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

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References

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  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]
  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]
  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]
  12. 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]
  13. K. Dholakia and P. Reece, "Optical micromanipulation takes hold," Nano Today 1, 18-27 (2006).
    [CrossRef]
  14. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).
  15. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
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    [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]
  19. M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, International Series of Monographs on Physics (Clarendon Press, Oxford, 1986).
  20. P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations (Springer-Verlag, 1992).
  21. M. Siler, M. Ser�??y, T. Ci�?zmar, and P. Zemanek, "Submicron particle localization using evanescent field," Proc. SPIE 5930, 59300R (2005).
  22. J. Ng and C. T. Chan. Private communication.
  23. J. Lekner, "Force on a scatterer in counter-propagating coherent beams," J. Opt. A 7, 238-248 (2005).
    [CrossRef]

2007

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]

2006

C. D. Mellor and C. D. Bain, "Array formation in evanescent waves," ChemPhysChem 7, 329-332 (2006).
[CrossRef]

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]

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]

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]

2005

M. Siler, M. Ser�??y, T. Ci�?zmar, and P. Zemanek, "Submicron particle localization using evanescent field," Proc. SPIE 5930, 59300R (2005).

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

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]

2004

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]

2002

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

1995

1991

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

1990

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]

1989

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

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]

1986

Alexander, D. R.

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]

Ashkin, A.

Bain, C. D.

Barton, J. P.

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]

Bjorkholm, J. E.

Burns, M. M.

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]

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

Carruthers, A. E.

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]

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

Chan, C. T.

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]

Chu, S.

Ci??zm???ar, T. ??

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]

Dholakia, K.

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]

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]

K. Dholakia and P. Reece, "Optical micromanipulation takes hold," Nano Today 1, 18-27 (2006).
[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]

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

Dziedzic, J. M.

Fennerty, T. A.

Fournier, J.-M.

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]

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

Golovchenko, J. A.

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]

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

Lekner, J.

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

Lin, Z. F.

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]

Mackowski, D. W.

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

McGloin, D.

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]

Mellor, C. D.

Metzger, N. K.

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]

Ng, J.

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]

Reece, P.

K. Dholakia and P. Reece, "Optical micromanipulation takes hold," Nano Today 1, 18-27 (2006).
[CrossRef]

Reece, P. J.

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]

Schaub, S. A.

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]

Sheng, P.

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]

Siler, M. ??

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]

M. Siler, M. Ser�??y, T. Ci�?zmar, and P. Zemanek, "Submicron particle localization using evanescent field," Proc. SPIE 5930, 59300R (2005).

Tatarkova, S. A.

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

Wright, E. M.

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]

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]

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]

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]

Xu, Y.-L.

Appl. Opt.

ChemPhysChem

C. D. Mellor and C. D. Bain, "Array formation in evanescent waves," ChemPhysChem 7, 329-332 (2006).
[CrossRef]

J. Appl. Phys.

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]

J. Opt. A

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

Nano Today

K. Dholakia and P. Reece, "Optical micromanipulation takes hold," Nano Today 1, 18-27 (2006).
[CrossRef]

New J. Phys.

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]

Opt. Express

Opt. Lett.

Phys. Rev. B

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]

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]

Phys. Rev. E

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]

Phys. Rev. Lett.

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

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

Proc. R. Soc. London, Ser. A

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

Proc. SPIE

M. Siler, M. Ser�??y, T. Ci�?zmar, and P. Zemanek, "Submicron particle localization using evanescent field," Proc. SPIE 5930, 59300R (2005).

Science

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]

Other

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

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. 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).

Supplementary Material (1)

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Figures (7)

Fig. 1.
Fig. 1.

(a) Schematic showing how the scattered intensity of light from a single particle (left) in a beam depends on 1/r, and therefore the gradient force on the second particle outside the beam depends on 1/d 3 for particle separation d. When both particles are exposed to the laser field (right), the modulation in the force scales with 1/d. (b) Simulation showing the field intensity distribution caused by interference of a scattered wave from a particle with the evanescent field of a single laser beam propagating left to right, showing the production of interference fringes which lead to optical binding.

Fig. 2.
Fig. 2.

The top graph shows the force acting on a single particle placed halfway between a bright and dark fringe, as a function of size parameter ka. A positive force indicates that the particle is attracted to the bright fringe. Two lines are shown for different polarization states. The first crossover occurs at ka=1.985. For full parameters used, see Table 1. It can be seen that for a range of values of ka the force is negative and and therefore particles are attracted to darker regions. The “crossover” sizes (where the particles switch from being attracted to light to dark) are indicated numerically on the graph. The lower two plots show simulated images showing how the interference fringes are distorted by the particle and that the particle sits on a bright fringe when small (a) and on a dark fringe when larger (b). The white circles indicate the particle size and location, and the size parameters for (a) and (b) are indicated on the graph.

Fig. 3.
Fig. 3.

Examples of chains of particles along the fringe direction. (a) is shown without scattered light, for clarity. (b) shows how the fringe pattern can be modified by the presence of the spheres. (c) Movie (1.3MB) shows a simulation demonstrating the switch in fringe affinity for 265nm radius particles as more particles are added to the system. The particles are free to move in three dimensions in this simulation (though they are of course physically constrained by the surface of the prism). [Media 1]

Fig. 4.
Fig. 4.

Force acting on the central particle of a chain of particles as a function of number of spheres in the chain. The forces are shown for both S and P laser polarizations for spheres of radius 260nm, and for P polarization for spheres of radius 280nm. A positive force indicates that the particle is attracted to the bright fringe, and a negative force a dark fringe. For full parameters used, see Table 1.

Fig. 5.
Fig. 5.

Scattered field intensity in the trapping plane for a particle with ka=2.5. The intensity is displayed on a logarithmic scale. (a) P-polarized beams and particle on a dark fringe. There is no scattering in the direction of the fringes when a particle lies on a dark fringe. (b) P-polarized beams and particle on a light fringe. A particle situated on a bright fringe formed by P-polarized light scatters relatively strongly in the direction of the fringes. This scattering enhances the intensity of the fringe that the particle lies on, which makes it more energetically favourable for other particles to be situated on that same fringe. (c) S-polarized beams and particle on a light fringe. There is very weak far-field scattering in the direction of the fringes (a dipole scatterer will not scatter in a direction parallel to the electric field polarization).

Fig. 6.
Fig. 6.

Force acting on the central of (2i+1) particles. We can compare this logarithmic plot to our Eq. 2, which predicted a logarithmic relationship. While we were correct in predicting that the force would continue to grow with the number of particles, rather than converging to a constant value, the equation has under-estimated the growth of the force. The reason for this is that the equation only used a simplistic treatment of the first-order scattered field. Also shown here is the force due only to the first-order scattered field (using the order-of-scattering method [16], we terminated the iteration at first order). The form of that curve is indeed well predicted by Eq. 2.

Fig. 7.
Fig. 7.

Examples of “broken hex” optically-bound structures with vacant fringes. (a) shows a snapshot from a Brownian dynamics simulation, and (b) shows an experimental image. In both cases the particle radius is 260nm. In the experimental image the vertical extent of the particle array is limited by the shape of the focused Gaussian beams.

Tables (1)

Tables Icon

Table 1. Parameters used in the calculations in this paper. In addition we refer to the particle radius a and the size parameter ka where k is the laser wavenumber in water.

Equations (2)

Equations on this page are rendered with MathJax. Learn more.

( 1 ± α r ) 2 ~ 1 ± 2 α r ,
F ~ 1 + i = 1 n 2 α id ~ 1 + 2 α ( ln n + γ ) d ,

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