Paraxial propagation in amorphous optical media with screw dislocation
aa r X i v : . [ phy s i c s . op ti c s ] F e b Paraxial propagation in amorphous optical media with screwdislocation
Leila Mashhadi, Mohammad Mehrafarin ∗ Physics Department, Amirkabir University of Technology, Tehran 15914, Iran
Abstract
We study paraxial beam propagation parallel to the screw axis of a dislocated amorphous mediumthat is optically weakly inhomogeneous and isotropic. The effect of the screw dislocation on thebeam’s orbital angular momentum is shown to change the optical vortex strength, rendering vortexannihilation or generation possible. Furthermore, the dislocation is shown to induce a weak biaxial anisotropy in the medium due to the elasto-optic effect, which changes the beam’s spin angularmomentum as well as causing precession of the polarization. We derive the equations of motion ofthe beam and demonstrate the optical Hall effect in the dislocated medium. Its application withregard to determining the Burgers vector as well as the elasto-optic coefficients of the medium isexplained.
PACS numbers: 42.25.Bs,03.65.Vf,42.25.Ja,42.25.Lc ∗ Electronic address: [email protected] . INTRODUCTION Paraxial beams with non-zero orbital angular momentum, which are exact eigenmodesof the paraxial wave equation [1], have been of considerable interest [2–4]. In addition tospin angular momentum (polarization), paraxial beams carry an intrinsic orbital angularmomentum that depends on their spatial structure. While the transfer of spin angularmomentum to matter has long been reported for optically anisotropic media [5], the transferof orbital angular momentum has newly triggered a wide range of applications. A notableexample is particle trapping [6], whereby particles are set into rotation by coupling withthe beam’s orbital angular momentum and are thus trapped. Such a coupling of the orbitalangular momentum with matter is possible in optically inhomogeneous isotropic media [7].Thus, a simultaneous independent coupling of both spin and orbital angular momentumwith matter is to be expected in a medium which is both optically inhomogeneous andanisotropic [8, 9].In the past decades, the geometric Berry phase [10] acquired by an optical beam (asthe Pancharatnam phase [11] or a spin redirection phase [12, 13]) has attracted extensiveattention. In particular, Bliokh et. al. [14–16] developed a modified geometrical opticsapproximation for optically weakly inhomogeneous isotropic media, which contained theRytov-Vladimirskii rotation law [17, 18] and the optical Magnus effect [19, 20] (also calledthe optical Hall effect) as manifestations of Berry effects. Such geometric Berry effectshave been also derived via the semiclassical wave packet approximation [21–23], which wasoriginally developed for the study of electronic spin transport in solids [24–26].By studying the propagation of paraxial beams in optically weakly inhomogeneousisotropic media, Bliokh [27] demonstrated that the optical Hall effect becomes more pro-nounced because of the contribution of the orbital angular momentum Hall effect due tothe orbit-orbit interaction. Here, we generalize his results for screw dislocated (amorphous)optical media. Mechanical deformation can result in dislocations that change the opticalproperties of a medium due to the elasto-optic effect. Such dislocations can be induced by thefocused beam itself (see e.g. [28, 29]). The study of beam transport provides an indirect wayto determine the dislocations as well as the elasto-optical properties of the medium. In thepresent work, we study paraxial beam propagation parallel to the screw axis of a dislocatedamorphous medium that is optically weakly inhomogeneous and isotropic. The paraxial2eam contains an optical dislocation, namely the optical vortex along its axis, which is atopological object on the wavefront the beam [30]. Regarding the medium as a continuum,we use the differential geometric theory of defects which is isomorphic to three dimensionalgravity [31, 32]. In this geometric approach, the effect of defects on the three dimensionalgeometry of the medium is incorporated into a metric. The effect of the screw dislocation onthe beam’s orbital angular momentum is, thus, shown to change the optical vortex strength,rendering vortex annihilation (for left-handed helical modes) or generation of non-integralvortices [33] possible. This is useful in generating fast switchable helical modes for opticalinformation encoding [9, 34]. Furthermore, the dislocation is shown to induce a weak biaxial anisotropy in the medium due to the elasto-optic effect. This anisotropy, which is felt morestrongly by beams that propagate close to the dislocation line, changes the beam’s spinangular momentum as well as causing precession of the polarization. Finally, we derive theequations of motion of the beam and demonstrate how beams with different values of spinand/or angular momentum split in the dislocated medium (the optical Hall effect). Becausethe beam’s singular vortex core can be observed with great accuracy, measurement of thesplittings is very reliable. Measuring beam splittings for opposite values of the polarizationand orbital angular momentum provides an indirect method for determining the Burgersvector as well as the elasto-optic coefficients of the medium. Determination of the former isdemonstrated in a simple example.
II. EFFECT OF SCREW DISLOCATION ON THE BEAM
Dislocation is a defect caused by the action of internal stress, which changes the physicalproperties of the medium, in particular, its optical properties. The displacement vectorfield, u , associated with a screw dislocation line oriented along the z -axis of the cylindricalcoordinates ( ρ, ϕ, z ) is u = (0 , , βϕ ), where β is the magnitude of Burgers vector dividedby 2 π . The corresponding strain tensor field S ij = ( ∂ i u j + ∂ j u i ) is, thus, given by S ϕz = S zϕ = β/ ρ , other components being zero. Regarding the medium as a continuum, we canuse the differential geometric theory of defects which is isomorphic to three dimensionalgravity [31, 32]. In this geometric approach, the effect of defects on the three dimensionalgeometry of the medium is incorporated into a metric. In particular, a screw dislocation,which corresponds to a singular torsion along the defect line, is described by the following3etric [35] ds = dρ + ρ dϕ + ( dz + βdϕ ) . (1)This metric carries torsion but no curvature and describes a screw dislocated medium.We consider a monochromatic paraxial beam with definite values of the spin and orbitalangular momentum, propagating in a weakly inhomogeneous isotropic medium where therefractive index n ( x ) varies adiabatically with position ( ∇ n → z as the paraxial direction, the beam’s electric field can be represented as in a homogeneousmedium according to E σlm ( ρ, ϕ, z ) = e σ E lm ( ρ, ϕ ) e ikz . (2)Here k = k n is the wave number, the adiabatic variation of which has been neglected ( k being the wave number in vacuum), e σ = 1 √ e ρ − iσ e ϕ )with σ = ± e ρ , e ϕ being the cylindrical unit vectors) are the orthonormal polarizationvectors corresponding, respectively, to right/left circular polarization and E lm ( ρ, ϕ ) = R l | m | ( ρ ) e imϕ R l | m | ; with l = 0 , , , . . . and m = − l, − l +1 , . . . , l ; being the radial solution of the Helmholtzequation ( σ and m represent the helicity and the z -component of the orbital angular mo-mentum of the photon, respectively (¯ h = 1)). In the presence of screw dislocation, theLaplacian operator ∇ appearing in the Helmholtz equation is to be replaced by ∂ z + 1 ρ ∂ ρ ( ρ∂ ρ ) + 1 ρ ( ∂ ϕ − β∂ z ) which is the Laplacian associated with the metric (1) of the dislocated medium. The solution(2), therefore, acquires an additional phase factor due to the coupling of torsion with angularmomentum, according to E lm ( ρ, ϕ ) = R l | m | ( ρ ) e i ( m + βk ) ϕ . (3)Furthermore, the relative permittivity tensor n δ ij acquires an anisotropic part ∆ ij due tothe strain field of the dislocation, where (see e.g. [36])∆ ij = − n p ijkl S kl ijkl being the elasto-optic coefficients of the medium. For amorphous media, where only twoindependent elasto-optic coefficients (customarily denoted by p and p ) exist, the screwdislocation yields ∆ ϕz = ∆ zϕ = − β ( p − p ) ρ n other components being zero. The principle refractive indices are, therefore, n ρ = n, n ϕ = n + β ( p − p )2 ρ n , n z = n − β ( p − p )2 ρ n to first order in the perturbation, which is generally sufficiently small. Hence, the elasto-opticperturbation induces a weak biaxial anisotropy in the medium, which is felt more stronglyby beams that propagate close to the dislocation line (small ρ ). Propagation of the beamthrough the dislocated medium, thus, introduces the phases k n ρ z and k n ϕ z in the linearpolarization states represented by e ρ and e ϕ , respectively. Hence, in (2), the polarizationvector should be rewritten as e σ = 1 √ e ρ − iσe ik ∆ nz e ϕ )where ∆ n ( ρ ) = n ϕ − n ρ = β ( p − p )2 ρ n is the induced birefringence. Equivalently, e σ = 1 √ e ρ − iσe iγ cot θ e ϕ ) (4)where cot θ = z/ρ and γ = βk ( p − p ) n .Equations (2) to (4) form our expression for the field of a paraxial beam propagating in anoptically weakly inhomogeneous and isotropic amorphous medium with screw dislocation.With the beam scalar product defined by [37]( E σlm | E σ ′ l ′ m ′ ) = Z Z E † σlm E σ ′ l ′ m ′ ρdρdϕ = δ σσ ′ δ mm ′ Z R ⋆l | m | R l ′ | m | πρdρ these beams form a complete orthonormal set, in terms of which an arbitrary field distri-bution can be constructed. The paraxial beam contains an optical dislocation, namely theoptical vortex of strength m along its axis, which is a topological object on the wavefrontthe beam [30]. The z -component of the beam’s orbital angular momentum (per photon) isgiven by the expectation value Z Z E ⋆lm ( − i∂ ϕ ) E lm ρdρdϕ = m + βk. m to m + βk owes itself to the torque exertedby the strain field of the dislocation. The effect of the screw dislocation on the orbitalangular momentum, therefore, is to change the optical vortex strength, rendering vortexannihilation (for negative values of m , i.e., left handed helical modes) or generation of non-integral vortices [33] possible. This is useful in generating fast switchable helical modes foroptical information encoding [9, 34]. Moreover, the z -component of the beam’s spin angularmomentum is [38] Z Z − i e z · E ⋆σlm × E σlm ρdρdϕ = − i e z · e ⋆σ × e σ = σ cos( γ cot θ ) . As expected [38], the spin component varies along the paraxial direction due to the inducedelasto-optic birefringence. Therefore, a screw dislocated medium exerts torques that changeboth the spin and orbital parts of the beam’s angular momentum. The spin change, beingattributed to the elasto-optic effect, is more pronounced for beams that propagate close tothe dislocation line. Note that, the z -component of the beam’s total angular momentum, j = σ cos( γ cot θ ) + m + βk reduces to the constant value σ + m in the absence of the screw dislocation ( β = 0), as itshould. III. PARAXIAL BEAM DYNAMICS IN THE DISLOCATED MEDIUM
The adiabatic variation of the wave number k , caused by the weak inhomogeneity, hasnegligible dynamical effect and was, therefore, ignored. However, the adiabatic variationof the beam direction (given by its wave vector k ) plays a geometric role with nontrivialconsequences for the beam dynamics. To incorporate this variation, we treat the coordinateframe used in the previous section as a local frame following the beam direction. As usual,the variation gives rise to a parallel transport law in the momentum space, defined by theBerry connection (gauge potential) A σσ ′ ( k ) = ( E σlm | − i ∇ k | E σ ′ lm ) . Using (2) to (4), we obtain A σσ ′ = [ jδ σσ ′ + iσ ( δ σσ ′ −
1) sin( γ cot θ )] cot θk e ϕ A = ( m + βk + ˆ Σ · h ) cot θk e ϕ (5)where ˆ Σ = (ˆ σ , ˆ σ , ˆ σ ) is the Pauli matrix vector and h ( θ ) = (0 , sin( γ cot θ ) , cos( γ cot θ )) . Equation (5) generalizes the result of [27] for dislocated (amorphous) media. The first twoterms are associated with the parallel transport of the beam’s transverse structure [27], whilethe last describes the parallel transport of the polarization vector along the beam. Note,in particular, the appearance of ˆ σ which shows the non-Abelian nature of the polarizationtransport, to be anticipated in view of the anisotropy [39]. The Berry curvature (gauge fieldstrength) associated with this connection is ( ˆ A × ˆ A = 0)ˆ B = ∇ k × ˆ A = − ( m + βk + ˆ Σ · D ) k k where D ( θ ) = d ( h cos θ ) /d (cos θ ).In the course of propagation, the paraxial beam evolves according to E σlm → e i ˆΘ E σlm ,where ˆΘ = Z C ˆ A · d k = ( m + βk )Θ + ˆ Σ · I (6)is the geometric Berry phase. Here C is the beam trajectory in momentum space, Θ = R C cos θdϕ is the Berry phase accumulated for σ = 1 and m = 0 in the absence of dislocationand I = R C h cos θdϕ = (0 , Im R C e iγ cot θ cos θdϕ, Re R C e iγ cot θ cos θdϕ ). The first term yieldsa rotation (through angle -Θ ) of the beam’s transverse structure about the direction ofpropagation, while the second yields a polarization precession about the direction of I . Sucha spin precession is characteristic of anisotropic media [39]. (In the absence of dislocation,the second term in (6) simply yields the phase factor e iσ Θ that leads to the well knownRytov rotation.)In view of the polarization evolution, the Berry curvature for a given beam is, therefore, B = ( e i ˆΘ e σ ) † ˆ B ( e i ˆΘ e σ ) = e † σ ˆ B ′ e σ where ˆ B ′ = e − i ˆΘ ˆ B e i ˆΘ = − ( m + βk + ˆ Σ ′ · D ) k k ˆ Σ ′ = exp( − i ˆ Σ · I ) ˆ Σ exp( i ˆ Σ · I ) . e − iG Ae iG = A − i [ G, A ] + ( − i )
2! [ G, [ G, A ]] + ( − i )
3! [ G, [ G, [ G, A ]]] + . . . and [ σ i , σ j ] = 2 iǫ ijk σ k , after some calculations we findˆ Σ ′ = ˆ Σ cos 2 I + I ( ˆ Σ · I ) I (1 − cos 2 I ) + ˆ Σ × I I sin 2 I where I = | I | = | R C e iγ cot θ cos θdϕ | . Henceˆ B ′ = − ( m + βk + ˆ Σ · D ′ ) k k where D ′ = D cos 2 I + I ( D · I ) I (1 − cos 2 I ) − D × I I sin 2 I. Therefore, B = − ( m + βk + σD ′ ) k k with D ′ = D cos 2 I + I ( D · I ) I (1 − cos 2 I )which reduces to the result of [27] in the absence of dislocation, namely, the field of amagnetic monopole of charge m + σ situated at the origin of the momentum space.The equations of motion of the beam in the presence of momentum space Berry curvaturehave been derived repeatedly for various particle beams (photons [14–16, 20–23], phonons[40–42] and electrons [24–26, 43]). We have˙ k = k ∇ ln n ˙ r = k k + B × ˙ k where dot denotes derivative with respect to the beam length. These differ from the standardray equations of the geometrical optics by the term involving the Berry curvature, whichyields the beam displacement δ r = − Z C ( m + βk + σD ′ ) k × d k k . (7)The displacement, which results in the splitting of beams with different polarizations and/ororbital angular momentums, is orthogonal to the beam direction and produces a current8cross the direction of propagation. This is the optical Hall effect in the dislocated mediumand generalizes the result of [27]. Because the beam’s singular vortex core can be observedwith great accuracy, measurement of the displacements is very reliable. Measuring δ r foropposite values of m and σ provides an indirect method for determining the Burgers vectoras well as the elasto-optic coefficients (or rather their difference p − p ) of the medium. IV. EXAMPLE: CIRCULAR WAVEGUIDE
We conclude by describing a simple situation that demonstrates the application of theresult (7) in determining the magnitude of the Burgers vector.Silicon, being transparent in the near infrared, has attracted extensive attention in op-toelectronic devices. Waveguides in silicon can serve as optical interconnects in siliconintegrated circuits, or distribute optical clock signals in a microprocessor. Fabrication ofsuch waveguides requires a core with a higher refractive index than that of crystalline sili-con. Amorphous silicon, with a higher refractive index in the near infrared, thus providesan interesting candidate for the core material [44]. Let us, therefore, consider an amorphoussilicon cylindrical waveguide with screw dislocation along its axis z . For a beam propagatinginside the waveguide along its circular cross section, the trajectory C in (7) is a circle ofradius k = 2 π/λ in the xy -plane ( θ = π/ I = 0 and D ′ = D = 1 so that (7)yields, for every one revolution of the beam, δ r = − [ b + λ ( m + σ )] e z where, of course, e z is the unit vector in the z -direction and b = 2 πβ is the magnitude ofthe Burgers vector. Therefore, for two beams of opposite polarization and orbital angularmomentum, we have a displacement (per unit revolution) in the negative z-direction givenby δz ± = b ± λ ( | m | + 1) . While δz + > δz − can take any value (including zero) by suitably tuning λ and | m | . Thusone can have the oppositely polarized beams emerging from the same end of the waveguide,with one lagging behind the other, or from opposite ends (see FIG. 1). Keeping the emergingdirection of the right-handed beam as the reference direction, the emerging direction of theleft handed beam changes at the threshold value λ ( | m | + 1) = b . With λ in the near infrared,9 IG. 1: Propagation of oppositely polarized beams inside a screw dislocated circular waveguide.(i) σ = 1 , m = | m | , (ii) σ = − , m = −| m | with λ ( | m | + 1) < b (left) and λ ( | m | + 1) > b (right). this provides an experimental method of determining Burgers vector magnitudes of the orderof micrometers. [1] B.E.A. Saleh and M.C. Teich, Fundamentals of photonics, Wiley, New York (1991).[2] L. Allen, M.W. Beijersbergen, R.J.C Spreeuw and J.P. Woerdman, Phys. Rev. A , 8185(1992).[3] L. Allen, M.J. Padgett and M. Babiker, Prog. Opt. , 291 (1999).[4] A. Maier, Nature (London) , 313 (2001).[5] R.A. Beth, Phys. Rev. , 115 (1936).[6] N.B. Simpson, L. Allen and M. J. Padgett, J. Mod. Opt. , 2485 (1996).[7] M.W. Beijersbergen, L. Allen, H.E.L.O. van der Veen and J.P. Woerdman, Opt. Commun. , 123 (1993).[8] B. Piccirillo and E. Santamato, Phys. Rev. E , 056613 (2004).[9] L. Marrucci, C. Manzo and D. Paparo, Phys. Rev. Lett. , 163905 (2006).[10] M.V. Berry, Proc. R. Soc. A , 45 (1984).
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