A ferroelectric liquid crystal spatial light modulator is used to generate up to 24 independently controllable traps in a holographic optical tweezers system using time-multiplexed Fresnel zone plates. For use in biological applications, helical zone plates are used to generate Laguerre-Gaussian laser modes. The high speed switching of the ferroelectric device together with recent advances in computer technology enable fast, smooth movement of traps that can be independently controlled in real time. This is demonstrated by the trapping and manipulation of yeast cells and fungal spores.
© 2006 Optical Society of America
Optical tweezers were introduced in the 1980’s  and are conceptually very simple. Single trap optical tweezers are capable of trapping and manipulating a single microscopic particle using the forces generated by a tightly focussed laser beam, typically using a microscope objective with a high numerical aperture. The same microscope objective is then used to image the trapped particle. This has applications in cell biology and colloid science . Dual trap tweezers  can be constructed by splitting the laser beam in two using a polarising beam splitter and mechanically steered galvo-mirrors can be used to move the resulting optical traps. If more than 2 traps are required, the only practical approaches are “time-sharing” a single laser beam  or the use of holographic optical tweezers . To perform time sharing, fast deflectors are used to move a single laser beam at high frequencies of up to several kHz between several trap sites.
Holographic optical tweezers use a spatial light modulator (SLM) placed in the optical path of the laser beam to generate a hologram . Typically, holographic diffraction gratings  are written to the SLM which are capable of generating fixed pattern arrays of points, each point corresponding to a laser focus. Such systems are not capable of independent control of individual traps. Other systems use computer generated holograms using various algorithms  to generate the required laser focuses. This is computationally very demanding especially where fast movement is required. Alternative approaches such as generalised phase contrast  or spatial multiplexing of Fresnel zone plates  greatly reduce the computational complexity to individually control multiple traps.
When using monochromatic laser light, a Fresnel zone plate  displayed on the SLM behaves as a lens which can form an optical trap when combined with a high power microscope objective. The laser focus can then be electronically translated in 3 dimensions by altering zone plate position and focal length. One drawback of binary devices such as our SLM compared to multilevel SLMs is their limitation to symmetric diffraction patterns. By using zone plates rather than diffraction gratings, the symmetric orders are moved to a different focal plane and can therefore be ignored. Since zone plates are mathematically simple, display and movement is not computationally demanding. To display overlapping images, time shared multiplexing of binary holograms is used which is made possible by the fast switching properties of the ferroelectric SLM .
The holographic optical system was constructed using a 24-bit ferroelectric liquid crystal microdisplay. This device is capable of trapping and independently manipulating up to 24 microscopic particles by using each bit plane as a binary hologram. Although holographic optical tweezers are not a new concept, the use of a multiplexed ferroelectric device is a more unusual approach when compared to existing holographic systems.
The software used to generate the holographic images was developed in java OpenGL. The resulting program was capable of very fast smooth animation. Fresnel zone plates were used for trapping colloidal particles and for biological samples, helical zone plates were used to generate Laguerre-Gaussian (LG) laser modes. Preliminary tests were performed on living yeast cells and fungal spores to determine the damage caused by trapping using standard Gaussian and LG modes with various different orders.
In , 3 independent images can be multiplexed on the FLC device using red green and blue channels. By reprogramming the SLM to display equal intensity bit planes, we are now able to exploit the full capability of the SLM by multiplexing a maximum of 24 images. This allows, for the first time, 24 independent time-multiplexed LG-beam traps with a holographic system. The advantage over existing systems is that the software can run smoothly and in real time on an average specification desktop computer. The system in  would require large computational power to control 24 independent LG traps without multiplexing and physical time sharing  would lack the versatility to switch between various modes of LG.
2. The FLC spatial light modulator and holographic tweezers setup
The main advantage of Ferroelectric (F-LCD) over Twisted nematic (TN-LCD) is the fast switching speed. The switching speed of F-LCD devices are typically 50-100 microseconds which is about a thousand times faster than a TN-LCD . One disadvantage is that voltages must be balanced in an F-LCD to prevent damage to the smectic liquid crystal. This is done by writing forward and inverse frames to the device at twice the bit plane rate and illuminating only on the positive frame. Although a continuous laser is used as illumination in the holographic optical tweezers, both the forward and inverse frames can be used since reversal of the zones has no effect on the intensity of the end result.
The ferroelectric device used is a reflective CRL Opto SXGA-R2-H1 microdisplay  which is commonly used in high-end commercial display devices. It contains an array of 1280 × 1024 reflective pixels with a pixel size of 13.62μm giving an active area of 17.43 × 13.95mm and has a colour frame rate of 60Hz giving a bit plane frame rate of 1440Hz. Illumination is via a polarising beam splitter and the orientation of the SLM determines the use of the device in amplitude or phase mode .
Figure 1 shows the full setup of the holographic optical tweezers  incorporating both bright field illumination and epifluorescence. The laser source used was a krypton ion laser wavelength 647. 1nm with power output of ~450mW. This is incident on two convex lenses (microscope objective and doublet lens shown in Fig. 1) which expand the incoming beam so that the diameter of the beam incident on the SLM makes full use of the size. The resulting beam was then passed into a ~ 2 × beam reducer for an optimal beam size for entering the 5mm rear aperture of the microscope objective. A dichroic mirror was used to separate the red laser light from the blue/green FITC fluorescence. An additional green filter is used to eliminate laser light reaching the camera.
3. Multiplexing and bit plane assignments
Each of the individual 24-bit planes can be used to display individual monochromatic images, in this case binary holograms. In a display device, for each pixel there are 3 channels; red green and blue each with 8-bit planes of increasing intensities (in powers of 2). These are addressed sequentially. Each bit plane is shown in order at a very fast speed of 1440Hz which gives the visual effect of a solid colour, this gives 224 ~ 16.8M different colour combinations. These different colours are formed by a fast switching RGB light source  and the different intensities displayed by showing each bit plane for a different amount of time. For the purposes of holographic optical tweezers, the SLM control unit was reprogrammed to display equal time bit planes. The device is now capable of multiplexing 24 independent binary holographic images just by receiving an input of a single image with the appropriate ‘colours’. The control of the individual zone plates is now purely a display issue. Each zone plate must be assigned an integer number of bit planes. There are 24 bit planes in total therefore, for the power to be evenly distributed between the traps the number of traps must be a factor of 24 so that each trap can use the same number of bit planes. This gives the possibility of 1,2,3,4,6,8,12, or 24 independent traps.
In order to achieve stable trapping each individual trap site must be revisited before the particle has had time to move by more than the trap radius . For static traps, the bead moves only by diffusion, but if the sample is moved this requirement becomes more stringent. Molloy  calculates the minimum acceptable frame rate for a high drag velocity of 200μmsec-1 for 2 traps to be 500Hz (for a typical sample of colloidal latex beads). This is easily obtainable with our SLM by alternating bit planes between traps. More traps require proportionally higher frame rates. Therefore 4 traps should be possible at this velocity but 24 traps will require reduced drag velocities.
4. Qualitative analysis of multiplexed output
To analyse the form of the multiplexed output the required laser focus(es) from the SLM were projected onto a fast photodiode. Figure 2(a) shows a single trap output where all 24 bit planes are identical. It can be seen that each bit plane appears as a single period of a small sine-like ripple and there are 24 of these per cycle. Each full cycle is indicated by a drop in intensity to zero output. As expected the period of a full cycle is ~ 16.6ms in accordance with the 60Hz frame rate specified.
Figure 2(b) shows 24 individual traps in the same vicinity but displaced slightly so that a redraw is required between traps. Each bit plane is now isolated by an inverted spike. Varying intensities shown are caused by these slight variations in position and quantisation effects on the display.
5. Laguerre-Gaussian laser modes
Laguerre-Gaussian (LG)  laser modes are commonly used in optical tweezers when trapping delicate biological samples that are easily damaged by excess laser power. In most cases, LG beams have improved trapping efficiency over standard Gaussian beams and produce significantly less heating as the laser power is not concentrated on one spot due to their doughnut shape . An LG optical field contains orbital angular momentum which is a result of a phase singularity. This produces a helical wave front as opposed to the plane wave front of a Gaussian mode. There are several ways to generate these laser modes, some involving modification to the laser itself. For holographic optical tweezers, the easiest and most obvious way is to modify the zone plates to generate the LG beams. These modified binary zone plates were first computer generated by Heckenberg et al by calculating the interference between a generalised LG laser mode and a plane wave . This results in the boundary between the 2 levels being given by:
where k is the wavenumber = 2π/λ, r is the radial distance in the cylindrical sense, R is the radius of curvature of the beam (the effective focal length). l is the ‘order’ of the LG laser mode, a parameter which can be used to control the size and definition of the doughnut shaped laser focus. l = 0 corresponds to a standard Fresnel zone plate. For each value of l, there are 2l distinct solutions, for example l = 3 has 6 distinct spirals e.g. n = 0,1,2,3,4,5 which generates a triple helix LG beam.
6. Rendering zone plates in JOGL and displaying
In java OpenGL (JOGL), graphics are rendered using convex polygons. These polygons are translated onto the display using a dedicated hardware graphics processing unit (GPU) commonly found on graphics cards today. This runs significantly faster since GPUs are typically very powerful and perform this task far more efficiently (than software rendering) while greatly reducing the demand on the computer’s CPU . To generate holograms, the pattern was divided into ring segments drawn on a black background with each segment approximated by a polygon. The more polygons per ring used, the more smooth the pattern appears as shown in Fig. 3. This gives the option of sacrificing zone plate quality (and possibly trapping efficiency) for speed.
To generate the l > 0 zone plates equation 1 was rearranged to give
This gives a relation between r and θ for the boundary between light and dark fringes. R was chosen to correspond to the same R as the l = 0 zone plate. To cover all the unique solutions, consecutive integer values of n were chosen from 0 to 2l and light fringes were drawn between the spirals given by n = 0 and 1, 2 and 3, 4 and 5 etc.
The JOGL program was written to allow mouse control of the individual traps. The interface allows switching between different numbers of traps and LG order. Figure 3(c) shows multiple zone plates as displayed on a standard computer monitor. This will appear differently on the SLM since a monochromatic laser is used and each bit plane is reprogrammed to give equal intensities. This demonstrates the way in which this device can combine bit planes from different colour channels since it shows 2l = 2 zone plates using 12 bit planes each.
7. Trapping of colloidal spheres
Figure 4 shows trapped particles of 1.7μm PMMA spheres with fluorophore: RITC and NBD-MAEM in decalin, viewed under a 100× microscope objective using fluorescence. This demonstrates the ability to independently control trapped particles. Figure 4(a) shows the results of 12 traps in a circle configuration. Figure 4(b) is a partial smiley face configuration showing 20 out of the 24 traps available. 24 stable traps could not be achieved due to insufficient laser power to overcome Brownian motion. This is also due to the small refractive index difference of Δn = 0.03 between particles and surroundings. It must be noted that this particular setup requires a large amount of laser power to compensate for the large power losses involved in a holographic system. Main losses include; non-optimal switching angle of the SLM and the use of binary instead of multilevel holograms, which distribute the laser power to unused orders of focal points.
The ability to control microscopic objects in three dimensions has many novel applications. Leach et al  demonstrates the use of holographic optical tweezers to form three dimensional crystal like structures and to rotate these about x, y and z axes. For our holographic system to control traps in three dimensions, the control program also allows variation in the focal length of each zone plate .
Figure 4(c) shows a line of 6 trapped particles in the same focal plane. Figure 4(d) shows the same 6 particles in a line but shifted in the z direction so that they lie on a slope. The relationship between Δz, the change in z-displacement and Δfzp, change in focal length of the zone plate to a good approximation, was calculated as where fobj = 2mm is the objective focal length, f 1 and f 2 are focal lengths of lens 1 (250mm) and lens 2 (120mm). Figure 4(d) has calculated z-displacements of each particle from the image plane superimposed. Considering that the diameter of the particles are 1.7 μm, this is consistent with what can be seen.
8. Laguerre-Gaussian trapping of biological samples
To demonstrate the ability of the holographic tweezers to handle sensitive biological samples, the tweezers were used to trap and control the positions of yeast cells of Sacchromyces cere-visiae and spores of Neurospora crassa, which are both important model systems in cell biology, genetics and molecular biology. Figure 5 shows four trapped yeast cells which were moved closer together and their configuration changed. Figure 6 shows six trapped yeast cells which were moved apart from each other. Ameborg et al  demonstrated a practical application of using a similar trap arrangement in which the growth of yeast cells of Hanseniaspora uvarum was stimulated by surrounding them with yeast cells of S. cerevisiae.
The issue of potentially damaging cells with laser radiation was assessed. Spores of N. crassa maintained under liquid medium were allowed to adhere to a cover glass surface allowing the plane of focus of the laser trap to be set directly at the centres of the adherent spores. Individual spores were irradiated with a laser beam with an output power of 200mW (which is sufficient for trapping), for variable periods of time and for l = 0 or 1, at 22°C. Once irradiated, the spores were then left for 4h, without laser exposure and then germination, as a measure of cell viability, was quantified. With continuous irradiation for periods <5min with 1=1, ~87% of the spores (n = 28) germinated. With similar periods of irradiation with 1 = 0, ~18% of the spores (n = 11) germinated. These results suggest that the LG mode of laser beam is less deleterious than the standard Gaussian mode.
A ferroelectric liquid crystal display device was used in generating holographic optical tweezers that are capable of 24 time multiplexed, independently controllable traps with Laguerre-Gaussian laser modes. Time-multiplexing permits multiple traps with minimal computational power since holograms for individual traps are pre-calculated. This allows for smooth movement and control over the traps in real time. This is demonstrated by trapping and manipulation of colloidal particles and biological cells. Preliminary results indicate that LG modes of order 1 = 1 are less deleterious than the standard Gaussian mode when manipulating fungal spores and yeast cells.
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