## Emergent properties in optically bound matter

Optics Express, Vol. 16, Issue 10, pp. 6921-6929 (2008)

http://dx.doi.org/10.1364/OE.16.006921

Acrobat PDF (354 KB)

### Abstract

Sub-micron particles have been observed to spontaneously form regular two-dimensional structures in counterpropagating evanescent laser fields. We show that collective properties of large numbers of optically-trapped particles can be qualitatively different to the properties of small numbers. This is demonstrated both with a computer model and with experimental results. As the number of particles in the structure is increased, optical binding forces can be sufficiently large to overcome the optical landscape imposed by the interference fringes of the laser beams and impose a different, competing structure.

© 2008 Optical Society of America

## 1. Introduction

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. M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, “Optical Binding,” Phys. Rev. Lett. **63**, 1233–1236 (1989). [CrossRef] [PubMed]

3. M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, “Optical Matter: Crystalization and binding in intense optical fields,” Science **249**, 749–754 (1990). [CrossRef] [PubMed]

4. N. K. Metzger, E. M. Wright, and K. Dholakia, “Theory and simulation of the Bistable behaviour of optically bound particles in the Mie size regime,” New J. Phys. **8**, 139 (2006). [CrossRef]

5. N. K. Metzger, K. Dholakia, and E. M. Wright, “Observation of Bistability and Hysteresis in optical binding of two dielectric spheres,” Phys. Rev. Lett. **96**, 068102 (2006). [CrossRef] [PubMed]

6. P. J. Reece, E. M. Wright, and K. Dholakia, “Experimental observation of modulation instability and Optical Spatial Soliton Arrays in soft condensed matter,” Phys. Rev. Lett. **98**, 203902 (2007). [CrossRef] [PubMed]

7. J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: numerical simulations,” Phys. Rev. B **72**, 085130 (2005). [CrossRef]

8. C. D. Mellor and C. D. Bain, “Array formation in evanescent waves,” ChemPhysChem **7**, 329–332 (2006). [CrossRef]

9. C. D. Mellor, T. A. Fennerty, and C. D. Bain, “Polarization effects in optically bound particle arrays,” Opt. Express **14**, 10079–10088 (2006). [CrossRef] [PubMed]

2. M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, “Optical Binding,” Phys. Rev. Lett. **63**, 1233–1236 (1989). [CrossRef] [PubMed]

10. S. A. Tatarkova, A. E. Carruthers, and K. Dholakia, “One-Dimensional Optically Bound Arrays of Microscopic Particles,” Phys. Rev. Lett. **89**, 283901 (2002). [CrossRef]

5. N. K. Metzger, K. Dholakia, and E. M. Wright, “Observation of Bistability and Hysteresis in optical binding of two dielectric spheres,” Phys. Rev. Lett. **96**, 068102 (2006). [CrossRef] [PubMed]

11. D. McGloin, A. E. Carruthers, K. Dholakia, and E. M. Wright, “Optically bound microscopic particles in one dimension,” Phys. Rev. E **69**, 021403 (2004). [CrossRef]

7. J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: numerical simulations,” Phys. Rev. B **72**, 085130 (2005). [CrossRef]

6. P. J. Reece, E. M. Wright, and K. Dholakia, “Experimental observation of modulation instability and Optical Spatial Soliton Arrays in soft condensed matter,” Phys. Rev. Lett. **98**, 203902 (2007). [CrossRef] [PubMed]

*r*, as observed in [7

7. J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: numerical simulations,” Phys. Rev. B **72**, 085130 (2005). [CrossRef]

12. T. Čižmár, M. Šiler, M. Šerý, P. Zemánek, V. Garcés-Chávez, and K. Dholakia, “Optical sorting and detection of submicrometer objects in a motional standing wave,” Phys. Rev. B **74**, 035105 (2006). [CrossRef]

13. K. Dholakia and P. Reece, “Optical micromanipulation takes hold,” Nano Today **1**, 18–27 (2006). [CrossRef]

9. C. D. Mellor, T. A. Fennerty, and C. D. Bain, “Polarization effects in optically bound particle arrays,” Opt. Express **14**, 10079–10088 (2006). [CrossRef] [PubMed]

9. C. D. Mellor, T. A. Fennerty, and C. D. Bain, “Polarization effects in optically bound particle arrays,” Opt. Express **14**, 10079–10088 (2006). [CrossRef] [PubMed]

**72**, 085130 (2005). [CrossRef]

## 2. Optical binding theory

*r*force relationship for optical binding. This relationship was derived in [3], but we present a more physically intuitive explanation here. Consider the scattered wave from a single particle. The amplitude of this wave decays as 1/

*r*and hence its intensity decays as 1/

*r*

^{2}. The gradient force on a second particle lying outside the illuminating beam is proportional to the gradient of intensity, and will therefore be proportional to 1/

*d*

^{3}, where

*d*is the distance between the two particles.

*α*/

*r*(where

*α*is a measure of the level of scattering by the particle). Thus their intensity varies as

*d*. Thus even for a one-dimensional chain, the magnitude of the forces on the central particle can grow indefinitely with the length of the chain. Consider a chain of 2

*n*+1 particles with spacing

*d*such that for a particle

*i*at position

*id*the scattered waves from every other particle are in phase. For large

*n*the strength of the forces on the central particle will scale as

*γ*=0.5772… is the Euler-Mascheroni constant [14, Eq. 6.1.3]. The force is a logarithmically-increasing function of

*n*. This is in contrast to an electrostatic interaction which would asymptotically approach a constant value at large

*n*.

16. Y.-L. Xu, “Electromagnetic Scattering by an Aggregate of Spheres,” Appl. Opt. **34**, 4573–4588 (1995). [CrossRef] [PubMed]

17. D. W. Mackowski, “Analysis of Radiative Scattering for Multiple Sphere Configurations,” Proc. R. Soc. London, Ser. A **433**, 599–614 (1991). [CrossRef]

18. J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of Net Radiation Force and Torque for a Spherical Particle Illuminated by a focused laser beam,” J. Appl. Phys. **66**, 4594–4602 (1989). [CrossRef]

*λ*.

**72**, 085130 (2005). [CrossRef]

**14**, 10079–10088 (2006). [CrossRef] [PubMed]

## 3. Fringe affinity of a chain of particles

23. J. Lekner, “Force on a scatterer in counter-propagating coherent beams,” J. Opt. A **7**, 238–248 (2005). [CrossRef]

*ka*, where

*k*is the wavenumber in water and

*a*is the particle size. When the electric field of the laser is in the trapping plane, this is designated “S” polarization. Similarly “P” polarization is when the polarization is parallel to the plane of incidence. The plot shows that a single, small particle is attracted to bright fringes. At larger radii the particle’s centre can instead be attracted to a dark region between fringes.

*between*two bright fringes [21], because two fringes are fairly well covered by the particle instead of one fringe being very well covered (see Fig. 2). We designate the radius at which the behaviour first switches from light-seeking to dark-seeking as the “crossover radius”.

*r*, but shows that the picture is complicated by multiple scattering: optical binding has become the dominant influence on the optical landscape, and the perturbation approach taken earlier is no longer entirely appropriate. Nevertheless, the argument used to arrive at Eq. 2 is a good starting point for understanding the origins of the long-range nature of the interaction.

**14**, 10079–10088 (2006). [CrossRef] [PubMed]

## 4. Conclusions

*r*relationship for optical binding, and have demonstrated its use in the limit of small perturbations. We have shown that a full Mie scattering model can be applied even where there is strong feedback through multiple scattering. We have analyzed a 1D case in detail, and shown that when extended to the 2D case our computer model agrees with experimental data.

## Acknowledgments

## 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. |

2. | M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, “Optical Binding,” Phys. Rev. Lett. |

3. | M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, “Optical Matter: Crystalization and binding in intense optical fields,” Science |

4. | N. K. Metzger, E. M. Wright, and K. Dholakia, “Theory and simulation of the Bistable behaviour of optically bound particles in the Mie size regime,” New J. Phys. |

5. | N. K. Metzger, K. Dholakia, and E. M. Wright, “Observation of Bistability and Hysteresis in optical binding of two dielectric spheres,” Phys. Rev. Lett. |

6. | P. J. Reece, E. M. Wright, and K. Dholakia, “Experimental observation of modulation instability and Optical Spatial Soliton Arrays in soft condensed matter,” Phys. Rev. Lett. |

7. | J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: numerical simulations,” Phys. Rev. B |

8. | C. D. Mellor and C. D. Bain, “Array formation in evanescent waves,” ChemPhysChem |

9. | C. D. Mellor, T. A. Fennerty, and C. D. Bain, “Polarization effects in optically bound particle arrays,” Opt. Express |

10. | S. A. Tatarkova, A. E. Carruthers, and K. Dholakia, “One-Dimensional Optically Bound Arrays of Microscopic Particles,” Phys. Rev. Lett. |

11. | D. McGloin, A. E. Carruthers, K. Dholakia, and E. M. Wright, “Optically bound microscopic particles in one dimension,” Phys. Rev. E |

12. | T. Čižmár, M. Šiler, M. Šerý, P. Zemánek, V. Garcés-Chávez, and K. Dholakia, “Optical sorting and detection of submicrometer objects in a motional standing wave,” Phys. Rev. B |

13. | K. Dholakia and P. Reece, “Optical micromanipulation takes hold,” Nano Today |

14. | M. Abramowitz and I. A. Stegun, |

15. | H. C. van de Hulst, |

16. | Y.-L. Xu, “Electromagnetic Scattering by an Aggregate of Spheres,” Appl. Opt. |

17. | D. W. Mackowski, “Analysis of Radiative Scattering for Multiple Sphere Configurations,” Proc. R. Soc. London, Ser. A |

18. | J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of Net Radiation Force and Torque for a Spherical Particle Illuminated by a focused laser beam,” J. Appl. Phys. |

19. | M. Doi and S. F. Edwards, |

20. | P. E. Kloeden and E. Platen, |

21. | M. Šiler, M. Šerý, T. Čižmár, and P. Zemánek, “Submicron particle localization using evanescent field,” |

22. | J. Ng and C. T. Chan Private communication. |

23. | J. Lekner, “Force on a scatterer in counter-propagating coherent beams,” J. Opt. A |

**OCIS Codes**

(160.4670) Materials : Optical materials

(170.4520) Medical optics and biotechnology : Optical confinement and manipulation

(240.6700) Optics at surfaces : Surfaces

(290.4020) Scattering : Mie theory

**ToC Category:**

Optical Trapping and Manipulation

**History**

Original Manuscript: April 2, 2008

Revised Manuscript: April 24, 2008

Manuscript Accepted: April 26, 2008

Published: April 30, 2008

**Virtual Issues**

Vol. 3, Iss. 6 *Virtual Journal for Biomedical Optics*

**Citation**

J. M. Taylor, L. Y. Wong, C. D. Bain, and G. D. Love, "Emergent properties in optically bound matter," Opt. Express **16**, 6921-6929 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-10-6921

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### References

- 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]
- M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, "Optical Binding," Phys. Rev. Lett. 63, 1233-1236 (1989). [CrossRef] [PubMed]
- M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, "Optical Matter: Crystalization and binding in intense optical fields," Science 249, 749-754 (1990). [CrossRef] [PubMed]
- N. K. Metzger, E. M. Wright, and K. Dholakia, "Theory and simulation of the Bistable behaviour of optically bound particles in the Mie size regime," New J. Phys. 8, 139 (2006). [CrossRef]
- N. K. Metzger, K. Dholakia, and E. M. Wright, "Observation of Bistability and Hysteresis in optical binding of two dielectric spheres," Phys. Rev. Lett. 96, 068102 (2006). [CrossRef] [PubMed]
- P. J. Reece, E. M. Wright, and K. Dholakia, "Experimental observation of modulation instability and Optical Spatial Soliton Arrays in soft condensed matter," Phys. Rev. Lett. 98, 203902 (2007). [CrossRef] [PubMed]
- J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, "Photonic clusters formed by dielectric microspheres: numerical simulations," Phys. Rev. B 72, 085130 (2005). [CrossRef]
- C. D. Mellor and C. D. Bain, "Array formation in evanescent waves," ChemPhysChem 7, 329-332 (2006). [CrossRef]
- C. D. Mellor, T. A. Fennerty, and C. D. Bain, "Polarization effects in optically bound particle arrays," Opt. Express 14, 10079-10088 (2006). [CrossRef] [PubMed]
- S. A. Tatarkova, A. E. Carruthers, and K. Dholakia, "One-Dimensional Optically Bound Arrays of Microscopic Particles," Phys. Rev. Lett. 89, 283901 (2002). [CrossRef]
- D. McGloin, A. E. Carruthers, K. Dholakia, and E. M. Wright, "Optically bound microscopic particles in one dimension," Phys. Rev. E 69, 021403 (2004). [CrossRef]
- T. Cizmar, M. Siler, M. Sery, P. Zemanek, V. Garces-Chavez, and K. Dholakia, "Optical sorting and detection of submicrometer objects in a motional standing wave," Phys. Rev. B 74, 035105 (2006). [CrossRef]
- K. Dholakia and P. Reece, "Optical micromanipulation takes hold," Nano Today 1, 18-27 (2006). [CrossRef]
- M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).
- H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
- Y.-L. Xu, "Electromagnetic Scattering by an Aggregate of Spheres," Appl. Opt. 34, 4573-4588 (1995). [CrossRef] [PubMed]
- D. W. Mackowski, "Analysis of Radiative Scattering for Multiple Sphere Configurations," Proc. R. Soc. London, Ser. A 433, 599-614 (1991). [CrossRef]
- J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of Net Radiation Force and Torque for a Spherical Particle Illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989). [CrossRef]
- M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, International Series of Monographs on Physics (Clarendon Press, Oxford, 1986).
- P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations (Springer-Verlag, 1992).
- M. Siler, M. Ser�??y, T. Ci�?zmar, and P. Zemanek, "Submicron particle localization using evanescent field," Proc. SPIE 5930, 59300R (2005).
- J. Ng and C. T. Chan. Private communication.
- J. Lekner, "Force on a scatterer in counter-propagating coherent beams," J. Opt. A 7, 238-248 (2005). [CrossRef]

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