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Creation and Manipulation of Three-Dimensional Optically Trapped Structures

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Science  10 May 2002:
Vol. 296, Issue 5570, pp. 1101-1103
DOI: 10.1126/science.1069571

Abstract

An interferometric pattern between two annular laser beams is used to construct three-dimensional (3D) trapped structures within an optical tweezers setup. In addition to being fully translatable in three dimensions, the trapped structure can be rotated controllably and continuously by introducing a frequency difference between the two laser beams. These interference patterns could play an important role in the creation of extended 3D crystalline structures.

At a microscopic level, transparent objects can be trapped and manipulated using the forces exerted by a tightly focused laser beam. This technique, known as “optical tweezers” (1, 2), has enabled major advances in numerous areas of science, including force detection measurements on biological samples, such as the determination of the elastic response of DNA (3). Recent work has also demonstrated the use of optical tweezers for developing optical micro-machines and micro-components (4–8). In parallel with this, the extension of optical tweezers to multiple beam sites to create two-dimensional particle arrays (9, 10) has been investigated. We take this technique a step further by creating vertical arrays of particles (stacking) in multiple trapping sites, forming the basis for creating 3D trapped structures.

Stacking of a small number of particles in standing-wave geometries (11) and Bessel light beams (12) has been observed experimentally, whereas Gauthier and Ashman theoretically predicted stacking in a Gaussian beam (13). Experimentally, we have observed controlled stacking of large numbers of particles in optical tweezers using a single Gaussian beam. By extending this to multiple trapping sites, formed in the interference pattern generated between two annular (Laguerre-Gaussian) light beams, we have created 3D trapped structures. Furthermore, we use the angular Doppler effect to achieve continuous and controlled rotation of the 3D structure.

The mechanism for creating particle stacks depends on whether the trapping laser propagates in the direction of gravity (standard tweezers) or against gravity (inverted tweezers). High refractive index silica particles in water tend to sink to the bottom of a sample cell. To create a stack of spheres in standard tweezers, one must therefore optically trap them to allow them to be lifted off the bottom of the sample cell (3D trapping). Once a particle is lifted up by the tweezing laser, the other particles align underneath it, creating a stack.

In an inverted optical tweezers geometry, the laser is focused at the top of the sample cell in such a way that it forms a cone, the widest cross section of which is at the bottom of the sample cell. Dielectric spheres are captured and guided upward within the cone to the focal region of the laser beam, aligning themselves in a vertical stack. The stack height in both geometries is limited by the divergence of the laser beam and the working distance of the microscope objective. Our experimental optical tweezers geometry uses a 200-mW laser, operating at 1064 nm, directed through a ×63 microscope objective (Numerical Aperture 0.95). In the inverted geometry, we have stacked as many as 16 spheres (5 μm in diameter) and have moved the chain as a whole across the sample slide. By tilting the tweezing laser beam, stacks of spheres have been inclined, and angles of 5° have been achieved. Figure 1 shows experimental data for stacking of particles in an inverted optical tweezers geometry. Combining the ability to stack particles with the use of multiple tweezing sites, using either inverted or standard optical tweezers, results in a technique for the creation of a 3D structure.

Figure 1

Assembly of a stack of six trapped particles (each 53 μm in diameter) in inverted tweezers. The stack is fully translatable across the sample slide (1 cm in diameter) at 200 μm/s using a tweezing laser power of 200 mW.

Single-ringed Laguerre-Gaussian (LG) light beams are described by the index l, which corresponds to the number of 2π cycles of phase around the mode circumference (14). Interfering an LG (l ≠ 0) beam collinearly with a plane wave produces a pattern consisting of l spiral arms that may be used for the rotation of trapped particles (8). However, the form of this pattern is determined by the mismatch in both the Gouy phase of each of the two beams and their respective wavefront curvatures, causing the pattern to exhibit marked azimuthal intensity variations as the pattern propagates away from a focus (15). We demonstrate an interference pattern that overcomes these limitations and propagates without change of form in space (apart from radial scaling), allowing the stacking of particles in each bright region of the pattern. Specifically, an LG beam of index l is interfered with its mirror image (index –l, opposite beam helicity), resulting in a spatial pattern of 2l spots arranged on the circumference of a ring (Fig. 2). Related patterns have been analyzed in other studies (16–18).

Figure 2

The interference pattern between two LG beams of opposite helicity but of the same beam parameters (l = 2 and an LG beam ofl = –2). The pattern is shown at the focal point and at various transverse planes up to twice the Rayleigh range (zR). Each of the constituent LG beams has the same Gouy phase and wavefront curvature, which allows the pattern to propagate without change of form.

The experimental setup for tweezing using the interference pattern is similar to that used in previous work (8). The LG laser beam is generated from a holographic element and passed through a Mach-Zehnder interferometer. A dove prism is placed in one arm of the interferometer, which creates a mirror image of the beam and inverts the handedness of the helical wavefront of the beam in that arm. The beams are superimposed collinearly at the output to give the interference pattern (Fig. 2). This pattern is then directed into an optical tweezers setup as described previously.

In addition to the lateral and vertical motion of the pattern, we can induce the pattern to rotate around its axis of symmetry using the angular Doppler effect (19). A frequency difference is induced between the two arms of the interferometer, resulting in the rotation of the whole interference pattern of spots at a frequency of Ω/l, where Ω is the rotation rate of a half-wave plate in one arm of the interferometer. The rotation of this pattern is a consequence of the helical wavefronts of each of the LG beams. The angular Doppler effect is a very simple and effective way to introduce the relatively small frequency difference required (from less than 1 Hz to 1 kHz) between two light beams for the controlled rotation of an interference pattern. This technique can be applied in any situation where slow motion of an interference pattern is desired, such as translating particles in simple linear interferometric tweezers (20) or creating a moving “conveyor belt” of dipole traps for deterministic delivery of ensembles of cold atoms (21).

The angular Doppler effect makes use of the fact that circularly polarized light reverses its handedness when passing through a half-wave plate. Thus a change in spin angular momentum of 2ℏ︀ per photon occurs as the light passes through this optical element. If the wave plate now rotates, energy can be exchanged between the rotating plate and the light beam, causing either an increase or decrease in the frequency of the light beam (22). In our system, a rotating half-wave plate is used, but this could be replaced by an electro-optic crystal to mimic this effect and therefore offers a system that would have no moving parts.

Different versions of the interference pattern between two LG beams of opposite helicity have been used for studies of various groups of trapped particles. The pattern is directed, in our tweezers geometry, into a sample cell containing 1-μm silica spheres dispersed in water. Using a pattern formed by superimposing an l = 1 beam onto an l = −1 beam (two spots), we have tweezed and rotated several rodlike particles including chromosomes. Groups of spheres have been rotated and simultaneously translated vertically through the sample slide, thus giving full 3D control (Fig. 3A). Interfering an l = 2 andl = –2 LG beam creates a 3D light intensity distribution suitable for the creation of cubic structures (Fig. 3B and Fig. 4A). By adjusting the sample stage, we were able to tweeze particles into the bright regions, forming four separate stacks, each with two spheres, creating a 3D cubic structure (Fig. 3B). In Fig. 3B, the light beam is blocked and the 3D structure (seen complete in the first frame) collapses to reveal the eight constituent 1-μm (in diameter) spheres. The structure is rotated using the angular Doppler effect (Fig. 3C). We also stacked additional particles in each bright region, creating different cubic structures. By controlling the number of spheres loaded in each bright region, we created asymmetric 3D structures (Fig. 4B). Our propagating interference pattern can be combined with a standing wave to offer discrete trapping sites in the axial direction (23). The creation and manipulation of artificial structures (such as those inFig. 4B) at or just below the micrometer level is of current interest in colloid physics. Our ensemble of trapped particles can act as a predetermined nucleus for subsequent growth of novel crystalline structures by self-assembly.

Figure 3

(A) A group of four 1-μm-diameter spheres, tweezed and lifted vertically (whilst simultaneously rotated) through the sample cell using a LG interference pattern with beams ofl = 1 and l = –1. View from above; only the top two spheres are visible. (B) A cubic structure created in optical tweezers with the interference pattern of Fig. 2. The laser beam is blocked and the cubic structure seen in the left-most frame collapses, revealing the eight constituent spheres of the structure (right-most frame). (C) The cubic structure of Fig. 3B is trapped in three dimensions. It is rotated using the angular Doppler effect at rates up to 2 Hz.

Figure 4

(A) The geometry of the interference pattern that leads to the data of Fig. 3, B and C. (B) Diagrams of various other trapped structures, including asymmetric structures, which we have assembled experimentally in our optical tweezers.

  • * To whom correspondence should be addressed. E-mail: kd1{at}st-and.ac.uk

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