Optical forces and torques in nonuniform beams of light
Abstract.
The spin angular momentum in an elliptically polarized beam of light plays several noteworthy roles in optical traps. It contributes to the linear momentum density in a nonuniform beam, and thus to the radiation pressure exerted on illuminated objects. It can be converted into orbital angular momentum, and thus can exert torques even on optically isotropic objects. Its curl, moreover, contributes to both forces and torques without spintoorbit conversion. We demonstrate these effects experimentally by tracking colloidal spheres diffusing in elliptically polarized optical tweezers. Clusters of spheres circulate deterministically about the beam's axis. A single sphere, by contrast, undergoes stochastic Brownian vortex circulation that maps out the optical force field.
Optical forces arising from the polarization and polarization gradients in vector beams of light constitute a new frontier for optical micromanipulation. Linearly polarized light has been used to orient birefringent objects in conventional optical tweezers (1); (2); (3) and circular polarization has been used to make them rotate (1); (4); (6); (7). More recently, optically isotropic objects also have been observed to circulate in circularly polarized optical traps (8); (9); (10), through a process described as spintoorbit conversion (11); (10); (13); (14). Here, we present a general formulation of the linear and angular momentum densities in vector beams of light that clarifies how the amplitude, phase and polarization profiles contribute to the forces and torques that such beams exert on illuminated objects. This formulation reveals that the curl of the spin angular momentum can exert torques on illuminated objects without contributing to the light's orbital angular momentum, and that this effect dominates spintoorbit conversion in circularly polarized optical tweezers. Predicted properties of polarizationdependent optical forces are confirmed through observations of a previously unreported mode of Brownian vortex circulation for an isotropic sphere in elliptically polarized optical tweezers.
The vector potential describing a monochromatic beam of light of angular frequency may be written as
(1) 
where is the realvalued amplitude, is the realvalued phase and is the complexvalued polarization vector at position . This description is useful for practical applications because , and may be specified independently, for example using holographic techniques (15); (16); (3); (17). Poynting's theorem then yields the timeaveraged momentum density
(2) 
where is the permeability of the medium and is the speed of light in the medium. The momentum density gives rise to the radiation pressure that the light exerts on illuminated objects and may be expressed in terms of the experimentally accessible parameters as
(3) 
where is the intensity and where
(4) 
is the spin angular momentum density in a beam of light with local helicity
(5) 
The projection of onto the propagation direction is related to the Stokes parameters of the beam (18) by . It achieves extremal values of and for right and leftcircularly polarized light, respectively.
The momentum density described by Eq. (3) gives rise to the radiation pressure experienced by objects that absorb or scatter light. Identifying with the radiation pressure on a particle is most appropriate in the Rayleigh limit, when the particle's size is no greater than the wavelength of light. In this limit, the three terms in may be interpreted as distinct mechanisms by which a beam of light exerts forces on illuminated objects.
The first two terms in Eq. (3) constitute the familiar phasegradient contribution to the radiation pressure (16). In this context, the second term accounts for the independent phase profiles that may be imposed on the real and imaginary components of the polarization in an elliptically polarized beam. Phase gradients have been used to create threedimensional optical force landscapes (16), such as knotted force fields (19) and true tractor beams (20). They also account for the orbital angular momentum density
(6) 
carried by helical modes of light (21); (22). In this context, the polarizationdependent term in Eq. (6) vanishes identically in linearly polarized light, but manifests spintoorbit conversion in elliptically polarized beams.
The third term in Eq. (3) describes how variations in spin angular momentum contribute to the linear momentum density in nonuniform beams of light. This spincurl term encompasses forces due to spatiallyvarying elliptical polarization and also those due to intensity variations in elliptically polarized beams. Streamlines of naturally loop around extrema in the beam's intensity. Spincurl forces thus tend to make illuminated objects circulate in the plane transverse to the direction of propagation. Observations of colloidal spheres circulating in beams of light with spatiallyvarying elliptical polarization (23); (10); (13) consequently have been interpreted as evidence that the curl of the polarization contributes to the light's orbital angular momentum. Equation (6), however, makes clear that the spincurl contribution to does not contribute in any way to . For the same reason, observations of opticallyinduced circulation in uniformly circularlypolarized optical traps (9); (8) need not imply spintoorbit conversion.
To illustrate these point, we consider the forces exerted on an optically isotropic colloidal sphere by elliptically polarized optical tweezers. We model the trap as an Gaussian beam of wavenumber brought to a focus with convergence angle by a lens of focal length and numerical aperture in a medium of refractive index . The beam's initial polarization is
(7) 
with a corresponding incident helicity along . The focused beam's vector potential may be expressed in cylindrical coordinates with the RichardsWolf integral formulation (24); (25),
(8) 
as a FourierBessel expansion
(9) 
with expansion coefficients (25)
(10)  
(11)  
(12) 
Streamlines of in a rightcircularly polarized optical tweezer (, ) are shown spiraling around the optical axis in Fig. 1(a).
A slice through the beam in the transverse plane indicated in Fig. 1(a) reveals the azimuthal component to the transverse momentum density that is plotted in Fig. (1b). The transverse momentum density may be resolved into two contributions
(13) 
arising from the spintoorbit and spincurl contributions to , respectively:
(14)  
(15) 
Both are proportional to the helicity of the incident beam, . They do not, however, contribute equally to the transverse component of the radiation pressure. At the focus of the circularlypolarized optical tweezer, for example, 79% of the transverse momentum density is due to the spincurl term and only 21% from spintoorbit conversion. More generally, both and vanish in the paraxial approximation; there is no spintoorbit conversion in weakly focused beams. The spincurl contribution, by contrast, persists in the paraxial limit.
We probe the properties of spindependent optical forces by measuring their influence on the motion of micrometerscale colloidal spheres. Our system consists of diameter polystyrene (PS) spheres (Polysciences, Lot # 586632) dispersed in water and trapped in optical tweezers whose helicity is controlled with a quarterwave plate. The isotropic dielectric spheres absorb very little light directly. By scattering light, however, they experience radiation pressure proportional to the local momentum density. Our optical tweezer is powered by up to 4 of laser light at a vacuum wavelength of (Coherent Verdi 5W). The elliptically polarized beam is relayed with a dichroic mirror to the input pupil of an objective lens (Nikon Plan Apo, , NA 1.4), which focuses the light into a trap. We account for the mirror's influence on the polarization by measuring the beam's Stokes parameters in the input plane of the objective lens. The sample is imaged using the same lens in conventional brightfield illumination, which passes through the dichroic mirror to a video camera (NEC TI324AII). Digitally recorded video is analyzed with standard methods of digital video microscopy (26) to measure the trajectory of a probe particle with 10 resolution at intervals.
The trajectory plotted in Fig. 1(c) was obtained for one of seven spheres trapped against a glass surface by a rightcircularlypolarized optical tweezer () powered by 1.5 . The opticallyassembled cluster, shown inset into Fig. 1(d), spans the region of the beam indicated in Fig. 1(b), and thus rotates about the beam axis at a rate of roughly . The data in Fig. 1(d) confirm the prediction of Eqs. (14) and (Optical forces and torques in nonuniform beams of light) that the rotation rate varies linearly with the degree of circular polarization.
The colloidal cluster circulates deterministically in the elliptically polarized optical tweezer because it continuously scatters light in regions where is substantial. A single sphere diffusing in an elliptically polarized optical tweezer, by contrast, explores the entire force landscape presented by the light. This includes regions near the optical axis where is predicted to vanish. Figure 1(e) shows the measured trajectory of one such sphere in a rightcircularlypolarized trap () powered by 0.05 . Opticallyinduced circulation is not immediately obvious in the noisy trajectory, which is shaded to indicate the passage of time. It becomes evident when the trajectory is compiled into a timeaveraged estimate (27) for the steadystate probability current
(16) 
which is plotted in Fig. 1(f). Here is the number of discrete samples, and is the kernel of an adaptive density estimator (27) whose width varies with the sampling density. The symbols in Fig. 1(f) are shaded by the estimated probability density
(17) 
for finding the particle near . Together, and confirm the prediction of Eqs. (14) and (Optical forces and torques in nonuniform beams of light) that circulation vanishes on the optical axis where the particle's probability density is greatest.
Taking care to measure from the center of circulation, the mean circulation rate may be estimated as
(18) 
Equation (18) improves upon the graphical method for estimating introduced in Ref. (28) by making optimal use of discretely sampled data (27). Because the single particle spends most of its time in a curlfree region of the optical force field, its circulation rate is substantially smaller than in the deterministic case. Even so, the data in Fig. 1(g) again are consistent with the prediction that scales linearly with .
The single particle's stochastic motion differs qualitatively from the cluster's deterministic circulation. Were it not for random thermal forces, the isolated sphere would remain at mechanical equilibrium on the optical axis. Thermal forces enable it to explore the optical force landscape, where it is advected by the spindependent contribution to the radiation pressure. This system, therefore constitutes an example of a Brownian vortex (29); (30), a stochastic machine that uses noise to transduce work out of a static nonconservative force field.
Unlike previous experimental demonstrations of Brownian vortexes (28); (29) the conservative restoring force in this system is transverse to the nonconservative contributions. Consequently, the particle's radial excursions are described by the Boltzmann distribution (30) where is the thermal energy scale at absolute temperature and is the particle's potential energy in the trap. The solenoidal part of the radiation pressure (30)
(19) 
then advects into the probability current , where is the particle's scattering crosssection and is its mobility. The first term in Eq. (19) may be neglected in a beam such as a optical tweezer that carries little or no orbital angular momentum. The data in Fig. 1(f) thus map out the spincurl force in the transverse plane. Moreover, because the circulation direction is determined unambiguously by the curl of , this system is a practical realization of a socalled trivial Brownian vortex, which has been proposed (30) but not previously demonstrated.
Formulating the optical momentum density in terms of experimentally accessible parameters clarifies the nature and origin of the forces that can be applied to microscopic objects using the radiation pressure in beams of light. This formulation confirms previous reports of forces arising from phase gradients (16) and demonstrates that phasegradient forces act independently of the state of polarization. The spincurl mechanism unifies forces arising from the curl of the polarization and forces due to intensity gradients in elliptically polarized beams. Because they induce circulatory motion, spincurl forces are easily misinterpreted as evidence for spintoorbit conversion. The spincurl density, however, does not contribute to the orbital angular momentum of the light. Spintoorbit conversion, by contrast, removes spin angular momentum from a beam of light and transmutes it into orbital angular momentum (11). The present formulation clarifies this mechanism, and reveals that spintoorbit conversion has played a secondary role in previous reports of opticallyinduced circulation. Using Eq. (3) as a guide, all three mechanisms now may be optimally leveraged to improve optical micromanipulation and the performance of lightdriven machines.
We acknowledge helpful discussions with Giovanni Milione. This work was supported by the National Science Foundation principally through Grant Number DMR0855741, and in part by Grant Number DMR0922680.
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