## Electromagnetic forces for an arbitrary optical trapping of a spherical dielectric

Optics Express, Vol. 14, Issue 26, pp. 13101-13106 (2006)

http://dx.doi.org/10.1364/OE.14.013101

Acrobat PDF (472 KB)

### Abstract

A double tweezers setup was employed to perform ultra sensitive force measurements and to obtain the full optical force curve as a function of radial position and wavelength. The light polarization was used to select either the transverse electric (TE), or transverse magnetic (TM), or both, modes excitation. Analytical solution for optical trapping force on a spherical dielectric particle for an arbitrary positioned focused beam is presented in a generalized Lorenz-Mie diffraction theory. The theoretical prediction of the theory agrees well with the experimental results. The algorithm presented here can be easily extended to other beam geometries and scattering particles.

© 2006 Optical Society of America

## 1. Introduction

10. A. Fontes, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, C. L. Cesar, and A. M. de Paula, “Double optical tweezers for ultrasensitive force spectroscopy in microsphere Mie scattering,” App. Phys. Lett. **87**, 221109 (2005). [CrossRef]

12. A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. de B. Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. **31**, 2477–2479 (2006). [CrossRef] [PubMed]

## 2. Theory

14. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A **19**, 1177–1179 (1979). [CrossRef]

15. 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]

*j*

_{n}(

*kr*) are spherical Bessel functions and

12. A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. de B. Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. **31**, 2477–2479 (2006). [CrossRef] [PubMed]

*x*-polarized truncated TEM

_{00}Gaussian beam focused at the (

*ρ*

_{o},

*ϕ*

_{o},

*z*

_{o}) cylindrical coordinates point, is given by the expression:

*a*

_{n}and

*b*

_{n}are the usual Lorenz-Mie coefficients that presents the observed Mie resonances.

## 3. Materials and methods

_{00}laser beam brought to diffraction limited focal spot with a large NA microscope objective (1.25NA 100x oil). We used the same oil immersion objective lens for focusing the trapping beam, the perturbing beam, and collecting the backscattered light [10

10. A. Fontes, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, C. L. Cesar, and A. M. de Paula, “Double optical tweezers for ultrasensitive force spectroscopy in microsphere Mie scattering,” App. Phys. Lett. **87**, 221109 (2005). [CrossRef]

18. E. Fallman and O. Axner, “Design for fully steerable dual-trap optical tweezers,” Appl. Opt. **36**, 2107–2113 (1997). [CrossRef] [PubMed]

## 4. Results and discussion

*n*

_{max}=

*kr*

_{o}[12

12. A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. de B. Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. **31**, 2477–2479 (2006). [CrossRef] [PubMed]

10. A. Fontes, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, C. L. Cesar, and A. M. de Paula, “Double optical tweezers for ultrasensitive force spectroscopy in microsphere Mie scattering,” App. Phys. Lett. **87**, 221109 (2005). [CrossRef]

**87**, 221109 (2005). [CrossRef]

## 5. Conclusions

## Acknowledgments

## References and links

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

2. | G. Gouesbet, B. Maheu, and G. Grehan, “Light scattering from a sphere arbitrarily located in a Gaussian beam using Bromwich formulation,” J Opt Soc Am A , |

3. | G. Gouesbet and G. Grehan, “Sur la generalization de la théorie de Lorenz-Mie,” J. Opt. |

4. | K. F. Ren, G. Gouesbet, and G. Grehan, “Integral localized approximation in generalized Lorenz-Mie theory,” Appl. Opt. |

5. | P. A. M. Neto and H. M. Nussenzveig, “Theory of optical tweezers,” Europhys. Lett. |

6. | A. Mazolli, P. A. M. Neto, and H. M. Nussenzveig, “Theory of trapping forces in optical tweezers,” Proc. R. Soc. A-Math. Phys. Eng. Sci. |

7. | J. A. Lock, “Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. I. Localized model description of an on-axis tightly focused laser beam with spherical aberration,” Appl. Opt. |

8. | J. A. Lock, “Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. II. On-axis trapping force,” Appl. Opt. |

9. | G. Knöner, S. Parkin, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Measurement of the Index of Refraction of Single Microparticles,” Phys. Rev. Lett. |

10. | A. Fontes, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, C. L. Cesar, and A. M. de Paula, “Double optical tweezers for ultrasensitive force spectroscopy in microsphere Mie scattering,” App. Phys. Lett. |

11. | A. Fontes, K. Ajito, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, A. M. de Paula, and C. L. Cesar, “Raman, hyper-Raman, hyper-Rayleigh, two-photon luminescence and morphology-dependent resonance modes in a single optical tweezers system,” Phys. Rev. E. |

12. | A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. de B. Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. |

13. | A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times Legendre polynomials class integrals,” J. Phys. A |

14. | L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A |

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

16. | L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge U. Press, 2006). |

17. | J. D. Jackson, Classical Electrodynamics (Wiley, 1999). |

18. | E. Fallman and O. Axner, “Design for fully steerable dual-trap optical tweezers,” Appl. Opt. |

**OCIS Codes**

(140.7010) Lasers and laser optics : Laser trapping

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

(180.0180) Microscopy : Microscopy

(260.1960) Physical optics : Diffraction theory

(260.2110) Physical optics : Electromagnetic optics

(290.4020) Scattering : Mie theory

**ToC Category:**

Trapping

**History**

Original Manuscript: October 10, 2006

Revised Manuscript: December 13, 2006

Manuscript Accepted: December 14, 2006

Published: December 22, 2006

**Virtual Issues**

Vol. 2, Iss. 1 *Virtual Journal for Biomedical Optics*

**Citation**

Antonio A. R. Neves, Adriana Fontes, Liliana de Y. Pozzo, Andre A. de Thomaz, Enver Chillce, Eugenio Rodriguez, Luiz C. Barbosa, and Carlos L. Cesar, "Electromagnetic forces for an arbitrary optical trapping of a spherical dielectric," Opt. Express **14**, 13101-13106 (2006)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-14-26-13101

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### References

- D. G. Grier, "A revolution in optical manipulation," Nature 424, 810-816 (2003). [CrossRef] [PubMed]
- G. Gouesbet, B. Maheu, G. Grehan, "Light scattering from a sphere arbitrarily located in a Gaussian beam using Bromwich formulation," J Opt Soc Am A, 5, 1427-1443 (1988). [CrossRef]
- G. Gouesbet and G. Grehan, "Sur la generalization de la théorie de Lorenz-Mie, " J. Opt. 13, 97-103 (1982). [CrossRef]
- K. F. Ren, G. Gouesbet, and G. Grehan, "Integral localized approximation in generalized Lorenz-Mie theory," Appl. Opt. 37, 4218-4225 (1998). [CrossRef]
- P. A. M. Neto, and H. M. Nussenzveig, "Theory of optical tweezers," Europhys. Lett. 50, 702-708 (2000). [CrossRef]
- A. Mazolli, P. A. M. Neto, and H. M. Nussenzveig, "Theory of trapping forces in optical tweezers," Proc. R. Soc. A-Math. Phys. Eng. Sci. 459, 3021-3041 (2003). [CrossRef]
- J. A. Lock, "Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. I. Localized model description of an on-axis tightly focused laser beam with spherical aberration," Appl. Opt. 43, 2532-2544 (2004). [CrossRef] [PubMed]
- J. A. Lock, "Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. II. On-axis trapping force," Appl. Opt. 43, 2545-2554 (2004). [CrossRef] [PubMed]
- G. Knöner, S. Parkin, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Measurement of the Index of Refraction of Single Microparticles," Phys. Rev. Lett. 97, 157402 (2006). [CrossRef] [PubMed]
- A. Fontes, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, C. L. Cesar, and A. M. de Paula, "Double optical tweezers for ultrasensitive force spectroscopy in microsphere Mie scattering," App. Phys. Lett. 87, 221109 (2005). [CrossRef]
- A. Fontes, K. Ajito, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, A. M. de Paula, and C. L. Cesar, "Raman, hyper-Raman, hyper-Rayleigh, two-photon luminescence and morphology-dependent resonance modes in a single optical tweezers system," Phys. Rev. E. 72, 012903 (2005). [CrossRef]
- A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. de B. Cruz, L. C. Barbosa, and C. L. Cesar, "Exact partial wave expansion of optical beams with respect to an arbitrary origin," Opt. Lett. 31, 2477-2479 (2006). [CrossRef] [PubMed]
- A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, "Analytical results for a Bessel function times Legendre polynomials class integrals," J. Phys. A 39, L293-L296 (2006). [CrossRef]
- L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19, 1177-1179 (1979). [CrossRef]
- 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]
- L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge U. Press, 2006).
- J. D. Jackson, Classical Electrodynamics (Wiley, 1999).
- E. Fallman and O. Axner, "Design for fully steerable dual-trap optical tweezers," Appl. Opt. 36, 2107-2113 (1997). [CrossRef] [PubMed]

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