## Optical binding mechanisms: a conceptual model for Gaussian beam traps

Optics Express, Vol. 17, Issue 17, pp. 15381-15389 (2009)

http://dx.doi.org/10.1364/OE.17.015381

Acrobat PDF (135 KB)

### 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

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]

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]

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 Dual-beam Optical Trap,” Opt. Express **16**, 9306–9317 (2008). [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**, 085,130 (2005). [CrossRef]

8. M. Kawano, J. T. Blakely, R. Gordon, and D. Sinton, “Theory of Dielectric Micro-sphere Dynamics in a Dual-beam 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. Karásek, O. Brzobohatý, and P. Zemánek, “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. Knöner, A.M. Bránczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical Tweezers Computational Toolbox,” J. Opt. A **9**, S196–S203 (2007). [CrossRef]

8. M. Kawano, J. T. Blakely, R. Gordon, and D. Sinton, “Theory of Dielectric Micro-sphere Dynamics in a Dual-beam Optical Trap,” Opt. Express **16**, 9306–9317 (2008). [CrossRef] [PubMed]

10. V. Karásek, O. Brzobohatý, and P. Zemánek, “Longitudinal Optical Binding of Several Spherical Particles Studied by the Coupled Dipole Method,” J. Opt. A **11**, 034,009 (2009). [CrossRef]

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]

**16**, 9306–9317 (2008). [CrossRef] [PubMed]

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

**a**

^{(i)}represents the net field incident on particle

*i*,

**a**

^{(i)}

_{ext}the zero-order incident field on particle

*i*due to the external laser field in the absence of other particles and s(

*i*) the scattered field from particle

*i*;

**F**

_{ji}is the translation matrix from a basis centered on particle

*j*to one centered on particle

*i*, and

**T**is the T-matrix representing the scattering properties of the particle, which is diagonal for spherical particles. This expression simply states that the field incident on a particle

*i*is the (coherent) sum of the laser field and the field scattered by all the other particles (although it should be noted that solving these coupled equations is not necessarily trivial, requiring inversion [17] or variational methods [18

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]

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]

*µ*m diameter particle illuminated by a single plane wave (Nd:YAG laser wavelength 806 nm in water; size parameter ~3.9). The dominant effect is a “focusing” of the light (in the limit of large particles we can consider the particle as a spherical lens within the framework of ray optics). Consider now the force on a second particle placed in this field. This is plotted in Fig. 1, which shows how two particles are stably bound in a Gaussian beam trap, but are not stably bound in counter-propagating plane waves. We can see that the force due to the light focused by the first particle causes the two particles to be repelled in the case of counter-propagating plane waves (Fig. 1(c)). When we consider counter-propagating Gaussian beams (Fig. 1(d)), through symmetry there is no net force on the center of mass of the particle pair, and the beams provide a broad background harmonic trapping potential. The particles will stabilize with a spacing which is largely determined by the balance of the repulsive force between the two particles and the harmonic trapping potential of the trap (Fig. 1(d)), as suggested in [2]. We emphasize that although we refer to a “focusing” of the light, we are far from the ray-optics regime, and it is not appropriate to use a ray-optics formula for the focal length, or to suggest that one particle will be bound “at the focus” formed by the other particle.

*net force*on a particle due to the two beams together can be non-negligible. Thus it is not really possible to state that either the gradient force or the scattering force will dominate under all circumstances, and the dominance will also to depend on the particle size.

*N*of trapped particles (indexed

*i*=1 to

*N*), for which we intend to explain the three main properties of the particle chains:

*N*.

*z*axis) is symmetric about the

*z*=0 plane, the force on particle i due to one beam is equal and opposite to the force on particle

*N*-

*i*+1 due to the other counter-propagating beam. This is true for any symmetric arrangement of particles, whether or not this is an equilibrium configuration. If

*f*

^{+}

_{i}(or

*f*

^{-}

_{i}) is the force on particle i due to the beam propagating in the +

*z*(or −

*z*) direction, then

*f*

^{+}

_{i}=

*f*

^{-}

_{N-i+1}.

*f*

^{+}

_{i}=

*f*

^{-}

_{i}), since by definition there must be no particle motion in equilibrium. Combining this with the previous requirement, we have

*f*

^{+}

_{i}=

*f*

^{+}

_{N-i+1}. In other words, the forces on the particles in the chain due to

*each individual beam*must be

*symmetric*about the center of the chain.

*forward*-scattering. As noted at the start of Section 2, this is a reasonable assumption for particles larger than the wavelength of light.

*i*is a function of the light intensity

*I*

^{(i)}

_{0}which would be found be at that point

*in the absence of*that given particle (the Born approximation; see Fig. 2).

*i*has the form

*I*

^{(i)}

_{0}×

*I*(

*z*-

*z*), where

_{i}*I*(

*z*-

*z*) is a fixed downstream intensity profile which applies to any particle at any position. Consequently, the force on particle

_{i}*i*+1 has the form

*I*

^{(i)}

_{0}×

*F*(

*z*-

*z*) for a fixed downstream force profile

_{i}*F*(

*z*-

*z*) (per unit incident intensity). Although the downstream force profile

_{i}*F*(

*z*-

*z*) is a

_{i}*function of*the intensity profile

*I*(

*z*-

*z*) (and its derivatives), the two are not necessarily

_{i}*proportional*. If they were, that would be equivalent to stating that scattering forces dominate over gradient forces. We do not make that assumption in our model, though. As mentioned earlier, we find in practice that, while for some particle sizes the scattering force dominates overwhelmingly, the gradient force can also be significant in some cases.

*decrease*if an additional particle is added onto either end of the chain. The justification for this is as follows. The force pushing inwards on what is now particle 2 in the chain has been increased (it was previously just the force due to the unperturbed laser beam; it is now enhanced by the additional light focused onto it by particle 1). Assuming there are

*some*losses along the length of the chain, then the force pushing outwards on what is now the last-but-one particle in the chain will also increase, but by a

*smaller amount*. Hence whenever additional particles are added to the chain, the inner ones will be pushed inwards, and so the inter-particle spacing between any given pair of particles in the chain will decrease. Equilibrium is then restored because the closer inter-particle spacing enhances the transmission efficiency, thereby further increasing the repulsive force on the last particle in the chain.

*f*as a function of

_{i}*z*and

_{i}*z*

_{i-1}, as follows:

*F*referred to in assumption 5), and the second term represents a background intensity due to the laser field, which is decreasing with distance from the beam waist (for tunable parameters

*α*,

*β*and

*γ*which depend on such things as the beam shape and the particle properties). We emphasize that the functional form of

*f*, and its parameters, have simply been selected empirically to give a reasonable approximation to the observed inter-particle force. If a closer agreement with the Mie scattering model was desired, a “hybrid” model could be used, in which

_{i}*f*is actually determined from the inter-particle forces for a pair of particles in a plane wave, calculated using Mie scattering theory. However, we have instead chosen to keep our model as elementary as possible.

_{i}*inwards*is weaker than the force pushing particle

*N outwards*, which means that the tendency will be for the outermost particles to move apart. The natural next step is therefore to allow the particle spacings to vary along the length of the chain, as they would do in real life in response to this repulsive force. As Fig. 4 shows, this approach allows a symmetric force profile to be produced, and leading to stable trapping of the chain with these slightly non-equilibrium spacings.

*i*. Particles towards the middle of the chain can be thought of as acting more like a (very inefficient) waveguide where the intensity is propagated from one particle to the next with some losses, which are compensated for by the re-focusing of additional background light. The intensity (and force) then drops again towards the end of the chain due to the increased particle spacings.

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]

*attractive*force between neighboring particles, and so the three-particle chain collapses once the compressive forces have pushed the particles close enough to enter this regime. Our implicit assumption that the inter-particle light forces are repulsive (it was assumed that radiation pressure will dominate) has broken down; near-field gradient force effects have come into play, producing a net

*attractive*force between the spheres at close ranges. There is no repulsive force to support the chain, and it collapses.

*13*], since the interaction here is largely a repulsive one, with stable chains only being formed due to the background harmonic potential of the trap). Here we have used a very simple model to successfully explain the trends of closer spacings as more particles are added to the chain, and of closer spacings in the center of a chain compared to near its edges.

*N*, and the wider spacing close to either end of the chain. It is

*only*through a simple model such as the one we have presented that the various complex effects in the experiment can be decoupled in order to understand

*why*optical binding occurs under these experimental conditions.

## Acknowledgement

## References and links

1. | M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, “Optical Matter: Crystalization and Binding in Intense Optical Fields,” Science |

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

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 |

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 |

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. | C. D. Mellor, T. A. Fennerty, and C. D. Bain, “Polarization Effects in Optically Bound Particle Arrays,” Opt. Express |

7. | J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic Clusters Formed By Dielectric Microspheres: Numerical Simulations,” Phys. Rev. B |

8. | M. Kawano, J. T. Blakely, R. Gordon, and D. Sinton, “Theory of Dielectric Micro-sphere Dynamics in a Dual-beam Optical Trap,” Opt. Express |

9. | J. M. Taylor, L. Y. Wong, C. D. Bain, and G. D. Love, “Emergent Properties in Optically Bound Matter,” Opt. Express |

10. | V. Karásek, O. Brzobohatý, and P. Zemánek, “Longitudinal Optical Binding of Several Spherical Particles Studied by the Coupled Dipole Method,” J. Opt. A |

11. | T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A.M. Bránczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical Tweezers Computational Toolbox,” J. Opt. A |

12. | M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical Manipulation of Nanoparticles: A Review,” J. Nanophoton. |

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

14. | D. McGloin, A. E. Carruthers, K. Dholakia, and E. M. Wright, “Optically Bound Microscopic Particles in One Dimension,” Phys. Rev. E |

15. | T. Čižmár, V. Kollárová, Z. Bouchal, and P. Zemánek, “Sub-micron Particle Organization by Self-imaging of Non-diffracting Beams,” New J. Phys. |

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

17. | C. Liang and Y. T. Lo, “Scattering by Two Spheres,” Radio Science |

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. |

**OCIS Codes**

(160.4670) Materials : Optical materials

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

(290.4020) Scattering : Mie theory

**ToC Category:**

Optical Trapping and Manipulation

**History**

Original Manuscript: July 15, 2009

Revised Manuscript: August 12, 2009

Manuscript Accepted: August 12, 2009

Published: August 14, 2009

**Virtual Issues**

Vol. 4, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

J. M. Taylor and G. D. Love, "Optical binding mechanisms: a conceptual model for Gaussian beam traps," Opt. Express **17**, 15381-15389 (2009)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-17-17-15381

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### References

- 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]
- 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).
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- M. Dienerowitz, M. Mazilu, and K. Dholakia, "Optical Manipulation of Nanoparticles: A Review," J. Nanophoton. 2, 021,875 (2008).
- M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, "Optical Binding," Phys. Rev. Lett. 63(12), 1233-1236 (1989). [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]
- 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]
- D.W. Mackowski, "Analysis of Radiative Scattering for Multiple Sphere Configurations," Proc. R. Soc. London, Ser. A 433, 599-614 (1991). [CrossRef]
- C. Liang and Y. T. Lo, "Scattering by Two Spheres," Radio Science 2, 1481-1495 (1967).
- 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]

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