In the late 1980s Ashkin demonstrated that a tightly focused laser beam could trap micron-sized particles within a surrounding fluid [1]. These “optical tweezers” have been widely used by biologists, for example to manipulate cells, to study the forces produced by individual muscle fibres [2], and to unwrap DNA [3]. Physicists have also applied optical tweezers to probe fundamental properties of light such as the transfer of its angular momentum to matter [4, 5].

Optical tweezers work on the principle that a focused laser beam gives rise to steep gradients in the electric field near the beam focus. Any dielectric particle within the vicinity of the focus will experience a corresponding gradient force directed towards the point of highest light intensity. If the laser power is a few mW or more and the numerical aperture of the objective lens is high, the gradient force can exceed the forces due to gravity and the scattering of the light. Immersing the particle in a fluid provides a damping force, thereby producing a stable optical trap known as optical tweezers. Optical tweezers are readily implemented by modifying high-magnification microscopes [6]; in addition to focusing the laser beam, such systems provide visualisation of the trapped objects.

As an alternative to trapping single objects, optical tweezers can be configured to trap multiple objects simultaneously. Initially, such multiple optical tweezers were implemented by dividing a single beam with beamsplitters [7], or by using beam scanners to rapidly re-position a single beam [8, 9]; in either case, the system complexity and cycle time limited the number of trapped objects in such systems to typically less then 10. More recently, optical tweezers were demonstrated where the local intensity maximum was not that of a focused beam but instead a region of constructive interference between two or more beams [10]. Such arrangements allow trapping of multiple objects [11], but the trapping pattern is restricted to intensity maxima of the interference pattern.

A fixed geometry of multiple traps can be produced using holographic optics [12], which transforms a single laser beam into an array of beams. Computer-controlled spatial light modulators (SLMs) have built on this approach, allowing the generation of multiple traps that can be reconfigured at video frame-rates or higher [13, 14]. For example, holographic optical tweezers have been used to create 2D arrays of trapped microparticles (using methods which have recently been reviewed in detail [15]). Often these techniques have used the Fourier-transform-based Gerchberg-Saxton algorithm, a technique that requires only a small number of iterations to obtain a hologram design giving the far-field intensity distribution of choice [16]. In addition to lateral displacements, an SLM can also introduce focal power, shifting the axial position of the trap away from the focal plane of the objective lens. Using this technique it has been possible to manipulate particles both laterally and axially over several 10s of microns and arrange them into simple three-dimensional (3D) configurations [13, 17, 18, 19]. The maximum size of the structures is limited by both the spatial resolution of the SLM and the out-of-focus performance of the microscope objective [20].

Here we demonstrate the creation and dynamic control of 3D crystal structures in holographic optical tweezers. The corresponding phase-hologram patterns are calculated using two different algorithms, one of which is itself new. Each algorithm has different strenghs; for example, one algorithms is fast enough for interactive use but restricted to symmetric structures, while the other algorithm is too slow for interactive use but allows arbitrary structures and improved trap performance.

The apparatus of the holographic-optical-tweezers system is based around a 1.3NA, ×100, microscope objective, used in an inverted geometry to give easy access to the sample cell (Fig. 1) [18]. The trapping laser is a frequency-doubled (532nm) Nd:YAG device with an output power in excess of one Watt and a near-diffraction-limited beam. The SLM is an optically addressed nematic-liquid-crystal device, configured to act as a phase hologram that uses 256 gray levels; we use it at an angle of incidence of approximately 5°. The SLM has near-VGA resolution without sharp pixelation, giving measured first-order diffraction efficiencies of approximately 40% [21]. The lack of pixelation specifically suppresses the loss of power into additional, widely-spaced, diffraction orders. As with gratings, limitations in the spatial resolution and phase linearity result in some of the light remaining in the zero order or being diffracted into higher orders. The resulting unwanted traps may be removed by using a spatial filter in an intermediate image plane (as we did in the case of figures 2 and 4, where we removed the zero order – typically about 40% of the light intensity – with a chrome-on-glass center stop in an intermediate Fourier plane), or by modifying the hologram design to displace the intensity distribution of interest away from the beam axis (as we did for figure 3). Typically we divide our SLM into 256×256 logical pixels (although the head of our SLM is not pixellated). This reduced resolution is well within the resolution limits of the SLM, sufficient to realize structures of the size and complexity demonstrated here, and allows faster calculation of the hologram patterns.

As stated, an important aspect of this paper is the use of new or modified/improved existing algorithms to calculate the required trapping pattern. The first algorithm is a simple – and to the best of our knowledge novel – approach to 3D hologram design that is applicable to structures symmetric about a centre point. In these simpler cases the inherent deficiency of the SLM can be utilised, and rather than removing the traps associated with the -1st diffraction order they are left in place. For example, a simple binary phase grating (phases alternating between 0 and π) gives two traps of equal intensity, symmetrically placed on either side the zero-order position with a separation proportional to the spatial frequency of the grating. More generally, if a Fresnel lens and a blazed grating are combined to produce a single trap (the +1st diffraction order) displaced in any direction from the zero order, its mirror trap (the -1st diffraction order) can be created by similarly converting the combined hologram pattern to binary. In this way, one trap can be split it into two traps, displaced in opposite directions from the original. These two traps can be further split in any direction by adding another binary hologram to produce four traps in a single plane. Finally, the addition of a third binary hologram results in eight traps. If the three split directions are chosen to be orthogonal to each other, the traps lie on the corners of a cuboid. This algorithm is fast enough that the hologram patterns can be calculated in real time (on a 2GHz desktop computer, the calculation of a 256×256 array takes about 0.1s), allowing interactive control over parameters such as 3D orientation (Euler angles) and scale. Figure 2 shows a sequence of video images, obtained from movie S1, with 8 silica spheres of 2µm diameter trapped at positions corresponding to the corners of a “tumbling” cube. In our case, practical limitations of the microscope and resolution of the SLM allow the unit cell size to be set between 4 and 20µm.

Often the target crystal structure will not be symmetric about a center point, and as a consequence a different technique for hologram design will be required. Accordingly, a second approach to our hologram design has been based upon the direct binary search (DBS) algorithm [22, 23]. An error function ε is defined that assesses the hologram’s performance in producing focused beams and low-intensity regions at arbitrary positions in a 3D volume. The error function can simply be the sum of the absolute differences between desired and generated field magnitudes at target point positions,

where N is the number of target positions and |U _T,i| are the desired field magnitudes at those points. The error function can also be more complex, involving for example weight factors or non-linear weight functions. A random pixel is selected, the error function is evaluated for all possible values of that pixel (in our case 16 phase levels) and the pixel is left at the value that gives the best performance. Repeated iterations of this process gradually lead to an optimized hologram design. It should be noted that other 3D hologram design algorithms, such as simulated annealing or genetic algorithms, may use the same error function and therefore produce essentially the same results. However, in our experience DBS is the fastest algorithm when compared to these alternatives and it produces at least equally accurate and efficient hologram designs. For the example shown in Figure 3 we used 18 target point positions. For these we set the desired target intensities (|U _T,i|²) to be 4.75% of the peak intensity in a single spot that would be produced by a clear lens pupil. The hologram was divided into 256×256 pixels with 16 phase levels. To achieve the desired target values ≈250000 iterations were required, i.e. each pixel was “visited” approximately four times. For each hologram design, this required ≈20s when the calculation was run on a standard single-CPU 2GHz PC. The theoretical diffraction efficiency of 4.75% could be improved, but in that case the uniformity between the 18 spots, which in our case was ≈0.5%, would become worse. The fraction of the light power illuminating the SLM that in practice is focused into one of these spots would be significantly lower: we estimate it to be roughly the product of the theoretical diffraction efficiency for that trap using a perfect hologram (4.75%), the first-order diffraction efficiency of the SLM (≈40%), and a factor that accounts for loss due to the different optical components, primarily the microscope objective (in our case of the order of 50%). Note that our experimental setup does not allow an easy measurement of this power.

 

Fig. 1. Schematic diagram of the optical tweezers and the SLM. The plane of the SLM, a, is imaged into the pupil of the microscope objective plane a*. The planes b and c are imaged into the focal region of the microscope.

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One specific problem with earlier work in optical tweezers has been that when two close traps are axially displaced but in the same lateral position, the on-axis intensity of the beams often results in trapped objects stacking above each other (and migrating into the same trap from which they cannot be separated). Although we do not do this here (our traps are sufficiently widely spaced to avoid stacking), this problem can be addressed directly in the DBS approach by defining the error function such that it not only favours bright foci at the desired trap positions (in 3D space), but also intensity zeros that separate individual traps. This increases the potential barrier between two traps, which in turn prevents stacking in many cases. A detailed characterisation of the improved trap performance is currently underway.

Using such an approach we have constructed both arbitrary configurations and crystal structures, both of which can be dynamically manipulated by pre-calculation of specific hologram sequences. Figure 3 shows 18 trapped silica spheres (each of diameter 2µm) positioned to form a diamond lattice with a unit cell size of 15µm. It also shows the intensity of the trapping light in one of these planes, which demonstrates the absence of spurious diffraction spots of an intensity 0s 2s 4s 6s sufficient to disturb the trapping. Note that the first and last planes have trapped spheres in the same lateral positions, but separated axially. As we have demonstrated previously, such crystal structures can be made permanent by setting them with a gel host after which the trapping laser can be turned off [19]. Figure 4 shows an example of dynamic control of a 3D structure – in this case 4 glass beads trapped in the corners of an imaginary tetrahedron – which uses a sequence of hologram patterns (one for every 2° of rotation – 60 holograms for the sequence shown) that were pre-calculated using a DBS algorithm.

 

Fig. 2. [3.1MB] Holograms and corresponding image sequence of the rotation of a cubic unit cell. The cell is constructed from 2µm silica beads, each separated by 5µm. The hologram patterns used to create the traps were calculated in real time. They are a combination of diffraction gratings and Fresnel lenses.

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Fig. 3. [1.3MB] Optically trapped diamond unit cell constructed from 18 beads of 2µm diameter and suspended in water. As the camera focus is moved through the structure (left to right), separate planes come into focus (top row). We used a static phase hologram (far right) to create the required 3D intensity distribution; the intensity cross-section in the top plane is shown (far left).

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Fig. 4. [2.9MB] Glass beads (2 µm diameter) trapped in the corners of an imaginary rotating tetrahedron. The phase-hologram patterns used for the generation of the corresponding optical traps are shown on the left; the centre column shows the corresponding video frames. A series of hologram patterns (including the ones shown) were pre-calculated using a direct-binary-search algorithm and then successively displayed on the SLM.

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We believe the generation and control of such pre-determined 3D 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, it is already known that hepatocytes cultured as a monolayer quickly loses specific functions, whilst those same cells in a quasi 3D collagen matrix (mimicking their natural structure) retain these same function in culture for periods of several weeks [24]. Tweezers could be used to engineer new tissue templates which could be allowed to gel or set before being removed from the tweezers. Utilising the ability to organize one or more cell types into spatially well-defined 3D arrays may help in tissue replacement therapies.