## Theory of dielectric micro-sphere dynamics in a dual-beam optical trap

Optics Express, Vol. 16, Issue 13, pp. 9306-9317 (2008)

http://dx.doi.org/10.1364/OE.16.009306

Acrobat PDF (808 KB)

### Abstract

We investigate the dynamics of an array of polystyrene micron-sized spheres in a dual-beam fiber-optic trap. Experimental results show non-uniform equilibrium particle spacing and spontaneous self-sustained oscillation for large particle numbers. Results are analyzed with a Maxwell-Stress Tensor method using the Generalized Multipole Technique, where hydrodynamic interactions between particles are included. The theoretical analysis matches well with the experimentally observed equilibrium particle spacing. The theory shows that an offset in the trapping beams is the underlying mechanism for the oscillations and influences both the oscillation frequency and the damping rate for oscillations. The theory presented is of general interest to other systems involving multi-particle optical interactions.

© 2008 Optical Society of America

## 1. Introduction

1. D. G. Grier, “A revolution in optical manipulation,” Nature **424**, 810–878 (2004). [CrossRef]

2. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. **24**, 156–159 (1970). [CrossRef]

3. A. Constable, J. Kim, J. Mervis, F. Zarinetchi, and M. Prentiss, “Demonstration of a fiber-optical lightforce trap,” Opt. Lett. **18**, 1867–1869 (1993). [CrossRef] [PubMed]

4. E. R. Lyons and G. J. Sonek, “Confinement and bistability in a tapered hemispherically lensed optical fiber trap,” Appl. Phys. Lett. **66**, 1584–1586 (1995). [CrossRef]

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

6. 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**, 1568–1574 (2003). [CrossRef]

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

12. H. C. Nagerl, W. Bechter, J. Eschner, F. Schmidt- Kaler, and R. Blatt, “Ion strings for quantum gates,” Appl. Phys. B **66**, 603–608 (1998). [CrossRef]

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

*et al.*using counter-propagating laser beams [5

**89**, 283901 (2002). [CrossRef]

*et al.*undertook experiments containing different particle sizes to investigate the dependency of equilibrium spacing on the particle size [6

6. 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**, 1568–1574 (2003). [CrossRef]

*et al.*proposed a numerical model based on a two-component approach, involving gradient and scattering forces, combined with the paraxial wave equation, where they calculated equilibrium spacing for a two- and three-particle system [7

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

*et al.*simulated bistability and hysteresis in optical binding between two particles [8

8. N. K. Metzger, K. Dholakia, and E. M. Wright, “Observation of Bistability and Hysteresis in Optical Binding of Two Dielectric Sphere,” Phys. Rev. Lett. **96**, 068102 (2006). [CrossRef] [PubMed]

11. N. K. Metzger, R. F. Marchington, M. Mazilu, R. L. Smith, K. Dholakia, and E. M. Wright, “Measurement of the Restoring Forces Acting on Two Optically Bound Particles from Normal Mode Correlations,” Phys. Rev. Lett. **98**, 068102 (2007). [CrossRef] [PubMed]

13. V. Karasek, K. Dholakia, and P. Zemanek, “Analysis of optical binding in one dimension,” Appl. Phys. B **84**, 149–156 (2006). [CrossRef]

14. J. Ng, C. T. Chan, and P. Sheng, “Strong optical force induced by morphology-dependent resonance,” Opt. Lett. **30**, 1956–1958 (2005). [CrossRef] [PubMed]

15. T. M. Grzegorczyk, B. A. Kemp, and J. A. Kong, “Trapping and binding of an arbitrary number of cylindrical particles in an in-plane electromagnetic field,” J. Opt. Soc. Am. A **23**, 2324–2330 (2006). [CrossRef]

**89**, 283901 (2002). [CrossRef]

*I*-particle array decreases as

*I*increases. Furthermore, the theory describes the onset of spontaneous oscillation for a critical number of particles, which has not been theoretically analyzed for a multi-particle array. To analyze the experimentally observed phenomena, optical forces acting on the trapped particles are numerically calculated using the Maxwell stress tensor with Generalized Multipole Technique (MST-GMT) [16, 17

17. K. Koba, H. Ikuno, and M. Kawano, “Numerical analysis of electromagnetic scattering from 3-D dielectric objects using the Yasuura method,” Electrical Engineering in Japan , **148** (2), 39–45, New York:Wiley (2004). [CrossRef]

## 2. Experiment

### 2.1 Chip fabrication and experimental setup

19. D. C. Duffy, J. C. McDonald, O. J. Schueller, and G. M. Whitesides, “Rapid Prototyping of Microfluidic Systems in Poly(dimethylsiloxane),” Anal. Chem. **70**, 4974–4984 (1998). [CrossRef] [PubMed]

^{2}vias where they come in contact with the perpendicular microfluidic channel used for particle delivery. The axial separation of the two cleaved fibers was set at 160 µm, whereas the layout and molding of the fiber vias ensured radial pre-alignment. Each fiber was coupled to a separate laser diode source, both operating at a wavelength of 980 nm and emitting 100 mW.

20. X. Ma, J. Q. Lu, R. S. Brock, K. M. Jacobs, P. Yang, and X. Hu, “Determination of complex refractive index of polystyrene microsphere from 370 to 1610 nm,” Phys. Med. Biol. **48**, 4165–4172 (2003). [CrossRef]

### 2.2 Measurement and imaging

### 2.3 Inhomogeneous spacing and oscillation

## 3. Optohydrodynamic theory

*a*trapped between opposing fibers separated by a distance

*D*

_{bf}. The divergent beams (DB1 and DB2) from the optical fibers (OF1 and OF2) are modeled by a Gaussian beam. To make the model realistic, we consider the case of transversely misaligned beam axes (

*D*

_{off}≠0), in addition to the case of perfectly aligned beam axes (

*D*

_{off}=0), where

*D*

_{off}is the distance between the beam axes (BA1 and BA2). As will be shown later, this misalignment results in spontaneous oscillation of the particle array.

### 3.1 Optical force calculation

**F**

_{i}acting on the

*i*

^{th}particle can be calculated by integrating the Maxwell stress tensor

**T⃡**over a closed surface S

_{M, i}which surrounds the particle [21]:

**v**

_{M}is a unit normal vector of the closed surface. The Maxwell stress tensor (MST),

**T⃡**is given by

**EE*** is a dyadic product,

**I⃡**is the unit tensor, and Re denotes the real part of. The electric and magnetic fields

**E**and

**H**are, respectively, given by

**E**

^{in}and

**H**

^{in}are the incident fields, i.e. divergent beams from the fiber facets, and

**E**

^{sc}and

**H**

^{sc}are the scattered field by particles.

**E**

^{sc}and

**H**

^{sc}numerically, we apply the Generalized Multipole Technique (GMT) [16, 17

17. K. Koba, H. Ikuno, and M. Kawano, “Numerical analysis of electromagnetic scattering from 3-D dielectric objects using the Yasuura method,” Electrical Engineering in Japan , **148** (2), 39–45, New York:Wiley (2004). [CrossRef]

**r**,

**E**

^{sc}

_{N}(

**r**), as

*I*is the total number of particles,

**r**

_{i}denotes a position

**r**in the

*i*th local coordinate system with origin O

_{i}located at the centre of the

*i*

^{th}particle, and

**E**

^{sc}

_{i,N}(

**r**

_{i}), is the scattered electric field from the

*i*

^{th}particle. Following the GMT, we expand

**E**

^{sc}

_{i,N}(

**r**

_{i}) as

**m**

^{(4)}

_{nm}and

**n**

^{(4)}

_{nm}are the Hankel function-based spherical vector wave functions (SVWFs) [22],

*N*is a truncation size of the expansion that is given depending on a desired accuracy of the solution, and

*a*and

^{i}_{nm}*b*are unknown coefficients to be determined. To calculate these coefficients, we also need to expand the transmitted field inside the

^{i}_{nm}*i*

^{th}particle,

**E**

^{tr}

_{i,N}(

**r**

_{i}), by SVWFs as

**m**

^{(1)}

_{nm}and

**n**

^{(1)}

_{nm}are Bessel function based SVWFs [22], and

*c*and

^{i}_{nm}*d*are unknown coefficients. Using Faraday’s law and the relation, ∇×

^{i}_{nm}**m**=

*k*

**n**and ∇×

**n**=

*k*

**m**, where

*k*is the wave number, the scattered magnetic field from the

*i*

^{th}particle,

**H**

^{sc}

_{i,N}(

**r**

_{i}), and the transmitted magnetic field inside the

*i*

^{th}particle,

**H**

^{tr}

_{i,N}(

**r**

_{i}), are given by

_{out}and Z

_{in}are the intrinsic impedance of surrounding medium and that inside the

*i*

^{th}particle, respectively.

*a*,

^{i}_{nm}*b*,

^{i}_{nm}*c*, and

^{i}_{nm}*d*are determined by matching the boundary conditions on the surface of the particles. Here we numerically match the boundary conditions in a least squares sense [17

^{i}_{nm}17. K. Koba, H. Ikuno, and M. Kawano, “Numerical analysis of electromagnetic scattering from 3-D dielectric objects using the Yasuura method,” Electrical Engineering in Japan , **148** (2), 39–45, New York:Wiley (2004). [CrossRef]

**v**

_{P}is unit normal vector on the particle surface S

_{P, j}. Minimization of the squared norm results in a

*K*×

*K*(

*K*=4

*IN*(

*N*+2)) matrix equation [17]. The discretization of the squared norm is important from a numerical point of view and the discretization rule can be found elsewhere [17

**148** (2), 39–45, New York:Wiley (2004). [CrossRef]

*N*=7-10. By integrating the MST on a closed surface S

_{M,i}outside the particles numerically, we can obtain the optical force acting on the

*i*

^{th}particle. As a closed surface S

_{M,i}, we choose a concentric sphere surface with radius b (>a) which surrounds only the

*i*

^{th}particle as shown in Fig. 2. In our simulation, we assume that the counter-propagating beams are mutually incoherent [6

6. 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**, 1568–1574 (2003). [CrossRef]

9. N. K. Metzger, E. M. Wright, W. Sibbett, and K. Dholakia, “Visualization of optical of microparticles using a femtosecond fiber optical trap,” Opt. Express **14**, 3677–3687 (2006). [CrossRef] [PubMed]

### 3.2 Dynamic simulation

**R**

_{i}and

**v**

_{i}are the center position and the velocity of the

*i*

^{th}particle, respectively. In our simulation, we approximate the velocity

**v**

_{i}using the Oseen tensor

**H⃡**[18],

**F**

_{j}is the optical force acting on the

*j*

^{th}particle and

*a*is the friction coefficient, η is the viscosity of the surrounding medium, δ

_{ij}is Kronecker delta, and

*r*=|

_{ij}**r**

_{ij}|=|

**r**

_{i}-

**r**

_{j}|. To evaluate Eq. (10) numerically, we use the Euler method, i.e. the position at time t=(

*m*+1)Δt is determined by

## 4. Numerical results and discussion

23. J. P. Barton and D. R. Alexander, “Fifth order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. **66**, 2800–2802 (1989). [CrossRef]

_{0}=3.0 µm and λ=980 µm in free space, respectively, and the distance between the two fibers is set at

*D*

_{bf}=160 µm. The radius and refractive index of the particles are 0.5 µm and 1.574, respectively, and the refractive index of the surrounding water is 1.33.

### 4.1 Equilibrium spacing of two- and three- particle arrays

24. FDTD Solutions, from Lumerical Solutions Inc., http://www.lumerical.com.

*D*

_{off}≠0).

*d*, is changed from 2 µm to 20 µm. A modulation was found in the force calculations, as reported previously [13

13. V. Karasek, K. Dholakia, and P. Zemanek, “Analysis of optical binding in one dimension,” Appl. Phys. B **84**, 149–156 (2006). [CrossRef]

*d*, is increased, the absolute value of the forces acting on particles 1 and 2 decrease and the forces take opposite sign after crossing the zero force line around

*d*=9.4 µm. For

*d*<9.4, particles repel each other, while for

*d*>9.4 µm, particles attract. For

*d*=9.4 µm, the net force acting on the particles is zero, and the calculated equilibrium spacing for the two particles is

*d*

_{2,cal}=9.4 µm. This spacing is in good agreement with the experimental result,

*d*

_{2,exp}=9.6 µm. In Fig. 3(b), forces acting on a three-particle array are shown, where the centre particle is fixed at

*z*=0 and the distance,

*d*, between this particle and the neighboring particles is varied. In this case, the calculated equilibrium spacing is around

*d*

_{3,cal}=8.3 µm and is also in good agreement with experimental result

*d*

_{3,exp}=8.0 µm. In agreement with Ref. [5

**89**, 283901 (2002). [CrossRef]

### 4.2 Equilibrium positions of an N-particle array

*D*

_{off}=0) is considered, where particle motion is confined to the z-axis.

*t*=0, the 1

^{st}particle is initially located at an equilibrium position,

*z*

_{1,1}=0, and a 2

^{nd}particle is introduced 20 µm away from the 1

^{st}particle, at

*z*=20 µm, to model an incoming particle. As the 2

^{nd}particle is introduced, the two particles shift in the negative zdirection and settle at their new equilibrium positions,

*z*

_{2,1}=-4.7 µm and

*z*

_{2,2}=4.7 µm, which yields an equilibrium spacing

*d*

_{2,dyn}=9.4 µm. This result is identical to results obtained in Section 4.1, where

*d*

_{2,cal}=9.4 µm. Once the two particles have settled, a 3

^{rd}particle is introduced 20 µm from the 2

^{nd}particle (

*z*=4.7+20 µm) at

*t*/η(T)/10

^{3}≈0.9 m

^{2}/N, where η(T) is the viscosity of water as a function of temperature T [25]. The two previously stable particles shift in the negative z-direction and settle to their new equilibrium positions,

*z*

_{3,1}=-8.3 µm,

*z*

_{3,2}=0 and

*z*

_{3,3}=8.3 µm, for the 1

^{st}, 2

^{nd}, and 3

^{rd}particles, respectively, resulting in an equilibrium spacing

*d*

_{3,dyn}=8.3 µm. This result also matches the value obtained in Section 4.1, where

*d*

_{3,cal}=8.3 µm. Extending this procedure to a 13-particle array, which was the critical number in the experiments that lead to array oscillations, equilibrium positions of a 13-particle array were calculated as shown in Fig. 4(b). At

*t*=0, 12 particles are initially located at their equilibrium positions and a 13

^{th}particle is introduced 20 µm from the 12

^{th}particle. As in the previous case, the 13 particles settle to their new equilibrium positions and the onset of the spontaneous oscillation does not occur for this model containing perfectly aligned beam axes. The results with and without HDI do not show significant deviation for a collinear dual-fiber arrangement; however, the results will be shown to differ significantly for offset beams. Even for perfect alignment, it is clear that the HDI has a stronger influence for more closely spaced particles, as expected from Eq. 12. For example, the deviation between the equilibrium-approach with and without HDI for 4 particles is greater than for just two as in Fig. 4(a).

*d*

_{av}, decreases in a nonlinear way as the total number of trapped particles,

*I*, increases. The calculated theoretical results agree well with those measured in the experiment. Fig. 5(b) shows that, for a given number of trapped particles in the array, the outer particle spacing,

*d*

_{out}, is larger than the inner spacing,

*d*

_{inn}, near the array centre, and that the difference between the outer and inner spacing increases with increasing particle numbers. Again, numerical and experimental results are in close agreement.

### 4.3 Dynamic simulation of the spontaneous oscillation

*D*

_{off}≠0) and show that the spontaneous oscillation observed in the experiment can be caused by this beam misalignment. We assume that the axis of a beam propagating in the positive z-direction lies on the z-axis and that propagating in the negative z-direction is located on a parallel axis with an offset in the positive x-direction,

*D*

_{off}, as shown in Fig. 2. Due to this beam offset, the particles can move in the x-z plane.

_{1}, z

_{1})=(

*D*

_{off}/2, 0), and a second particle is introduced at (x

_{2}, z

_{2})=(

*D*

_{off}, 20 µm) to model an incoming particle. Once the second particle is added, the two particles move towards their new equilibrium positions and settle as described previously. After settling, a third particle is added and this procedure is iterated for up to 7 particles.

26. A. Ashkin, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. **11**, 288–290 (1986). [CrossRef] [PubMed]

26. A. Ashkin, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. **11**, 288–290 (1986). [CrossRef] [PubMed]

27. F. J. Garcia de Abajo, “Collective oscillations in optical matter,” Opt. Express **15**, 11082–11094 (2007). [CrossRef]

*D*

_{off}, as a function of the number

*I*, and quantitative estimates were made by fitting the trajectories to the damping relation

*A*cos(Ω

*t*)exp(-Λ

*t*)+

*B*. Results of the angular frequency Ω and the damping coefficient Λ as a function of

*I*are shown in Fig. 7, together with the results without HDI for reference. The threshold for self-sustained oscillation occurs when Λ becomes negative, which can be seen for the 7 particle case with a 5 µm offset. It is also clear that the oscillation frequency and the damping coefficient are larger when hydrodynamic interaction is included.

## 5. Conclusion

*I*-particle array, the inner inter-particle spacing is smaller than the outer spacing, and the other is a nonlinear decrease of the averaged equilibrium spacing as the number of trapped particles increases. In addition, distinct spontaneous oscillations occurred for a large number of particles. The theory reproduced the features of the experiments by using a Maxwell stress tensor with the GMT while including hydrodynamic interactions. It was shown that a transverse misalignment of beam axes is the mechanism responsible for spontaneous self-sustained oscillation in our dual-beam trap.

## Acknowledgment

## References and links

1. | D. G. Grier, “A revolution in optical manipulation,” Nature |

2. | A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. |

3. | A. Constable, J. Kim, J. Mervis, F. Zarinetchi, and M. Prentiss, “Demonstration of a fiber-optical lightforce trap,” Opt. Lett. |

4. | E. R. Lyons and G. J. Sonek, “Confinement and bistability in a tapered hemispherically lensed optical fiber trap,” Appl. Phys. Lett. |

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

6. | 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 |

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

8. | N. K. Metzger, K. Dholakia, and E. M. Wright, “Observation of Bistability and Hysteresis in Optical Binding of Two Dielectric Sphere,” Phys. Rev. Lett. |

9. | N. K. Metzger, E. M. Wright, W. Sibbett, and K. Dholakia, “Visualization of optical of microparticles using a femtosecond fiber optical trap,” Opt. Express |

10. | M. Guillon, O. Moine, and B. Stout, “Longitudinal optical binding of high optical contrast microdroplets in air,” Phys. Rev. Lett. |

11. | N. K. Metzger, R. F. Marchington, M. Mazilu, R. L. Smith, K. Dholakia, and E. M. Wright, “Measurement of the Restoring Forces Acting on Two Optically Bound Particles from Normal Mode Correlations,” Phys. Rev. Lett. |

12. | H. C. Nagerl, W. Bechter, J. Eschner, F. Schmidt- Kaler, and R. Blatt, “Ion strings for quantum gates,” Appl. Phys. B |

13. | V. Karasek, K. Dholakia, and P. Zemanek, “Analysis of optical binding in one dimension,” Appl. Phys. B |

14. | J. Ng, C. T. Chan, and P. Sheng, “Strong optical force induced by morphology-dependent resonance,” Opt. Lett. |

15. | T. M. Grzegorczyk, B. A. Kemp, and J. A. Kong, “Trapping and binding of an arbitrary number of cylindrical particles in an in-plane electromagnetic field,” J. Opt. Soc. Am. A |

16. | C. Hafner, The generalized multiple multipole technique for computational electromagnetics, (Boston: Artech, 1990). |

17. | K. Koba, H. Ikuno, and M. Kawano, “Numerical analysis of electromagnetic scattering from 3-D dielectric objects using the Yasuura method,” Electrical Engineering in Japan , |

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

19. | D. C. Duffy, J. C. McDonald, O. J. Schueller, and G. M. Whitesides, “Rapid Prototyping of Microfluidic Systems in Poly(dimethylsiloxane),” Anal. Chem. |

20. | X. Ma, J. Q. Lu, R. S. Brock, K. M. Jacobs, P. Yang, and X. Hu, “Determination of complex refractive index of polystyrene microsphere from 370 to 1610 nm,” Phys. Med. Biol. |

21. | J. D. Jackson, |

22. | J. A. Stratton, |

23. | J. P. Barton and D. R. Alexander, “Fifth order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. |

24. | FDTD Solutions, from Lumerical Solutions Inc., http://www.lumerical.com. |

25. | I. H. Shames, |

26. | A. Ashkin, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. |

27. | F. J. Garcia de Abajo, “Collective oscillations in optical matter,” Opt. Express |

**OCIS Codes**

(140.7010) Lasers and laser optics : Laser trapping

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

(290.5850) Scattering : Scattering, particles

**ToC Category:**

Optical Trapping and Manipulation

**History**

Original Manuscript: April 22, 2008

Revised Manuscript: May 29, 2008

Manuscript Accepted: June 5, 2008

Published: June 9, 2008

**Virtual Issues**

Vol. 3, Iss. 7 *Virtual Journal for Biomedical Optics*

**Citation**

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)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-13-9306

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### References

- D. G. Grier, "A revolution in optical manipulation," Nature 424, 810-878 (2004). [CrossRef]
- A. Ashkin, "Acceleration and trapping of particles by radiation pressure," Phys. Rev. Lett. 24, 156-159 (1970). [CrossRef]
- A. Constable, J. Kim, J. Mervis, F. Zarinetchi, and M. Prentiss, "Demonstration of a fiber-optical light-force trap," Opt. Lett. 18, 1867-1869 (1993). [CrossRef] [PubMed]
- E. R. Lyons and G. J. Sonek, "Confinement and bistability in a tapered hemispherically lensed optical fiber trap," Appl. Phys. Lett. 66, 1584-1586 (1995). [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]
- 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, 1568-1574 (2003). [CrossRef]
- D. McGloin, A. E. Carruthers, K. Dholakia, and M. Wright, "Optically bound microscopic particles in one dimension," Phys. Rev. E 69, 021403 (2004). [CrossRef]
- N. K. Metzger, K. Dholakia, and E. M. Wright, "Observation of Bistability and Hysteresis in Optical Binding of Two Dielectric Sphere," Phys. Rev. Lett. 96, 068102 (2006). [CrossRef] [PubMed]
- N. K. Metzger, E. M. Wright, W. Sibbett, and K. Dholakia, "Visualization of optical of microparticles using a femtosecond fiber optical trap," Opt. Express 14, 3677-3687 (2006). [CrossRef] [PubMed]
- M. Guillon, O. Moine, and B. Stout, "Longitudinal optical binding of high optical contrast microdroplets in air," Phys. Rev. Lett. 96, 143902 (2006). [CrossRef] [PubMed]
- N. K. Metzger, R. F. Marchington, M. Mazilu, R. L. Smith, K. Dholakia, and E. M. Wright, "Measurement of the Restoring Forces Acting on Two Optically Bound Particles from Normal Mode Correlations," Phys. Rev. Lett. 98, 068102 (2007). [CrossRef] [PubMed]
- H. C. Nagerl, W. Bechter, J. Eschner, F. Schmidt- Kaler, and R. Blatt, "Ion strings for quantum gates," Appl. Phys. B 66, 603-608 (1998). [CrossRef]
- V. Karasek, K. Dholakia, and P. Zemanek, "Analysis of optical binding in one dimension," Appl. Phys. B 84, 149-156 (2006). [CrossRef]
- J. Ng, C. T. Chan, and P. Sheng, "Strong optical force induced by morphology-dependent resonance," Opt. Lett. 30, 1956-1958 (2005). [CrossRef] [PubMed]
- T. M. Grzegorczyk, B. A. Kemp, and J. A. Kong, "Trapping and binding of an arbitrary number of cylindrical particles in an in-plane electromagnetic field," J. Opt. Soc. Am. A 23, 2324-2330 (2006). [CrossRef]
- C. Hafner, The Generalized Multiple Multipole Technique for Computational Electromagnetics, (Boston: Artech, 1990).
- K. Koba, H. Ikuno, and M. Kawano, "Numerical analysis of electromagnetic scattering from 3-D dielectric objects using the Yasuura method," Electrical Engineering in Japan, 148 (2), 39-45, New York:Wiley (2004). [CrossRef]
- M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Oxford Press, Oxford, 1994).
- D. C. Duffy, J. C. McDonald, O. J. Schueller and G. M. Whitesides, "Rapid prototyping of Microfluidic Systems in Poly(dimethylsiloxane)," Anal. Chem. 70, 4974-4984 (1998). [CrossRef] [PubMed]
- X. Ma, J. Q. Lu, R. S. Brock, K. M. Jacobs, P. Yang and X. Hu, "Determination of complex refractive index of polystyrene microsphere from 370 to 1610 nm," Phys. Med. Biol. 48, 4165-4172 (2003). [CrossRef]
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