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Using holographic optical tweezers we create crystal-like structures over a scale of 10s microns. Real-time calculation of the required holographic pattern allows us to scale or rotate the structures about arbitrary axes. Once the desired structure is formed we can make it permanent by gelling the surrounding fluid.
I. Introduction
Optical tweezers work on a very simple principle. They use the gradient force produced by a light field to trap and hold micron-sized objects. For a dielectric particle the gradient force acts to attract the particle to the point of highest intensity. The intensity gradient, the corresponding gradient force, and consequently the trapping capability of the light increases with the degree of focussing. Hence, to produce the smallest possible focus, optical tweezers are usually implemented using a diffraction-limited laser beam, coupled into a high-magnification microscope objective. Conveniently, the microscope objective both focuses the laser beam and images any trapped objects. In general, the particles are suspended in a fluid that not only offers partial buoyancy but also provides the damping force essential for a stable trap.
Optical tweezers were first described in the late 1980s by Ashkin [1], who trapped small particles at the focus of a laser beam. Following their first demonstration, the number of applications of optical tweezers has grown and a number of commercial systems are available [2]. In biological fields, optical tweezers have measured the pN-forces necessary to stretch individual muscle fibres, have uncoiled DNA, and have manipulated individual cells. In the physical sciences, tweezers have found themselves at the centre of experiments studying the linear and angular momentum of light and being used as drives for micromachines [3].

Fig. 1. The rotating cube. Eight 2mm diameter silica spheres trapped at the corners of a cube.

Over the last few years, the commercial availability of spatial light modulators (SLMs), which act as reconfigurable phase holograms, has revolutionised optical tweezers. Unlike a data projector, which produces an image using a pixellated device to selectively attenuate the transmitted light, a phase-holographic component simply redirects the light to the desired location. Not only is this inherently more optically efficient, but it allows the light to be brought to be focussed in different planes. Within the context of optical tweezers, a spatial light modulator placed into the beam allows a single laser to be efficiently divided to produce simultaneous multiple optical traps that can be positioned anywhere within a 3 dimensional volume [4]. Previous approaches to multiple traps use scanning mirrors or other beam deflectors, but such systems require time-multiplexing between traps and can only position the traps within a single plane. Interference patterns between two or more beams have also been used to create three-dimensional arrays of traps [5], but this approach is limited to relatively simple interference patterns.
This newsletter details our recent research in trapping multiple particles in three dimensions. By using existing and developing new hologram-design algorithms, we have created crystal-like structures within optical tweezers.
II. Multiple Trapping and Hologram Design
The key to successful trapping of multiple objects is the hologram design transferred to the SLM. The ideal result is a design giving multiple beams without introducing aberrations that enlarge the focus spot and reduce the trapping strength. Additionally, when two traps come close together there is a risk that one trap may capture the particle held in the other. This is a particular problem if the displacement is axial, in which case great care needs to be taken to avoid unwanted stacking of the particles. We have used three different algorithms, which offer different trade-offs between complexity of achievable trap arrangements and computational effort.

Fig. 2. Eighteen 2mm diameter silica spheres trapped at the lattice cites of a diamond unit cell.

The first approach, used for simple, symmetric patterns, is effectively the summation of familiar optical components. For example, a two-level phase hologram of a Fresnel lens, combined with two crossed phase gratings, will produce eight traps positioned at the corners of a cuboid. Figure 1 shows a video sequence of eight, 2µm diameter silica spheres, trapped at the corners of a cube. This cube can be rotated or scaled by altering the hologram appropriately. At video resolution, such calculations can be carried out at video frame rates. In our case, practical limitations in the design of the microscope objective and to the resolution of the SLM limit the unit-cell size to 4mm to 20mm. Note that at certain times in the rotation cycle, spheres are trapped immediately above one another separated only in the axial direction. Contrary to what one might expect, the high numerical aperture of the system means that the distant trap is not influenced by the ‘shadow” cast by the object in the near trap. We have also shown that such structures can be made permanent by using a suspension fluid that after a fixed time turns into gel.
For more complex, non-symmetric designs we use a direct binary search algorithm. Here we specify both regions of high (i.e. the trap) and low intensity (i.e. the trap separation) in the target beam and define a figure of merit corresponding to the hologram’s performance at meeting these criteria. We then use an iterative process to optimise the design of the hologram, which simply consists of making a random change and keeping or discarding it depending on whether or not the hologram’s performance has improved. Figure 2 is an example of a structure created using this method. It shows silica spheres trapped at the lattice sites of a diamond unit cell — 18 particles trapped in 5 planes.
The third approach to hologram design is an extension to the two-dimensional Fourier transform based Gerchberg-Saxton algorithms. In original form this allows the general specification of a target intensity distribution in one plane. Although this plane can be deformed, the trapping of two axially displaced particles is problematic. By incorporating an additional beam-propagation steps into the algorithm we can define unrelated trapping patterns in multiple planes. Although the algorithm is significantly slower than the original Gerchberg-Saxton algorithm, it is still possible to calculate video resolution holograms two or three times a second. Figure 3 shows a video sequence of a general manipulation in 3 dimensions where the positions of individual particles are independently controlled.

Fig. 3. Image sequence showing general manipulation in 3 dimensions where the positions of individual particles are independently controlled.

III. Optical Tweezers System
As is typical, our tweezers system is based upon an inverted microscope with a 1.3NA ¥100, oil immersion objective lens. The SLM is introduced into the beam path and is imaged onto the entrance aperture of the objective lens such that lateral and axial displacements of the optical trap correspond to the SLM displaying a diffraction grating and Fresnel lens respectively. A combination of the spatial resolution of the SLM and the aberrations within the object lens limit the maximum lateral and axial displacements to approximately 30 and 20 microns respectively.
IV. Conculsions
We have outlined a technique to create three dimensional crystal-like structures within optical tweezers that can be scaled or rotated about arbitrary axes. We anticipate that the generation and control of such pre-determined three dimensional crystal structures, over length scales over several 10s of microns will have significant potential in fields as diverse as photonic crystal construction, creation of metrological standards within nanotechnology and the seeding of tissue growth. For example, by organizing one or more cell types into spatially defined 3D arrays, tweezers could perhaps be used to engineer templates for tissue replacement therapies.
[1] A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, “Observation of a single-beam gradient force optical trap for dielectric particles”, Opt. Lett., 11, 288-290, (1986)
[2] Arryx Inc. (Chicago, USA), Cell robotics International Inc. (Alberqureque, USA), P.A.L.M GmbH (Bernried, Germany)
[3] J. E. Molloy, M. J. Padgett, “Lights, action: optical tweezers”, Cont. Phys., 43, 241-258, (2002)
[4] D. G. Grier, “A revolution in optical manipulation”, Nature, 424, 810-816, (2003)
[5] M. P. MacDonald, L. Paterson , K. Volke-Sepulveda, J. Arlt, W. Sibbett, K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures”, Science, 296, 1101-1103, (2002)

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