Active landscapes: conveyors and ratchets

While static optical trap arrays act as filters or prisms for externally driven dispersions, dynamic arrays are useful for inducing motion. Dynamic holographic optical traps do not move continuously, as do optical tweezers scanned with moving mirrors (27) or traps created by the generalized phase contrast method (28). Rather, one pattern of traps dissolves into another as the DOE encoding the first is replaced by that encoding the second. If trapped objects diffuse slowly enough, they still can be passed from trap to trap by rapidly updating the phase hologram. Viscous relaxation, in this case, plays the role in active holographic transport that persistence of vision plays in cartoon animation to provide the appearance of continuous motion.

Sequences of overlapping trapping patterns can dynamically organize mesoscopic objects into arbitrary three-dimensional configurations, and reorganize them quasi-continuously (29,4,5,12,16,7,19). Periodically cycled sequences of as few as three holograms can induce complicated patterns of motion over large areas through a process called optical peristalsis (30). Here, an object is transferred forward from one manifold of traps in a given pattern to the next by two or more intervening trapping patterns whose manifolds bridge the gap. The sequence of patterns breaks spatiotemporal symmetry and ensures that motion proceeds in the intended direction. Unlike interactive manipulation that requires an individual particle to be captured and its path to be calculated, optical peristalsis operates over the entire field of view, directing and orienting objects automatically through small sequences of precalculated holograms, much like an optical conveyor belt.

This process also provides a means to implement a so-called thermal ratchet (31), in which diffusing particles' random Brownian motion is rectified into a directed flux by a time-evolving potential energy landscape. Unlike conventional motors and deterministic processes such as optical peristalsis whose performance is degraded by random fluctuations, thermal ratchets are stochastic machines and require noise to operate. Most proposed models for thermal ratchets exploit a space-filling spatially asymmetric potential energy landscape. Breaking spatial symmetry is not enough to eke a flux out of fluctuations for a system in equilibrium. As part of a sequence of states driving the system out of equilibrium, however, it can help to break diffusion's spatiotemporal symmetry and thereby induce motion. This works even if the landscape itself has no overall slope and thus exerts no net force.

Figure 6: A thermal ratchet implemented with holographic optical tweezers. (a) The focused light from a $20 \times 5$ array with manifolds separated by $L = 3.8~\ensuremath{\unit{\mu m}}\xspace$. (b) A dispersion of 1.58  $\unit{\mu m}$ diameter spheres interacting with the array. (c) After repeated displacements of the array by $L/3$ and $2L/3$, with each step lasting $T = 5~\unit{sec}$, all the spheres are translated to the right. (d) The transport velocity $v$ as a function of dwell time shows flux reversal as the cycle rate increases.

A regular array of holographic optical tweezers, such as that shown in Fig. 6, presents a potential energy landscape that neither fills space nor breaks spatial symmetry. Even the individual traps are locally symmetric potential wells. Nevertheless, translating the array first by one third of a lattice constant, then by two, and then returning it to its initial state creates a discrete-state traveling ratchet (32) whose time evolution breaks spatiotemporal symmetry. The resulting motion differs from that induced by optical peristalsis in a way that leads to additional applications.

If the lattice constant $L$ is comparable to the traps' effective widths (24), then the traveling ratchet reduces to an example optical peristalsis, and particles are deterministically translated along the displacement direction. Increasing the separation causes particles trapped in one state to be left behind in a flat and featureless region of the potential energy landscape in the next state. They must diffuse to the nearest manifold of traps before they can be localized and transported. If the time $\tau$ required for the particles to diffuse across the potential energy plateau is shorter than the duration $T$ of each state, then most particles are transported forward. On the other hand, if the particles diffuse too slowly, they can miss the forward-going wave and may end up instead being transported backward (32). Such flux reversal as a function of cycle time $T$ and trap separation $L$ is a hallmark of thermal ratchet operation, and is clearly seen in the data in Fig. 6.

Flux reversal in microfabricated thermal ratchets already has been exploited for separating DNA and other macromolecules on the basis of their diffusivity (33). The holographically implemented variant complements optical fractionation by permitting automatic sorting in situ. Variants of the holographic optical ratchet exploit more subtle symmetries to achieve simpler operation (34) or more sophisticated configurations of tweezers to optimize sorting.