Tractor Beams in the Rayleigh Limit

Aaron Yevick, David B. Ruﬀner, and David G. Grier Department of Physics and Center for Soft Matter Research, New York University, New York, NY 10003

A tractor beam is a travelling wave that transports illuminated objects back to its source, op- posite to the wave’s direction of propagation, along its entire length. The requisite retrograde force arises when an object scatters the wave’s momentum density downstream into the direction of propagation, and then recoils upstream by conservation of momentum. Achieving this condition imposes constraints on the structure of the wave, which we elucidate in the Rayleigh limit, when the wavelength exceeds the size of the object. Continuously propagation-invariant modes such Bessel beams do not satisfy these conditions at dipole order in the multipole expansion, and so cannot serve as general-purpose long-ranged tractor beams. Modes with discrete propagation invariance, however, can act as ﬁrst-order tractor beams. We demonstrate this by introducing a class of minimal solenoidal waves together with a set of design criteria that distinguish tractor beams that pull objects from repulsor beams that push them.

I. INTRODUCTION can be avoided by creating tractor beams from modes that display discrete propagation invariance. We illustrate this by introducing a family of minimal solenoidal Radiation pressure commonly is understood to push tractor-beam modes that exert retrograde forces as first- illuminated objects downstream along the direction of order photokinetic effects. light’s propagation. The recent realization [1–4] that ap- propriately structured traveling waves can pull objects upstream consequently has excited considerable interest. II. PHOTOKINETIC FORCES Retrograde forces arise when an illuminated object scatters more of the wave’s momentum into the downstream direction than otherwise would have been present [5–7]. The electric and magnetic fields in a monochromatic The object then recoils upstream to conserve momentum. beam of light at frequency ω may be projected into Carte- This happens, for example, when a small particle enters sian components, the diverging wave downstream of the focus in an optical 3 X trap, redirects light into the forward direction, and then E(r, t) = E (r) e−iωt ˆ and (1a) recoils toward the focus [5–7]. The trapping beam’s di- j j j=1 vergence, however, inherently limits how far it can pull i an object upstream. B(r, t) = − ∇ × E(r, t) (1b) ω Achieving long-range retrograde transport requires a 3 X non-diffracting or weakly-diffracting mode. A plane wave = B (r) e−iωt ˆ , (1c) cannot serve in this capacity because all of its momen- j j j=1 tum is directed along the axis of propagation. Remark- ably enough, some other non-diffracting traveling waves where Ej(r) and Bj(r) are the complex-valued electric [1–4,8] have the requisite characteristics. Such modes and magnetic field amplitudes along the Cartesian coor- have come to be known as tractor beams because of their dinate ˆj. For comparison with experimental implemen- resemblance to the popular science fiction trope [9]. tations, it is convenient to express these field components Previously identified tractor-beam modes were discov- in terms of real-valued amplitudes and phases, ered empirically [1,2,8]. Few have been demonstrated iφj (r) experimentally [2,8, 10], and most of these demonstra- Ej(r) = uj(r) e , (2) tions have relied on fine tuning of material properties [8, 10]. Here, we present an analytical theory for op- and similarly for Bj(r). Expressing optical forces in tical tractor beams that applies in the simplifying, yet terms of uj(r) and φj(r) yields insights that are useful practically important limit that the transported object for designing and optimizing beams of light for optical is smaller than the wavelength of light, the so-called micromanipulation, including tractor beams. Rayleigh regime. This theoretical framework expresses the forces experienced by an illuminated object in terms of the experimentally controllable amplitude and phase A. First-order photokinetic forces profiles of a vector mode of light. It reveals that the most commonly discussed tractor-beam mode [1,3,4, 11] Photokinetic forces arise from light’s ability to polarize is inherently undermined by its continuous propagation small objects. For simplicity and clarity, we will confine invariance; such beams can pull only under exceptional our discussion to optically isotropic objects whose dimen- circumstances, and then only weakly. These drawbacks sions are small enough relative to the wavelength of light 2 that they may be treated as point-like scatterers. In this also from interference between different types and orders so-called Rayleigh regime, the light’s leading-order effect of multipole scattering [3]. The multipole polarizabili- is to induce electric and magnetic dipole moments, ties that parametrize these higher-order terms generally become rapidly smaller with increasing multipole order, p(r, t) = αeE(r, t) and (3a) particularly for objects that are smaller than the wave- m(r, t) = αmB(r, t), (3b) length of light. For Rayleigh particles, the leading-order corrections to the first-order dipole forces arise from the respectively, where αe and αm are the object’s electric interference between the scattered electric and magnetic and magnetic dipole polarizabilities. dipole fields, and from the induced electric quadrupole The induced dipole moments described by Eq. (3) ex- moment. The dipole-interference term in particular pro- perience time-averaged forces in gradients of the electric vides a mechanism for creating propagation-invariant [12] and magnetic [13] fields, tractor beams. Dipole scattering by a small isotropic particle gives rise  3  1 X ∗  to a second-order force of the form [3, 13] F e(r) = < pj(r, t)∇Ej (r, t) and (4a) 2   k4 j=1 F (r) = − < {α α∗ E(r, t) × B∗(r, t)} . (6) em 12π c e m  3  0 1 X  F (r) = < m (r, t)∇B∗(r, t) . (4b) Expressing this in terms of the fields’ amplitudes and m 2 j j j=1  phases yields four distinctive contributions, 3 k3 X Although Eq. (4) completely describes optical dipole F (r) = = {α α∗ } ∇ u2(r) (7a) em 24π e m j forces, how to control these forces is more clearly revealed 0 j=1 when they are presented in terms of the amplitudes and 3 k3 X the phase of the fields. Expressed in these terms, the − < {α α∗ } u2(r)∇φ (r) (7b) electric dipole force has the form [14, 15] 12π e m j j 0 j=1 3 3 k3 1 X 1 X ∗ (a) F (r) = α0 ∇ u2(r) + α00 u2(r)∇φ (r), (5) − < {αeαm} ∇ · T (r) (7c) e 4 e j 2 e j j 12π0 j=1 j=1 k3 − = {α α∗ } ∇ · T(s)(r), (7d) 0 00 e m where αe and αe are the real and imaginary parts of αe, 12π0 respectively. where is the permeability of the medium and where we Equation (5) and the analogous expression for the have introduced the antisymmetric (a) and symmetric magnetic dipole force constitute the first-order photoki- (s) components of the Maxwell stress tensor, netic forces. The first term on the right-hand side of (a) Eq. (5) describes the manifestly conservative intensity- Tij (r) = ui(r)uj(r) sin φj(r) − φi(r) and (8a) gradient force responsible for trapping by single-beam T (s)(r) = u (r)u (r) cosφ (r) − φ (r). (8b) optical traps such as optical tweezers [5]. The second de- ij i j j i scribes a non-conservative contribution to radiation pres- Equation (7) is the complete result for the dipole- sure that is directed by gradients of the phase [14, 16, 17]. interference contribution, including both electric and Recognizing that these phase gradients define the local magnetic contributions. The force it describes tends to 00 00 wave vector [18] and that αe and αm are positive for be weaker than the first-order dipole force described by 3 conventional materials, the phase-gradient term tends to Eq. (5) because k |αe| 1 in the Rayleigh regime. Even drive objects downstream along the beam of light. so, each term in Eq. (7) has interesting implications for Light’s polarization plays remarkably little role in first- optical micromanipulation. order photokinetic effects. This can be appreciated be- The intensity-gradient term in Eq. (7a) either strength- cause phase differences among the Cartesian components ens or weakens the first-order trapping force depending ∗ that determine the degree of circular polarization are ab- on material parameters that affect the sign of = {αeαm}. sent from Eq. (5). Even so, polarization-dependent pho- ∗ > Because < {αeαm} ∼ 0 for conventional materials, the tokinetic effects are well documented in the experimen- phase-gradient term in Eq. (7b) describes a retrograde tal literature [19–21] and emerge naturally in numerical force that could provide the basis for a tractor beam studies [19, 22–24]. Such effects can be attributed to were it not generally weaker than the first-order phase- higher-order photokinetic forces. gradient term from Eq. (5). Equations (7c) and (7d) describe optical forces with properties distinct from those of dipole forces, and thus B. Higher-order photokinetic forces create new avenues for optical micromanipulation. We can interpret Eq. (7c) by noting that Higher-order photokinetic effects result both from s(r) = = {E∗(r, t) × E(r, t)} (9a) forces exerted on higher-order induced multipoles and 2ω 3 is the time-averaged spin angular momentum density car- a tractor beam is the Bessel beam [34–37], which origi- ried by the beam of light [18, 25–27]. From this, we ob- nally was proposed for this application in the context of tain the general result, acoustic waves [1]. ω Bessel beams can be generated from the longitudinal ∇ · T(a)(r) = ∇ × s(r), (9b) vector potential which identifies Eq. (7c) as a force arising from the curl Am,α(r) = um,α Am,α(r)ˆz, where (12a) of the spin angular momentum density in a nonuniform imθ ikz cos α Am,α(r) = Jm(kr sin α) e e (12b) beam of light [21]. Spin-curl forces have been a subject of active research [15, 21, 27–31] and here may be iden- is the scalar cylindrical harmonic that is characterized tified as a second-order photokinetic effect. They may by convergence angle α and an integer winding number be directed along the axis of propagation, and thus can m. The overall amplitude, um,α, sets the intensity of the contribute to the retrograde forces in a tractor beam. beam. Equation (12) is expressed in cylindrical coordi- The remaining term in Eq. (7d) describes a comple- nates, r = (ρ, θ, z), aligned with the propagation direc- mentary curl-free force field directed along the local axis tion,z ˆ. The fields in a Bessel beam may be obtained of linear polarization. This term also can give rise to from Am,α(r) as [34, 35] a retrograde force in longitudinally polarized traveling waves of the kind observed in experiments on crossed ETE (r, t) = ∇ × A (r) e−iωt, (13) plane waves [8]. m,α m,α Induced-quadrupole contributions to the optical force with the magnetic field following from Eq. (1b). Equa- also arise at second order in the multipole expansion tion (13) describes a transverse electric (TE) Bessel beam and so may be comparable in strength to the dipole- that is equivalent to the form presented in [3]. The cor- interference force. This contribution has the form [3] responding transverse magnetic (TM) mode can be ob-   tained by exchanging the electric and magnetic fields in 1 X  Eq. (13). Bessel beams with more general polarizations F (r) = < Q (r, t)∇∂ E∗(r, t) , (10) q jk j k may be obtained as linear superpositions of these TE and 4   j,k TM modes. The range of non-diffracting propagation is where the induced quadrupole moment is proportional to csc α, and so is longer for beams with smaller convergence angles, α [37, 38]. 1 Because the Bessel beams form a complete orthogonal Q (r, t) = γ [∂ E (r, t) + ∂ E (r, t)] (11) jk 2 e j k k j basis, any propagation-invariant mode can be expressed as a linear superposition of Bessel beams. Those super- for a neutral particle with scalar quadrupole suscepti- positions comprising Bessel beams with the same con- bility, γe. The induced-quadrupole force is symmetric vergence angle, α, display continuous propagation invari- under exchange of the Cartesian components of E(r, t). ance. Any such non-diffracting beam lacks axial intensity Consequently, it does not depend on the spin angular 2 gradients as a matter of definition, so that ∂zu (r) = 0. momentum of the light, and does not contribute to spin- The remaining phase-gradient contribution to the axial dependent photokinetic effects. dipole force is purely repulsive for any material with Second-order forces are weaker than first-order forces 00 00 αe > 0 and αm > 0. From this, we conclude that no for Rayleigh particles. Higher multipole contributions continuously propagation-invariant beam of light will be weaker still. We will focus, therefore, on iden- can serve as a general-purpose first-order tractor tifying characteristics of propagation-invariant modes of beam. light that are useful for creating first-order tractor beams. Non-diffracting modes nonetheless may act as tractor beams for particles whose scattering properties favor retrograde forces from higher-order contributions [3]. To III. PROPAGATION-INVARIANT TRACTOR illustrate this, we consider the photokinetic force exerted BEAMS by TE Bessel beams and contrast this with the performance of TM Bessel beams. The axial component of the Achieving long-ranged retrograde transport without force then has the form active intervention [32] requires a tractor beam to be based on a weakly-diffracting or propagation-invariant 1 J 2 (kr sin α) F (r) · zˆ = k u2 m cos α f(α, x), (14) mode. Ideally, such a mode would carry retrograde mo- 2 m,α r2 mentum [33] along its entire length, and thus exert retrograde radiation pressure. It is sufficient, however, for the where x = kr sin α and where f(α, x) is a polarization- axial component of the beam’s momentum density to be dependent factor describing the particle’s coupling to the smaller than the plane-wave limit. In that case, an illumi- light. The axial force vanishes in the strongly converging nated particle can scatter additional momentum down- limit because the Bessel beam becomes a standing wave stream, and thus recoil upstream. The archetype for such at α = π/2. 4

The coupling factor for a TE-polarized Bessel beam IV. FIRST-ORDER SOLENOIDAL TRACTOR due to dipole and dipole-interference contributions is BEAMS

3 Relaxing the condition of continuous propagation in- TE 2 00 00 k ∗ f (α, x) = gm(x) αe + αm − < {αeαm} variance in favor of discrete propagation invariance, 6π0 2 2 2 00 2 2 + sin α [x − gm(x)] αm, (15) u (r, θ, z + ∆z) = u (r, θ, z), (19) creates opportunities for strong, long-ranged, general- with the mode-dependent geometric factor purpose tractor beams that operate at first order. Axial intensity gradients in a discretely propagation invariant 0 2 2 2 2 Jm (x) beam can establish retrograde forces that counteract re- gm(x) = m + x 2 . (16) Jm (x) pulsive phase-gradient forces. This is the principle by which an optical conveyor establishes an axial array of The equivalent expression for TM modes is obtained by traps [32]. Achieving continuous retrograde transport in such a beam requires a continuous path through the transposing αe and αm. beam along which the net optical force acts in the retro- The electric and magnetic quadrupole contributions grade direction. Optical conveyors do not have this prop- to the axial force are strictly repulsive, with the elec- erty, but waves displaying screw invariance can. These tric quadrupole repulsion being somewhat weaker in TM- include spiral waves [39] and solenoidal beams [2]. Al- polarized modes. Quadrupole contributions therefore do though solenoidal waves have been described and their not help to create a second-order tractor beam. efficacy as tractor beams has been demonstrated experi- Considering, for the moment, only those contributions mentally [2], the nature of their force fields has not been to the axial force arising from dipole terms, the direction explored previously. Applying the theory of photokinetic TE of motion depends on the sign of f (α, x), and therefore effects reveals that solenoidal modes can act as first-order on the particle’s position within the beam. Retrograde tractor beams. forces appear at positions x that satisfy Discretely propagation-invariant beams can be created by superposing two mutually coherent Bessel beams with g2 (x) sin2 α α00 different convergence angles. Because we are interested in m < m . (17) 2 k3 ∗ 00 2 00 long-range transport, we choose component Bessel beams x < {αeαm} − αe − cos α αm 6π0 with small convergence angles, which may be described in the paraxial approximation as TEM modes with uniform The absence of solutions in the limit of small α demon- polarization, ˆ. The superposition then has the form strates that weakly converging Bessel beams generally act as repulsors. More strongly converging modes can E(r, t) = [um,αAm,α(r, t) + un,βAn,β(r, t)]. ˆ (20) act as tractor beams if the radial component of the force vanishes at positions that satisfy Eq. (17). This occurs, The resulting axial interference pattern is spatially pe- for example, in TE modes with m = ±1 along the princi- riodic, as required, and stationary. Superposing modes pal intensity maximum at r = 0. Such beams can stably with the same winding number creates optical conveyors pull an object upstream if the convergence angle satisfies [32, 40, 41]. Superposing modes with different winding numbers creates a solenoidal wave. We will order the α00 k3 α0 contributions with |m| > |n|. The original description sin2 α > 1 + e − α00 + α0 m . (18) α00 6π e e α00 of solenoidal waves [2] involved superpositions of large m 0 m numbers of Bessel modes. Being composed of just two modes, the solenoidal waves presented here may be con- Because Eq. (17) neglects the quadrupole contribution, sidered minimal realizations of the genus. it constitutes a necessary condition for a Bessel beam to We ensure that a solenoidal mode traps particles at a act as a second-order tractor beam, but is not a sufficient radius r = R by choosing the convergence angles α and condition. β so that extremum p of mode m overlaps extremum q of 2 Numerical studies involving all orders of multipole mode n. This condition ensures ∂r |E(r, t)| = 0 along scattering confirm that Bessel beams can act as tractor r = R. Equation (5) then ensures thatr ˆ · F e(r)|r=R = 0. beams for some objects that are larger than the wave- The intensity maximum at r = R acts as a potential length of light [3,4, 11]. The present considerations energy well along the radial direction for light-seeking 0 demonstrate that the necessary pulling force does not particles with αe > 0. It is achieved by setting arise at first order, but rather emerges from a competition between first-order repulsion and higher-order ret- sin α sin β 1 0 = 0 = , (21) rograde forces. Because such tractor beams require com- jm,p jn,q kR paratively steep convergence angles, moreover, they do 0 0 not lend themselves to long-range transport. where x = jm,p is the p-th root of Jm(x) = 0. 5

integer winding numbers. The 2`-fold sign changes in u(r), as defined in Eq. (23b), eliminate the transverse vortices that otherwise would form in a helical beam with non-integer winding number [43, 44]. The transverse intensity distribution is proportional to u2(r) and has a number of maxima equal to |m − n|. These maxima spiral once aroundz ˆ each time the axial position z advances by ∆z. The spiral is right-handed for positive values of ∆z. For simplicity, we will consider this wave’s influence on dielectric Rayleigh particles, for which the electric dipole force described by Eq. (5) dominates the magnetic dipole force and all other multipole contributions. We then define the locus, rs(z) = (R, θs, z), of the solenoidal trap to be the set of points along which intensity-gradient forces compensate the axial component of the radiation pressure, Fz(rs) = 0. This condition is satisfied along the spiral curve z θ (z) = 2π + θ , (24a) s ∆z 0 FIG. 1. (color online) (a) Calculated intensity distribution of a right-handed solenoidal tractor beam with a vacuum wave- with angular offset, length of 532 nm and design parameters m = −5 and n = −4, 00 α = 0.418 rad and β = 0.627 rad, propagating along +ˆz 2 −1 αe cos α + cos β θ0 = tan 0 . (24b) through water. (b) Computed trajectory of a 50 nm-diameter m − n αe cos α − cos β silica sphere moving along the tractor beam in (a). The color bar indicates the local intensity, u2(r). The path that a particular object takes through the beam runs parallel to this. Such solutions exist for small objects regardless of the light-scattering properties en- Interference is optimized if the two components con- coded in αe. Whether or not these solutions act as traps tribute equally to the amplitude, u0, at r = R. This depends on whether or not the particle also is stably establishes a relationship between the amplitudes trapped in the radial direction. Small dielectric particles are stably trapped along the bright locus described 0 0 1 by Eq. (23); reflecting and absorbing particles might not uα,mJm(j ) = uβ,nJn(j ) = u0. (22) m,p n,q 2 be. A solenoidal trap acts as a tractor beam if the az- The resulting superposition has the form imuthal component of the radiation pressure, Fθ(rs), E(r, t) = u(r) eiφ(r)e−iωtx,ˆ (23a) drives the trapped particle into a region where the axial component of the force, Fz(rs), points in the retrograde direction, F (r ) ∂ F (r)| < 0. This condition is with amplitude and phase profiles along r = R of θ s θ z r=rs satisfied if m − n h z i u(r)| = u cos θ − 2π and (23b) r=R 0 2 ∆z n cos α > m cos β. (25) cos α + cos β Flipping the sense of this inequality transforms a tractor φ(r)|r=R = `θ + kz. (23c) 2 beam into a repulsor that pushes objects downstream This is a helical mode that carries orbital angular mo- rather than pulling them upstream. mentum [42] proportional to its winding number, Equation (25) is the principal design requirement for creating a general-purpose first-order tractor beam. m + n With this constraint, Eq. (21) then requires q > p. Min- ` = . (23d) 2 imizing α and β to maximize pulling range then requires q = p + 1. Maximizing the pulling force by maximizing The axial pitch of the solenoid’s spiraling intensity dis- the intensity along the solenoid’s principal spiral then tribution is yields p = 1 and q = 2 as optimal choices. 2π m − n Equations (21), (22) and (25) constitute previously ∆z = . (23e) k cos α − cos β lacking design criteria for general-purpose first-order tractor beams. As described, they establish conditions Interestingly, solenoidal waves with half-integer wind- on the individual Bessel components’ convergence angles, ing numbers do not differ qualitatively from those with α and β, and winding numbers, m and n. Equivalently, 6 these conditions may be expressed in terms of the radius, cost of additional complexity. For some applications, R, pitch, ∆z, winding number, `, and non-diffracting solenoidal waves based on other helical modes, such as range of the desired tractor beam. Previously, functional Laguerre-Gaussian beams, may be preferable. solutions had to be sought numerically [2]. Beyond its immediate application to designing trac- The example of a solenoidal tractor beam presented tor beams, the theory of photokinetic effects offers addi- in Fig.1(a) was designed according to these princi- tional benefits for optical micromanipulation. Express- ples. Figure1(b) shows the computed trajectory of a ing optical forces in terms of experimentally accessible 50 nm-diameter silica sphere moving through water in quantities helps to elucidate the origin of experimen- its force field. Because solenoidal tractor beams oper- tally observed phenomena. The polarization-dependent ate at dipole order, they should be inherently stronger contribution from Eq. (7d), for example, explains why than higher-order tractor beams based on continuously linearly polarized optical tweezers exert anisotropic in- invariant modes. plane forces [16, 23]. The circulation of optically isotropic Accounting for the magnetic field’s contribution to the spheres in circularly polarized optical tweezers similarly optical force shifts θ0 by an amount that depends on may be identified with the spin-curl force from Eq. (7c). the relative magnitudes of αe and αm. The net effect is Polarization-dependent size selectivity in the force cre- to strengthen the pulling force, at least for conventional ated by crossed plane waves [8] can be explained as aris- materials. ing from competition between the phase-gradient terms Polarization also can enhance a solenoidal tractor in Eq. (5) and Eq. (7b). Such observations provide guid- beam’s performance. For example, Eq. (7c) shows that ance for designing and optimizing new modes of optical a solenoid beam created with right-circularly polarized micromanipulation, including tractor beams. light has a transverse spin-curl force of the form The development presented here directly addresses the forces experienced by objects that are substantially 3 k 0 0 00 00 rˆ ˆ 2 smaller than the wavelength of light. For larger particles, F sc(r) = (αeαm + αe αm) ∂θ − θ∂r u (r). 24π0 r the forces described by Eqs. (5) and (7) are augmented (26) and possibly dominated by higher-order multipole contri- When applied to a right-handed solenoidal tractor beam, butions. For Rayleigh particles, we expect the first-order this increases Fθ(r) for r < R and decreases it for r > forces described by Eq. (5) to be substantially stronger R. By moderating the trapped particle’s speed in this than any higher-order contributions, including those de- way, F sc(r) contributes to a kinematic trap that tends scribed by Eq. (7). Applying these considerations to the to localize an object near r = R as it is transported by particular case of tractor beams, we conclude that Bessel the beam. beams generally push small objects whereas appropri- ately designed solenoidal waves can pull them.

V. DISCUSSION ACKNOWLEDGMENTS The two-component solenoidal tractor beams described here illustrate the process by which ﬁrst-order This work was supported by the National Science tractor beams may be designed and optimized using the Foundation under Grant No. DMR-1305875. AY ac- theory of photokinetic eﬀects. More sophisticated super- knowledges support from a NASA Space Technology Re- positions [2] may have superior properties, but at the search Fellowship under Award No. NNX15AQ40H.

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