Effect of refractive index difference

4.10. Modelling the size dependence 136

Figure 4.34: Radial force and potential energy curves for a 6.84µm silica sphere in a Bessel beam. Notice the fact that at a radial displacement of zero the radial force is positive and the potential energy curve has an unstable equilibrium point.

Figure 4.35: Trajectory of a 6.84µm silica particle stably trapped in an offset core equilibrium position. This trajectory represents a video sequence that was over 10 minutes long.

Figure 4.36: Radial force colourmap for sphere radii ranging from 0 to 30µm, close to the beam core. Positive force regimes repeat with a spatial periodicity equivalent to the typical ring spacing of the Bessel beam.

Figure 4.37: Increasing the refractive index to that of polystyrene (n= 1.55) has little effect on the shape of the force colourmap, although the overall magnitude of the forces increase.

4.11. Conclusion 138

As can be seen from figure 4.37, increasing the refractive index of the sphere in our model to that of polystyrene (n = 1.55) has a negligible effect on the shape of the colourmap when compared with the force colourmap for silica particles (figure 4.33). As refractive index increases the zones of positive radial force become slightly larger, however in general for similarly sized spheres the radial forces and associated potential energy landscapes they experience retain very similar shapes. This is consistent with the findings of Papagiakoumou

et al.[51], in which cells of different composition (refractive index) but similar size and shape were different were found to behave similarly for a given Bessel beam.

4.11

Conclusion

I have verified claims made in the earlier work of Tatarkovaet al.[27], observing the directed Brownian transport of microscopic particles by using a tilted Bessel beam, where the tilt was generated by longitudinal displacement of one of the collimation lenses. I observed experimentally that explicit tilt in the transverse intensity profile is not actually required for net particle migration towards the beam core. To prove this, I developed a ray-optics model based on previous work by Karen Volke-Sepulveda which turned out to be very effective, correctly predicting all the observed behaviour of particles in my system.

Based on this work, it was found that Bessel beams could be used to sort silica spheres by size. This differential behaviour is entirely consistent with the ray-optics model. Later, my colleagues and I were able to demonstrate the successful sorting of biological cells using a Bessel beam. This was the first reported case of passive optical sorting that did not require an applied fluid flow. We had particular success sorting human blood cells with this approach, due largely to the radically different shapes of erythrocytes and lymphocytes.

4.12

Future work

The work presented in this chapter charts a journey that began with an attempt to create a model thermodynamic system, based around a washboard potential, and ended with the successful demonstration for a new method for the static, passive optical sorting of

microscopic particles.

Work is ongoing on a computational model to back up experiemental evidence that tilted Bessel beams can be generated by varying the divergence of the Gaussian beam illuminating the back of the axicon, and this will be complemented with a more detailed experimental analysis.

It would be satisfying to complete the early study of particle dynamics on an optical washboard potential in rigorous detail. This could perhaps be realised using a tilted Bessel beam or by using optical landscapes generated using other means, such as with an SLM. For a full and comprehensive study, a completely automated system would have to be finished to allow the collection of large data sets. Tightly controlled electrostatic clamping should be used to limit the effects of proximity to the sample walls. If a Bessel beam is to be used, the highest possible quality optics should be used to ensure that the beam’s transverse intensity profile is circularly symmetric and is completely independent of azimuthal variations.

It is interesting that our ray-optics correctly predicts the general behaviour of particles in a size regime where it is not strictly appropriate. This phenomenon deserves further investigation, and work is currently underway comparing the predictions of the ray optics model with those obtained from more rigorous calculations.

The optical sorting work could progress principally in two directions. Firstly, it would be interesting to look at how a static optical sorting configuration could be incorporated into a microfluidic lab-on-a-chip environment. Secondly, the system described in this chapter uses a fixed arrangement that can produce only one type of Bessel beam. For a truly dynamic sorting system, able to sort two arbitrary particle species, reconfigurability is required. It may be possible to achieve this with hard optics using the method described by Vaicaitis [9] or by replacing part of the optical train with a re-configurable device such as an SLM. In fact, if a device such as an SLM were to be used one is no longer restricted to using circularly symmetric landscapes to enable sorting, and I explore this concept in the next chapter.

Bibliography

[1] J. Durnin. Exact solutions for nondiffracting beams. i. the scalar theory. Journal of the Optical Society of America A, 4:651–654, 1987.

[2] J. Durnin, J.J. Miceli, and J.H. Eberly. Diffraction-free beams.Physical Review Letters, 58:1499–1501, 1987.

[3] J. Durnin, J.J. Miceli, and J.H. Eberly. Comparison of bessel and gaussian beams.

Optics Letters, 13:79–80, 1988.

[4] J. Arlt. Applications of Laguerre-Gaussian beams and Bessel beams to both nonlinear optics and atom optics. PhD thesis, University of St Andrews, December 1999. [5] J. Arlt, K. Dholakia, J. Soneson, and E.M. Wright. Optical dipole traps and atomic

waveguides based on bessel light beams. Physical Review A, 63:063602, 2001.

[6] C.A. McQueen, J. Arlt, and K. Dholakia. An experiment to study a nondiffracting light beam. American Journal of Physics, 67:912–915, 1999.

[7] G. Indebetouw. Nondiffracting optical fields: some remarks on their analysis and syn- thesis. Journal of the Optical Society of America A, 6:150–152, 1989.

[8] A.J. Cox and D.C. Dibble. Nondiffracting beam from a spatially filtered fabry-perot resonator. Journal of the Optical Society of America A, 9:282–286, 1992.

[9] V. Vaicaitis and S. Paulikas. Formation of bessel beams with continuously variable cone angle. Optical and Quantum Electronics, 35:1065–1071, 2003.

[10] A. Vasara, J. Turunen, and A.T. Friberg. Realization of general nondiffracting beams with computer-generated holograms. Journal of the Optical Society of America A, 6:1748 1754, 1989.

[11] J.A. Davis, E. Carcole, and D.M. Cottrell. Intensity and phase measurements of non- diffracting beams generated with a magneto-optic spatial light modulator. Applied Optics, 35:593–598, 1996.

[12] J.A. Davis, E. Carcole, and D.M. Cottrell. Nondiffracting interference patterns gener- ated with programmable spatial light modulators. Applied Optics, 35:599 602, 1996. [13] N. Chattrapiban, E.A. Rogers, D. Cofield, W.T. Hill, and R. Roy. Generation of

nondiffracting bessel beams by use of a spatial light modulator.Optics Letters, 28:2183– 2185, 2003.

[14] D. McGloin and K. Dholakia. Bessel beams: diffraction in a new light. Contemporary Physics, 46:15–28, 2005.

[15] K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia. Orbital angular momentum of a high-order bessel light beam. Journal of Optics B, 4:S82–S89, 2002.

[16] V. Garces-Chavez, D. McGloin, M.J. Padgett, W. Dultz, H. Schmitzer, and K. Dho- lakia. Observation of the transfer of the local angular momentum density of a multi- ringed light beam to an optically trapped particle. Physical Review Letters, 91:093602, 2003.

[17] V. Garces-Chavez, K. Volke-Sepulveda, S. Chvez-Cerda, W. Sibbett, and K. Dholakia. Transfer of orbital angular momentum to an optically trapped low-index particle.Phys- ical Review A, 66:063402, 2002.

[18] V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia. Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam. Nature, 419:145–147, 2002.

BIBLIOGRAPHY 142

[19] V. Garces-Chavez, D. McGloin, M.D. Summers, A. Fernandez-Nieves, G.C. Spalding, G. Crisobal, and K. Dholakia. Reconstruction of optical angular momentum after distortion in amplitude, phase and polarization. Journal of Optics A, 6:S235, 2004. [20] D. McGloin, V. Garces-Chavez, and K. Dholakia. Interfering bessel beams for optical

micromanipulation. Optics Letters, 28:657–659, 2003.

[21] J. Arlt and K. Dholakia. Generation of high-order bessel beams by use of an axicon.

Optics Communications, 177:297–301, 2000.

[22] D. McGloin, G.C. Spalding, H. Melville, W. Sibbett, and K. Dholakia. Three- dimensional arrays of optical bottle beams.Optics Communications, 225:215–222, 2003. [23] D.P. Rhodes. Experimental Studies of Cold Atom Guiding Using Hollow Light Beams.

PhD thesis, University of St Andrews, 2006.

[24] M.D. Summers, J.P. Reid, and D. McGloin. Optical guiding of aerosol droplets.Optics Express, 14:6373–6380, 2006.

[25] J.A. Davis, E. Carcole, and D.M. Cottrell. Range-finding by triangulation with non- diffracting beams. Applied Optics, 35:2159–2161, 1996.

[26] Y. Lin, W. Seka, J. H. Eberly, H. Huang, and D. L. Brown. Experimental investigation of bessel beam characteristics. Applied Optics, 31:2708–2713, 1992.

[27] S. Tatarkova, W. Sibbett, and K. Dholakia. Brownian particle in an optical potential of the washboard type. Physical Review Letters, 91:038101, 2003.

[28] F. Marchesoni. Conceptual design of a molecular shuttle. Physics Letters A, 237:126– 130, 1998.

[29] S. Lee and D.G. Grier. Flux reversal in a two-state symmetric optical thermal ratchet.

Physical Review E, 71:060102, 2005.

[30] S. Lee, K. Ladavac, M. Polin, and D.G. Grier. Observation of flux reversal in a sym- metric optical thermal ratchet. Physical Review Letters, 94:110601, 2005.

[31] S. Lee and D. Grier. One-dimensional optical thermal ratchets. Journal of Physics: Condensed Matter, 17:S3685–S3695, 2006.

[32] A.E. Martirosyan, C. Altucci, C. de Lisio, A. Porzio, S. Solimeno, and V. Tosa. Fringe pattern of the field diffracted by axicons. Journal of the Optical Society of America A, 21:770–776, 2004.

[33] K. Volke-Sepulveda, S. Chavez-Cerda, V. Garces-Chavez, and K. Dholakia. Three- dimensional optical forces and transfer of orbital angular momentum from multiringed light beams to spherical microparticles. Journal of the Optical Society of America B, 21:1749–1757, 2004.

[34] R. Gussgard, T. Lindmo, and I. Brevik. Calculation of the trapping force in a strongly focused laser beam. Journal of the Optical Society of America B, 9:1922, 1992. [35] A. Ashkin. Forces of a single-beam gradient laser trap on a dielectric sphere in the ray

optics regime. Biophysical Journal, 61:569–582, 1992.

[36] F. Marchesoni. Comment on stochastic resonance in washboard potentials. Physics Letters A, 231:61–64, 1997.

[37] J.K. Bhattacharjee and K. Banerjee. Kramers time in bistable potentials. Journal of Physics A, 22:L1141–L1146, 1989.

[38] H.A. Kramers. Brownian motion in a field of force and the diffusion model of chemical reactions. Physica, 7:284, 1940.

[39] L.I. McCann, M. Dykman, and B. Golding. Thermally activated transitions in a bistable three-dimensional optical trap. Nature, 402:785, 1999.

[40] S.L. Reck-Peterson, A. Yildiz, A.P. Carter, A. Gennerich, N. Zhang, and R.D. Vale. Single molecule analysis of dynein processivity and stepping behavior. Cell, 126:335– 348, 2006.

[41] E. R. Dufresne, T.M. Squires, M.P. Brenner, and D.G. Grier. Hydrodynamic coupling of two brownian spheres to a planar surface. Physical Review Letters, 85:3317–3320, 2000.

BIBLIOGRAPHY 144

[42] L. Belloni. Colloidal interactions. Journal of Physics: Condensed Matter, 12:R549– R587, 2000.

[43] Y. Han and D. G. Grier. Confinement-induced colloidal attractions in equilibrium.

Physical Review Letters, 91:038302, 2003.

[44] Z. Chen, A. Taflove, and V. Backman. Highly efficient optical coupling and transport phenomena in chains of dielectric microspheres. Optics Letters, 31:389–391, 2006. [45] A. Simon and A. Libchaber. Escape and synchronization of a brownian particle.Phys-

ical Review Letters, 68:3375–3378, 1992.

[46] J. Happel and H. Brenner. Low Reynolds Number Hydrodynamics. Kluwer, 1991. [47] A. Pralle, E.L. Florin, E.H.K. Steizer, and J.K.H. Horber. Local viscosity probed by

photonic force microscopy. Applied Physics A, 66:S71–S73, 1998.

[48] G.M. Kepler and S. Fraden. Video microscopy study of the potential energy of a colloidal particle confined between two plates. Langmuir, 10:2501 – 2506, 1994. [49] M.P. MacDonald, G.C. Splading, and K. Dholakia. Microfluidic sorting in an optical

lattice. Nature, 426:421, 2003.

[50] L. Paterson, E. Papagiakoumou, G. Milne, V. Garces-Chavez, S. A. Tatarkova, W. Sib- bett, F.J. Gunn-Moore, P.E. Bryant, A.C. Riches, and K. Dholakia. Light-induced cell separation in a tailored optical landscape. Applied Physics Letters, 87:123901, 2005. [51] E. Papagiakoumou, L. Paterson, G. Milne, V. Garces-Chavez, W. Sibbett, K. Dholakia,

and A. Riches. Optical separation of cell populations in a bessel beam. Biomedical Optics (submitted), 2007.

[52] Microscopix.

www.microscopix.co.uk.

[53] S.C. Grover, R.C. Gauthier, and A.G. Skirtach. Analysis of the behaviour of erythro- cytes in an optical trapping system. Optics Express, 7:533, 2000.

Optical Sorting and

Micromanipulation Using

Re-configurable Optical Devices

5.1

Introduction

In this chapter I review progress that I have made in optical sorting using re-configurable devices. The most impressive results were obtained using an acousto-optic deflector (AOD). I begin this chapter by looking at the principles that underlie typical AOD design and operation, before going on to describe our own implementation of the technology. I discuss how we have recently employed an AOD in a new technique for optical sorting. I then compare these results with an attempt to conduct a similar experiment using a spatial light modulator (SLM), a reconfigurable device that has already been discussed extensively in chapter 2. Finally, I describe how I used an SLM to create landscapes for static optical sorting.

5.2. Principles of acousto-optic deflector design 146