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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 10 — May. 9, 2011
  • pp: 9708–9713
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Radiation force of highly focused Lorentz-Gauss beams on a Rayleigh particle

Yunfeng Jiang, Kaikai Huang, and Xuanhui Lu  »View Author Affiliations


Optics Express, Vol. 19, Issue 10, pp. 9708-9713 (2011)
http://dx.doi.org/10.1364/OE.19.009708


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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

Recently, a new type of optical beam called Lorentz-Gauss beam has attracted a great deal of interest. The existence of Lorentz-Gauss beam is demonstrated both in theory and in experiment. In theory, the Lorentz-Gauss beam is proved a closed-form solution of the paraxial wave equation [1

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

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

]; in experiment, the Lorentz-Gauss beam can be realized by certain double heterojunction lasers [3

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

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

]. The characteristics and applications of Lorentz-Gauss beams have been investigated [5

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

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

Using the Cartesian coordinate system, whose origin is the center of the beam, we can represent the electric field of the Lorentz-Gauss beam at the input plane as [1

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]

]:
E0=A0wxwy1[1+(x0/wx)2][1+(y0/wy)2]exp[ik(x02+y02)2q0],
(1)
where x0 and y0 is the coordinates in the input plane; wx and wy are beam widths of the Lorentz part;k=2π/λis the wave number,q0=iπw02/λis the radius of curvature, λ is the wavelength, w0 is the waist of the Gaussian part; A0 is a constant.

Under paraxial approximation, the electric field of Lorentz-Gauss beam that passes through an ABCD optical system can be expressed as [1

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

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

]:
E(x,y,z)=iA0π24λBexp(ikz)exp[ik(x2+y2)2q](Vx++Vx)(Vy++Vy),
(2)
where z is the axial distance, q=(Aq0+B)/(Cq0+D)is the radius of curvature at the output plane, and
Vj±=exp[ikG2B(wj±ijG)]{1erf[ikG2B(wj±ijG)]},
(3)
where j stands for x or y, G=A+B/q0, erf(x) is the error function.We consider the Lorentz-Gauss beam propagating through a lens as shown in Fig. 1
Fig. 1 The sketch of the optical system, in which the Lorentz-Gauss beam is highly focused by the lens.
, so the transfer matrix for this system is:
[ABCD]=[z1fsz1f+f+z11fsf+1],
(4)
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, w0 = 10mm, wx = wy, and the input power P = 1W.

In Fig. 2
Fig. 2 The intensity distribution in the x direction of the Lorentz-Gauss beam (LGB) and the fundamental Gaussian beam (GB) at (a) the input plane; (b) the focus plane. w0 is also the beam waist of GB and wx = 10mm for LGB.
, the intensity distribution of the Lorentz-Gauss beam is compared with the fundamental Gaussian beam in the x direction. We can see that when the two types of beams possess the same power and w0, 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.

As is well known, the Rayleigh particle, whose radius is much smaller than the wavelength, can be treated as a point dipole in the light fields. And the polarisabilityαfor 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]

]:
α=4πa3εpεmεp+2εm,
(5)
where a is the radius of the particle,εpandεmare dielectric functions of the particle and the medium surrounding the particle, respectively. So the gradient force Fgrad and the scattering force Fsca can be calculated by [24

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

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

]:
Fgrad=14ε0εmRe(α)|E2|,
(6)
Fsca=ε0εm3k412π|α2||E2|,
(7)
where ε0 is the dielectric constant in vacuum. In our following investigation, we assume the a = 30nm, the refractive index of the particle np = 1.59 (i.e., glass) and the refractive index of the surrounding medium nm = 1.33 (i.e., water), soεp=np2, εm=nm2.

We plot distributions of the transverse gradient force Fgrad,x, the longitudinal gradient force Fgrad,z, the scattering force Fsca exerted on the dielectric particles in Fig. 3
Fig. 3 The transverse gradient force ((a)-(c)) produced by highly focused Lorentz-Gauss beam with different wx at different positions z1, and the Longitudinal gradient force ((d)-(f)) and the scattering force ((g) and (h)) at different transverse positions x. The calculation parameters are w0 = 10mm, f = 10mm. GB is the fundamental Gaussian beam with beam waist w0.
. For simplicity, we only investigate the radiation force distribution in the 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 wx increases: Fgrad,x in the focus plane and Fgrad,z increase with wx, but Fgrad,x near the focus plane (z1 = 2um) decreases as wx increase. From Figs. 3(e) and 3(f), we can find that the Lorentz-Gauss beam could stably trap the particle at 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 wx results in small scattering force, which is of benefit to trapping.

3. Analysis of trapping stability

For stably trapping, the gradient force must be large enough to overcome the scattering force, the Brownian force and the gravity of the particle [10

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]

]. The Brownian force, which describes the influence of the Brownian motion, can be calculated by the expression [26

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

]:
|Fb|=12πηakBT,
(8)
where η is the viscosity of the ambient, which is 8.0 × 10−4 Pa·s at T = 300K; a is the radius and kB is the Boltzmann constant.

We plot the change of the magnitude of all the forces in Fig. 4
Fig. 4 Comparison of Fgrad,xm, Fgrad,zm, Fb and Fg (a) with different wx, while the other parameters are f = 10 mm, a = 30 nm, w0 = 10 mm, and (b) with different particle radius a, while the other parameters are wx = 10 mm, f = 10 mm,w0 = 10 mm.
, where Fgrad,xm is maximum transverse gradient force, Fgrad,zm is maximum longitudinal gradient force, Fscam is maximum scattering force, Fb is the Brownian force, Fg is the gravity. We can see that the gravity of the particle could be neglected comparing with the gradient force. In Fig. 4(a), it is found that the particle can be stably trapped when 1mm<wx<20mm under the conditions that f=10 mm, a=30nm, w0=10mm; if wx 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<a<50nm under the conditions that wx=10mm, f=10mm, w0=10mm. When 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

In summary, we have studied the radiation force on a dielectric spherical particle produced by the highly focused Lorentz-Gauss beam in the Rayleigh scattering regime. We have also investigated the effect of the beam width of the Lorentz part wx on the radiation force. It is found that the gradient force at the focus plane increases as wx increases, but decreases at the planes near the focus plane. So the Lorentz Gauss beam with a small wx offers advantage over the beam with a great wx 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 wx and radius a for effective trapping with the Lorentz-Gauss beam. Our results are interesting and useful for particle trapping.

Acknowledgments

This work was supported by National Nature Science Foundation of China (Grant No. 10974177, 10874012) and the program of International S&T Cooperation of China (Grant No. 2010DFA04690).

References and links

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]

6.

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

7.

G. Q. Zhou, “Nonparaxial propagation of a Lorentz-Gauss beam,” J. Opt. Soc. Am. B 26(1), 141–147 (2009). [CrossRef]

8.

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]

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]

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]

11.

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]

12.

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]

13.

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]

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 413(6857), 748–752 (2001). [CrossRef] [PubMed]

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. 97(5), 058301 (2006). [CrossRef] [PubMed]

16.

C. Day, “Optical trap resolves the stepwise transfer of genetic information from DNA to RNA,” Phys. Today 59(1), 26–27 (2006). [CrossRef]

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 16(7), 4991–4999 (2008). [CrossRef] [PubMed]

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 363(5-6), 502–506 (2007). [CrossRef]

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.) 119(10), 477–480 (2008). [CrossRef]

20.

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]

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. 32(11), 1393–1395 (2007). [CrossRef] [PubMed]

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 17(3), 1753–1765 (2009). [CrossRef] [PubMed]

23.

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

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]

26.

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

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

  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]
  6. G. Q. Zhou, “Focal shift of focused truncated Lorentz-Gauss beam,” J. Opt. Soc. Am. A 25(10), 2594–2599 (2008). [CrossRef]
  7. G. Q. Zhou, “Nonparaxial propagation of a Lorentz-Gauss beam,” J. Opt. Soc. Am. B 26(1), 141–147 (2009). [CrossRef]
  8. 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]
  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]
  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]
  11. 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]
  12. 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]
  13. 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]
  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 413(6857), 748–752 (2001). [CrossRef] [PubMed]
  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. 97(5), 058301 (2006). [CrossRef] [PubMed]
  16. C. Day, “Optical trap resolves the stepwise transfer of genetic information from DNA to RNA,” Phys. Today 59(1), 26–27 (2006). [CrossRef]
  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 16(7), 4991–4999 (2008). [CrossRef] [PubMed]
  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 363(5-6), 502–506 (2007). [CrossRef]
  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.) 119(10), 477–480 (2008). [CrossRef]
  20. 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]
  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. 32(11), 1393–1395 (2007). [CrossRef] [PubMed]
  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 17(3), 1753–1765 (2009). [CrossRef] [PubMed]
  23. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988). [CrossRef]
  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]
  26. 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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