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Tractor Beams in the Rayleigh Limit

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Tractor Beams in the Rayleigh Limit 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 mini- mal 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 illus- trate 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 scat- ters 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 ! 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−i!t ^ 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) = − r × E(r; t) (1b) ! Achieving long-range retrograde transport requires a 3 X non-diffracting or weakly-diffracting mode. A plane wave = B (r) e−i!t ^ ; (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 ^j. 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 8 3 9 1 <X ∗ = to a second-order force of the form [3, 13] F e(r) = < pj(r; t)rEj (r; t) and (4a) 2 : ; k4 j=1 F (r) = − < fα α∗ E(r; t) × B∗(r; t)g : (6) em 12π c e m 8 3 9 0 1 <X = F (r) = < m (r; t)rB∗(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) = = fα α∗ g r 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 − < fα α∗ g u2(r)rφ (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 r u2(r) + α00 u2(r)rφ (r); (5) − < fαeαmg r · T (r) (7c) e 4 e j 2 e j j 12π0 j=1 j=1 k3 − = fα α∗ g r · 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 jαej 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 = fαeαmg. sent from Eq. (5). Even so, polarization-dependent pho- ∗ > Because < fαeαmg ∼ 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) = = fE∗(r; t) × E(r; t)g (9a) forces exerted on higher-order induced multipoles and 2! 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].
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