We demonstrate the use of a spatial light modulator (SLM) to facilitate the trapping of particles in three-dimensional structures through time-sharing. This method allows particles to be held in complex, three-dimensional configurations using cycling of simple holograms. Importantly, we discuss limiting factors inherent in current phase only SLM design for applications in both optical tweezing and atom trapping.
©2003 Optical Society of America
Optical tweezers can trap and manipulate microscopic particles through the interaction of light and matter. Ashkin demonstrated guiding and trapping using radiation pressure forces  and this was subsequently followed by the discovery that a focused light beam, by refraction, can hold a microscopic particle in three dimensions near the beam focus . Since then optical tweezers have evolved and become an important and versatile tool in both the physical and life sciences. Holographic optical tweezers have facilitated the use of multiple trapping sites, greatly extending the applications of optical tweezers . Multiple optical trapping sites in a two-dimensional plane can give insights into magnetic flux line pinning , particle dynamics on a washboard potential  and phase transitions in two dimensions . The ability to extend optical tweezers to create arbitrary structures in three dimensions is a current significant challenge; its solution will enable the next generation of experiments in biological and colloidal science.
Holograms allow a single laser beam to be transformed into a number of independent traps. Encoding computer generated holographic patterns onto a spatial light modulator (SLM) allows us to tie this advantage with the reconfigurability and dynamic control offered by these devices. In this paper we demonstrate the use of a nematic SLM to create three-dimensional arrangements of microscopic particles through time-sharing. This technique relies on the fast update speeds of the SLM providing us with a simple, powerful and versatile method of constructing three-dimensional structures.
Sending a stream of holograms to the SLM allows us to create dynamic patterns and hence we can time-share between trap sites. This means that we can create, for example, a dual beam trap by switching between the holograms for each of the individual trap sites. If the frequency of this switching is high enough then the particle diffusion time will be too low to enable it to leave the trap site before the trapping light returns to that position. Time-sharing is a well established technique for creating arrays of trap sites, usually relying on acousto-optic deflection techniques (AOD) [7, 8] to provide the beam deflection and rapid switching. The use of AODs provides the ability to trap many tens of particles since the switching time of these devices is very high (e.g., kilohertz). However, although there has been some work  using time-sharing AOD techniques to create three-dimensional structures, this has been limited to two planes created one after the other. Time-sharing with an SLM has previously been used to create multiple trap sites in two dimensions using a ferroelectric SLM . Here the multiple bit-planes of the ferroelectric device were used to create both static and dynamic holograms simultaneously. These ferroelectric devices have relatively high refresh rates but low efficiency compared to the nematic devices used here.
Holograms can be used to produce, among other things, an array of trap sites. The holograms are calculated using the Gerchberg-Saxton (GS) algorithm [11–13] as shown in . The trap sites created by these holograms all lie in the same xy plane (the focal plane of the objective assuming the incoming light beam is collimated). We can also create a lens function, for display on the SLM, according to:
where Φz() is the phase modulation imposed on the beam, denotes a radial position in the diffractive optic element’s (DOE) aperture, z is the desired displacement of the optical trap(s) relative to the focal plane in an optical train (including relay optics and objective lens) with effective focal length f and λ is the wavelength. The lens function has the effect of moving the beam focus in z, either in the positive or negative direction. Adding the lens to another hologram will therefore simply translate the generated pattern.
Three-dimensional arrangements of particles have been demonstrated by other groups using SLMs. However, the methods used to achieve these structures differ from those shown here. For instance Curtis et al.  also demonstrate the use of a Fresnel lens, but with a different lens applied to each individual trap site. A range of ±5µm in z is achieved using a single shot hologram but the calculations required are more complex than those used here. Bingelyte et al.  have successfully demonstrated the creation of three-dimensional structures using a Fresnel lens with multiple focal planes. Using this method they have managed to move two spheres around each other in the x, y or z planes in a tumbling motion. We also note that recent work by Leach et al.  using binary search algorithms offers powerful methods for the creation of three-dimensional structures.
In this paper we demonstrate that, through the use of lensed holograms, we can create three-dimensional arrays via time-sharing. In doing so we illustrate some important issues regarding the speed and efficiency of dynamic holographic tweezers and their applications, not only for optical tweezers but also in atom optics .
An optical tweezers setup (as shown in Fig. 1) was used for this experiment, with the laser beam expanded to fill the active area of the SLM. This allowed us to control and update the beam pattern as described previously. In order to demonstrate the accessibility of this three-dimensional trapping technique we used two different setups with two different phase only SLMs. The first SLM is a Hamamatsu PPMX8267 optimised for light of wavelength 532nm. The light from the SLM was passed into an Olympus IX71 research microscope, which acted as an inverted tweezer setup (similar to that in fig. 1 but with the objective below the sample). The second SLM is a Boulder Nonlinear Systems 5128 optimised for light of wavelength 1064nm. The light from this SLM was passed through a homemade tweezer system as shown in figure 1. The Hamamatsu SLM is optically addressed and has a 768×768 pixel display with 255 phase levels. The Boulder SLM is electrically addressed and has a 512×512 pixel display with 128 phase levels, although a non-linear response reduces the effective number of phase levels significantly. The refresh rate of the Boulder is nominally higher than the Hamamatsu, 75Hz compared to 10Hz .
As outlined above we are using a time-share method to allow us to trap particles in three-dimensional configurations. The holograms are calculated for each plane and cycled. For the Hamamatsu this is done by creating an MPEG file with each frame of the movie corresponding to a plane in the structure. The computer interface used with the Boulder SLM allows us to run the images in sequence directly and at a chosen frequency.
For the experiments using the Hamamatsu SLM 2µm silica spheres suspended in a water and detergent solution were trapped. The first configuration consisted of just two particles in separate planes, as shown in Fig. 2(a). More complicated configurations were also created, as shown in Figs. 2(b) and (c).
The limitations to the patterns that can be created are due to the refresh rate of the SLM. As the number of planes is increased then the dark time at any one plane increases and the hologram no longer has sufficient power to trap the particles. The trapping planes are separated by 3µm from the centre of one sphere to the centre of one in the next plane. As more planes are attempted the particles can be seen to be dropped and picked up again as the holograms cycle. This prevents good trapping from occurring in more than two planes in this system. However, complex patterns in two planes, such as the pyramids shown in Figs. 2(b) and (c) are achievable.
For the Boulder system 2.3µm silica spheres suspended in water were used. These were successfully trapped in configurations as shown in Fig. 3. Further to this it was possible to trap in up to six separate planes, as shown in Fig. 4.
The higher refresh rate of the Boulder SLM means that it is possible to trap in a larger number of planes than with the Hamamatsu system, as shown in Fig. 4. This should allow the creation of more complex patterns and structures. As the number of trap sites is increased the trapping potential decreases, therefore the particles move slightly between successive cycles of the holograms.
The higher refresh rate of the Boulder SLM is advantageous as the number of holograms being used is increased. However, despite our Boulder SLM being rated as running at 75Hz and possibly even greater, in practice this was not the case. The results shown above (Fig. 4) were achieved at approximately 10Hz and we found that trapping in more than six planes was difficult. The reason for this is that the SLM is designed for use with light of wavelength 1064nm, therefore the liquid crystal layer is quite thick. This thickness impedes the response time of the SLM resulting in ‘clipping’, see Fig. 5. A slower refresh rate of around 10Hz is therefore the maximum frequency achievable for this SLM. If we run the SLM at higher frequencies the full phase modulation is not available, the efficiency drops and the holograms are not properly formed. This currently places severe restrictions on their use in any sort of dynamic application. Although we believe that the use of a wavelength requiring a thinner liquid crystal layer would allow a greater range of possible refresh rates, many optical tweezer experiments use near-infrared light and therefore for any such experiment requiring dynamic reconfigurability the use of SLMs is limited. This result also has implications for applications in atom optic experiments discussed in our earlier paper . In this paper it is assumed that the electrically addressed SLMs will provide fast refresh rates to facilitate their use with cold atoms. The problems encountered with rise times and clipping could reduce the effectiveness of these SLMs in this field.
As we might expect based on the loss of full phase modulation at high refresh rates the achievable displacement of the trap sites is decreased when we are time sharing. Using a static hologram to displace a particle in z, with the lens function, it is possible to move through a range in excess of ±8µm comfortably. Time-sharing reduces this range to only ±3µm.
The techniques available for creating three-dimensional structures in optical tweezers are currently limited by the refresh rates of the SLMs. Previously interference patterns have allowed for the creation and rotation of crystal type arrangements . Spatial light modulators have also been used to create three-dimensional structures [14–16]. However, this was done using a single hologram, although time-sharing has been demonstrated for two-dimensional arrays using a ferroelectric SLM . In this paper we have shown that this technique may be extended into three dimensions using a sampling lensing effect generated by the SLM. Our main conclusion is that operating in the near-infrared SLMs have consistently lower refresh rates than specified by the manufacturers. This has obvious disadvantages for the applications demonstrated in this paper where the higher refresh rate prevents particles from being dropped by a time-varying trap site. However, it looks likely that phase only SLMs will see improvements over the next few years and as such their use in applications discussed may become both viable and desirable.
D.M. is a Royal Society University Research Fellow. G.C.S. is supported by an award from the Research Corporation and by the National Science Foundation through Grant No. DMR-0216631. This work is supported by the UK’s EPSRC.
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