## Radiation force of highly focused Lorentz-Gauss beams on a Rayleigh particle |

Optics Express, Vol. 19, Issue 10, pp. 9708-9713 (2011)

http://dx.doi.org/10.1364/OE.19.009708

Acrobat PDF (919 KB)

### Abstract

The radiation force of highly focused Lorentz-Gauss beams (LG beam) on a dielectric sphere in the Rayleigh scattering regime is theoretically studied. The numerical results show that the Lorentz-Gauss beam can be used to trap particles with the refractive index larger than that of the ambient. The radiation force distribution has been studied under different beam widths of the Lorentz part. The trapping stability under different conditions is also analyzed.

© 2011 OSA

## 1. Introduction

1. O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. **8**(5), 409–414 (2006). [CrossRef]

2. M. A. Bandres and J. C. Gutiérrez-Vega, “Cartesian beams,” Opt. Lett. **32**(23), 3459–3461 (2007). [CrossRef] [PubMed]

3. W. P. Dumke, “Angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. **11**(7), 400–402 (1975). [CrossRef]

4. A. Naqwi and F. Durst, “Focusing of diode laser beams: a simple mathematical model,” Appl. Opt. **29**(12), 1780–1785 (1990). [CrossRef] [PubMed]

5. A. Torre, W A B. Evans, O. E. Gawhary, and S. Severini, “Relativistic Hermite polynomials and Lorentz beams,” J. Opt. A, Pure Appl. Opt. **10**(11), 115007 (2008). [CrossRef]

9. G. Q. Zhou and X. X. Chu, “Average intensity and spreading of a Lorentz-Gauss beam in turbulent atmosphere,” Opt. Express **18**(2), 726–731 (2010). [CrossRef] [PubMed]

## 2. Radiation force produced by the Lorentz-Gauss beam

1. O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. **8**(5), 409–414 (2006). [CrossRef]

*x*and

_{0}*y*is the coordinates in the input plane;

_{0}*w*and

_{x}*w*are beam widths of the Lorentz part;

_{y}*λ*is the wavelength,

*w*is the waist of the Gaussian part;

_{0}*A*is a constant.

_{0}1. O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. **8**(5), 409–414 (2006). [CrossRef]

6. G. Q. Zhou, “Focal shift of focused truncated Lorentz-Gauss beam,” J. Opt. Soc. Am. A **25**(10), 2594–2599 (2008). [CrossRef]

*z*is the axial distance,

*j*stands for

*x*or

*y*,

*x*) is the error function.We consider the Lorentz-Gauss beam propagating through a lens as shown in Fig. 1 , so the transfer matrix for this system is:where

*z*

_{1}is the axial distance between the focus plane and the output plane,

*s*is the axial distance between the input plane and the lens,

*f*is the focal length. Substituting Eq. (4) into Eq. (2), we can get the distribution of the electric field at the output plane. In our paper, we choose

*s*= 100mm,

*f*= 10 mm,

*λ*= 1064nm,

*w*= 10mm,

_{0}*w*=

_{x}*w*, and the input power

_{y}*P*= 1W.

*x*direction. We can see that when the two types of beams possess the same power and

*w*, the maximum intensity of the Lorentz-Gauss beam is greater than that of the Gaussian beam at the input plane; but this relationship reverses at the focus plane.

_{0}*α*for the point dipole in SI units is [23

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

*a*is the radius of the particle,

24. Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. **124**(5-6), 529–541 (1996). [CrossRef]

25. M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics **2**(1), 021875 (2008). [CrossRef]

*a*= 30nm, the refractive index of the particle

*n*= 1.59 (i.e., glass) and the refractive index of the surrounding medium

_{p}*n*= 1.33 (i.e., water), so

_{m}*x*direction; the force distribution in other transverse directions can be obtained by analogy. From Figs. 3(a), 3(d) and 3(g), we can see that a Rayleigh particle whose refractive index is larger than that of the ambient can be trapped at the focus point by the highly focused Lorentz-Gauss beam, because there are stable equilibrium points in Figs. 3(a) and 3(d). From Fig. 3, we can also see that the force distribution of Lorentz-Gauss beam gradually approaches to that of the fundamental Gaussian beam as

*w*increases:

_{x}*w*, but

_{x}*z*= 2um) decreases as

_{1}*w*increase. From Figs. 3(e) and 3(f), we can find that the Lorentz-Gauss beam could stably trap the particle at

_{x}*x*= 0.25um, but the fundamental Gaussian beam could not. When

*x*= 0.5um, neither the two beams could trap the particle. So we can conclude that using Lorentz-Gauss beam instead of Gaussian beam, the trapping stability for the Rayleigh dielectric particle will be better in the trapping region except at the focal point. From Figs. 3(g) and 3(h), it is also found that the Lorentz-Gauss beam with small

*w*results in small scattering force, which is of benefit to trapping.

_{x}## 3. Analysis of trapping stability

10. 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**(5), 288–290 (1986). [CrossRef] [PubMed]

26. K. Okamoto and S. Kawata, “Radiation force exerted on subwavelength particles near a nanoaperture,” Phys. Rev. Lett. **83**(22), 4534–4537 (1999). [CrossRef]

*η*is the viscosity of the ambient, which is 8.0 × 10

^{−4}Pa·s at

*T*= 300K;

*a*is the radius and

*k*is the Boltzmann constant.

_{B}*w*<20mm under the conditions that

_{x}*f*=10 mm,

*a*=30nm,

*w*=10mm; if

_{0}*w*is too small, the Brownian force will be too strong. In Fig. 4(b), it is found that the particle can be stably trapped when 10nm<

_{x}*a*<50nm under the conditions that

*w*=10mm,

_{x}*f*=10mm,

*w*=10mm. When

_{0}*a*is too small, the disturbance is mainly from the Brownian motion; but when

*a*is greater than 27nm, the disturbance is mainly from the scattering force. However, trapping for even larger particles with Lorentz-Gauss beam could not be analyzed by the theory of Rayleigh approximation, so alternative theories must be applied, which deserves our further investigations.

## 4. Conclusion

*w*on the radiation force. It is found that the gradient force at the focus plane increases as

_{x}*w*increases, but decreases at the planes near the focus plane. So the Lorentz Gauss beam with a small

_{x}*w*offers advantage over the beam with a great

_{x}*w*and the Gaussian beam while trapping particles at the planes near the focus plane. Finally, the analysis of trapping stability shows that it is necessary to choose suitable

_{x}*w*and radius

_{x}*a*for effective trapping with the Lorentz-Gauss beam. Our results are interesting and useful for particle trapping.

## Acknowledgments

## References and links

1. | O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. |

2. | M. A. Bandres and J. C. Gutiérrez-Vega, “Cartesian beams,” Opt. Lett. |

3. | W. P. Dumke, “Angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. |

4. | A. Naqwi and F. Durst, “Focusing of diode laser beams: a simple mathematical model,” Appl. Opt. |

5. | A. Torre, W A B. Evans, O. E. Gawhary, and S. Severini, “Relativistic Hermite polynomials and Lorentz beams,” J. Opt. A, Pure Appl. Opt. |

6. | G. Q. Zhou, “Focal shift of focused truncated Lorentz-Gauss beam,” J. Opt. Soc. Am. A |

7. | G. Q. Zhou, “Nonparaxial propagation of a Lorentz-Gauss beam,” J. Opt. Soc. Am. B |

8. | G. Q. Zhou, “Propagation of a partially coherent Lorentz-Gauss beam through a paraxial ABCD optical system,” Opt. Express |

9. | G. Q. Zhou and X. X. Chu, “Average intensity and spreading of a Lorentz-Gauss beam in turbulent atmosphere,” Opt. Express |

10. | 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. | A. A. Ambardekar and Y. Q. Li, “Optical levitation and manipulation of stuck particles with pulsed optical tweezers,” Opt. Lett. |

12. | P. Zemánek and C. J. Foot, “Atomic dipole trap formed by blue detuned strong Gaussian standing wave,” Opt. Commun. |

13. | S. M. Block, L. S. B. Goldstein, and B. J. Schnapp, “Bead movement by single kinesin molecules studied with optical tweezers,” Nature |

14. | D. E. Smith, S. J. Tans, S. B. Smith, S. Grimes, D. L. Anderson, and C. Bustamante, “The bacteriophage straight phi29 portal motor can package DNA against a large internal force,” Nature |

15. | L. Oroszi, P. Galajda, H. Kirei, S. Bottka, and P. Ormos, “Direct measurement of torque in an optical trap and its application to double-strand DNA,” Phys. Rev. Lett. |

16. | C. Day, “Optical trap resolves the stepwise transfer of genetic information from DNA to RNA,” Phys. Today |

17. | M. Dienerowitz, M. Mazilu, P. J. Reece, T. F. Krauss, and K. Dholakia, “Optical vortex trap for resonant confinement of metal nanoparticles,” Opt. Express |

18. | C. L. Zhao, L. G. Wang, and X. H. Lu, “Radiation forces on a dielectric sphere produced by highly focused hollow Gaussian beams,” Phys. Lett. A |

19. | C. L. Zhao, L. G. Wang, and X. H. Lu, “Radiation forces of highly focused Bessel-Gaussian beams on a dielectric sphere,” Optik (Stuttg.) |

20. | Q. W. Zhan, “Radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams,” J. Opt. A, Pure Appl. Opt. |

21. | L. G. Wang, C. L. Zhao, L. Q. Wang, X. H. Lu, and S. Y. Zhu, “Effect of spatial coherence on radiation forces acting on a Rayleigh dielectric sphere,” Opt. Lett. |

22. | C. L. Zhao, Y. J. Cai, X. H. Lu, and H. T. Eyyuboğlu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express |

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

24. | Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. |

25. | M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics |

26. | K. Okamoto and S. Kawata, “Radiation force exerted on subwavelength particles near a nanoaperture,” Phys. Rev. Lett. |

**OCIS Codes**

(140.3300) Lasers and laser optics : Laser beam shaping

(140.7010) Lasers and laser optics : Laser trapping

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

**ToC Category:**

Optical Trapping and Manipulation

**History**

Original Manuscript: April 14, 2011

Manuscript Accepted: April 28, 2011

Published: May 3, 2011

**Virtual Issues**

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

**Citation**

Yunfeng Jiang, Kaikai Huang, and Xuanhui Lu, "Radiation force of highly focused Lorentz-Gauss beams on a Rayleigh particle," Opt. Express **19**, 9708-9713 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-10-9708

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### References

- O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8(5), 409–414 (2006). [CrossRef]
- M. A. Bandres and J. C. Gutiérrez-Vega, “Cartesian beams,” Opt. Lett. 32(23), 3459–3461 (2007). [CrossRef] [PubMed]
- W. P. Dumke, “Angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. 11(7), 400–402 (1975). [CrossRef]
- A. Naqwi and F. Durst, “Focusing of diode laser beams: a simple mathematical model,” Appl. Opt. 29(12), 1780–1785 (1990). [CrossRef] [PubMed]
- A. Torre, W A B. Evans, O. E. Gawhary, and S. Severini, “Relativistic Hermite polynomials and Lorentz beams,” J. Opt. A, Pure Appl. Opt. 10(11), 115007 (2008). [CrossRef]
- G. Q. Zhou, “Focal shift of focused truncated Lorentz-Gauss beam,” J. Opt. Soc. Am. A 25(10), 2594–2599 (2008). [CrossRef]
- G. Q. Zhou, “Nonparaxial propagation of a Lorentz-Gauss beam,” J. Opt. Soc. Am. B 26(1), 141–147 (2009). [CrossRef]
- G. Q. Zhou, “Propagation of a partially coherent Lorentz-Gauss beam through a paraxial ABCD optical system,” Opt. Express 18(5), 4637–4643 (2010). [CrossRef] [PubMed]
- G. Q. Zhou and X. X. Chu, “Average intensity and spreading of a Lorentz-Gauss beam in turbulent atmosphere,” Opt. Express 18(2), 726–731 (2010). [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(5), 288–290 (1986). [CrossRef] [PubMed]
- A. A. Ambardekar and Y. Q. Li, “Optical levitation and manipulation of stuck particles with pulsed optical tweezers,” Opt. Lett. 30(14), 1797–1799 (2005). [CrossRef] [PubMed]
- P. Zemánek and C. J. Foot, “Atomic dipole trap formed by blue detuned strong Gaussian standing wave,” Opt. Commun. 146(1-6), 119–123 (1998). [CrossRef]
- S. M. Block, L. S. B. Goldstein, and B. J. Schnapp, “Bead movement by single kinesin molecules studied with optical tweezers,” Nature 348(6299), 348–352 (1990). [CrossRef] [PubMed]
- D. E. Smith, S. J. Tans, S. B. Smith, S. Grimes, D. L. Anderson, and C. Bustamante, “The bacteriophage straight phi29 portal motor can package DNA against a large internal force,” Nature 413(6857), 748–752 (2001). [CrossRef] [PubMed]
- L. Oroszi, P. Galajda, H. Kirei, S. Bottka, and P. Ormos, “Direct measurement of torque in an optical trap and its application to double-strand DNA,” Phys. Rev. Lett. 97(5), 058301 (2006). [CrossRef] [PubMed]
- C. Day, “Optical trap resolves the stepwise transfer of genetic information from DNA to RNA,” Phys. Today 59(1), 26–27 (2006). [CrossRef]
- M. Dienerowitz, M. Mazilu, P. J. Reece, T. F. Krauss, and K. Dholakia, “Optical vortex trap for resonant confinement of metal nanoparticles,” Opt. Express 16(7), 4991–4999 (2008). [CrossRef] [PubMed]
- C. L. Zhao, L. G. Wang, and X. H. Lu, “Radiation forces on a dielectric sphere produced by highly focused hollow Gaussian beams,” Phys. Lett. A 363(5-6), 502–506 (2007). [CrossRef]
- C. L. Zhao, L. G. Wang, and X. H. Lu, “Radiation forces of highly focused Bessel-Gaussian beams on a dielectric sphere,” Optik (Stuttg.) 119(10), 477–480 (2008). [CrossRef]
- Q. W. Zhan, “Radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams,” J. Opt. A, Pure Appl. Opt. 5(3), 229–232 (2003). [CrossRef]
- L. G. Wang, C. L. Zhao, L. Q. Wang, X. H. Lu, and S. Y. Zhu, “Effect of spatial coherence on radiation forces acting on a Rayleigh dielectric sphere,” Opt. Lett. 32(11), 1393–1395 (2007). [CrossRef] [PubMed]
- C. L. Zhao, Y. J. Cai, X. H. Lu, and H. T. Eyyuboğlu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express 17(3), 1753–1765 (2009). [CrossRef] [PubMed]
- B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988). [CrossRef]
- Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 (1996). [CrossRef]
- M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics 2(1), 021875 (2008). [CrossRef]
- K. Okamoto and S. Kawata, “Radiation force exerted on subwavelength particles near a nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 (1999). [CrossRef]

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