## Analytical prediction of stable optical trapping in optical vortices created by three TE or TM plane waves

Optics Express, Vol. 15, Issue 13, pp. 8010-8020 (2007)

http://dx.doi.org/10.1364/OE.15.008010

Acrobat PDF (238 KB)

### Abstract

A closed-form analytical expression of the force on an infinite lossless dielectric cylinder due to multiple plane wave incidences is proposed. The formula for a TE polarization is derived and completes our previous work which was limited to TM polarizations. A unified form of the analytical expression of the force is proposed and used to study the curvature of the one dimensional potential of an optical lattice created by the interference of three plane waves. It is shown that the points of zero curvature yield optical vortices which can be used to stably trap particles of particular sizes and index contrasts with the background. Under these circumstances, the trajectories of the particles can be assimilated to spirals whose centers correspond to the points of undetermined phase in the optical landscape.

© 2007 Optical Society of America

## 1. Introduction

1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A **45**, 8185–8189 (1992). [CrossRef] [PubMed]

2. J. F. Nye and J. V. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Prof. R. Soc. Lond. A **409**, 21–36 (1987). [CrossRef]

3. D. G. Grier, “A revolution in optical manipulation,” Nature **424**, 810–816 (2003). [CrossRef] [PubMed]

4. M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. **6**, 259–268 (2004). [CrossRef]

5. L. Paterson, E. Papagiakoumou, G. Milne, V. Garcés-Chávez, S. A. Tatarkova, W. Sibbett, F. J. Gunn-Moore, P. E. Bryant, A. C. Riches, and K. Dholakia, “Light-induced cell separation in a tailored optical landscape,” Appl. Phys. Lett. **87**, 123901 (2005). [CrossRef]

6. A. Jesacher, S. Fürhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, “Holographic optical tweezers for object manipulations at an air-liquid surface,” Opt. Express **14**, 6342–6352 (2006). [CrossRef] [PubMed]

7. K. T. Gahagan and J. G. A. Swartzlander, “Optical vortex trapping of particles,” Opt. Lett. **21**, 827–829 (1996). [CrossRef] [PubMed]

8. 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]

7. K. T. Gahagan and J. G. A. Swartzlander, “Optical vortex trapping of particles,” Opt. Lett. **21**, 827–829 (1996). [CrossRef] [PubMed]

9. D. Cojoc, V. Garbin, E. Ferrari, L. Businaro, F. Romanato, and E. Di Fabrizio, “Laser trapping and micro-manipulation using optical vortices,” Microelectron. Eng. **78–79**, 125–131 (2005). [CrossRef]

10. J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. **207**, 169–175 (2002). [CrossRef]

11. M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, “Optical Matter: Crystallization and Binding in Intense Optical Fields,” Science **249**, 749–754 (1990). [CrossRef] [PubMed]

12. J.-M. Fournier, G. Boer, G. Delacrétaz, P. Jacquot, J. Rohner, and R. Salathé, “Building Optical Matter with Binding and Trapping Forces,” Proc. SPIE **5514**, 309–317 (2004). [CrossRef]

*x̂*direction obtained from the force). In order to show this, we first derive a closed-form analytical expression of the force on an infinite lossless dielectric cylinder subject to multiple TE (magnetic field parallel to the axis of the cylinder) plane waves. This derivation completes our previous work [13

13. T. M. Grzegorczyk and J. A. Kong, “Analytical expression of the force due to multiple TM plane wave incidences on an infinite dielectric cylinder,” J. Opt. Soc. Am. B **24**, 644–652 (2006). [CrossRef]

14. J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Optics Commun. **198**, 21–27 (2001). [CrossRef]

## 2. Analytical expression of the force

*θ*=

*π*/2,

*ϕ*) impinge on an infinite lossless dielectric cylinder of radius

*a*and permittivity

*ε*, in a background medium of permittivity

_{p}*ε*. The interference of the multiple in-plane incidences creates an optical landscape which in turn induces a force on the cylindrical particle. This force has so far been computed either numerically using techniques such as the discrete-dipole approximation [15

15. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. **333**, 848–872 (1988). [CrossRef]

16. P. C. Chaumet and M. Nieto-Vesperinas, “Coupled dipole method determination of the electromagnetic force on particle over a flat dielectric substrate,” Phys. Rev. B **61**, 14119–14127 (2000). [CrossRef]

17. T. M. Grzegorczyk, B. A. Kemp, and J. A. Kong, “Stable optical trapping based on optical binding forces,” Phys. Rev. Lett. **96**, 113903 (2006). [CrossRef] [PubMed]

18. 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]

19. D. Maystre and P. Vincent, “Making photonic crystals using trapping and binding optical forces on particles,” J. of Opt. A: Pure Appl. Opt. **8**, 1059–1066 (2006). [CrossRef]

20. B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Optical momentum transfer to absorbing Mie particles,” Phys. Rev. Lett. **97**, 133902 (2006). [CrossRef] [PubMed]

13. T. M. Grzegorczyk and J. A. Kong, “Analytical expression of the force due to multiple TM plane wave incidences on an infinite dielectric cylinder,” J. Opt. Soc. Am. B **24**, 644–652 (2006). [CrossRef]

13. T. M. Grzegorczyk and J. A. Kong, “Analytical expression of the force due to multiple TM plane wave incidences on an infinite dielectric cylinder,” J. Opt. Soc. Am. B **24**, 644–652 (2006). [CrossRef]

*a*

_{n}

^{(M)s}into Eqs. (2) and using the Wronskian to obtain

*k*is the wave vector inside the particle and

_{p}*D*

^{TE}_{n}is given by

*x̂*component of the force, we write

*F*= (

_{x}^{TE}*πε*/2)ℜ(

*A*+

*B*) where, after some algebra,

*k̂*=

_{i}**k**

_{i}/|

**k**

_{i}|), and

*i*and #

*j*, where Φ

_{ij}= (

**k**

_{i}-

**k**

_{j}) ∙

*ρ*+ (

*ϕ*+

_{i}*ϕ*)/2. Naturally, (7) reduces to (6) when

_{j}*i*=

*j*in a straightforward manner.

**24**, 644–652 (2006). [CrossRef]

*p*) as

*ρ*is the position of the particle in the (

*xy*) plane, (

*p*) =TE, TM indicates the polarization,

*E*

^{(p)}is the amplitude of the incident polarization, while (

**k**

_{i},

*ϕ*) and (

_{i}**k**

_{j},

*ϕ*) are the two wavevectors and incident angles of the two plane waves with |

_{j}**k**

_{i},| = |

**k**

_{j}| =

*k*=

*k*

_{0}√

*ε*,

*k*=

_{p}*k*

_{0}√

*ε*,

_{p}*k*

_{0}= 2

*π*/λ. Under the notation of (8), the

*x*and

*y*components of the force are given by

*F*

_{xij}^{(p)}= ℜ (

*F*

_{ij}^{(p)}) and

*F*

_{yij}^{(p)}= ℑ (

*F*

_{ij}^{(p)}), where ℜ (∙) and ℑ (∙) denote the real and imaginary part operators, respectively.

21. P. Zemánek, V. Karásek, and A. Sasso, “Optical forces acting on Rayleigh particle placed into interference field,” Optics Commun. **240**, 401–415 (2004). [CrossRef]

22. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. **61**, 569–582 (1992). [CrossRef] [PubMed]

*F*

^{(p)}=

*F*

_{ii}^{(p)}), the force is in the same direction as the incident wave, the force vanishes if there is no permittivity contrast, it is always positive and pushing the particle; in the case of a Gaussian beam (written as a superposition of weighted plane waves), the operating mode of an optical tweezer can be demonstrated and more generally, the longitudinal and transverse forces can be shown to be separately controlled by the size and the permittivity contrast, respectively [13

**24**, 644–652 (2006). [CrossRef]

*ρ*. The subsequent study on the pseudo-potential is therefore in essence identical for both TE or TM polarizations.

## 3. Pseudo-potential and zero curvature points

*x̂*direction) created by the interference of three plane waves in an analytical manner. The plane waves are supposed to have identical amplitudes and their wavevectors are separated by an angle of 2

*π*/3, yielding an hexagonal interference pattern of high and low intensity regions. These regions can either attract or repel particles, as function of their size and permittivity contrast compared to the background medium. The attraction or repulsion can be quantitatively studied by analyzing the curvature of the pseudo-potential, obtained by a simple derivative of (8) with respect to

*x*, which can be here obtained analytically. This process is simplified by combining the symmetric force terms as

*k*= (

_{ijx}**k**

_{i}-

**k**

_{j}) ∙

*x̂*. A positive (negative) curvature yields a stable (unstable) trap due to a pseudo-potential well (mountain). Various curvatures as function of particle size are shown in Fig. 2 for both TE and TM polarizations, at two highly symmetric locations in the lattice:

*ρ*

_{0}= (0,0) and

*ρ*

_{1}= (2λ/(3√3

*ε*), 0), the latter point being denoted by the right-most cross in Fig. 3 and corresponding to a phase singularity as we shall illustrate hereafter. Although Fig. 2 only shows the curvature along the

*x*direction, which is not sufficient to conclude on the efficiency of the two-dimensional trapping at either

*ρ*

_{0}or

*ρ*

_{1}, it is straightforward to extend the study to other directions and unambiguously characterize the trapping properties in two dimensions. It can be seen from Fig. 2 that at the origin, the curvature is positive for small particles under a TM illumination. This is a direct illustration of the fact that small dielectric particles are well trapped in high intensity regions due to the gradient force, as it has already been demonstrated multiple times using the Rayleigh approximation. Interestingly, the same particles are not stably trapped in the high intensity regions for a TE polarization (note that the optical landscape refers to the magnetic field in this case). A dual behavior occurs at

*ρ*

_{1}for both polarizations.

*ρ*

_{0}presented in Fig. 2) are independent of frequency if the electrical size of the particle remains unchanged,

*i. e*.

*k*

_{0}

*a*= constant: at the origin the sine term vanishes and the cosine term reduces to unity while the other dependencies are

*K*

^{(p)}~

*k*

_{0}, Λ

^{(p)}~ 1/k

_{0}

^{2}, and

*γ*

_{n}

^{(p)}~

*k*

_{0}

^{2}. With the additional

*k*term,

_{ijx}*∂*

^{2}

*U*

^{(p)}/

*∂x*

^{2}becomes independent of frequency at

*ρ*

_{0}. This property allows us to make a connection with the results presented in [18

18. 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]

*a*= 0.15λ and

*a*= 0.3λ, respectively, although the results were obtained at a different wavelengths. The previous argument can be generalized to the location

*ρ*

_{1}to show that in this case, only the crossing points in Fig. 2 are independent of frequency.

*ρ*

_{0}is obtained when the curvature of the corresponding pseudo-potential well is positive and maximum, which corresponds to the positive maxima of the solid (dashed) curves for the TM (TE) polarization. The first such sizes are

*a*~ 0.142λ and

*a*~ 0.293λ for a TM and TE incidence, respectively. Not only these points correspond to the steepest well at ρ

_{0}but they also correspond to the steepest mountain at

*ρ*

_{1}, thus combining the best attraction by the high intensity regions to the best repulsion by the low intensity regions. Correspondingly, it is also possible to trap in the low intensity regions by selecting the negative extrema at

*ρ*

_{0}or the positive extrema at

*ρ*

_{1}. In two dimensions, however, this trapping is not symmetric because of the asymmetry of the three incident waves so that the pseudo-potential well is steeper in some directions than in others (data not shown for brevity).

*ρ*

_{0}and

*ρ*

_{1}. The force distribution for a TM polarization at

*a*= 0.2264λ (the first crossing point) is illustrated by the arrows on the right side of Fig. 3, while the background represent the incident intensity. It is clearly seen that neither the high intensity regions nor the low intensity regions are attractive or repulsive, but that optical vortices are formed around

*ρ*

_{1}. The left side of Fig. 3 shows the phase of the field, where singularities are clearly seen at

*ρ*

_{1}symmetry points, yielding optical vortices. These vortices, however, are only effective at trapping particles if the curvature of the pseudo-potential is close to zero, as we show subsequently.

*ρ*

_{1}when the phase singularity points induce an attractive force on the particle. This can be quantified by looking at a circle centered at

*ρ*

_{1}and requiring the force in the normal direction to be pointing inward (toward

*ρ*

_{1}). Such a circle is represented in Fig. 3, with a radius arbitrarily chosen of 140 nm, along which we compute the force and the normal component as

*F̂*∙

^{TM}*n̂*where

*n̂*is the outward pointing normal vector to the circle. A positive (negative) value of the dot product indicates a normal force pointing away from (toward)

*ρ*

_{1}, the negative value yielding a spiral-like attractor. The results are presented in Fig. 4 for two sizes of particles. The first one corresponds to the zero curvature of the pseudo-potential,

*i.e*.

*a*= 0.2264λ, and shows that various locations around the circle experience different directions of normal force. Such configuration is therefore not adequate for an inward spiraling motion, and a larger size of particle needs to be chosen (a smaller size would yield a less attractive

*ρ*

_{1}point). The minimum value obtained for which the dot product is always negative all around the circle is

*a*= 0.243λ, which is the second curve shown in Fig. 4. The motion corresponding to these two sizes of particles is illustrated in Fig. 3. The first trajectory (marked ‘1’) corresponds to

*a*= 0.2264λ and is seen to spiral away from the singular phase points. Eventually, the particle takes a triangular path between the three closest high intensity points, exactly following the force flow shown on the right side of Fig. 3. Although theoretically the particle can be trapped at

*ρ*

_{0}, Brownian motion will prevent such a trapping to occur in experiments. The second trajectory illustrated in Fig. 3 (marked ‘2’) corresponds to

*a*= 0.243λ. As predicted in Fig. 4, the normal force is here always attractive resulting in an inward spiral-like motion toward the point of low intensity (a symmetric point to

*ρ*

_{1}), resulting in a stable trap. The final position exactly corresponds to the phase singular point, as predicted.

## 4. Conclusion

**24**, 644–652 (2006). [CrossRef]

## Appendix

*ϕ*, and ϕ

_{i}_{j}are the angles of incidence of two plane waves, the total force being obtained by adding all (

*i*,

*j*) contributions. In addition, the space dependency exp(i

**k**

_{i}, ∙

*ρ*) is also added to each

*a*

_{n}

^{(M)}coefficient. The equations become

*k*(i.e.

_{p}a*J*

_{n}=

*J*

_{n}(

*k*)). Further simplifications require the conversion of the sums to have the same limits (a sum from ∑

_{p}a_{n}= ∑

_{n=-∞}

^{+∞}is converted to ∑

_{n=0}

^{+∞}by adding the symmetric terms (n,-n-1) and conversely), the proper combination of terms from (14a) and (14b), and the use of identities on the derivative of the Bessel functions. We finally obtain (for a single incidence for example):

## References and links

1. | L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A |

2. | J. F. Nye and J. V. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Prof. R. Soc. Lond. A |

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

4. | M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. |

5. | L. Paterson, E. Papagiakoumou, G. Milne, V. Garcés-Chávez, S. A. Tatarkova, W. Sibbett, F. J. Gunn-Moore, P. E. Bryant, A. C. Riches, and K. Dholakia, “Light-induced cell separation in a tailored optical landscape,” Appl. Phys. Lett. |

6. | A. Jesacher, S. Fürhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, “Holographic optical tweezers for object manipulations at an air-liquid surface,” Opt. Express |

7. | K. T. Gahagan and J. G. A. Swartzlander, “Optical vortex trapping of particles,” Opt. Lett. |

8. | 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. |

9. | D. Cojoc, V. Garbin, E. Ferrari, L. Businaro, F. Romanato, and E. Di Fabrizio, “Laser trapping and micro-manipulation using optical vortices,” Microelectron. Eng. |

10. | J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. |

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

12. | J.-M. Fournier, G. Boer, G. Delacrétaz, P. Jacquot, J. Rohner, and R. Salathé, “Building Optical Matter with Binding and Trapping Forces,” Proc. SPIE |

13. | T. M. Grzegorczyk and J. A. Kong, “Analytical expression of the force due to multiple TM plane wave incidences on an infinite dielectric cylinder,” J. Opt. Soc. Am. B |

14. | J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Optics Commun. |

15. | B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. |

16. | P. C. Chaumet and M. Nieto-Vesperinas, “Coupled dipole method determination of the electromagnetic force on particle over a flat dielectric substrate,” Phys. Rev. B |

17. | T. M. Grzegorczyk, B. A. Kemp, and J. A. Kong, “Stable optical trapping based on optical binding forces,” Phys. Rev. Lett. |

18. | 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 |

19. | D. Maystre and P. Vincent, “Making photonic crystals using trapping and binding optical forces on particles,” J. of Opt. A: Pure Appl. Opt. |

20. | B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Optical momentum transfer to absorbing Mie particles,” Phys. Rev. Lett. |

21. | P. Zemánek, V. Karásek, and A. Sasso, “Optical forces acting on Rayleigh particle placed into interference field,” Optics Commun. |

22. | A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. |

**OCIS Codes**

(020.7010) Atomic and molecular physics : Laser trapping

(260.2110) Physical optics : Electromagnetic optics

(290.4020) Scattering : Mie theory

**ToC Category:**

Trapping

**History**

Original Manuscript: April 24, 2007

Revised Manuscript: June 4, 2007

Manuscript Accepted: June 6, 2007

Published: June 12, 2007

**Virtual Issues**

Vol. 2, Iss. 7 *Virtual Journal for Biomedical Optics*

**Citation**

Tomasz M. Grzegorczyk and Jin Au Kong, "Analytical prediction of stable optical trapping in optical vortices created by
three TE or TM plane waves," Opt. Express **15**, 8010-8020 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-13-8010

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### References

- L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992). [CrossRef] [PubMed]
- J. F. Nye and J. V. Hajnal, "The wave structure of monochromatic electromagnetic radiation," Proc. R. Soc. London Ser. A 409, 21-36 (1987). [CrossRef]
- D. G. Grier, "A revolution in optical manipulation," Nature 424, 810-816 (2003). [CrossRef] [PubMed]
- M. V. Berry, "Optical vortices evolving from helicoidal integer and fractional phase steps," J. Opt. A: Pure Appl. Opt. 6, 259-268 (2004). [CrossRef]
- L. Paterson, E. Papagiakoumou, G. Milne, V. Garcés-Chávez, S. A. Tatarkova, W. Sibbett, F. J. Gunn-Moore, P. E. Bryant, A. C. Riches, and K. Dholakia, "Light-induced cell separation in a tailored optical landscape," Appl. Phys. Lett. 87, 123901 (2005). [CrossRef]
- A. Jesacher, S. Fürhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, "Holographic optical tweezers for object manipulations at an air-liquid surface," Opt. Express 14, 6342-6352 (2006). [CrossRef] [PubMed]
- K. T. Gahagan and J. G. A. Swartzlander, "Optical vortex trapping of particles," Opt. Lett. 21, 827-829 (1996). [CrossRef] [PubMed]
- 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]
- D. Cojoc, V. Garbin, E. Ferrari, L. Businaro, F. Romanato, and E. Di Fabrizio, "Laser trapping and micromanipulation using optical vortices," Microelectron. Eng. 78-79, 125-131 (2005). [CrossRef]
- J. E. Curtis, B. A. Koss, and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002). [CrossRef]
- M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, "Optical Matter: crystallization and binding in intense optical fields," Science 249, 749-754 (1990). [CrossRef] [PubMed]
- J.-M. Fournier, G. Boer, G. Delacrétaz, P. Jacquot, J. Rohner, and R. Salathé, "Building optical matter with binding and trapping forces," Proc. SPIE 5514, 309-317 (2004). [CrossRef]
- T. M. Grzegorczyk and J. A. Kong, "Analytical expression of the force due to multiple TM plane wave incidences on an infinite dielectric cylinder," J. Opt. Soc. Am. B 24, 644-652 (2006). [CrossRef]
- J. Masajada and B. Dubik, "Optical vortex generation by three plane wave interference," Opt. Commun. 198, 21-27 (2001). [CrossRef]
- B. T. Draine, "The discrete-dipole approximation and its application to interstellar graphite grains," Astrophys. J. 333, 848-872 (1988). [CrossRef]
- P. C. Chaumet and M. Nieto-Vesperinas, "Coupled dipole method determination of the electromagnetic force on particle over a flat dielectric substrate," Phys. Rev. B 61, 14119-14127 (2000). [CrossRef]
- T. M. Grzegorczyk, B. A. Kemp, and J. A. Kong, "Stable optical trapping based on optical binding forces," Phys. Rev. Lett. 96, 113903 (2006). [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]
- D. Maystre and P. Vincent, "Making photonic crystals using trapping and binding optical forces on particles," J. Opt. A: Pure Appl. Opt. 8, 1059-1066 (2006). [CrossRef]
- B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, "Optical momentum transfer to absorbing Mie particles," Phys. Rev. Lett. 97, 133902 (2006). [CrossRef] [PubMed]
- P. Zemanek, V. Karasek, and A. Sasso, "Optical forces acting on Rayleigh particle placed into interference field," Opt. Commun. 240, 401-415 (2004). [CrossRef]
- A. Ashkin, "Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime," Biophys. J. 61, 569-582 (1992). [CrossRef] [PubMed]

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