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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 21 — Oct. 10, 2011
  • pp: 20736–20742
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FDTD simulation of trapping nanowires with linearly polarized and radially polarized optical tweezers

Jing Li and Xiaoping Wu  »View Author Affiliations


Optics Express, Vol. 19, Issue 21, pp. 20736-20742 (2011)
http://dx.doi.org/10.1364/OE.19.020736


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Abstract

In this paper a model of the trapping force on nanowires is built by three dimensional finite-difference time-domain (FDTD) and Maxwell stress tensor methods, and the tightly focused laser beam is expressed by spherical vector wave functions (VSWFs). The trapping capacities on nanoscale-diameter nanowires are discussed in terms of a strongly focused linearly polarized beam and radially polarized beam. Simulation results demonstrate that the radially polarized beam has higher trapping efficiency on nanowires with higher refractive indices than linearly polarized beam.

© 2011 OSA

1. Introduction

Optical tweezers have become important and commonly used tools in the field of physics and life science for trapping and moving objects ranging in size from tens of nanometres to tens of micrometres [1

1. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). [CrossRef] [PubMed]

,2

2. A. Ashkin, “History of optical trapping and manipulation of small-neutral particle, atoms, and molecules,” IEEE J. Sel. Top. Quantum Electron. 6(6), 841–856 (2000). [CrossRef]

] since their introduction in 1986 [3

3. 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]

]. Recently, Optical tweezers have been used to trap and process nanowires and other nanomaterials to build new types of micro-nano devices [4

4. P. J. Pauzauskie, A. Radenovic, E. Trepagnier, H. Shroff, P. D. Yang, and J. Liphardt, “Optical trapping and integration of semiconductor nanowire assemblies in water,” Nat. Mater. 5(2), 97–101 (2006). [CrossRef] [PubMed]

6

6. R. Agarwal, K. Ladavac, Y. Roichman, G. H. Yu, C. M. Lieber, and D. G. Grier, “Manipulation and assembly of nanowires with holographic optical traps,” Opt. Express 13(22), 8906–8912 (2005). [CrossRef] [PubMed]

]. However, most materials used for nanowires have high refractive indices, which cause difficulty in trapping by optical tweezers.

The radially polarized beam is a kind of center hollow beam, and the energy of the beam is distributed at the fringe of the beam, which is an advantage for the reduction of scattering force on the nanowire and the improvement of gradient force. In addition, the radial polarization also contributes to the improvement of gradient force. Here the trapping capacity on nanowires with the radially polarized beam is simulated and compared with that of conventionally linearly polarized beam.

For the diameter and length of nanowires in the simulation model both are between 0.1λ~10λ (λ is beam wavelength). Direct solutions of Maxwell’s equations are suitable to get the electromagnetic field, and then the force on a nanowire can be calculated from the Maxwell stress tensor or Poynting’s vector method. In this regard the T-matrix method has been used to simulate laser trapping of nanorods with a circularly polarized beam [7

7. P. B. Bareil and Y. L. Sheng, “Angular and position stability of a nanorod trapped in an optical tweezers,” Opt. Express 18(25), 26388–26398 (2010). [CrossRef] [PubMed]

] and linear nanostructures composed of identical nano-spheres with a linearly polarized beam [8

8. F. Borghese, P. Denti, R. Saija, M. A. Iatì, and O. M. Maragò, “Radiation torque and force on optically trapped linear nanostructures,” Phys. Rev. Lett. 100(16), 163903 (2008). [CrossRef] [PubMed]

]. The three dimensional finite-difference time-domain (FDTD) method is also a robust approach to solve Maxwell’s equations. Although this method is time-consuming, it is very powerful with the potential to model arbitrary shaped objects and different traps. Therefore, in our work a three dimensional FDTD is used to setup a numerical simulation model.

2. Calculation

2.1 Setting of beam

The vector spherical wave functions (VSWFs) are a complete and orthogonal set of solutions to the vector Helmholtz equation [7

7. P. B. Bareil and Y. L. Sheng, “Angular and position stability of a nanorod trapped in an optical tweezers,” Opt. Express 18(25), 26388–26398 (2010). [CrossRef] [PubMed]

,9

9. D. C. Benito, S. H. Simpson, and S. Hanna, “FDTD simulations of forces on particles during holographic assembly,” Opt. Express 16(5), 2942–2957 (2008). [CrossRef] [PubMed]

]. For tightly focused beams, the fifth-order Gaussian beam description provides a significantly improved solution to Maxwell’s equations in comparison with commonly used paraxial Gaussian beam descriptions [10

10. J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66(7), 2800–2802 (1989). [CrossRef]

]. In order to accurately simulate the transmission form of electromagnetic field, the VSWFs are adopted to express fifth order Gaussian beams as incident light [9

9. D. C. Benito, S. H. Simpson, and S. Hanna, “FDTD simulations of forces on particles during holographic assembly,” Opt. Express 16(5), 2942–2957 (2008). [CrossRef] [PubMed]

13

13. G. J. Hu, J. Li, Q. Long, T. Tao, G. X. Zhang, and X. P. Wu, “FDTD numerical simulation of the trapping force of microsphere in single optical tweezers,” Acta Phys. Sin. 60, 30301 (2011).

].

Einc=n=1m=nn[amnRgMmn(kr,θ,φ)+bmnRgNmn(kr,θ,φ)]
(1)
Hinc=jεμn=1m=nn[bmnRgMmn(kr,θ,φ)+amnRgNmn(kr,θ,φ)]
(2)

In the above formulae k is wave number, amnandbmnare shape coefficients, RgMmnandRgNmn are vector wave functions based on spherical coordinates. In the case of x linearly polarized beam, m = 1 and −1,

a1n=a1n=b1n=b1n=(i)n+1[4π(2n+1)]1/2g5,n
(3)
g5,n=exp[s2(n1)(n+2){1+(n1)(n+2)s4[3(n1)(n+2)s2]+            (n1)2(n+2)2s8[105(n1)(n+2)s2+0.5(n1)2(n+2)2s4]},  
(4)
s=1/kω0
(5)

Here g5,n are coefficients for fifth order Davis beam [9

9. D. C. Benito, S. H. Simpson, and S. Hanna, “FDTD simulations of forces on particles during holographic assembly,” Opt. Express 16(5), 2942–2957 (2008). [CrossRef] [PubMed]

], s is the dimensionless beam shape parameter, ω0 is the beam waist radius, n is the number of spherical wave. As n≥30, the numerical error is negligible [12

12. S. H. Simpson and S. Hanna, “Numerical calculation of interparticle forces arising in association with holographic assembly,” J. Opt. Soc. Am. A 23(6), 1419–1431 (2006). [CrossRef] [PubMed]

], so n = 30.

In the case of radially polarized beam,

Eincr=Eincrinc
(6)
Hincr=Hincrinc
(7)

Here rinc is radial position vector matrix in the incident plane of the beam. The TM components of formulas (6) and (7) represent a radially polarized beam [14

14. N. Passilly, R. de Saint Denis, K. Aït-Ameur, F. Treussart, R. Hierle, and J.-F. Roch, “Simple interferometric technique for generation of a radially polarized light beam,” J. Opt. Soc. Am. A 22(5), 984–991 (2005). [CrossRef] [PubMed]

,15

15. T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Forces in optical tweezers with radially and azimuthally polarized trapping beams,” Opt. Lett. 33(2), 122–124 (2008). [CrossRef] [PubMed]

]. Figure 1
Fig. 1 The normalized intensity distribution in the focal plane. (a)linearly polarized beam; (b)the radial cross section of linearly polarized beam; (c)radially polarized beam; (d)the radial cross section of radially polarized beam.
shows the normalized intensity distribution in the focal plane of linearly polarized beam and radially polarized beam respectively.

2.2. Calculation of optical trapping force

The electromagnetic force on a nanowire in a medium can be obtained by the Maxwell stress tensor.

F=S<T>dS
(8)

T is Maxwell stress tensor,

T=12Re[εEE*+μHH*12(εE2+μH2)n].
(9)

The bracket < > denotes time averaged value, dS is the unit normal to the nanowire’s surface, ε is the permittivity of the background medium, μ is the permeability of the background medium, n is the normal vector of the nanowire’s outer surface, E* and H* represent the complex conjugates of the electric and magnetic field.

3. Results

Simulations were performed on the computer with dual-core CPU E5200 and 4.0G Memory. Matlab language is used for programming. In the model the size of the lattice can be 250 × 250 × 220, the cell size is not greater than λ/20, PML (Perfectly Matched Layer) absorbing boundary is adopted. Figure 2
Fig. 2 Computational domain.
demonstrates a calculation region for single optical tweezers to trap a nanowire in water with the beam propagating along the z axis, in the z direction. The radial direction r is perpendicular to the z axis. The nanowire has a cylindrical shape with a radius of 60nm and a height of 2100nm, its refractive index is 1.6. The trapping beam with a wavelength of 600nm, a waist radius of 300nm and a power of 10mW is linearly polarized and radially polarized respectively.

As single optical tweezers trap a nanowire, the nanowire orients itself to the optical axis of the beam (Fig. 2). The axial force efficiency Qz (Q = Fc/(np), c is the speed of light, F is the trapping force, n is the refractive index of the background medium, p is the beam power.) and radial force efficiency Qr are shown in Fig. 3
Fig. 3 Axial (a) and radial (b) forces on nanowire with refractive index of 1.6 as a function of nanowire’s displacement from the focus. The asterisk represents linearly polarized beam, the circle denotes radially polarized beam.
. In the following figures the abscissa denotes the displacement of the geometrical center of nanowire from the focus and the ordinate represents force efficiency. Figure 3(a) shows that there is little difference between the maximal axial forces of the linearly polarized beam and these of the radially polarized beam, the axial trapping positions in both beams are located behind the focus. The maximal radial forces of the radially polarized beam are obviously greater than those of the linearly polarized beam (Fig. 3(b)), the radial forces of the radially polarized beam near the focus appear to fluctuate and there is more than one radial equilibrium position existing.

As the nanowire’s refractive index is 2.5, the obtained forces are shown in Fig. 4
Fig. 4 Forces on nanowire with refractive index of 2.5.
. It can be seen that the axial forces and radial forces of radially polarized beams increase greatly compared with those of linearly polarized beams. It is due to the hollow beam and polarization singularity of radially polarized beams, which decrease radiation pressure forces on nanowire and improve gradient forces.

As the nanowire’s refractive index is 3.0, it is infeasible for a linearly polarized beam to trap nanowires along axial direction in the vicinity of the focus (shown in Fig. 5
Fig. 5 Axial forces on nanowire with refractive index of 3.0.
). However, trapping is achievable for radially polarized beams and the axial trapping position is located 1.5µm behind the focus. The axial trapping potential corresponding to the area under the curve Qz (z) is defined as U(z)=Fzdz=(np/c )Qzdz [7

7. P. B. Bareil and Y. L. Sheng, “Angular and position stability of a nanorod trapped in an optical tweezers,” Opt. Express 18(25), 26388–26398 (2010). [CrossRef] [PubMed]

], which represents the energy required for nanowires to escape from the trap. Although the axial trapping potential of radially polarized beam in Fig. 5 is of pronounced asymmetry with respect to the equilibrium position Qz = 0, the energy barrier in the beam propagating direction is significantly larger than the thermal energy kBT (kB is the Boltzmann constant, T is the temperature of the background medium, T = 300K).Therefore, it is difficult for nanowires to escape from the trap by the Brownian motion.

4. Conclusions

The three dimensional FDTD method and Maxwell stress tensor are used to setup simulation model and calculate the trapping force. By using VSWFs the transmission fields are obtained with linearly polarized and radially polarized beams in the vicinity of focus point. The trapping capacities on nanoscale-diameter nanowires based on both linearly polarized beam and radially polarized beam are simulated and compared with each other. When radially polarized beam is adopted for lower refractive index nanowire, the multiple trapping equilibrium positions beyond the focal plane exist. With the increase of the refractive indices of nanowires, the axial and radial forces of the radially polarized beam both increase greatly. The radially polarized beam, compared with the linearly polarized beam, demonstrates higher trapping efficiency on the higher refractive index nanowire. It is shown that the radially polarized beam is suitable for trapping those higher refractive index nanowires.

Acknowledgments

We are grateful to the National Natural Science Foundation of China (Grant Nos. 50975271 and 91023049) for financial support of this work. We thank the reviewers for their advice on the paper and thank Fred Firstbrook for critical reading.

References and links

1.

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). [CrossRef] [PubMed]

2.

A. Ashkin, “History of optical trapping and manipulation of small-neutral particle, atoms, and molecules,” IEEE J. Sel. Top. Quantum Electron. 6(6), 841–856 (2000). [CrossRef]

3.

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]

4.

P. J. Pauzauskie, A. Radenovic, E. Trepagnier, H. Shroff, P. D. Yang, and J. Liphardt, “Optical trapping and integration of semiconductor nanowire assemblies in water,” Nat. Mater. 5(2), 97–101 (2006). [CrossRef] [PubMed]

5.

T. Yu, F. C. Cheong, and C. H. Sow, “The manipulation and assembly of CuO nanorods with line optical tweezers,” Nanotechnology 15(12), 1732–1736 (2004). [CrossRef]

6.

R. Agarwal, K. Ladavac, Y. Roichman, G. H. Yu, C. M. Lieber, and D. G. Grier, “Manipulation and assembly of nanowires with holographic optical traps,” Opt. Express 13(22), 8906–8912 (2005). [CrossRef] [PubMed]

7.

P. B. Bareil and Y. L. Sheng, “Angular and position stability of a nanorod trapped in an optical tweezers,” Opt. Express 18(25), 26388–26398 (2010). [CrossRef] [PubMed]

8.

F. Borghese, P. Denti, R. Saija, M. A. Iatì, and O. M. Maragò, “Radiation torque and force on optically trapped linear nanostructures,” Phys. Rev. Lett. 100(16), 163903 (2008). [CrossRef] [PubMed]

9.

D. C. Benito, S. H. Simpson, and S. Hanna, “FDTD simulations of forces on particles during holographic assembly,” Opt. Express 16(5), 2942–2957 (2008). [CrossRef] [PubMed]

10.

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66(7), 2800–2802 (1989). [CrossRef]

11.

G. Gouesbet, J. A. Lock, and G. Gréhan, “Partial-wave representations of laser beams for use in light-scattering calculations,” Appl. Opt. 34(12), 2133–2143 (1995). [CrossRef] [PubMed]

12.

S. H. Simpson and S. Hanna, “Numerical calculation of interparticle forces arising in association with holographic assembly,” J. Opt. Soc. Am. A 23(6), 1419–1431 (2006). [CrossRef] [PubMed]

13.

G. J. Hu, J. Li, Q. Long, T. Tao, G. X. Zhang, and X. P. Wu, “FDTD numerical simulation of the trapping force of microsphere in single optical tweezers,” Acta Phys. Sin. 60, 30301 (2011).

14.

N. Passilly, R. de Saint Denis, K. Aït-Ameur, F. Treussart, R. Hierle, and J.-F. Roch, “Simple interferometric technique for generation of a radially polarized light beam,” J. Opt. Soc. Am. A 22(5), 984–991 (2005). [CrossRef] [PubMed]

15.

T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Forces in optical tweezers with radially and azimuthally polarized trapping beams,” Opt. Lett. 33(2), 122–124 (2008). [CrossRef] [PubMed]

OCIS Codes
(140.7010) Lasers and laser optics : Laser trapping
(350.4855) Other areas of optics : Optical tweezers or optical manipulation

ToC Category:
Optical Trapping and Manipulation

History
Original Manuscript: June 21, 2011
Revised Manuscript: August 21, 2011
Manuscript Accepted: September 16, 2011
Published: October 4, 2011

Virtual Issues
Vol. 6, Iss. 11 Virtual Journal for Biomedical Optics

Citation
Jing Li and Xiaoping Wu, "FDTD simulation of trapping nanowires with linearly polarized and radially polarized optical tweezers," Opt. Express 19, 20736-20742 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-21-20736


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References

  1. D. G. Grier, “A revolution in optical manipulation,” Nature424(6950), 810–816 (2003). [CrossRef] [PubMed]
  2. A. Ashkin, “History of optical trapping and manipulation of small-neutral particle, atoms, and molecules,” IEEE J. Sel. Top. Quantum Electron.6(6), 841–856 (2000). [CrossRef]
  3. 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]
  4. P. J. Pauzauskie, A. Radenovic, E. Trepagnier, H. Shroff, P. D. Yang, and J. Liphardt, “Optical trapping and integration of semiconductor nanowire assemblies in water,” Nat. Mater.5(2), 97–101 (2006). [CrossRef] [PubMed]
  5. T. Yu, F. C. Cheong, and C. H. Sow, “The manipulation and assembly of CuO nanorods with line optical tweezers,” Nanotechnology15(12), 1732–1736 (2004). [CrossRef]
  6. R. Agarwal, K. Ladavac, Y. Roichman, G. H. Yu, C. M. Lieber, and D. G. Grier, “Manipulation and assembly of nanowires with holographic optical traps,” Opt. Express13(22), 8906–8912 (2005). [CrossRef] [PubMed]
  7. P. B. Bareil and Y. L. Sheng, “Angular and position stability of a nanorod trapped in an optical tweezers,” Opt. Express18(25), 26388–26398 (2010). [CrossRef] [PubMed]
  8. F. Borghese, P. Denti, R. Saija, M. A. Iatì, and O. M. Maragò, “Radiation torque and force on optically trapped linear nanostructures,” Phys. Rev. Lett.100(16), 163903 (2008). [CrossRef] [PubMed]
  9. D. C. Benito, S. H. Simpson, and S. Hanna, “FDTD simulations of forces on particles during holographic assembly,” Opt. Express16(5), 2942–2957 (2008). [CrossRef] [PubMed]
  10. J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys.66(7), 2800–2802 (1989). [CrossRef]
  11. G. Gouesbet, J. A. Lock, and G. Gréhan, “Partial-wave representations of laser beams for use in light-scattering calculations,” Appl. Opt.34(12), 2133–2143 (1995). [CrossRef] [PubMed]
  12. S. H. Simpson and S. Hanna, “Numerical calculation of interparticle forces arising in association with holographic assembly,” J. Opt. Soc. Am. A23(6), 1419–1431 (2006). [CrossRef] [PubMed]
  13. G. J. Hu, J. Li, Q. Long, T. Tao, G. X. Zhang, and X. P. Wu, “FDTD numerical simulation of the trapping force of microsphere in single optical tweezers,” Acta Phys. Sin.60, 30301 (2011).
  14. N. Passilly, R. de Saint Denis, K. Aït-Ameur, F. Treussart, R. Hierle, and J.-F. Roch, “Simple interferometric technique for generation of a radially polarized light beam,” J. Opt. Soc. Am. A22(5), 984–991 (2005). [CrossRef] [PubMed]
  15. T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Forces in optical tweezers with radially and azimuthally polarized trapping beams,” Opt. Lett.33(2), 122–124 (2008). [CrossRef] [PubMed]

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