## Gradient forces on double-negative particles in optical tweezers using Bessel beams in the ray optics regime |

Optics Express, Vol. 18, Issue 23, pp. 24287-24292 (2010)

http://dx.doi.org/10.1364/OE.18.024287

Acrobat PDF (956 KB)

### Abstract

Gradient forces on double negative (DNG) spherical dielectric particles are theoretically evaluated for *v*-th Bessel beams supposing geo-metrical optics approximations based on momentum transfer. For the first time in the literature, comparisons between these forces for double positive (DPS) and DNG particles are reported. We conclude that, contrary to the conventional case of positive refractive index, the gradient forces acting on a DNG particle may not reverse sign when the relative refractive index *n* goes from |*n*| > 1 to |*n*| < 1, thus revealing new and interesting trapping properties.

© 2010 OSA

## 1. Introduction

1. A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science **235**(4795), 1517–1520 (1987). [CrossRef] [PubMed]

2. R. W. Steubing, S. Cheng, W. H. Wright, Y. Numajiri, and M. W. Berns, “Laser induced cell fusion in combination with optical tweezers: the laser cell fusion trap,” Cytometry **12**(6), 505–510 (1991). [CrossRef] [PubMed]

3. M. W. Berns, W. H. Wright, B. J. Tromberg, G. A. Profeta, J. J. Andrews, and R. J. Walter, “Use of a laser-induced optical force trap to study chromosome movement on the mitotic spindle,” Proc. Natl. Acad. Sci. U.S.A. **86**(12), 4539–4543 (1989). [CrossRef] [PubMed]

4. V. Emiliani, D. Cojoc, E. Ferrari, V. Garbin, C. Durieux, M. Coppey-Moisan, and E. Di Fabrizio, “Wave front engineering for microscopy of living cells,” Opt. Express **13**(5), 1395–1405 (2005). [CrossRef] [PubMed]

5. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. **61**(2), 569–582 (1992). [CrossRef] [PubMed]

6. G. Gouesbet, G. Gréhan, and B. Maheu, “Localized interpretation to compute all the coefficients *g _{n}^{m}* in the generalized Lorenz-Mie theory,” J. Opt. Soc. Am. A

**7**(6), 998–1007 (1990). [CrossRef]

*n*=

*n*/

_{p}*n*(

_{m}*n*and

_{p}*n*are, respectively, the refractive index of the particle and of the surrounding medium) is positive and higher than one, it is to be expected that the particle be directed towards high intensity regions of the beam, whereas if

_{m}*n*is higher than zero but less than unity, the contrary take place: the particle is directed away from these regions. Finally, if

*n*>> 1, scattering forces (parallel to the optical axis of the beam) can make optical trapping inefficient or even impossible to be achieved, and other schemes, such as two counter-propagating beams, must be used [7

7. A. van der Horst, P. D. J. van Oostrum, A. Moroz, A. van Blaaderen, and M. Dogterom, “High trapping forces for high-refractive index particles trapped in dynamic arrays of counterpropagating optical tweezers,” Appl. Opt. **47**(17), 3196–3202 (2008). [CrossRef] [PubMed]

*n*>> 1 case, even gradient forces (perpendicular to the optical axis of the beam) can reverse sign, making that previous scheme useless [8

8. L. A. Ambrosio and H. E. Hernández-Figueroa, “Inversion of gradient forces for high refractive index particles in optical trapping,” Opt. Express **18**(6), 5802–5808 (2010). [CrossRef] [PubMed]

*n*or, equivalently,

_{p}*n*, were negative? In a recent paper [9

9. L. A. Ambrosio and H. E. Hernández-Figueroa, “Trapping double negative particles in the ray optics regime using optical tweezers with focused beams,” Opt. Express **17**(24), 21918–21924 (2009). [CrossRef] [PubMed]

*n*|, and a repulsive pattern for incident angles above some critical angle (see, e.g., Fig. 2 of ref [9

9. L. A. Ambrosio and H. E. Hernández-Figueroa, “Trapping double negative particles in the ray optics regime using optical tweezers with focused beams,” Opt. Express **17**(24), 21918–21924 (2009). [CrossRef] [PubMed]

10. L. A. Ambrosio and H. E. Hernández-Figueroa, “Fundamentals of negative refractive index optical trapping: forces and radiation pressures exerted by focused Gaussian beams using the generalized Lorenz-Mie theory,” Biomed. Opt. Express (to be published). [PubMed]

*v*-th order Bessel beam

*J*(.), using geometrical optics. Multi-ringed beams offers several advantages over focused beams, such as the simultaneous trapping of several biological particles of arbitrary shapes and sizes [11

_{v}11. J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. **197**(4-6), 239–245 (2001). [CrossRef]

12. V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. **85**(18), 4001–4003 (2004). [CrossRef]

13. V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature **419**(6903), 145–147 (2002). [CrossRef] [PubMed]

## 2. Theoretical analysis

*z*, meaning that all rays impinging the spherical particle comes from –

*z*. In the ray optics regime, we can assign a specific power

*P*to each of these rays according to a Bessel beam profile, as follows:(

*r*,

*θ*,

*ϕ*) being spherical coordinates relative to the center of the particle,

*k*the radial wavenumber of the beam and

_{ρ}*ρ*

_{0}and

*ϕ*

_{0}defined according to Fig. 1.

*R*and

*T*are the Fresnel coefficients of reflection and transmission, respectively,

*θ*the incident angle of the incident ray and

_{i}*θ*the angle of the transmitted ray [9

_{t}9. L. A. Ambrosio and H. E. Hernández-Figueroa, “Trapping double negative particles in the ray optics regime using optical tweezers with focused beams,” Opt. Express **17**(24), 21918–21924 (2009). [CrossRef] [PubMed]

*θ*+ 2

_{i}*θ*, because all infinite reflected/transmitted rays follow the inverted Snell’s law, due to

_{t}*n*being negative [14

_{p}14. V. G. Veselago, “The Electrodynamics of Substances with Simultaneously Negative Values of ε and μ,” Sov. Phys. Usp. **10**(4), 509–514 (1968). [CrossRef]

**17**(24), 21918–21924 (2009). [CrossRef] [PubMed]

*k*= 83261.3 m

_{ρ}^{−1}. A dielectric, homogeneous, isotropic and linear spherical particle with radius

*a*= 10λ is assumed throughout in all simulations. These parameters were chosen in order to ensure the ray optics requirements and that the scalar power formula approximation, given by (1), is completely within its range of validity.

*ϕ*

_{0}= −π (

*F*=

_{g}*F*),

_{x}*v*= 0,

*n*= 1.33 and several values of

_{m}*n*=

*n*/

_{p}*n*. Because of

_{m}*ϕ*

_{0}, negative

*F*means an attractive force. The intensity profile of the beam is also plotted as a solid line. Note the common repulsive/attractive pattern according to the intensity of the beam and its reversion when

_{x}*n*=

*n*/

_{p}*n*goes from

_{m}*n*> 1 to 0 <

*n*< 1. For

*n*> 1, points of stable equilibrium occurs at

*ρ*

_{0}≈0, 46, 84 μm and all subsequent high intensity regions of the beam (not seen in the picture), whereas for 0 <

*n*< 1, those points are

*ρ*

_{0}≈30, 67 μm and so forth. It is obvious that

*F*would be zero for all

_{x}*ρ*

_{0}whenever

*n*= 1 (matched case).

*F*≠ 0 for

_{x}*n*= −1, this curve is also presented for completeness. In Figs. 4 and 5 , results for

*F*over a DPS and a DNG particle are shown for

_{x}*v*= 3, respectively. In all cases, positive values of

*F*means that this force is repulsive, and vice-versa. To ensure that our theoretical proposal is adequate, all simulations were compared with those obtained by means of the generalized Lorenz-Mie theory (GLMT) with the integral localized approximation [15,16]. Good agreement was achieved.

_{x}*n*| < 1 and |

*n*| ≥ 1 and compare it with the DPS case, as we have done before for a focused Gaussian beam (Fig. 2 of ref [9

**17**(24), 21918–21924 (2009). [CrossRef] [PubMed]

*ρ*

_{0}< Δρ (although we have assumed

*ϕ*

_{0}= −π, due to symmetry and the optical regime adopted, the choice of

*ϕ*

_{0}is irrelevant). If the overall presence of rays with an attractive nature relative to this axis, i.e., towards the optical axis, overcomes the repulsive effect (away from the optical axis), then the DNG particle will be directed towards lower values of

*ρ*

_{0}. The same qualitative analysis is valid for

*ρ*

_{0}> Δρ.

*ρ*

_{0}< Δρ and all subsequent regions where the total gradient force is positive: for a zero-order Bessel beam and the parameters chosen, the DNG particle will always be directed towards low intensity regions of the beam (nulls of intensity) and there it will remain trapped, regardless of |

*n*| being higher or lower than one. Analogous considerations can be made for Fig. 5, where

*v*= 3. Differently from what was observed for a Gaussian beam [12

12. V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. **85**(18), 4001–4003 (2004). [CrossRef]

*F*as a function of both ρ

_{x}_{0}and

*n*for the zero-order Bessel beam used before, as shown in Fig. 6 , reveals that it is also possible to have attractive forces towards high intensity regions (bright annular disks) of this multi-ringed beam even for DNG particles.

*P*(

*r*,

*θ*,

*ϕ*) and the radius

*a*of the DNG particle, it can be pushed against or toward high intensity regions of the incident beam, regardless of |

*n*| being higher or less than one.

## Acknowledgements

## References and links

1. | A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science |

2. | R. W. Steubing, S. Cheng, W. H. Wright, Y. Numajiri, and M. W. Berns, “Laser induced cell fusion in combination with optical tweezers: the laser cell fusion trap,” Cytometry |

3. | M. W. Berns, W. H. Wright, B. J. Tromberg, G. A. Profeta, J. J. Andrews, and R. J. Walter, “Use of a laser-induced optical force trap to study chromosome movement on the mitotic spindle,” Proc. Natl. Acad. Sci. U.S.A. |

4. | V. Emiliani, D. Cojoc, E. Ferrari, V. Garbin, C. Durieux, M. Coppey-Moisan, and E. Di Fabrizio, “Wave front engineering for microscopy of living cells,” Opt. Express |

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

6. | G. Gouesbet, G. Gréhan, and B. Maheu, “Localized interpretation to compute all the coefficients 7(6), 998–1007 (1990). [CrossRef] |

7. | A. van der Horst, P. D. J. van Oostrum, A. Moroz, A. van Blaaderen, and M. Dogterom, “High trapping forces for high-refractive index particles trapped in dynamic arrays of counterpropagating optical tweezers,” Appl. Opt. |

8. | L. A. Ambrosio and H. E. Hernández-Figueroa, “Inversion of gradient forces for high refractive index particles in optical trapping,” Opt. Express |

9. | L. A. Ambrosio and H. E. Hernández-Figueroa, “Trapping double negative particles in the ray optics regime using optical tweezers with focused beams,” Opt. Express |

10. | L. A. Ambrosio and H. E. Hernández-Figueroa, “Fundamentals of negative refractive index optical trapping: forces and radiation pressures exerted by focused Gaussian beams using the generalized Lorenz-Mie theory,” Biomed. Opt. Express (to be published). [PubMed] |

11. | J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. |

12. | V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. |

13. | V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature |

14. | V. G. Veselago, “The Electrodynamics of Substances with Simultaneously Negative Values of ε and μ,” Sov. Phys. Usp. |

15. | L. A. Ambrosio, and H. E. Hernández-Figueroa are preparing a manuscript to be called “Integral localized approximation description of ordinary Bessel beams in the generalized Lorenz-Mie theory and application to optical forces.” |

16. | L. A. Ambrosio, and H. E. Hernández-Figueroa, “Integral localized approximation description of |

**OCIS Codes**

(080.0080) Geometric optics : Geometric optics

(350.3618) Other areas of optics : Left-handed materials

(160.3918) Materials : Metamaterials

(350.4855) Other areas of optics : Optical tweezers or optical manipulation

**ToC Category:**

Optical Trapping and Manipulation

**History**

Original Manuscript: August 6, 2010

Revised Manuscript: October 22, 2010

Manuscript Accepted: October 28, 2010

Published: November 5, 2010

**Virtual Issues**

Vol. 6, Iss. 1 *Virtual Journal for Biomedical Optics*

**Citation**

Leonardo A. Ambrosio and Hugo E. Hernández-Figueroa, "Gradient forces on double-negative particles in optical tweezers using Bessel beams in the ray optics regime," Opt. Express **18**, 24287-24292 (2010)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-18-23-24287

Sort: Year | Journal | Reset

### References

- A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235(4795), 1517–1520 (1987). [CrossRef] [PubMed]
- R. W. Steubing, S. Cheng, W. H. Wright, Y. Numajiri, and M. W. Berns, “Laser induced cell fusion in combination with optical tweezers: the laser cell fusion trap,” Cytometry 12(6), 505–510 (1991). [CrossRef] [PubMed]
- M. W. Berns, W. H. Wright, B. J. Tromberg, G. A. Profeta, J. J. Andrews, and R. J. Walter, “Use of a laser-induced optical force trap to study chromosome movement on the mitotic spindle,” Proc. Natl. Acad. Sci. U.S.A. 86(12), 4539–4543 (1989). [CrossRef] [PubMed]
- V. Emiliani, D. Cojoc, E. Ferrari, V. Garbin, C. Durieux, M. Coppey-Moisan, and E. Di Fabrizio, “Wave front engineering for microscopy of living cells,” Opt. Express 13(5), 1395–1405 (2005). [CrossRef] [PubMed]
- A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61(2), 569–582 (1992). [CrossRef] [PubMed]
- G. Gouesbet, G. Gréhan, and B. Maheu, “Localized interpretation to compute all the coefficients gnm in the generalized Lorenz-Mie theory,” J. Opt. Soc. Am. A 7(6), 998–1007 (1990). [CrossRef]
- A. van der Horst, P. D. J. van Oostrum, A. Moroz, A. van Blaaderen, and M. Dogterom, “High trapping forces for high-refractive index particles trapped in dynamic arrays of counterpropagating optical tweezers,” Appl. Opt. 47(17), 3196–3202 (2008). [CrossRef] [PubMed]
- L. A. Ambrosio and H. E. Hernández-Figueroa, “Inversion of gradient forces for high refractive index particles in optical trapping,” Opt. Express 18(6), 5802–5808 (2010). [CrossRef] [PubMed]
- L. A. Ambrosio and H. E. Hernández-Figueroa, “Trapping double negative particles in the ray optics regime using optical tweezers with focused beams,” Opt. Express 17(24), 21918–21924 (2009). [CrossRef] [PubMed]
- L. A. Ambrosio and H. E. Hernández-Figueroa, “Fundamentals of negative refractive index optical trapping: forces and radiation pressures exerted by focused Gaussian beams using the generalized Lorenz-Mie theory,” Biomed. Opt. Express (to be published). [PubMed]
- J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197(4-6), 239–245 (2001). [CrossRef]
- V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85(18), 4001–4003 (2004). [CrossRef]
- V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002). [CrossRef] [PubMed]
- V. G. Veselago, “The Electrodynamics of Substances with Simultaneously Negative Values of ε and μ,” Sov. Phys. Usp. 10(4), 509–514 (1968). [CrossRef]
- L. A. Ambrosio, and H. E. Hernández-Figueroa are preparing a manuscript to be called “Integral localized approximation description of ordinary Bessel beams in the generalized Lorenz-Mie theory and application to optical forces.”
- L. A. Ambrosio, and H. E. Hernández-Figueroa, “Integral localized approximation description of v-th order Bessel beams in the generalized Lorenz-Mie theory and applications to optical trapping,” in Proceedings of PIERS2011in Marrakesh (to be published).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.