r (cid:19) + 1 r ∂ V∂φ + ∂ V∂z = 0 (50)(51)Using (48) in (50), it comes that after some algebra that: d vdr + 1 r dvdr − (cid:18) m r + k (cid:19) v = 0 (52)Performing the change in variable X ↔ k r , we can rearrange the previous expression to get: X d vdX + X dvdX − (cid:0) m + X (cid:1) v = 0 (53)The solutions of this equation are the modified Bessel functions I ± m ( X ) and K m ( X ). As the electric field involvesthe divergence of the scalar potential, it is natural to retain only the K m ( X ) functions so that the electromagneticfield vanishes at infinity. Therefore, pluging v ( r ) = K m ( k r ) in eqs (42)-(44) and using the ansatz (45)-(47), we areled to: a r ( r ) = ik ω (cid:18) K m +1 ( kr ) + K m − ( kr ) + 2 akr (cid:19) (54) a φ ( r ) = iω (cid:18) ar p r + β − im K m ( kr ) r (cid:19) (55) a z ( r ) = iω (cid:18) − βar + ikK m ( kr ) (cid:19) (56)(57)The orbital angular momentum is defined from the vector potential and the electric field by [25]: ~L = X j =1 µ c ˆ d xE j (cid:16) ~x ∧ ~ ∇ (cid:17) A j (58)The volume density of angular momentum, which at the point ~R , is given by ~M = ~R × ~S. (59)A straightforward calculation gives ~M = − zβa r ~e ˆ r − a p r + β r ~e ˆ ϕ + βa r ~e ˆ z . (60)It is well-known that with a unit system in which c = 1, the linear momentum density identifies with the Poyntingvector. Thus, the ratio between the flux of angular momentum to that of energy across the surface z = const is givenby: L/P = ´ ∞ δ dr ´ πδ rdϕM z ´ ∞ δ dr ´ πδ rdϕS z = β ´ ∞ δ dxx ´ ∞ δ √ x x dx where δ is an ultraviolet cut-off corresponding to a core structure. In smectic liquid crystals, δ is of the order of theaverage thickness of a layer [16], whereas in a cosmological context, δ is about the inverse of the energy scale at whichthe symmetry-breaking phase transition occurs [17]. Then after simple manipulations, it comes that L/P = β = m/ω (61)The solution (26) has therefore a well-defined orbital angular momentum. It has to be emphasized that even if thisresult is derived from a simple solution of Maxwell’s equations, it corresponds to what is obtained for realistic lasermodes (Laguerre-Gaussian beams [18]).Before closing this section a few remarks on the conservation laws are in order. It is interesting to notice that thePoynting vector (31) obeys the conservation law ~ ∇ · ~S = 0. Furthermore, the transversal part of the Poynting vectoris also divergence-free. So, the beam intensity distribution does not change in the plane perpendicular to the directionof propagation. In other words, it is a non-diffracting beam ([19]). Also, the radial and azimuthal components ofthe angular momentum density are symmetric about the z -axis. This implies that integration over the beam profileleaves only the ~e ˆ z component. This is easily seen by writing ~M in terms of its Cartesian components while keepingthe cylindrical coordinates. IV. CONCLUSION
In this work, we investigated some features of electrodynamics in the neighborhood of a screw dislocation. Fromthe geometric treatment of topological defects, it appears that the torsion induced by the dislocation couples tothe electromagnetic field in two ways. First, it is responsible for a quantization of the modes, for which the allowedfrequencies depend only on the value of Burgers vector. This may be of prime interest for several potential applicationssuch as defect sounding or X-ray filters or even the design on the heat rectifier devices [26], due to the periodicity of thescrew dislocation. Second, the torsion forces the Poynting vector to spiral along the direction of propagation, possiblyendowing the electromagnetic field with an orbital angular momentum. Such property is relevant in observationalcosmology as a signature of cosmic strings, but it also provides an alternate approach to design optical tweezers froma simple (and tunable) waveguide effect.One of the main interests of this work is that it can be generalized to other kinds of defects. Indeed, the differentialforms formalism provides the general process of dealing with electromagnetism in non-trivial background geometries.Other kinds of line defects (edge dislocations, disclinations) and even distributions of defects can be treated this way,and one may expect strong couplings between the quantized modes in this last case. This will be the object of a nextpaper.
Acknowledgments
F.M. is grateful to LEMTA for financial support during his stay there and CNPq, CAPES (Brazilian agencies) forfinancial support. E.P. is grateful to FAPEAL and CNPq (Brazilian agencies) for financial support. The authorsthank Pablo Vaveliuk and Dragi Karevski for fruitful discussions related to part III.
APPENDIX: Hodge duality
In cylindrical coordinates, the 1-form basis writes: B = { d t, d r, d ϕ, d z } (A-1)In electrodynamics, the components of the Faraday 2-form that accounts for the field write as [21]: F = 12 F µν d x µ ∧ d x ν , (A-2)where d x µ ∧ d x ν are elements of the 2-form basis. B = { d ϕ ∧ d z, d z ∧ d r, d r ∧ d ϕ, d r ∧ d t, d ϕ ∧ d t, d z ∧ d t } . (A-3)To translate the usual Maxwell’s equations in terms of differential forms, it is convenient to introduce the Hodge staroperator ⋆ . This latter acts on a p -form in n -dimensional space and turns it into the ( n − p )-form that somehowcompletes the volume n -form. Given the product of two p-forms ρ and ψ defined on an oriented n -manifold describedby metric g µν , then the Hodge star operator is defined as: ρ ∧ ⋆ψ = q | det ( g µν ) | h ρ, ψ i d x ∧ .. d x n − p (A-4)Taking into account metric (2), the action of ⋆ on the 2-forms of B is then: ⋆ ( d ϕ ∧ d z ) = − r d r ∧ d t, (A-5) ⋆ ( d z ∧ d r ) = − r + β r d ϕ ∧ d t − βr d z ∧ d t, (A-6) ⋆ ( d r ∧ d ϕ ) = − βr d ϕ ∧ d t − r d z ∧ d t, (A-7) ⋆ ( d r ∧ d t ) = r d ϕ ∧ d z, (A-8) ⋆ ( d ϕ ∧ d t ) = 1 r d z ∧ d r − βr d r ∧ d ϕ, (A-9) ⋆ ( d z ∧ d t ) = r + β r d r ∧ d ϕ − βr d r ∧ d z. (A-10)On the elements of the 3-form basis B , the action of Hodge’s star operator is ⋆ ( d r ∧ d ϕ ∧ d t ) = − r d z − βr d ϕ, (A-11) ⋆ ( d z ∧ d r ∧ d t ) = − r + β r d ϕ − βr d z, (A-12) ⋆ ( d ϕ ∧ d z ∧ d t ) = − r d r, (A-13) ⋆ ( d r ∧ d ϕ ∧ d z ) = − r d t. (A-14) [1] A. Muthukrishnan and C.R. Stroud, J. Opt. B: Quantum semi-classical Opt. , S73 (2002). [2] M. Babiker, C.R. Bennett, D.L. Andrews, and L.C.D. Romero, Phys. Rev. Lett. , 143601 (2002).[3] A. Alexandrescu, D. Cojoc, and E. Di Fabrizio, Phys. Rev. Lett. , 243001 (2006).[4] S. Thanvanthri, K.T. Kapale and J.P. Dowling, Phys. Rev. A , 053825 (2008).[5] L. Marrucci, C. Manzo and D. Paparo, Phys. Rev. Lett. , 163905 (2006).[6] C. S´atiro and F. Moraes, Eur. Phys. J. E , 173 (2006).[7] M.-F. Achard, M. Kleman, Yu. A. Nastishin and H.-T. Nguyen, Eur. Phys. J. E , 37 (2005).[8] Frank R.N. Nabarro and John P. Hirth (Editors) Dislocations in Solids , Volume 12, Elsevier, Amsterdam (2004).[9] D. V. Gal’tsov and P. S. Letelier, Phys. Rev. D , 4273 (1993).[10] M. Kleman, Points, Lignes, Parois dans les fluides anisotropes et les solides cristallins , Edition de Physique, Paris (1977).[11] R. A. Puntigam and H. H. Soleng, Class. Quantum Grav. , 1129 (1997).[12] M. O. Katanaev and I. V. Volovich, Ann. Phys. , 1 (1992).[13] D. Sahoo, A.K. Arora and R. Kesavamoorthy, J. Phys. C:Solid State Phys. , 1687 (1983).[14] C. Furtado, V.B. Bezerra, F. Moraes, Europhys. Lett. , 1 (2000).[15] P.A.M. Dirac, Proc. R. Soc. A