Hands in the Microscopic World

Figure 3: Two- and three-dimensional configurations of optical tweezers created from a single laser beam with a computer-designed hologram. of a single beam's wavefront. (a), (b), and (c): Thirty-six water-borne polystyrene spheres, 800 nm in diameter, are trapped in a plane and reconfigured with dynamic trapping patterns, from Ref. (1). (d) and (e): Two 1  $\unit$$\mu$m diameter silica spheres being moved past each other in two different planes, from Ref. (24). The spheres' different appearance results from their different heights relative to the microscope's focal plane. (f), (g), and (h): Seven 1  $\unit$$\mu$m diameter silica spheres being moved up and down through seven different planes, from Ref. (1).

Figure 2 schematically depicts an optical tweezer system in which a strongly converging objective lens focuses beams of laser light into optical traps. A collimated beam passing straight into the lens' input pupil comes to a focus in the middle of the objective's focal plane, where it forms a trap. Sweeping the angle of incidence translates the trap across the field of view. Diverging the beam causes it to focus downstream of the focal plane, while converging it moves the focus upstream.

Translating an optical trap, creating multiple optical traps, and then converting these into multifunctional optical traps all are greatly facilitated by first forming an image of the lens' input pupil with the telescope in Fig. 2. Any beam passing through the pupil's image, centered at point A in Fig. 2, also passes through the actual pupil and forms a trap. Tilting the beam as it passes through the image scans the optical trap. A single rapidly scanned optical tweezer can trap multiple particles by dwelling briefly on each before moving on to the next (26,27,25). The extent and complexity of such time-shared trapping patterns is limited by the time required to reposition multiple wandering particles. Scanned optical tweezers, furthermore, are restricted to the lens' focal plane. Even so, scanned optical tweezers have proved extremely useful for organizing planar assemblies of colloidal particles (28), for testing new ideas in statistical mechanics (29), and for measuring macromolecular interactions (30).

Placing a diffractive beamsplitter at point A converts a single input beam into several beams, each of which forms a separate optical trap. Such a beamsplitter can be implemented as a computer-generated hologram, and the resulting trapping patterns have come to be known as holographic optical tweezers (HOTs) (32,31). To see how this works, consider multiple beams all passing simultaneously through point A on their way to being focused into optical traps. Their superposition creates a distinctive interference pattern centered at point A. Imprinting this pattern onto the wavefronts of a single input laser beam transforms the one beam into the several, and thus forms the same pattern of optical traps.

The input beam's electric field, $E(\ensuremath{\vec \rho}\xspace ) \, \exp( i \varphi(\ensuremath{\vec \rho}\xspace )) \, \hat \epsilon$, around point A is characterized by a real-valued amplitude, $E(\ensuremath{\vec \rho}\xspace )$, and phase, $\varphi(\ensuremath{\vec \rho}\xspace )$, both of which depend on position transverse to the optical axis, and a polarization vector $\hat{\epsilon}$ describing the field's orientation. A multi-beam interference hologram generally would modify both the amplitude and the phase of the input beam, with the amplitude modifications diverting power away from the optical traps. Fortunately, a variety of iterative optimization algorithms have been developed (1,24,32) to create equivalent holographic beamsplitters that modify only the phase of the input beam. Such a phase-only diffractive optical element (DOE), also known as a kinoform, was used to create the 400 optical traps shown in Fig. 2.

The full utility of holographic optical tweezers is realized when a computer-addressed spatial light modulator (SLM) is used to project sequences of trap-forming kinoforms in real time. An SLM imposes a prescribed amount of phase shift at each pixel in an array by varying the local optical path length. Typically, this is accomplished by controlling the local orientation of molecules in a layer of liquid crystal, although arrays of microelectromechanical (MEMS) mirrors also are becoming available for SLM applications. Slightly displacing the traps from one pattern to the next transfers particles along arbitrary three-dimensional trajectories (33,1,24), animating matter with light in much the same way that cartoons animate light with matter. Figure 3 shows this principle in action.

In a variation on this theme, the generalized phase contrast (GPC) technique converts a pattern of phase modulation across an SLM's face directly into the corresponding intensity modulation in the objective lens' focal plane (34), and thus creates arbitrary planar trapping patterns. The conversion involves an annular phase plate similar to that used in phase contrast microscopy. This approach bypasses the need to calculate holograms, and thus is extremely efficient. The limited spatial resolution of existing SLMs currently limits GPC to creating lateral traps, rather than three-dimensional optical tweezers, but still has proved useful for rapidly organizing small objects in thin samples (35).

Even static arrays of optical traps have exciting and surprising applications. For example, an array of traps can continuously sort fluid-borne particles, playing much the same role as the gel in gel electrophoresis. The array sorts particles on the basis of their differing affinities for optical traps and an externally applied driving force. Inclining a regular array with respect to the driving force deflects the selected fraction so that it can be collected separately from the other, undeflected fraction (36). Unlike most sorting techniques that operate on discrete batches of samples, optical fractionation works continuously, and can be dynamically optimized by adjusting the wavelength, intensity and geometry of the trap array. Moreover, because optical fractionation relies on object's abilities to hop from potential well to potential well, it is exponentially sensitive to particle size, and so promises unparalleled size resolution (37).

An array of traps also may be viewed as a tailor-made potential energy landscape for interacting colloidal particles. How strongly interacting systems evolve on modulated substrate potentials is a classic problem in statistical physics, and colloid in modulated optical fields constitutes a rare model system whose microscopic interactions can be measured and controlled as its macroscopic thermodynamic properties unfold (38,39). Insights obtained from studying optically modulated colloid are relevant to such analogous systems as atoms adsorbed on crystal surfaces, electrons passing through charge density waves and two-dimensional electron gases, magnetic flux quanta passing through defects in type-II superconductors, and motor proteins translating along filaments in living cells. Early studies demonstrated that modulation along even one direction can freeze a two-dimensional colloidal fluid (41,40). Deeper modulation actually melts the substrate-induced crystal (42) by suppressing inter-row coupling (43). More recent studies have demonstrated still other intriguing behavior such as rotational melting in an array of multiply-occupied traps (39,44) and have shed new light on the mechanisms by which magnetic flux lines invade superconductors (38,39).

Time-varying potential energy landscapes created with dynamic optical traps promise new insights into molecular motors' operation by providing a powerful experimental system with which to study thermal ratchets (29) and related models in non-equilibrium statistical mechanics (45). Once perfected, such ratchet potentials also will be useful for dynamically sorting mesoscopic objects and transporting them through tiny integrated laboratories for processing and testing (46).

All such studies provide valuable insights into how nature creates and exploits hierarchically organized structures. Once these principles are understood, they will be extraordinarily useful for creating new materials and devices to order. Until then, many of the most interesting three-dimensionally structured functional systems can be assembled using large numbers of optical tweezers operating in concert. Indeed, optical tweezers have an essentially unique ability to construct three-dimensional heterostructures with features ranging in size from a few nanometers to a few millimeters. The real power of this approach only becomes apparent when optical trapping is combined with other techniques to create permanent structures with embedded functionality.