Optical Peristalsis

Brian A. Koss

David G. Grier

James Franck Institute and Institute for Biophysical Dynamics
The University of Chicago, Chicago, IL 60637

Abstract.

We describe an efficient method for transporting and rearranging mesoscopic objects in three dimensions using short repetitive sequences of holographic optical trapping patterns. Material transport in this process is analogous to peristaltic pumping, with the configurations of optical traps mimicking the states of a physical peristaltic pump. Optical peristalsis can transport large numbers of small particles rapidly and determinstically through complex three-dimensional patterns. The same system also can be used to study transport in a variety of model thermal ratchets.

pacs: 87.80.-y, 05.60.Cd, 87.80.Cc, 42.40.Jv

Understanding and controlling transport in mesoscopic systems is a central theme in biophysics and nanotechnology. Nature has evolved an array of protein-based molecular motors responsible for transporting submicrometer-scale payloads within living cells and for generating motion at cellular and larger scales. Although molecular motors' operation is not yet fully understood, a general consensus is emerging around a class of models known as thermal ratchets (1) in which transport results from symmetry-breaking in either the temporal evolution or spatial configuration of a fluctuating potential energy landscape. Experimental realizations of thermal ratchet models (2); (3) have been inspired both by an interest in understanding natural processive motors, and also by the increasingly pressing technological need to transport materials on ever smaller length scales (3).

Figure 1. Dynamic holographic optical tweezers. A computer-designed phase modulation $\varphi(\vec{r})$ is imprinted onto the wavefronts of a TEM ${}_{{00}}$ laser beam by a spatial light modulator (SLM). Diffracted beams are relayed by a telescope to an objective lens that focuses each into an optical tweezer in the sample volume. The ten linear manifolds of traps, denoted by arrows, were imaged by placing a mirror in the focal plane.

This Letter describes a class of deterministic diffusionless ratchets created with the holographic optical tweezer technique (4); (5); (6) whose operation we term optical peristalsis. The potential energy landscapes in an optical peristalsis system consist of discrete optical traps projected by computer-generated holograms, each of which is creates a symmetric potential well for small volumes of matter. Replacing one trap-forming hologram with another updates the positions of the traps and can be used to move large numbers of particles independently in three dimensions (5); (6). Previous studies of controlled transport using multiple optical traps have required each particle to be transported interactively (5); (6); (7). By contrast, optical ratchets eliminate the need for adaptive control in many cases. Because the individual traps are projected onto an otherwise featureless potential energy landscape in viscously damped systems, transport driven by optical peristalsis constitutes “force-free” directed motion (1).

Most theoretical studies and all previous experimental realizations of ratchet-driven transport have relied upon spatial asymmetry to induce unidirectional motion (1). By contrast, each of our optically-created potential energy landscapes is both locally and globally symmetric. Optical peristalsis induces transport by breaking spatiotemporal symmetry in sequences of at least three states (8). Cycling through a sequence of states in this manner also brings to mind the operation of mechanical peristaltic pumps, hence the technique's name. Unlike conventional peristaltic pumps that drive linear flows, optical peristalsis also can be adapted to move mesoscopic matter through very complex three-dimensional patterns. Straightforward extensions of this approach yield non-deterministic thermal ratchets capable of demonstrating both flux reversal as a function of cycling frequency and also fractionation in heterogeneous samples (8).

Each of our traps is an optical tweezer (9) that uses forces exerted by a strongly focused beam of light to trap small objects. Stable trapping generally requires a viscous medium capable of dissipating the trapped objects' kinetic energy (10). Dissipation similarly is required for force-free motion in thermal ratchets (1).

Our apparatus, shown schematically in Fig. 1, uses a liquid crystal phase-only spatial light modulator (Hamamatsu X7550 PAL-SLM) (11) to diffract a single TEM ${}_{{00}}$ laser beam at 532 nm into a fan-out of beams, each with individually specified characteristics (6). The SLM can impose 150 discrete levels of phase delay between 0 and $2\pi~\text{radians}$ at each 40 $\mathrm{\upmu}$ $\mathrm{m}$ wide phase pixel in a $480\times 480$ array, and thus can be used to create quite general phase holograms. Each diffracted beam is relayed to the input pupil of a $100\times$ NA 1.4 oil immersion objective (Zeiss S-Plan Apo) mounted in an inverted optical microscope (Zeiss Axiovert S100TV) and focused into a discrete optical trap. Our methods for calculating holograms are described in Ref. (6) and enable us to project several hundred optical tweezers into a volume of roughly $100~\mathrm{\upmu}\mathrm{m}\times 100~\mathrm{\upmu}\mathrm{m}\times 30~\mathrm{\upmu}\mathrm{m}$ (6). The optical train, including computer-designed diffraction grating, has an overall efficiency of roughly 30 percent. The microscope's objective lens also can be used to create images of trapped objects. These images pass through a dichroic mirror to an attached video camera and are recorded on video tape before being digitized and analyzed.

Figure 2. Principle of optical peristalsis. (a) through (d) show one cycle through three distinct states of a spatially symmetric optical ratchet. A particle in one manifold is transferred to the next by the end of the cycle.

Figure 2 schematically depicts how optical peristalsis works. At the outset of each cycle, a particle is localized on one (or more) optical traps in a manifold, or locus, of optical traps. Several such manifolds form a pattern that covers some or all of the field view. In Fig. 2(a), two manifolds are represented schematically by two Gaussian potential wells. Actual trapping patterns, such as the example in Fig. 1, may consist of several hundred traps. Replacing the first pattern with another whose manifolds are slightly displaced transfers the object onto the nearest manifold in the new pattern. Subsequently turning on a third pattern transfers the object once again. In the final step, the first pattern is turned on once again, with the particle having been transferred to the next manifold. Repeating the same cycle of three patterns passes the object deterministically from manifold to manifold until it reaches the end of the pattern. Because the pattern can cover the entire field of view, transporting a particle in this manner does not require active tracking.

If the potential wells in consecutive patterns overlap, then three patterns should suffice to ensure objects move deterministically in the intended direction, as shown in Fig. 2. Practical considerations may limit the number of optical tweezers that can be projected in a single pattern, so that more patterns will be required to complete a cycle. Even if the manifolds in consecutive patterns are disjoint, particles can be translated in a desired direction by biasing their natural Brownian motion (1). Breaking spatial symmetry through the temporal sequence of trapping patterns suffices to induce directional motion in a statistical sense, although the direction of the motion also will depend on the particles' diffusivity.

Figure 3. Examples of particle manipulation with optical peristalsis. (a) Linear motion driven by a sequence of translated patterns, each resembling the example in Fig. 1. Arrows denoting the ten manifolds' positions correspond with those in Fig. 1. (b) Linear transport incorporating both compression and rarefaction. (c) Radial transport in three-stage expanding circular manifolds. (d) Controlled rotation.

Figure 3 shows examples of optical peristalsis in action. The samples in these demonstrations consist of colloidal silica spheres 1.58 $\mathrm{\upmu}$ $\mathrm{m}$ in diameter (Duke Scientific Lot 24169) dispersed in a layer of water between a glass microscope slide and a #1 glass coverslip. Each sphere is tracked to within 30 nm in 1/30 s increments using standard video analysis techniques (12). The data in Fig. 3 include trajectories accumulated over several minutes from particles diffusing into the field of view.

Figure 3(a) demonstrates rapid linear transport driven by a four-pattern cycle in which each pattern consists of parallel linear manifolds. The focused traps from one of these patterns appear in in Fig. 1. Each trap was powered with roughly $500~\mathrm{\upmu}\mathrm{W}$ of light, yielding a typical well depth of $V_{0}\approx 10~k_{B}T$ and width of $\sigma=1.5~\mathrm{\upmu}\mathrm{m}$ (13). Manifolds were separated by $L=20~\mathrm{\upmu}\mathrm{m}$ , and the patterns were cycled at 1 Hz, yielding a mean transport speed of $7~\mathrm{\upmu}\mathrm{m}\mathrm{/}\mathrm{s}$ . Much higher speeds should be possible with higher repetition rates.

Curved manifolds yield useful generalizations. The data in Fig. 3(b) demonstrate that samples can be concentrated or dispersed by an appropriate choice of manifold geometry. The former is useful for driving samples into orifices, while the latter is useful for mixing.

Closed manifolds of traps, typified by the circular patterns in Figs. 3(c) and (d) have still other applications. Scaling the manifolds' diameters in each step creates the radial flux shown in Fig. 3(c). An outward flux empties a region in the middle of a sample. Being able to create and maintain a void within a dense heterogeneous suspension suggests a possible novel approach to microculturing living cells in bacterial colonies without special sample handling facilities (14). It also makes possible studies of the interactions and dynamics of colloidal particles in the middle of dense suspensions using techniques developed for isolated pairs (12); (15). Reversing the sequence creates an inward flux useful for concentrating small objects in the field of view or for centering larger samples.

Figure 3(d) demonstrates controlled rotation of a sample mediated by a radial pattern of linear manifolds rotating about a common axis. This approach is useful for orienting macroscopic samples, and is easily generalized to three-dimensional rotations (6).

These examples have all addressed ratchet-driven motions in an otherwise quiescent system. Competition with an external force such as viscous drag due to flowing fluid or field-induced electrokinetic forces could lead to substantially different transport properties for objects with different physical characteristics, and therefore to new methods for fractionation. Large static arrays of holographic optical tweezers already have demonstrated a capacity for continuously fractionating fluid-borne materials (13). When combined with active transport through optical peristalsis, such optical fractionation could prove even more rapid and selective.

Our implementation of optical peristalsis incorporates both imaging and the flexibility afforded by a computer-addressed spatial light modulator. Comparable transport can be achieved by cycling mechanically through a sequence of microfabricated diffractive optical elements (4). In cases where imaging is not required, optical peristalsis requires just an objective lens, a sequence of holograms, and a collimated laser source.

This work was supported principally by the MRSEC Program of the NSF through Grant number DMR-9880595. Additional support was provided by Arryx, Inc. and the W. M. Keck Foundation. The PAL-SLM was made available by Hamamatsu, Inc.

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