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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 8 — Apr. 19, 2004
  • pp: 1665–1670
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Interactive application in holographic optical tweezers of a multi-plane Gerchberg-Saxton algorithm for three-dimensional light shaping

Gavin Sinclair, Jonathan Leach, Pamela Jordan, Graham Gibson, Eric Yao, Zsolt John Laczik, Miles J. Padgett, and Johannes Courtial  »View Author Affiliations


Optics Express, Vol. 12, Issue 8, pp. 1665-1670 (2004)
http://dx.doi.org/10.1364/OPEX.12.001665


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Abstract

Phase-hologram patterns that can shape the intensity distribution of a light beam in several planes simultaneously can be calculated with an iterative Gerchberg-Saxton algorithm [T. Haist et al., Opt. Commun. 140, 299 (1997)]. We apply this algorithm in holographic optical tweezers. This allows us to simultaneously trap several objects in individually controllable arbitrary 3-dimensional positions. We demonstrate the interactive use of our approach by trapping microscopic spheres and moving them into an arbitrary 3-dimensional configuration.

© 2004 Optical Society of America

1. Introduction

Optical tweezers [1

1. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986). [CrossRef] [PubMed]

] have recently undergone a revolution made possible by the advent of commercially available spatial light modulators (SLMs) [2

2. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003). [CrossRef] [PubMed]

]. Optical tweezers comprise a tightly focussed laser beam, which can “trap” microscopic objects at its focus. This can happen through one of various mechanisms; for example, dielectric materials (e.g. biological cells, silica spheres) experience a force towards the strong electric field at the focus where the laser is brightest. Optical tweezers have been used very widely, for example to manipulate individual biological cells [3

3. A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared-laser beams,” Nature 330, 769–771 (1987). [CrossRef] [PubMed]

] and to probe fundamental effects like the rotation of microscopic objects induced by the orbital angular momentum of the laser light [4

4. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct Observation of Transfer of Angular Momentum to Absorbtive Particles from a Laser Beam with a Phase Singularity,” Phys. Rev. Lett. 75, 826–829 (1995). [CrossRef] [PubMed]

].

Many applications require the simultaneous trapping of more than one object in different positions. For example, recently objects optically trapped in 2-dimensional (2D) patterns were used as a pump [5

5. A. Terray, J. Oakey, and D. W. M. Marr, “Microfluidic control using colloidal devices,” Science 296, 1841–1844 (2002). [CrossRef] [PubMed]

] for microfluidic applications. The required configurations of bright spots (foci) can be created by placing a suitable (computer-generated) hologram in the laser beam. In the form of SLMs, such holograms become reconfigurable – in our case, a computer can turn our SLM into any arbitrary phase hologram (within resolution limits). Here we describe and demonstrate experimentally the application of an algorithm by which the phase hologram pattern required to create arbitrary 3D configurations of traps can be calculated [6

6. T. Haist, M. Schönleber, and H. J. Tiziani, “Computer-generated holograms from 3D-objects written on twisted-nematic liquid crystal displays,” Opt. Commun. 140, 299–308 (1997). [CrossRef]

]. In our implementation, the algorithm is fast enough to be used interactively.

2. Holographic optical tweezers

Optical tweezers that comprise a hologram (static or – as in the case of an SLM – reconfigurable) to shape the trapping beam are called holographic optical tweezers (HOTs). The hologram is almost always a phase hologram (the case we consider here), as intensity holograms decrease the power in the trapping beam. HOTs allow the beam to be split up into more than one trap at different positions, and to change the shape of the trap (e.g., [4

4. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct Observation of Transfer of Angular Momentum to Absorbtive Particles from a Laser Beam with a Phase Singularity,” Phys. Rev. Lett. 75, 826–829 (1995). [CrossRef] [PubMed]

, 7

7. J. Arlt and M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: an optical bottle beam,” Opt. Lett. 25, 191–193 (2000). [CrossRef]

, 8

8. V. Bingelyte, J. Leach, J. Courtial, and M. J. Padgett, “Optically controlled three-dimensional rotation of microscopic objects,” Appl. Phys. Lett. 82, 829–831 (2003). [CrossRef]

]) or traps [9

9. J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002). [CrossRef]

].

HOTs first used static holograms, for example to shape the trapping beam into a doughnut mode possessing orbital angular momentum that makes trapped particles rotate [4

4. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct Observation of Transfer of Angular Momentum to Absorbtive Particles from a Laser Beam with a Phase Singularity,” Phys. Rev. Lett. 75, 826–829 (1995). [CrossRef] [PubMed]

]; to split the focus in arbitrary 2D patterns of foci and trap particles there [15

15. E. R. Dufresne and D. G. Grier, “Optical tweezer arrays and optical substrates created with diffractive optics,” Rev. Sci. Instr. 69, 1974–1977 (1998). [CrossRef]

]; or to shape the focus into a “light bubble” that should allow trapping of metallic particles [7

7. J. Arlt and M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: an optical bottle beam,” Opt. Lett. 25, 191–193 (2000). [CrossRef]

]. Computer-controlled holograms were first used in optical tweezers to trap several objects simultaneously with interactive control over the individual trap positions [16

16. M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. 24, 608–610 (1999). [CrossRef]

]. The individual traps could also be shaped (for example into doughnuts), but the trapping positions were restricted to a single transverse plane, that is to 2 dimensions. The corresponding phase hologram pattern was calculated simply by superimposing in a specific way the holograms that create the individual shaped and displaced traps. This method was computationally very fast, but it also created unwanted “ghost” traps. Beam shaping that led to higher-quality traps was soon achieved by using a phase-contrast technique [17

17. P. C. Mogensen and J. Glückstad, “Dynamic array generation and pattern formation for optical tweezers,” Opt. Commun. 175, 75–81 (2000). [CrossRef]

]. As before, trapping positions were restricted to a plane, the traps could be individually shaped, and the trap positions could be controlled interactively. Interactive control over 3D trapping positions in HOTs was first achieved by generalising the algorithm used in ref. [16

16. M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. 24, 608–610 (1999). [CrossRef]

] to 3 dimensions, that is by adding lens holograms to the individual trap holograms before superimposing [18

18. J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, “Multi-functional optical tweezers using computer-generated holograms,” Opt. Commun. 185, 77–82 (2000). [CrossRef]

]. This worked surprisingly well, but again suffered from ghost traps. This problem was reduced by the use of an iterative algorithm – a modification of the Gerchberg-Saxton (GS) algorithm – for the calculation of the hologram patterns [9

9. J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002). [CrossRef]

]. This allowed dynamic simultaneous trapping of large numbers (e.g., 200) of objects.

Here we use a different modification of the GS algorithm [6

6. T. Haist, M. Schönleber, and H. J. Tiziani, “Computer-generated holograms from 3D-objects written on twisted-nematic liquid crystal displays,” Opt. Commun. 140, 299–308 (1997). [CrossRef]

] to calculate the hologram patterns for the light shaping required 3D optical trapping. This GS algorithm is well suited for use in holographic optical tweezers, as it offers a combination of speed and quality: the algorithm quickly calculates a useful hologram pattern during its first iteration and refines this pattern during subsequent iterations. We demonstrate interactive simultaneous control over the 3D positions of several trapped objects.

3. Gerchberg-Saxton algorithm for intensity shaping in one plane

The GS algorithm [19

19. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

] was originally developed to infer an electron beam’s phase distribution in a transverse plane, P, from intensity distributions in plane P and in a second plane, Q. It can also be applied to light, specifically to find a phase distribution that turns a given intensity distribution IP in plane P into a desired intensity distribution IQ in plane Q (both IP and IQ are functions of x and y). We consider here the common case of plane P being the plane of the phase hologram, IP being the intensity distribution of the illuminating laser beam, and plane Q being the far field.

Consider starting with a beam with the given intensity distribution IP and flat phase in plane P. The complex amplitude (i.e. phase and intensity) of the beam at Q, AQ, is fully determined by the complex amplitude at P, AP: AQ is simply the Fourier transform of AP. Usually the intensity distribution at Q, |AQ|2, is not the desired one, namely IQ. Naïvely, this can be “fixed” by replacing the intensity distribution at Q with IQ, while keeping the phase distribution. This change to AQ in turn affects AP – the inverse Fourier transform of AQ -, and now the intensity distribution at P is not the given one, IP, any longer. Again, we replace the “wrong” intensity with the “correct” one (in this case IP) while keeping the new phase. The process is repeated, that is the beam is calculated in alternate planes and the intensity there replaced with the correct one. The process converges, and usually within a few iterations the phase distribution at P, when imposed (with a phase hologram) on a beam with intensity IP in plane P, produces a good approximation to intensity IQ in plane Q. This GS algorithm is also known as error reduction algorithm [20

20. V. Soifer, V. Kotlyar, and L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Taylor & Francis Ltd, London, 1997).

], as each iteration decreases the errors between the desired and actual intensity distribution [21

21. B. Lin and N. C. Gallagher, “Convergence of a spectrum shaping algorithm,” Appl. Opt. 13, 2470–2471 (1974). [CrossRef]

]. Figure 1 a shows a flowchart representation of the GS algorithm.

Fig. 1. Flow chart for the GS algorithm for intensity shaping (a) in one plane, Q, and (b) in several planes, shown here for 2 planes, Q 1 and Q 2. The arrows marked PQi and QiP respectively indicate the mathematical operations that transform the amplitude profile in plane P into that plane Qi and vice versa (usually forward and inverse Fourier transforms). The current amplitude profile is always represented in terms of its intensity, I(x, y), and phase, Φ(x, y); the desired intensity profiles in planes P and Qi are respectively labelled IP and IQi. The circled phase profile (in the plane P) converges to the phase-hologram pattern.

4. … and in multiple planes

We are especially concerned with the optical-tweezers-inspired problem of generating patterns of bright spots in different planes. In the 2D algorithm described above, the phase distribution in plane Q is retained while the intensity distribution is replaced with the desired one. It is useful here to think of this step in terms of placing tiny mirrors of variable reflectivity in the positions of the spots; the reflectivity of each mirror is changed so that the power reflected from that mirror is proportional to the desired intensity of the corresponding bright spot.

This image is very easily extended to spot patterns in any number of planes: the tiny mirrors are again at the corresponding spot positions, which are now no longer all in the far-field plane but in any 3D arrangement. At the hologram, the amplitude distributions of the light reflected from each individual mirror simply add up. The remainder of the procedure remains unchanged: the resulting intensity at the hologram is replaced with that of the illuminating beam (while not altering the phase), this new beam is then again bounced off the tiny mirrors, and so on. This algorithm was first described in Ref. [6

6. T. Haist, M. Schönleber, and H. J. Tiziani, “Computer-generated holograms from 3D-objects written on twisted-nematic liquid crystal displays,” Opt. Commun. 140, 299–308 (1997). [CrossRef]

]. Figure 1(b) shows a flow diagram of the algorithm.

The multi-plane algorithm now requires the light to be propagated not just into the far field and back, but into any arbitrary plane. This is not difficult (see, for example, [22

22. E. A. Sziklas and A. E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain. 2: Fast Fourier transform method,” Appl. Opt. 14, 1874–1889 (1975). [CrossRef] [PubMed]

]), but it is significantly more time-consuming than a single Fourier transform.

Figure 2 shows a light beam designed to have intensity cross sections in the shape of the characters “1”, “2” and “3” in different transverse planes. The intensity distributions produced by the hologram are clearly recognisable as the desired characters. However, they are not the desired uniform-intensity characters, but have instead the speckle-structure common to many beams shaped by pixellated holograms. In addition, the light forming the desired pattern in one plane is also present in all the other planes, although less intense.

This algorithm is well suited for application within interactive HOTs because it calculates useful holograms for the creation of point patterns in a single iteration: each iteration is mathematically very similar to the fast and simple lens-superposition algorithm in Ref. [18

18. J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, “Multi-functional optical tweezers using computer-generated holograms,” Opt. Commun. 185, 77–82 (2000). [CrossRef]

]. However, compared to a repetition of the lens-superposition algorithm for use in dynamic HOTs, the iterative multi-plane GS algorithm offers greater flexibility (each trapping focus can, for example, be shaped into a ring by simply changing the target intensity profile accordingly) and – particularly for shaped foci – subsequent iterations improve the hologram pattern instead of calculating the same pattern repeatedly.

Fig. 2. Beam shaped in three transverse planes. The main series of images shows modelled intensity distributions (over a 7mm×7mm area) in transverse planes at various distances z behind the hologram. The wavelength is λ=633nm. At z=20cm, 50cm and 100cm the intensity takes on the shape of the characters “1”, “2” and “3”, respectively. The phase hologram (calculated with the multi-plane GS algorithm; a grayscale representation is shown) is illuminated by a collimated Gaussian beam. The additional images show experimental results, which were obtained by illuminating a commercial SLM (Boulder Nonlinear Systems 512×512 SLM system), acting as the shown phase hologram, with the widened beam from a HeNe laser.

5. Interactive experimental demonstration in optical tweezers

To use our algorithm interactively, we developed a computer program that runs continuously through the loop shown in Fig. 1(b), while the desired intensity distributions in arbitrarily many transverse planes, as well as the number of these planes and their z positions, can be altered interactively. In our software, the intensity distributions consist of a number of points, whose number and brightness can be controlled over the keyboard and whose positions can be moved by mouse control; altering any of the trap positions (or even adding/deleting traps or trapping planes) does not reset the hologram pattern; instead the GS-algorithm loop simply keeps running with a changed target intensity pattern consisting of bright points in the new trap positions. The SLM always displays the latest phase distribution in the hologram plane. Our software does not use any further optimizations.

Figure 3 demonstrates the application in optical tweezers. It shows two examples of 3D trapping and rearrangement of multiple 2µm-diameter silica spheres. In the first example four spheres were moved out of a planar array of 13 spheres (all optically trapped at all times) and arranged into a line in a different plane. The trapping positions of the four spheres were changed interactively, one at a time. In the second example 6 spheres are manipulated individually and as two groups of 3 spheres; at one point several traps are directly above other traps without the trapped spheres being caught by the trap above or below. In that example the sequence of hologram patterns was pre-calculated. The optical tweezers arrangement was based on a NA 1.3, ×100, Zeiss Plan Neofluar oil immersion microscope objective in an inverted geometry with a Laser Quantum Excel 1500 frequency-doubled Nd:YVO4 laser, supplying a 1.5W beam at 532nm. The laser beam was expanded and reflected off a computer-controlled phase-only SLM based on a Hamamatsu PAL-SLM X7665. A 2.9GHz dual-processor computer calculated the desired holographic patterns and displayed them on the SLM. With the hologram pattern and the beam amplitude represented on a 256×256 grid, each iteration of the algorithm took ≈0.5s – fast enough for interactive use.

Fig. 3. [2.1MB], [1.6MB] Interactive (a) and pre-programmed (b) 3-dimensional (3D) rearrangement of optically trapped 2µm-diameter glass spheres. The 3D positions of all the spheres are indicated underneath each video frame of the view through the microscope during different stages of the re-arrangement; all spheres (moving or stationary) were optically trapped at all times. In example a the positions of the individual spheres were changed interactively in the computer program, which then calculated the required hologram patterns in real time. This took about 3 minutes (the movie is speeded-up by a factor of ≈5); the speed was limited by the time it took to enter the new positions into the program. In (b) 100 hologram patterns, which had been pre-calculated with the multiplane-GS algorithm, were displayed on the SLM over a period of 15s (the times at which individual frames were taken are indicated at the top of each frame; the movie runs in real time).

6. Conclusions

We have successfully applied the multi-plane GS algorithm in HOTs. This approach allows full control of the trap positions in 3D, and can run fast enough on current computers to be used interactively.

Acknowledgments

Thanks to Gabriel Spalding for very useful discussions. ZJL and JC are Royal Society University Research Fellows. We gratefully acknowledge funding from the UK Engineering and Physical Sciences Research Council (EPSRC) and from the Glasgow-Strathclyde Synergy Fund.

References and links

1.

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986). [CrossRef] [PubMed]

2.

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003). [CrossRef] [PubMed]

3.

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared-laser beams,” Nature 330, 769–771 (1987). [CrossRef] [PubMed]

4.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct Observation of Transfer of Angular Momentum to Absorbtive Particles from a Laser Beam with a Phase Singularity,” Phys. Rev. Lett. 75, 826–829 (1995). [CrossRef] [PubMed]

5.

A. Terray, J. Oakey, and D. W. M. Marr, “Microfluidic control using colloidal devices,” Science 296, 1841–1844 (2002). [CrossRef] [PubMed]

6.

T. Haist, M. Schönleber, and H. J. Tiziani, “Computer-generated holograms from 3D-objects written on twisted-nematic liquid crystal displays,” Opt. Commun. 140, 299–308 (1997). [CrossRef]

7.

J. Arlt and M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: an optical bottle beam,” Opt. Lett. 25, 191–193 (2000). [CrossRef]

8.

V. Bingelyte, J. Leach, J. Courtial, and M. J. Padgett, “Optically controlled three-dimensional rotation of microscopic objects,” Appl. Phys. Lett. 82, 829–831 (2003). [CrossRef]

9.

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002). [CrossRef]

10.

J. E. Molloy and M. J. Padgett, “Lights, action: optical tweezers,” Contemp. Phys. 43, 241–258 (2002). [CrossRef]

11.

K. Visscher, G. J. Brakenhoff, and J. J. Kroll, “Micromanipulation by multiple optical traps created by a single fast scanning trap integrated with the bilateral confocal scanning laser microscope,” Cytometry 14, 105–114 (1993). [CrossRef] [PubMed]

12.

J. E. Molloy, J. E. Burns, J. C. Sparrow, R. T. Tregear, J. Kendrick-Jones, and D. C. S. White, “Single-molecule mechanics of heavy-meromyosin and S1 interacting with rabbit or drosophila actins using optical tweesers,” Biophys. J. 68, S298–S305 (1995).

13.

T. Ota, S. Kawata, T. Sugiura, M. J. Booth, M. A. A. Neil, R. Juškaitis, and T. Wilson, “Dynamic axial-position control of a laser-trapped particle by wave-front modification,” Opt. Lett. 28, 465–467 (2003). [CrossRef] [PubMed]

14.

D. J. Cho, S. T. Thurman, J. T. Donner, and G. M. Morris, “Characteristics of a 128×128 liquid-crystal spatial light modulator for wave-front generation,” Opt. Lett. 23, 969–971 (1998). [CrossRef]

15.

E. R. Dufresne and D. G. Grier, “Optical tweezer arrays and optical substrates created with diffractive optics,” Rev. Sci. Instr. 69, 1974–1977 (1998). [CrossRef]

16.

M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. 24, 608–610 (1999). [CrossRef]

17.

P. C. Mogensen and J. Glückstad, “Dynamic array generation and pattern formation for optical tweezers,” Opt. Commun. 175, 75–81 (2000). [CrossRef]

18.

J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, “Multi-functional optical tweezers using computer-generated holograms,” Opt. Commun. 185, 77–82 (2000). [CrossRef]

19.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

20.

V. Soifer, V. Kotlyar, and L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Taylor & Francis Ltd, London, 1997).

21.

B. Lin and N. C. Gallagher, “Convergence of a spectrum shaping algorithm,” Appl. Opt. 13, 2470–2471 (1974). [CrossRef]

22.

E. A. Sziklas and A. E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain. 2: Fast Fourier transform method,” Appl. Opt. 14, 1874–1889 (1975). [CrossRef] [PubMed]

OCIS Codes
(050.1970) Diffraction and gratings : Diffractive optics
(090.1760) Holography : Computer holography
(090.2890) Holography : Holographic optical elements
(170.4520) Medical optics and biotechnology : Optical confinement and manipulation
(230.4110) Optical devices : Modulators
(230.6120) Optical devices : Spatial light modulators

ToC Category:
Research Papers

History
Original Manuscript: February 25, 2004
Revised Manuscript: March 29, 2004
Published: April 19, 2004

Citation
Gavin Sinclair, Jonathan Leach, Pamela Jordan, Graham Gibson, Eric Yao, Zsolt Laczik, Miles Padgett, and Johannes Courtial, "Interactive application in holographic optical tweezers of a multi-plane Gerchberg-Saxton algorithm for three-dimensional light shaping," Opt. Express 12, 1665-1670 (2004)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-8-1665


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References

  1. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, �??Observation of a single-beam gradient force optical trap for dielectric particles,�?? Opt. Lett. 11, 288�??290 (1986). [CrossRef] [PubMed]
  2. D. G. Grier, �??A revolution in optical manipulation,�?? Nature 424, 810�??816 (2003). [CrossRef] [PubMed]
  3. A. Ashkin, J. M. Dziedzic, and T. Yamane, �??Optical trapping and manipulation of single cells using infrared-laser beams,�?? Nature 330, 769�??771 (1987). [CrossRef] [PubMed]
  4. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, �??Direct Observation of Transfer of Angular Momentum to Absorbtive Particles from a Laser Beam with a Phase Singularity,�?? Phys. Rev. Lett. 75, 826�??829 (1995). [CrossRef] [PubMed]
  5. A. Terray, J. Oakey, and D. W. M. Marr, �??Microfluidic control using colloidal devices,�?? Science 296, 1841�??1844 (2002). [CrossRef] [PubMed]
  6. T. Haist, M. Schönleber, and H. J. Tiziani, �??Computer-generated holograms from 3D-objects written on twisted-nematic liquid crystal displays,�?? Opt. Commun. 140, 299�??308 (1997). [CrossRef]
  7. J. Arlt and M. J. Padgett, �??Generation of a beam with a dark focus surrounded by regions of higher intensity: an optical bottle beam,�?? Opt. Lett. 25, 191�??193 (2000). [CrossRef]
  8. V. Bingelyte, J. Leach, J. Courtial, and M. J. Padgett, �??Optically controlled three-dimensional rotation of microscopic objects,�?? Appl. Phys. Lett. 82, 829�??831 (2003). [CrossRef]
  9. J. E. Curtis, B. A. Koss, and D. G. Grier, �??Dynamic holographic optical tweezers,�?? Opt. Commun. 207, 169�??175 (2002). [CrossRef]
  10. J. E. Molloy and M. J. Padgett, �??Lights, action: optical tweezers,�?? Contemp. Phys. 43, 241�??258 (2002). [CrossRef]
  11. K. Visscher, G. J. Brakenhoff, and J. J. Kroll, �??Micromanipulation by multiple optical traps created by a single fast scanning trap integrated with the bilateral confocal scanning laser microscope,�?? Cytometry 14, 105�??114 (1993). [CrossRef] [PubMed]
  12. J. E. Molloy, J. E. Burns, J. C. Sparrow, R. T. Tregear, J. Kendrick-Jones, and D. C. S. White, �??Single-molecule mechanics of heavy-meromyosin and S1 interacting with rabbit or drosophila actins using optical tweesers,�?? Biophys. J. 68, S298�??S305 (1995).
  13. T. Ota, S. Kawata, T. Sugiura, M. J. Booth, M. A. A. Neil, R. Juškaitis, and T. Wilson, �??Dynamic axial-position control of a laser-trapped particle by wave-front modification,�?? Opt. Lett. 28, 465�??467 (2003). [CrossRef] [PubMed]
  14. D. J. Cho, S. T. Thurman, J. T. Donner, and G. M. Morris, �??Characteristics of a 128�?128 liquid-crystal spatial light modulator for wave-front generation,�?? Opt. Lett. 23, 969�??971 (1998). [CrossRef]
  15. E. R. Dufresne and D. G. Grier, �??Optical tweezer arrays and optical substrates created with diffractive optics,�?? Rev. Sci. Instr. 69, 1974�??1977 (1998). [CrossRef]
  16. M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, �??Optical particle trapping with computer-generated holograms written on a liquid-crystal display,�?? Opt. Lett. 24, 608�??610 (1999). [CrossRef]
  17. P. C. Mogensen and J. Glückstad, �??Dynamic array generation and pattern formation for optical tweezers,�?? Opt. Commun. 175, 75�??81 (2000). [CrossRef]
  18. J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, �??Multi-functional optical tweezers using computer-generated holograms,�?? Opt. Commun. 185, 77�??82 (2000). [CrossRef]
  19. R. W. Gerchberg and W. O. Saxton, �??A practical algorithm for the determination of the phase from image and diffraction plane pictures,�?? Optik 35, 237�??246 (1972).
  20. V. Soifer, V. Kotlyar, and L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Taylor & Francis Ltd, London, 1997).
  21. B. Lin and N. C. Gallagher, �??Convergence of a spectrum shaping algorithm,�?? Appl. Opt. 13, 2470�??2471 (1974). [CrossRef]
  22. E. A. Sziklas and A. E. Siegman, �??Mode calculations in unstable resonators with flowing saturable gain. 2: Fast Fourier transform method,�?? Appl. Opt. 14, 1874�??1889 (1975). [CrossRef] [PubMed]

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