Abstract

Optical binding interactions between laser-trapped spherical microparticles are familiar in a wide range of trapping configurations. Recently it has been demonstrated that these experiments can be accurately modeled using Mie scattering or coupled dipole models. This can help confirm the physical phenomena underlying the inter-particle interactions, but does not necessarily develop a conceptual understanding of the effects that can lead to future predictions. Here we interpret results from a Mie scattering model to obtain a physical description which predict the behavior and trends for chains of trapped particles in Gaussian beam traps. In particular, it describes the non-uniform particle spacing and how it changes with the number of particles. We go further than simply demonstrating agreement, by showing that the mechanisms “hidden” within a mathematically and computationally demanding Mie scattering description can be explained in easily-understood terms.

© 2009 Optical Society of America

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References

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  1. 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]
  2. S. A. Tatarkova, A. E. Carruthers, and K. Dholakia, "One-Dimensional Optically Bound Arrays of Microscopic Particles," Phys. Rev. Lett. 89(28), 283,901 (2002).
  3. W. Singer, M. Frick, S. Bernet, and M. Ritsch-Marte, "Self-organized Array of Regularly Spaced Microbeads in a Fiber-optical Trap," J. Opt. Soc. Am. B 20(7), 1568-1574 (2003).
    [CrossRef]
  4. N. K. Metzger, E. M. Wright, W. Sibbett, and K. Dholakia, "Visualization of Optical Binding of Microparticles Using a Femtosecond Fiber Optical Trap," Opt. Express 14(8), 3677-3687 (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, 068,102 (2006).
    [CrossRef]
  6. C. D. Mellor, T. A. Fennerty, and C. D. Bain, "Polarization Effects in Optically Bound Particle Arrays," Opt. Express 14, 10,079-10,088 (2006).
    [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, 085,130 (2005).
    [CrossRef]
  8. M. Kawano, J. T. Blakely, R. Gordon, and D. Sinton, "Theory of Dielectric Micro-sphere Dynamics in a Dualbeam Optical Trap," Opt. Express 16, 9306-9317 (2008).
    [CrossRef] [PubMed]
  9. 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).
    [CrossRef] [PubMed]
  10. V. Kar´asek, O. Brzobohat´y, and P. Zem´anek, "Longitudinal Optical Binding of Several Spherical Particles Studied by the Coupled Dipole Method," J. Opt. A 11, 034,009 (2009).
    [CrossRef]
  11. T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Kn¨oner, A.M. Br’anczyk, N. R. Heckenberg, and H. Rubinsztein- Dunlop, "Optical Tweezers Computational Toolbox," J. Opt. A 9, S196-S203 (2007).
    [CrossRef]
  12. M. Dienerowitz, M. Mazilu, and K. Dholakia, "Optical Manipulation of Nanoparticles: A Review," J. Nanophoton. 2, 021,875 (2008).
  13. M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, "Optical Binding," Phys. Rev. Lett. 63(12), 1233-1236 (1989).
    [CrossRef]
  14. D. McGloin, A. E. Carruthers, K. Dholakia, and E. M. Wright, "Optically Bound Microscopic Particles in One Dimension," Phys. Rev. E 69, 021,403 (2004).
    [CrossRef]
  15. T. C? iz?ma’r, V. Kolla’rova´, Z. Bouchal, and P. Zema’nek, "Sub-micron Particle Organization by Self-imaging of Non-diffracting Beams," New J. Phys. 8, 43 (2006).
    [CrossRef]
  16. D.W. Mackowski, "Analysis of Radiative Scattering for Multiple Sphere Configurations," Proc. R. Soc. London, Ser. A 433, 599-614 (1991).
    [CrossRef]
  17. C. Liang and Y. T. Lo, "Scattering by Two Spheres," Radio Science 2, 1481-1495 (1967).
  18. K. A. Fuller and G. W. Kattawar, "Consummate Solution to the Problem of Classical Electromagnetic Scattering by an Ensemble of Spheres I: Linear Chains," Opt. Lett. 13(2), 90-92 (1988).
    [CrossRef]

2008 (3)

2007 (1)

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Kn¨oner, A.M. Br’anczyk, N. R. Heckenberg, and H. Rubinsztein- Dunlop, "Optical Tweezers Computational Toolbox," J. Opt. A 9, S196-S203 (2007).
[CrossRef]

2006 (4)

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, 068,102 (2006).
[CrossRef]

C. D. Mellor, T. A. Fennerty, and C. D. Bain, "Polarization Effects in Optically Bound Particle Arrays," Opt. Express 14, 10,079-10,088 (2006).
[CrossRef]

T. C? iz?ma’r, V. Kolla’rova´, Z. Bouchal, and P. Zema’nek, "Sub-micron Particle Organization by Self-imaging of Non-diffracting Beams," New J. Phys. 8, 43 (2006).
[CrossRef]

N. K. Metzger, E. M. Wright, W. Sibbett, and K. Dholakia, "Visualization of Optical Binding of Microparticles Using a Femtosecond Fiber Optical Trap," Opt. Express 14(8), 3677-3687 (2006).
[CrossRef]

2005 (1)

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, "Photonic Clusters Formed By Dielectric Microspheres: Numerical Simulations," Phys. Rev. B 72, 085,130 (2005).
[CrossRef]

2004 (1)

D. McGloin, A. E. Carruthers, K. Dholakia, and E. M. Wright, "Optically Bound Microscopic Particles in One Dimension," Phys. Rev. E 69, 021,403 (2004).
[CrossRef]

2003 (1)

1991 (1)

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

1990 (1)

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 (1)

M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, "Optical Binding," Phys. Rev. Lett. 63(12), 1233-1236 (1989).
[CrossRef]

1988 (1)

1967 (1)

C. Liang and Y. T. Lo, "Scattering by Two Spheres," Radio Science 2, 1481-1495 (1967).

Bain, C. D.

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).
[CrossRef] [PubMed]

C. D. Mellor, T. A. Fennerty, and C. D. Bain, "Polarization Effects in Optically Bound Particle Arrays," Opt. Express 14, 10,079-10,088 (2006).
[CrossRef]

Bernet, S.

Blakely, J. T.

Br’anczyk, A.M.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Kn¨oner, A.M. Br’anczyk, N. R. Heckenberg, and H. Rubinsztein- Dunlop, "Optical Tweezers Computational Toolbox," J. Opt. A 9, S196-S203 (2007).
[CrossRef]

Brzobohat´y, O.

V. Kar´asek, O. Brzobohat´y, and P. Zem´anek, "Longitudinal Optical Binding of Several Spherical Particles Studied by the Coupled Dipole Method," J. Opt. A 11, 034,009 (2009).
[CrossRef]

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(12), 1233-1236 (1989).
[CrossRef]

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, 021,403 (2004).
[CrossRef]

S. A. Tatarkova, A. E. Carruthers, and K. Dholakia, "One-Dimensional Optically Bound Arrays of Microscopic Particles," Phys. Rev. Lett. 89(28), 283,901 (2002).

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, 085,130 (2005).
[CrossRef]

Dholakia, K.

M. Dienerowitz, M. Mazilu, and K. Dholakia, "Optical Manipulation of Nanoparticles: A Review," J. Nanophoton. 2, 021,875 (2008).

N. K. Metzger, E. M. Wright, W. Sibbett, and K. Dholakia, "Visualization of Optical Binding of Microparticles Using a Femtosecond Fiber Optical Trap," Opt. Express 14(8), 3677-3687 (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, 068,102 (2006).
[CrossRef]

D. McGloin, A. E. Carruthers, K. Dholakia, and E. M. Wright, "Optically Bound Microscopic Particles in One Dimension," Phys. Rev. E 69, 021,403 (2004).
[CrossRef]

S. A. Tatarkova, A. E. Carruthers, and K. Dholakia, "One-Dimensional Optically Bound Arrays of Microscopic Particles," Phys. Rev. Lett. 89(28), 283,901 (2002).

Dienerowitz, M.

M. Dienerowitz, M. Mazilu, and K. Dholakia, "Optical Manipulation of Nanoparticles: A Review," J. Nanophoton. 2, 021,875 (2008).

Fennerty, T. A.

C. D. Mellor, T. A. Fennerty, and C. D. Bain, "Polarization Effects in Optically Bound Particle Arrays," Opt. Express 14, 10,079-10,088 (2006).
[CrossRef]

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(12), 1233-1236 (1989).
[CrossRef]

Frick, M.

Fuller, K. A.

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(12), 1233-1236 (1989).
[CrossRef]

Gordon, R.

Heckenberg, N. R.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Kn¨oner, A.M. Br’anczyk, N. R. Heckenberg, and H. Rubinsztein- Dunlop, "Optical Tweezers Computational Toolbox," J. Opt. A 9, S196-S203 (2007).
[CrossRef]

Kar´asek, V.

V. Kar´asek, O. Brzobohat´y, and P. Zem´anek, "Longitudinal Optical Binding of Several Spherical Particles Studied by the Coupled Dipole Method," J. Opt. A 11, 034,009 (2009).
[CrossRef]

Kattawar, G. W.

Kawano, M.

Kn¨oner, G.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Kn¨oner, A.M. Br’anczyk, N. R. Heckenberg, and H. Rubinsztein- Dunlop, "Optical Tweezers Computational Toolbox," J. Opt. A 9, S196-S203 (2007).
[CrossRef]

Liang, C.

C. Liang and Y. T. Lo, "Scattering by Two Spheres," Radio Science 2, 1481-1495 (1967).

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, 085,130 (2005).
[CrossRef]

Lo, Y. T.

C. Liang and Y. T. Lo, "Scattering by Two Spheres," Radio Science 2, 1481-1495 (1967).

Loke, V. L. Y.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Kn¨oner, A.M. Br’anczyk, N. R. Heckenberg, and H. Rubinsztein- Dunlop, "Optical Tweezers Computational Toolbox," J. Opt. A 9, S196-S203 (2007).
[CrossRef]

Love, G. D.

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]

Mazilu, M.

M. Dienerowitz, M. Mazilu, and K. Dholakia, "Optical Manipulation of Nanoparticles: A Review," J. Nanophoton. 2, 021,875 (2008).

McGloin, D.

D. McGloin, A. E. Carruthers, K. Dholakia, and E. M. Wright, "Optically Bound Microscopic Particles in One Dimension," Phys. Rev. E 69, 021,403 (2004).
[CrossRef]

Mellor, C. D.

C. D. Mellor, T. A. Fennerty, and C. D. Bain, "Polarization Effects in Optically Bound Particle Arrays," Opt. Express 14, 10,079-10,088 (2006).
[CrossRef]

Metzger, N. K.

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, 068,102 (2006).
[CrossRef]

N. K. Metzger, E. M. Wright, W. Sibbett, and K. Dholakia, "Visualization of Optical Binding of Microparticles Using a Femtosecond Fiber Optical Trap," Opt. Express 14(8), 3677-3687 (2006).
[CrossRef]

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, 085,130 (2005).
[CrossRef]

Nieminen, T. A.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Kn¨oner, A.M. Br’anczyk, N. R. Heckenberg, and H. Rubinsztein- Dunlop, "Optical Tweezers Computational Toolbox," J. Opt. A 9, S196-S203 (2007).
[CrossRef]

Ritsch-Marte, M.

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, 085,130 (2005).
[CrossRef]

Sibbett, W.

Singer, W.

Sinton, D.

Stilgoe, A. B.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Kn¨oner, A.M. Br’anczyk, N. R. Heckenberg, and H. Rubinsztein- Dunlop, "Optical Tweezers Computational Toolbox," J. Opt. A 9, S196-S203 (2007).
[CrossRef]

Tatarkova, S. A.

S. A. Tatarkova, A. E. Carruthers, and K. Dholakia, "One-Dimensional Optically Bound Arrays of Microscopic Particles," Phys. Rev. Lett. 89(28), 283,901 (2002).

Taylor, J. M.

Wong, L. Y.

Wright, E. M.

N. K. Metzger, E. M. Wright, W. Sibbett, and K. Dholakia, "Visualization of Optical Binding of Microparticles Using a Femtosecond Fiber Optical Trap," Opt. Express 14(8), 3677-3687 (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, 068,102 (2006).
[CrossRef]

D. McGloin, A. E. Carruthers, K. Dholakia, and E. M. Wright, "Optically Bound Microscopic Particles in One Dimension," Phys. Rev. E 69, 021,403 (2004).
[CrossRef]

Zem´anek, P.

V. Kar´asek, O. Brzobohat´y, and P. Zem´anek, "Longitudinal Optical Binding of Several Spherical Particles Studied by the Coupled Dipole Method," J. Opt. A 11, 034,009 (2009).
[CrossRef]

J. Nanophoton. (1)

M. Dienerowitz, M. Mazilu, and K. Dholakia, "Optical Manipulation of Nanoparticles: A Review," J. Nanophoton. 2, 021,875 (2008).

J. Opt. A (1)

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Kn¨oner, A.M. Br’anczyk, N. R. Heckenberg, and H. Rubinsztein- Dunlop, "Optical Tweezers Computational Toolbox," J. Opt. A 9, S196-S203 (2007).
[CrossRef]

J. Opt. Soc. Am. B (1)

New J. Phys. (1)

T. C? iz?ma’r, V. Kolla’rova´, Z. Bouchal, and P. Zema’nek, "Sub-micron Particle Organization by Self-imaging of Non-diffracting Beams," New J. Phys. 8, 43 (2006).
[CrossRef]

Opt. Express (4)

Opt. Lett. (1)

Phys. Rev. B (1)

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, "Photonic Clusters Formed By Dielectric Microspheres: Numerical Simulations," Phys. Rev. B 72, 085,130 (2005).
[CrossRef]

Phys. Rev. E (1)

D. McGloin, A. E. Carruthers, K. Dholakia, and E. M. Wright, "Optically Bound Microscopic Particles in One Dimension," Phys. Rev. E 69, 021,403 (2004).
[CrossRef]

Phys. Rev. Lett. (2)

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, 068,102 (2006).
[CrossRef]

M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, "Optical Binding," Phys. Rev. Lett. 63(12), 1233-1236 (1989).
[CrossRef]

Proc. R. Soc. London, Ser. A (1)

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

Radio Science (1)

C. Liang and Y. T. Lo, "Scattering by Two Spheres," Radio Science 2, 1481-1495 (1967).

Science (1)

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 (2)

S. A. Tatarkova, A. E. Carruthers, and K. Dholakia, "One-Dimensional Optically Bound Arrays of Microscopic Particles," Phys. Rev. Lett. 89(28), 283,901 (2002).

V. Kar´asek, O. Brzobohat´y, and P. Zem´anek, "Longitudinal Optical Binding of Several Spherical Particles Studied by the Coupled Dipole Method," J. Opt. A 11, 034,009 (2009).
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

Force on a 1 µm particle (size parameter ~3.9) in a two particle system, as a function of inter-particle spacing. (a) Force on the downstream (right-hand) one of a pair of such particles illuminated with a single right-going plane wave; (b) as (a) but for a 25mW Gaussian beam; (c) repulsive force between the pair of particles when two counter-propagating plane waves are used; (d) two counter-propagating Gaussian beams, showing the stable spacing of ~8 µm. Note the modulation due to backscattered light, which has little effect on the general trend of the binding behavior. The broad harmonic potential introduced by the use of Gaussian beams has altered curve (c) to give curve (d), which is outwardly similar but which has a stable inter-particle spacing at around 7µm (marked with an arrow).

Fig. 2.
Fig. 2.

A simple Born approximation model uses the intensity of a right-going Gaussian beam in the absence of particle 3 (whose location is indicated with an arrow in (c)) to determine the force f 3 on that particle. (a) shows the field for 3 particles exposed to a single Gaussian beam; (b) shows the field in the absence of particle 3 and (c) plots this field (all data in this figure was generated using a full Mie scattering model).

Fig. 3.
Fig. 3.

Example of how the force (arbitrary units) on particle i in a chain of 8 particles due only to the right-going beam varies along the chain if the inter-particle spacing is constant. Mie scattering model (red, unbroken line) and our simple ansatz model (green, dotted line). Both plots show similar trends; neither is symmetric with respect to the center of the chain. This means that, when both beams are considered, there will not be a net force of zero on a given particle, and so for this imposed uniform spacing the system will not be in equilibrium.

Fig. 4.
Fig. 4.

Example of how a force profile like those in Fig. 3 (arbitrary units) can be made symmetric by altering inter-particle spacings. Now, in contrast to Fig. 3, when both beams are considered there will be a net force of zero on each particle, and so the system will be in equilibrium.

Fig. 5.
Fig. 5.

Collapse of a longer chain, where a shorter chain would be stable. Parameters have been selected to give an extreme case where a two-particle chain is stable but a three-particle chain is not (parameters as [3], but with 1.9 µm diameter spheres). The curves “1 of 2” and “2 of 2” show the forces due to the right-going beam for the particles in a two-particle chain. Since the force on the second particle is greater than the force on the first particle over a range of around 5–12 µm inter-particle spacing, there is a stable trapped configuration with an inter-particle spacing of about 12 µm (indicated with an arrow) when net effect of both beams is considered. The curves “1 of 3” and “3 of 3” show the same forces on the end particles of a three-particle chain. Since there is no spacing for which the force on the third particle is greater than the force on the first particle, the chain will collapse. Even though there is some enhancement of the force on the third particle over what it would be in the absence of the other particles, this is not enough to overcome the compressive force due to the effect of the beam on the first particle.

Equations (4)

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a(i)=aext(i)+ΣjiFji.s(j)
s(i)=T . a(i)
a(i)=aext(i)+j<iFjis(j)
fi(zi,zi1)=ezizi1α×fi1+(βI0γzi)

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