Universal, strong and long-ranged trapping by optical conveyors
Optical conveyors are active tractor beams that selectively transport illuminated objects either upstream or downstream along their axes. Formed by the coherent superposition of coaxial Bessel beams, an optical conveyor features an axial array of equally spaced intensity maxima that act as optical traps for small objects. We demonstrate through measurements on colloidal spheres and numerical calculations based on the generalized Lorenz-Mie theory that optical conveyors’ interferometric structure endows them with trapping characteristics far superior to those of conventional optical tweezers. Optical conveyors form substantially stiffer traps and can transport a wider variety of materials over a much longer axial range.
pacs:(350.4855) Optical tweezers or optical manipulation; (090.1760) Computer holography; (120.4880) Optomechanics.
I Introduction: optical conveyors
Long a staple of science fiction , tractor beams are traveling waves that transport illuminated objects back to their sources, opposite to the direction of energy flow. Such retrograde transport has been demonstrated in optical trapping experiments using collimated solenoid beams , interfering plane waves , and optical conveyors created with coaxial Bessel beams [4, 5, 6]. Among the more surprising proposals arising from these initial studies is that optical conveyors might act as universal tractor beams, transporting small objects at uniform speed regardless of their composition . This contrasts with the performance of conventional optical tweezers, which can only trap bright-seeking objects over a limited domain of size, shape and composition [7, 8, 9]. Another surprising suggestion is that optical conveyors might exceed the trapping stiffness even of diffraction-limited optical tweezers because their intensity actually vanishes at regular intervals along the optical axis .
Here, we report the results of a head-to-head comparison between the trapping characteristics of optical tweezers and optical conveyors involving both experimental measurements on micrometer-scale spheres and also numerical evaluations of optical forces using the generalized Lorenz-Mie theory. In confirming the superior performance of optical conveyors, these studies also enable us to establish the axial range over which optical conveyors can usefully transport material, and provide guidance for developing long-ranged tractor beams.
The electric field of a monochromatic optical conveyor of angular frequency , linearly polarized along and propagating along through a medium of refractive index , is the superposition of two Bessel beams [5, 6, 4],
each of which may be described as a conical superposition of plane waves ,
|is the polarization of a plane wave incident on the optical axis at polar angle and azimuthal angle , and where|
is the corresponding wave vector. In the paraxial approximation, which is appropriate for long-range tractor beams, Eq. (2) reduces to
where is related to the Bessel beam’s cone angle by . The approximate expression in Eq. (3) differs from the exact expression in Eq. (2) by terms involving higher-order Bessel functions[20, 23], which vanish on the optical axis. The numerical results developed in Sec. II.1 are based on Eq. (2). Analytical results developed in Sec. II.2 are more readily obtained from the approximation in Eq. (3).
The two Bessel beams comprising an optical conveyor share the same amplitude , frequency and polarization along , but differ in their relative phase, , and also in their axial wave numbers, , which are reduced from the plane wave value, , by the dimensionless factor . Here, is the speed of light in vacuum. The superposition is usefully characterized by the mean convergence factor and the difference . The upper limit of an optical conveyor’s range, , is set by the non-diffracting range of the most strongly converging Bessel beam ,
where is the radius of the beam’s aperture. Longer ranges can be achieved with values of approaching 1.
Figures 1(a) and 1(b) show the measured  three-dimensional intensity distribution of optical conveyors with equal values of and differing values of that were projected with the holographic optical trapping technique [12, 4]. These beams were powered by a linearly polarized diode-pumped solid-state laser (Coherent Verdi) operating at a vacuum wavelength of that was shaped by a liquid crystal spatial light modulator (Holoeye Pluto) before being projected with a microscope objective lens (Nikon Plan-Apo, , numerical aperture 1.4, oil immersion). The mode projected by this method is not the simple superposition of Bessel beams, but rather incorporates contributions from small range of axial wave numbers around . Equation (1) therefore should be considered an idealized model for the actual beam.
The intensity distribution, , has maxima at axial positions
each of which can act as an optical trap for a small object. Here, is the permittivity of space. Varying as a function of time moves these extrema, and thus conveys trapped objects along the beam. Increasing reduces the spacing between maxima, as shown in Figs. 1(a) and 1(b), and so provides control over the optical force profile.
II Trap stiffness
We previously have proposed  that optical conveyors should make better traps than conventional optical tweezers because their intensity vanishes altogether between maxima. The data in Figs. 1(c) and 1(d) demonstrate this to be true. Figure 1(c) shows the measured trajectory of a colloidal silica sphere diffusing through water in one of the potential energy wells of the static optical conveyor from Fig. 1(b). The optical conveyor has a peak intensity of , as measured with imaging photometry. Holographic characterization [13, 14, 15] reveals the sphere’s radius to be , and its refractive index to be . Holographic tracking [13, 16] yields the sphere’s position with 1 nm precision in-plane and 3 nm resolution axially [17, 15] at 16.7 ms intervals. Figure 1(d) shows the same sphere diffusing in a conventional optical tweezer with the same peak intensity projected by the same instrument.
In both cases, the trapped particle explores the optical force landscape under the influence of random thermal forces. The optical conveyor restricts the particle’s axial excursions to less than half the range of the optical tweezer, resulting in a nearly isotropic trajectory. This suggests that an optical conveyor makes a substantially stiffer trap, even though the 2.8:1 aspect ratio of the optical tweezer’s trajectory approaches the theoretical limit for a diffraction-limited Gaussian trap .
Modeling the traps as cylindrically symmetric harmonic potential energy wells,
we may estimate the transverse and axial trap stiffness, and from the particle’s trajectory using thermal fluctuation analysis [19, 21]. These measurements are performed for fixed values of the relative phase, , so that the optical traps do not move during the measurement. For the trajectory in Fig. 1(c), we obtain and , and an anisotropy of . The equivalent results for the optical tweezer are , and . The optical conveyor performs as well as the optical tweezer in the transverse direction and is nearly twice as stiff along the axis.
The performance of single beam optical traps typically is limited by their axial trapping ability. Figure 1(e) shows how the optical conveyor’s stiffness varies with for fixed , and compares this with the performance of a diffraction-limited optical tweezer. Highlighted points correspond to the data from Figs. 1(c) and 1(d). The optical conveyor’s axial stiffness exceeds that of a diffraction-limited optical tweezer by as much as a factor of two. This advantage is emphasized by the ratio between axial and transverse stiffness plotted in Fig. 1(f). Dashed horizontal dashed lines represent the measured and theoretical maximum performance of a Gaussian optical tweezer. Optical conveyors exceed this performance for , which corresponds to an axial period less than 6.7 .
II.1 Generalized Lorenz-Mie theory
Optical conveyors’ superior trapping performance is consistent with predictions of generalized Lorenz-Mie theory [22, 23, 6, 24, 25], which are plotted as continuous curves in Figs. 1(e) and 1(f). For these calculations, each Bessel beam is expanded as a series,
in the vector spherical harmonics, and , that constitute the natural basis for transverse electric (TE) and transverse magnetic (TM) waves, respectively . The expansion coefficients, and , have been reported previously for individual Bessel beams ; they can be superimposed according to Eq. (1) to create an optical conveyor with specified values of , and relative phase .
The associated expansion of the scattered field,
|is obtained by matching boundary conditions for the electric and magnetic fields at the sphere’s surface [23, 24, 25, 27, 28, 29]. The expansion coefficients,|
are related to the expansion coefficients of the incident field by the standard Mie coefficients, and , , which depend on the radius and refractive index of the sphere.
The superposed incident and scattered fields, , contribute to the Maxwell stress tensor, , whose integral over a closed surface provides an estimate for the optically-induced force on a sphere centered at :
where is the unit normal to the surface enclosing the sphere. In practice, the integral is computed directly from the combined expansion coefficients using established techniques [7, 8, 9, 30]. This calculation must also be repeated for each value of the relative phase .
A trap’s position, , for a given value of is calculated numerically as a solution of . The traps’s effective stiffness along then is calculated as
Results for the predicted transverse and axial stiffness are plotted in Fig. 1(e) for conveyor beams with and ranging up to 0.13, and for values of the sphere’s radius and refractive index obtained with holographic microscopy. The curves are scaled to a maximum intensity of , which is less than the measured value presumably because holographically projected Bessel beams are not uniformly bright. This discrepancy does not affect the computed ratio of stiffnesses in Fig. 1(f), which also agree well with experimental results, with no adjustable parameters.
II.2 Dipole approximation
Generalized Lorenz-Mie theory is useful for computing the forces on specific particles in particular traps. To assess trends in optical traps’ capabilities, we invoke the dipole approximation in which the time-averaged force [31, 32],
is proportional to the object’s polarizability . The polarizability of a dielectric sphere is related to its size and refractive index by the Clausius-Mossotti-Draine relation 
where is the Lorentz-Lorenz factor. Absorptivity increases the imaginary part of . Conductivity contributes an imaginary part to . The dipole approximation typically applies in the Rayleigh limit, for particles much smaller than the wavelength of light.
An object near the axis of an optical conveyor experiences an axial force
and a transverse force
where , and where we have assumed . These results are obtained by averaging over times long compared with the optical cycle but short compared with variations in . The particle is trapped where the force vanishes, which occurs at axial positions that are displaced from the intensity maxima by an amount that depends on the particle’s light-scattering properties,
which is strictly positive. An optical conveyor thus can trap and transport any dipolar particle, regardless of its light-scattering characteristics. This universal, material-independent trapping capability does not require feedback  or fine tuning of the beam’s properties . Decreasing the inter-trap separation by increasing enhances intensity gradients and thus increases trap stiffness.
Picking the largest possible value of to optimize axial trapping is not necessarily the best strategy. An optical conveyor’s transverse stiffness,
can vanish or even change sign as increases.
The dependence of trap stiffness on particle size predicted by Eqs. (12), (16) and (17) is plotted as dashed curves in Fig. 2 for silica spheres in an optical conveyor with and . Results from the dipole approximation agree well with the Lorenz-Mie results that are plotted as solid curves in Fig. 2, at least for . The Lorenz-Mie predictions, in turn, agree quantitatively with results from Fig. 1, which are plotted as discrete points.
The dipole approximation severely underestimates the optical conveyor’s stiffness for , and suggests that the particle studied in Fig. 1 would not have been stably trapped in the transverse direction. Rather than displaying a single crossover to instability, the generalized Lorenz-Mie result displays limited domains of instability for particles in particular size ranges. Results from the dipole approximation therefore are useful for establishing strict lower bounds on the performance of optical conveyors.
For example, requiring stable transverse trapping () according to Eq. (17) establishes an upper bound on and, through Eq. (4), a lower bound on the range over which an optical conveyor can transport small objects:
The practically accessible range of transport may be substantially greater than for particles larger than the wavelength of light.
II.3 Comparison with optical tweezers
Conventional optical tweezers do not share optical conveyors’ universal trapping ability. To show this, we model an optical tweezer as a focused Gaussian beam whose axial electric field profile is [36, 37]
where is the Rayleigh range of a Gaussian beam converging at distance from an aperture of radius , and where is the Gouy phase. Because we are interested in long-ranged axial transport, we assume . The axial component of the associated force,
forms a trap only for particles satisfying
This condition is most easily satisfied in strongly converging beams for which is small. The condition on is qualitatively consistent with numerical studies [7, 8, 9] which describe the difficulty of trapping high index or absorbing particles with laser tweezers.
III Transport range
The ultimate rate of an optical conveyor is limited by the non-diffracting range of the constituent Bessel beams. Equation (21), by contrast, establishes an inherent upper limit on an optical tweezer’s range
At ranges beyond , radiation pressure overwhelms trapping forces due to axial intensity gradients and ejects the particle. The upper bound of an optical tweezer’s range therefore is smaller than the lower bound of an optical conveyor’s. This means that optical conveyors can transport objects over substantially longer axial ranges, and at much lower numerical apertures. The price for long-ranged transport is paid in the strength of an optical conveyor’s traps.
Figure 3(a) is a volumetric reconstruction of an optical conveyor with and that was projected with a objective lens (Nikon Plan-Apo, numerical aperture 1.4 oil immersion). Holographic projection limits the conveyor’s range to , and thus suggests an effective numerical aperture of 0.43 given the convergence angle of . The holographically measured trajectory plotted in Fig. 3(b) shows this optical conveyor trapping and transporting a 1.5 -diameter silica sphere over 66 . The hologram at the bottom of this figure was recorded when the sphere was located at the position indicated by the sphere in Fig. 3(b). This measurement demonstrates that an optical conveyor can transport objects along its entire length, even at low numerical aperture.
Figure 3(c) shows how the traps’ stiffness falls off with range. The solid and dashed curves are Lorenz-Mie calculations of the axial trap stiffness of optical conveyors and optical tweezers, respectively, and are scaled for a peak intensity of . One set of curves is calculated for a 1.5 silica sphere, and agrees reasonably well with experimental results for the optical tweezer in Fig. 1(c), the optical conveyor in Fig. 1(d) and the optical conveyor in Fig. 3(a). The other set is calculated in the Rayleigh regime at , and agrees quantitatively with the dipole result from Eq. (20).
Despite the dipole prediction from Eq. (13), the force experienced by a large particle in an optical conveyor can be purely repulsive. Lorenz-Mie calculations reveal that additional radiation pressure due to off-axis scattering can overwhelm trapping forces due to axial intensity gradients for particles larger than the wavelength of light. Although a conveyor can be projected with a range exceeding this limit, it will not be able to transport large particles in the retrograde direction, and so will not act as a tractor beam.
The dependence of the trapping force on range for small particles shows no such crossover from stable trapping to repulsion. While the optical tweezer is inherently limited to the optical conveyor extends indefinitely, albeit with a stiffness that falls off as . The maximum trapping force in this regime falls off as . Both the force and the stiffness scale with the intensity of the beam, and therefore with the laser power. These considerations demonstrate that optical conveyors are viable candidates for long-ranged tractor beams, particularly for objects that are smaller than the wavelength of light.
Interference endows optical conveyors with trapping characteristics surpassing those of conventional single-beam optical traps. Most notably, optical conveyors are universal traps that can hold and transport small objects regardless of their light scattering properties. Under conditions where optical tweezers also are effective, moreover, optical conveyors make stiffer traps, particularly in the axial direction.
Optical conveyors act as universal traps because their intensity vanishes at points along the optical axis [5, 6, 4]. This ensures that retrograde forces arising from intensity gradients can counteract radiation pressure to hold illuminated objects in place. These traps can be positioned any placed along the propagation-invariant range of the conveyor by varying the relative phase of the constituent Bessel beams. As a consequence, optical conveyors can transport objects over substantially larger axial distances than conventional optical traps [5, 6, 4], which are inherently limited by radiation pressure.
Further improvements in performance almost certainly can be realized by appropriately structuring the intensity and phase gradients [38, 39, 40] along the projected beam. The range and trapping strength also should improve in optical conveyors created with radially polarized light [41, 42, 43]. These insights apply also to other interferometrically structured beams of light, such as solenoidal waves . Both as tractor beams, and also as ordinary optical traps, interferometrically structured beams of light offer clear benefits for optical micromanipulation.
This work was supported principally by the National Science Foundation through grant number DMR-1305875, and in part by a grant from NASA.
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