Universal, strong and long-ranged trapping by optical conveyors

David B. Ruffner; David G. Grier

doi:10.1364/OE.22.026834

1. Introduction: optical conveyors

Long a staple of science fiction [1], 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 [2], interfering plane waves [3], and optical conveyors created with coaxial Bessel beams [4–6]. 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 [4]. 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 [7–9]. 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 [4].

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 ω, linearly polarized along x̂ and propagating along ẑ through a medium of refractive index n_m, is the superposition of two Bessel beams [4–6],

E (r, t) = \frac{1}{2} E_{0} e^{- i ω t} [b_{1} (k r) + e^{i φ (t)} b_{2} (k r)],

each of which may be described as a conical superposition of plane waves [6],

b_{j} (k r) = \int_{0}^{2 π} \hat{ε} (θ_{j}, ϕ) e^{i k (θ_{j}, ϕ) \cdot r} d ϕ,

where

\hat{ε} (θ, ϕ) = cos ϕ \hat{θ} + sin ϕ \hat{ϕ}

is the polarization of a plane wave incident on the optical axis at polar angle θ and azimuthal angle ϕ, and where

k (θ, ϕ) = k (sin θ cos ϕ \hat{x} + sin θ sin ϕ \hat{y} + cos θ \hat{z})

is the corresponding wave vector. In the paraxial approximation, which is appropriate for long-range tractor beams, Eq. (2) reduces to

b_{j} (k r) \approx J_{0} (\sqrt{1 - η_{j}^{2}} k r) e^{i η_{j} k z} \hat{x},

where η_j is related to the Bessel beam’s cone angle by η_j = cosθ_j. 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. 2.1 are based on Eq. (2). Analytical results developed in Sec. 2.2 are more readily obtained from the approximation in Eq. (3).

The two Bessel beams comprising an optical conveyor share the same amplitude E₀, frequency ω and polarization along x̂, but differ in their relative phase, φ(t), and also in their axial wave numbers, η_jk, which are reduced from the plane wave value, k = n_mω/c, by the dimensionless factor η_j ∈ (0, 1]. Here, c is the speed of light in vacuum. The superposition is usefully characterized by the mean convergence factor η = (η₁ + η₂)/2 and the difference Δη = |η₁ − η₂|. The upper limit of an optical conveyor’s range, R, is set by the non-diffracting range of the most strongly converging Bessel beam [10],

R \leq A cot θ_{1} = A \frac{η_{1}}{\sqrt{1 - η_{1}^{2}}}

where A is the radius of the beam’s aperture. Longer ranges can be achieved with values of η approaching 1.

Figures 1(a) and 1(b) show the measured [11] three-dimensional intensity distribution of optical conveyors with equal values of η and differing values of Δη that were projected with the holographic optical trapping technique [4, 12]. These beams were powered by a linearly polarized diode-pumped solid-state laser (Coherent Verdi) operating at a vacuum wavelength of λ = 532nm that was shaped by a liquid crystal spatial light modulator (Holoeye Pluto) before being projected with a microscope objective lens (Nikon Plan-Apo, 100×, 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 η_jk [4]. Equation (1) therefore should be considered an idealized model for the actual beam.

Fig. 1 (a) Experimental reconstruction of an optical conveyor η = 0.8, Δη = 0.04, and (b) η = 0.8, Δη = 0.086. (c) Measured trajectory of a 1.5 μm-diameter silica sphere trapped in one of the intensity maxima in (b). (d) Trajectory of the same particle trapped in a conventional optical tweezer projected by the same instrument with the same peak intensity. (e) Measured transverse and axial stiffness as a function Δη. (f) Stiffness ratio, k_z/k_r, as a function of Δη. Solid curves in (e) and (f) represent predictions of the Lorenz-Mie theory. Shaded bands represent uncertainty in the measured size and refractive index of the trapped silica sphere. Highlighted plot symbols represent results from (c) and (d). Dotted horizontal lines represent the measured performance of the optical tweezer. The dashed horizontal line in (f) shows the theoretical limit for optical tweezer performance. The shaded region above this line represents the optical conveyor’s superior performance for optical microma-nipulation.

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The intensity distribution, $I (r, t) = \frac{1}{2} n_{m} ε_{0} c {| E (r, t) |}^{2}$ , has maxima at axial positions

z_{n} (t) = \frac{2 π n + φ (t)}{Δ η k},

each of which can act as an optical trap for a small object. Here, ε₀ is the permittivity of space. Varying φ(t) as a function of time moves these extrema, and thus conveys trapped objects along the beam. Increasing Δη reduces the spacing between maxima, as shown in Figs. 1(a) and 1(b), and so provides control over the optical force profile.

2. Trap stiffness

We previously have proposed [4] 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 79 mW/μm², as measured with imaging photometry. Holographic characterization [13–15] reveals the sphere’s radius to be a_p = 0.730 ± 0.005μm, and its refractive index to be n_p = 1.424 ± 0.005. Holographic tracking [13, 16] yields the sphere’s position with 1 nm precision in-plane and 3 nm resolution axially [15, 17] 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 [18].

Modeling the traps as cylindrically symmetric harmonic potential energy wells,

U (r) = \frac{1}{2} k_{r} r^{2} + \frac{1}{2} k_{z} z^{2},

we may estimate the transverse and axial trap stiffness, k_r and k_z from the particle’s trajectory using thermal fluctuation analysis [19, 21]. These measurements are performed for fixed values of the relative phase, φ(t), so that the optical traps do not move during the measurement. For the trajectory in Fig. 1(c), we obtain k_r = 2.2 ± 0.1pN/μm and k_z = 0.89 ± 0.03pN/μm, and an anisotropy of

\sqrt{k_{r} / k_{z}} = 1.57 \pm 0.04

. The equivalent results for the optical tweezer are k_r = 2.8 ± 0.4pN/μm, k_z = 0.41 ± 0.08pN/μm and

\sqrt{k_{r} / k_{z}} = 2.6 \pm 0.4

. 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 Δη for fixed η, 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 Δη > 0.06, which corresponds to an axial period less than 6.7 μm.

2.1. Generalized Lorenz-Mie theory

Optical conveyors’ superior trapping performance is consistent with predictions of generalized Lorenz-Mie theory [6,22–25], which are plotted as continuous curves in Figs. 1(e) and 1(f). For these calculations, each Bessel beam is expanded as a series,

b_{j} (r) = \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} [a_{m n} (θ_{j}) M_{n m}^{(1)} (k r) + b_{m n} (θ_{j}) N_{n m}^{(1)} (k r)],

in the vector spherical harmonics,

M_{n m}^{(1)} (k r)

and

N_{n m}^{(1)} (k r)

, that constitute the natural basis for transverse electric (TE) and transverse magnetic (TM) waves, respectively [26]. The expansion coefficients, a_mn(θ_j) and b_mn(θ_j), have been reported previously for individual Bessel beams [22]; they can be superimposed according to Eq. (1) to create an optical conveyor with specified values of η, Δη and relative phase φ(t).

The associated expansion of the scattered field,

\begin{array}{l} E_{s} (r, t) = E_{0} e^{- i ω t} \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} {[r_{m n} (θ_{1}) + e^{i φ (t)} r_{m n} (θ_{2})] M_{n m}^{(3)} (k r) \\ + [s_{m n} (θ_{1}) + e^{i φ (t)} s_{m n} (θ_{2})] N_{n m}^{(3)} (k r)}, \end{array}

is obtained by matching boundary conditions for the electric and magnetic fields at the sphere’s surface [23–25, 27–29]. The expansion coefficients,

r_{m n} (θ_{j}) = - a_{n} a_{m n} (θ_{j}) and

s_{m n} (θ_{j}) = - b_{n} b_{m n} (θ_{j}),

are related to the expansion coefficients of the incident field by the standard Mie coefficients, a_n and b_n, [27], which depend on the radius and refractive index of the sphere.

The superposed incident and scattered fields, E(r, t) + E_s(r, t), contribute to the Maxwell stress tensor, T(r, t), whose integral over a closed surface provides an estimate for the optically-induced force on a sphere centered at r:

F (r, t) = \oint_{S} \hat{n} \cdot T (r^{'}, t) d r^{'},

where n̂ is the unit normal to the surface S enclosing the sphere. In practice, the integral is computed directly from the combined expansion coefficients using established techniques [7–9, 30]. This calculation must also be repeated for each value of the relative phase φ(t).

A trap’s position, r₀(t), for a given value of φ(t) is calculated numerically as a solution of F(r₀(t), t) = 0. The traps’s effective stiffness along r̂_ν then is calculated as

k_{ν} = - \partial_{ν} {F_{ν} (r, t) |}_{r = r_{0} (t)} .

Results for the predicted transverse and axial stiffness are plotted in Fig. 1(e) for conveyor beams with η = 0.8 and Δη 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 41 mW/μm², 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.

2.2. 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],

F (r, t) = \frac{1}{2} ℜ {α_{e} \sum_{ν = 1}^{3} E_{ν} (r, t) \nabla E_{ν}^{*} (r, t)},

is proportional to the object’s polarizability α_e = α′_e + iα″_e. The polarizability of a dielectric sphere is related to its size and refractive index by the Clausius-Mossotti-Draine relation [33]

α_{e} = \frac{4 π ε_{0} n_{m}^{2} K a_{p}^{3}}{1 - i \frac{2}{3} K k^{3} a_{p}^{3}},

where

K = (n_{p}^{2} - n_{m}^{2}) / (n_{p}^{2} + 2 n_{m}^{2})

is the Lorentz-Lorenz factor. Absorptivity increases the imaginary part of α_e. Conductivity contributes an imaginary part to K. 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

\frac{F_{z} (r, t)}{E_{0}^{2}} \approx - \frac{1}{4} α_{e}^{'} Δ η sin (Φ (z, t)) + α_{e}^{″} η {cos}^{2} (\frac{1}{2} Φ (z, t))

and a transverse force

\frac{F_{r} (r, t)}{E_{0}^{2}} \approx α_{e}^{'} \frac{1}{2} (1 - η^{2} - \frac{1}{4} Δ η^{2}) k r {cos}^{2} (\frac{1}{2} Φ (z, t)) + \frac{1}{4} α_{e}^{″} η Δ η k r sin (Φ (z, t)),

where Φ(z, t) = Δηkz − φ(t), and where we have assumed kr ≪ 1. These results are obtained by averaging over times long compared with the optical cycle but short compared with variations in φ(t). The particle is trapped where the force vanishes, which occurs at axial positions Z_n(t) that are displaced from the intensity maxima by an amount that depends on the particle’s light-scattering properties,

Z_{n} (t) - z_{n} (t) = \frac{2}{Δ η k} {tan}^{- 1} (\frac{α_{e}^{″}}{α_{e}^{'}} \frac{2 η}{Δ η}) .

Equations (10) and (16) then yield the traps’ axial stiffness,

k_{z} = \frac{1}{4} | α_{e}^{'} | k E_{0}^{2} Δ η^{2},

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 [34] or fine tuning of the beam’s properties [35]. Decreasing the inter-trap separation by increasing Δη enhances intensity gradients and thus increases trap stiffness.

Picking the largest possible value of Δη to optimize axial trapping is not necessarily the best strategy. An optical conveyor’s transverse stiffness,

k_{r} = k_{z} \frac{α_{e}^{' 2} (1 - η^{2} - \frac{1}{4} Δ η^{2}) - 2 α_{e}^{″ 2} η^{2}}{\frac{1}{2} α_{e}^{' 2} Δ η^{2} + 2 α_{e}^{″ 2} η^{2}},

can vanish or even change sign as Δη 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 η = 0.8 and Δη = 0.086. Results from the dipole approximation agree well with the Lorenz-Mie results that are plotted as solid curves in Fig. 2, at least for ka_p ≤ 1. The Lorenz-Mie predictions, in turn, agree quantitatively with results from Fig. 1, which are plotted as discrete points.

Fig. 2 (a) Trap stiffness as a function of particle size for silica spheres in the optical conveyor from Fig. 1(b). Predictions from Lorenz-Mie theory are plotted as solid curves, and the corresponding results in the dipole approximation are plotted as dashed curves. Discrete points show experimental results obtained from the data in Fig. 1(c).

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The dipole approximation severely underestimates the optical conveyor’s stiffness for ka_p > 1, 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 (k_r > 0) according to Eq. (17) establishes an upper bound on Δη and, through Eq. (4), a lower bound on the range over which an optical conveyor can transport small objects:

R_{⊥} \leq \frac{A}{2} (\frac{α_{e}^{'}}{α_{e}^{″}} - \frac{α_{e}^{″}}{α_{e}^{'}}) < R .

The practically accessible range of transport may be substantially greater than R_⊥ for particles larger than the wavelength of light.

2.3. 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]

E_{G} (z, t) = E_{0} \frac{z_{R}}{\sqrt{z^{2} + z_{R}^{2}}} e^{i k z} e^{i ζ (z)} e^{- i ω t} \hat{ε},

where z_R ≈ 2R²/(kA²) is the Rayleigh range of a Gaussian beam converging at distance R from an aperture of radius A, and where ζ(z) = tan⁻¹(z/z_R) is the Gouy phase. Because we are interested in long-ranged axial transport, we assume R > A. The axial component of the associated force,

\frac{F_{G} (z)}{E_{0}^{2}} = - \frac{1}{2} z_{R}^{2} \frac{z α_{e}^{'} - k (z^{2} - z_{R}^{2}) α_{e}^{″}}{{(z^{2} + z_{R}^{2})}^{2}},

forms a trap only for particles satisfying

{(\frac{α_{e}^{'}}{α_{e}^{″}})}^{2} > 4 k z_{R} (k z_{R} - 1) .

This condition is most easily satisfied in strongly converging beams for which z_R is small. The condition on α_e is qualitatively consistent with numerical studies [7–9] which describe the difficulty of trapping high index or absorbing particles with laser tweezers.

3. 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

R_{G} = \frac{A}{2} \sqrt{1 + \sqrt{1 + {(\frac{α_{e}^{'}}{α_{e}^{″}})}^{2}}} .

At ranges R beyond R_G, 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 η = 0.96 and Δη = 0.04 that was projected with a 60× objective lens (Nikon Plan-Apo, numerical aperture 1.4 oil immersion). Holographic projection limits the conveyor’s range to R = 70μm, and thus suggests an effective numerical aperture of 0.43 given the convergence angle of θ = 19°. The holographically measured trajectory plotted in Fig. 3(b) shows this optical conveyor trapping and transporting a 1.5 μm-diameter silica sphere over 66 μm. 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.

Fig. 3 (a) Experimental reconstruction of an optical conveyor with η = 0.96 and Δη = 0.04. (b) Trajectory of a 1.5 μm silica sphere transported by the same optical conveyor over a range of 66 μm. (c) Axial stiffness as a function of transport range. Solid curves show Lorenz-Mie predictions for optical conveyors transporting large (1.5 μm-diameter) and small (0.036 μm-diameter, ka_p = 0.5) silica spheres through water. Dashed curves show corresponding results for optical tweezers. Discrete symbols show results for the optical tweezer in Fig. 1(c) and the optical conveyors in Figs. 1(d) and 3(a).

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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 41 mW/μm². One set of curves is calculated for a 1.5 μm 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 ka_p = 0.5, 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 R/A < 9 the optical conveyor extends indefinitely, albeit with a stiffness that falls off as R⁻⁴. The maximum trapping force in this regime falls off as R⁻². 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.

4. Conclusions

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 [4–6]. 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 [4–6], which are inherently limited by radiation pressure.

Further improvements in performance almost certainly can be realized by appropriately structuring the intensity and phase gradients [38–40] along the projected beam. The range and trapping strength also should improve in optical conveyors created with radially polarized light [41–43]. These insights apply also to other interferometrically structured beams of light, such as solenoidal waves [2]. Both as tractor beams, and also as ordinary optical traps, interferometrically structured beams of light offer clear benefits for optical micromanipulation.

Acknowledgments

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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Universal, strong and long-ranged trapping by optical conveyors

Abstract

1. Introduction: optical conveyors

2. Trap stiffness

2.1. Generalized Lorenz-Mie theory

2.2. Dipole approximation

2.3. Comparison with optical tweezers

3. Transport range

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (3)

Equations (26)

Optics Express