Abstract

We present finite-difference time-domain (FDTD) calculations of the forces and torques on dielectric particles of various shapes, held in one or many Gaussian optical traps, as part of a study of the physical limitations involved in the construction of micro- and nanostructures using a dynamic holographic assembler (DHA). We employ a full 3-dimensional FDTD implementation, which includes a complete treatment of optical anisotropy. The Gaussian beams are sourced using a multipole expansion of a fifth order Davis beam. Force and torques are calculated for pairs of silica spheres in adjacent traps, for silica cylinders trapped by multiple beams and for oblate silica spheroids and calcite spheres in both linearly and circularly polarized beams. Comparisons are drawn between the magnitudes of the optical forces and the Van der Waals forces acting on the systems. The paper also considers the limitations of the FDTD approach when applied to optical trapping.

© 2008 Optical Society of America

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References

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  1. A. Ashkin, "Acceleration and trapping of particles by radiation pressure," Phys. Rev. Lett. 24, 156-159 (1970).
    [CrossRef]
  2. E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets and D. G. Grier, "Computer-generated holographic optical tweezer arrays," Rev. Sci. Instrum. 72, 1810-1816 (2001).
    [CrossRef]
  3. J. E. Curtis, B. A. Koss and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002).
    [CrossRef]
  4. Y. Roichman and D. G. Grier, "Holographic assembly of quasicrystalline photonic heterostructures," Opt. Express 13, 5434-5439 (2005).
    [CrossRef] [PubMed]
  5. J. E. Curtis and D. G. Grier, "Modulated optical vortices," Opt. Lett. 28, 872-874 (2003).
    [CrossRef] [PubMed]
  6. C. H. J. Schmitz, K. Uhrig, J. P. Spatz and J. E. Curtis, "Tuning the orbital angular momentum in optical vortex beams," Opt. Express 14, 6604-6612 (2006).
    [CrossRef] [PubMed]
  7. D. A. White, "Vector finite element modeling of optical tweezers," Comput. Phys. Commun. 128, 558-564 (2000).
    [CrossRef]
  8. N. V. Voshchinnikov and V. G. Farafonov, "Optical properties of spheroidal particles," Astrophys. Space Sci. 204, 19-86 (1993).
    [CrossRef]
  9. M. I. Mishchenko, L. D. Travis and D.W. Mackowski, "T-matrix computations of light scattering by nonspherical particles: A review," J. Quant. Spectrosc. Radiat. Transfer 55, 535-575 (1996).
    [CrossRef]
  10. T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg and A. I. Bishop, "Numerical modelling of optical trapping," Comput. Phys. Commun. 142, 468-471 (2001).
    [CrossRef]
  11. S. H. Simpson and S. Hanna, "Numerical calculation of interparticle forces arising in association with holographic assembly," J. Opt. Soc. Am. A 23, 1419-1431 (2006).
    [CrossRef]
  12. S. H. Simpson and S. Hanna, "Optical trapping of spheroidal particles in Gaussian beams," J. Opt. Soc. Am. A 24, 430-443 (2007).
    [CrossRef]
  13. S. H. Simpson, D. C. Benito and S. Hanna, "Polarization-induced torque in optical traps," Phys. Rev. A 76, 408 (2007).
    [CrossRef]
  14. D. W. Zhang, X. C. Yuan, S. C. Tjin and S. Krishnan, "Rigorous time domain simulation of momentum transfer between light and microscopic particles in optical trapping," Opt. Express 12, 2220-2230 (2004).
    [CrossRef] [PubMed]
  15. A. R. Zakharian, M. Mansuripur and J. V. Moloney, "Radiation pressure and the distribution of electromagnetic force in dielectric media," Opt. Express 13, 2321-2336 (2005).
    [CrossRef] [PubMed]
  16. R. C. Gauthier, "Computation of the optical trapping force using an FDTD based technique," Opt. Express 13, 3707-3718 (2005).
    [CrossRef] [PubMed]
  17. K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas Propag. Mag. 14, 302-307 (1966).
    [CrossRef]
  18. A. Taflove and S. C. Hagness, Computational Electrodynamics The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, Inc, Norwood, MA, 2005).
  19. S. A. Schelkunoff, Electromagnetic Waves (Van Nostrand, New York, 1943).
  20. P. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields, vol. 12 of International Series of Monographs in Electromagnetic Waves, 1st ed. (Pergamon Press, London, 1966).
  21. L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (John Wiley & Sons, New York, 1985).
  22. O. Moine and B. Stout, "Optical force calculations in arbitrary beams by use of the vector addition theorem," J. Opt. Soc. Am. B 22, 1620-1631 (2005).
    [CrossRef]
  23. J. A. Pereda, A. Vegas, and A. Prieto, "An improved compact 2D fullwave FDFD method for general guided wave structures," Microwave Opt. Technol. Lett. 38, 331-335 (2003).
    [CrossRef]
  24. S. H. Simpson and S. Hanna, "Analysis of the effects arising from the near-field optical microscopy of homogeneous dielectric slabs," Opt. Commun. 196, 17-31 (2001).
    [CrossRef]
  25. S. H. Simpson and S. Hanna, "Scanning near-field optical microscopy of metallic features," Opt. Commun. 256, 476-488 (2005).
    [CrossRef]
  26. 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, 8906-8912 (2005).
    [CrossRef] [PubMed]
  27. M. K. Liu, N. Ji, Z. F. Lin, and S. T. Chui, "Radiation torque on a birefringent sphere caused by an electromagnetic wave," Phys. Rev. E 72, 610 (2005).
    [CrossRef]
  28. V. L. Y. Loke, T. A. Nieminen, S. J. Parkin, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "FDFD/T-matrix hybrid method," J. Quant. Spectrosc. Radiat. Transfer 106, 274-284 (2007).
    [CrossRef]

2007

S. H. Simpson, D. C. Benito and S. Hanna, "Polarization-induced torque in optical traps," Phys. Rev. A 76, 408 (2007).
[CrossRef]

V. L. Y. Loke, T. A. Nieminen, S. J. Parkin, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "FDFD/T-matrix hybrid method," J. Quant. Spectrosc. Radiat. Transfer 106, 274-284 (2007).
[CrossRef]

S. H. Simpson and S. Hanna, "Optical trapping of spheroidal particles in Gaussian beams," J. Opt. Soc. Am. A 24, 430-443 (2007).
[CrossRef]

2006

2005

2004

2003

J. E. Curtis and D. G. Grier, "Modulated optical vortices," Opt. Lett. 28, 872-874 (2003).
[CrossRef] [PubMed]

J. A. Pereda, A. Vegas, and A. Prieto, "An improved compact 2D fullwave FDFD method for general guided wave structures," Microwave Opt. Technol. Lett. 38, 331-335 (2003).
[CrossRef]

2002

J. E. Curtis, B. A. Koss and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002).
[CrossRef]

2001

E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets and D. G. Grier, "Computer-generated holographic optical tweezer arrays," Rev. Sci. Instrum. 72, 1810-1816 (2001).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg and A. I. Bishop, "Numerical modelling of optical trapping," Comput. Phys. Commun. 142, 468-471 (2001).
[CrossRef]

S. H. Simpson and S. Hanna, "Analysis of the effects arising from the near-field optical microscopy of homogeneous dielectric slabs," Opt. Commun. 196, 17-31 (2001).
[CrossRef]

2000

D. A. White, "Vector finite element modeling of optical tweezers," Comput. Phys. Commun. 128, 558-564 (2000).
[CrossRef]

1996

M. I. Mishchenko, L. D. Travis and D.W. Mackowski, "T-matrix computations of light scattering by nonspherical particles: A review," J. Quant. Spectrosc. Radiat. Transfer 55, 535-575 (1996).
[CrossRef]

1993

N. V. Voshchinnikov and V. G. Farafonov, "Optical properties of spheroidal particles," Astrophys. Space Sci. 204, 19-86 (1993).
[CrossRef]

1970

A. Ashkin, "Acceleration and trapping of particles by radiation pressure," Phys. Rev. Lett. 24, 156-159 (1970).
[CrossRef]

1966

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas Propag. Mag. 14, 302-307 (1966).
[CrossRef]

Agarwal, R.

Ashkin, A.

A. Ashkin, "Acceleration and trapping of particles by radiation pressure," Phys. Rev. Lett. 24, 156-159 (1970).
[CrossRef]

Benito, D. C.

S. H. Simpson, D. C. Benito and S. Hanna, "Polarization-induced torque in optical traps," Phys. Rev. A 76, 408 (2007).
[CrossRef]

Bishop, A. I.

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg and A. I. Bishop, "Numerical modelling of optical trapping," Comput. Phys. Commun. 142, 468-471 (2001).
[CrossRef]

Chui, S. T.

M. K. Liu, N. Ji, Z. F. Lin, and S. T. Chui, "Radiation torque on a birefringent sphere caused by an electromagnetic wave," Phys. Rev. E 72, 610 (2005).
[CrossRef]

Curtis, J. E.

Dearing, M. T.

E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets and D. G. Grier, "Computer-generated holographic optical tweezer arrays," Rev. Sci. Instrum. 72, 1810-1816 (2001).
[CrossRef]

Dufresne, E. R.

E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets and D. G. Grier, "Computer-generated holographic optical tweezer arrays," Rev. Sci. Instrum. 72, 1810-1816 (2001).
[CrossRef]

Farafonov, V. G.

N. V. Voshchinnikov and V. G. Farafonov, "Optical properties of spheroidal particles," Astrophys. Space Sci. 204, 19-86 (1993).
[CrossRef]

Gauthier, R. C.

Grier, D. G.

Hanna, S.

S. H. Simpson and S. Hanna, "Optical trapping of spheroidal particles in Gaussian beams," J. Opt. Soc. Am. A 24, 430-443 (2007).
[CrossRef]

S. H. Simpson, D. C. Benito and S. Hanna, "Polarization-induced torque in optical traps," Phys. Rev. A 76, 408 (2007).
[CrossRef]

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

S. H. Simpson and S. Hanna, "Scanning near-field optical microscopy of metallic features," Opt. Commun. 256, 476-488 (2005).
[CrossRef]

S. H. Simpson and S. Hanna, "Analysis of the effects arising from the near-field optical microscopy of homogeneous dielectric slabs," Opt. Commun. 196, 17-31 (2001).
[CrossRef]

Heckenberg, N. R.

V. L. Y. Loke, T. A. Nieminen, S. J. Parkin, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "FDFD/T-matrix hybrid method," J. Quant. Spectrosc. Radiat. Transfer 106, 274-284 (2007).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg and A. I. Bishop, "Numerical modelling of optical trapping," Comput. Phys. Commun. 142, 468-471 (2001).
[CrossRef]

Ji, N.

M. K. Liu, N. Ji, Z. F. Lin, and S. T. Chui, "Radiation torque on a birefringent sphere caused by an electromagnetic wave," Phys. Rev. E 72, 610 (2005).
[CrossRef]

Koss, B. A.

J. E. Curtis, B. A. Koss and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002).
[CrossRef]

Krishnan, S.

Ladavac, K.

Lieber, C. M.

Lin, Z. F.

M. K. Liu, N. Ji, Z. F. Lin, and S. T. Chui, "Radiation torque on a birefringent sphere caused by an electromagnetic wave," Phys. Rev. E 72, 610 (2005).
[CrossRef]

Liu, M. K.

M. K. Liu, N. Ji, Z. F. Lin, and S. T. Chui, "Radiation torque on a birefringent sphere caused by an electromagnetic wave," Phys. Rev. E 72, 610 (2005).
[CrossRef]

Loke, V. L. Y.

V. L. Y. Loke, T. A. Nieminen, S. J. Parkin, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "FDFD/T-matrix hybrid method," J. Quant. Spectrosc. Radiat. Transfer 106, 274-284 (2007).
[CrossRef]

Mackowski, D.W.

M. I. Mishchenko, L. D. Travis and D.W. Mackowski, "T-matrix computations of light scattering by nonspherical particles: A review," J. Quant. Spectrosc. Radiat. Transfer 55, 535-575 (1996).
[CrossRef]

Mansuripur, M.

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis and D.W. Mackowski, "T-matrix computations of light scattering by nonspherical particles: A review," J. Quant. Spectrosc. Radiat. Transfer 55, 535-575 (1996).
[CrossRef]

Moine, O.

Moloney, J. V.

Nieminen, T. A.

V. L. Y. Loke, T. A. Nieminen, S. J. Parkin, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "FDFD/T-matrix hybrid method," J. Quant. Spectrosc. Radiat. Transfer 106, 274-284 (2007).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg and A. I. Bishop, "Numerical modelling of optical trapping," Comput. Phys. Commun. 142, 468-471 (2001).
[CrossRef]

Parkin, S. J.

V. L. Y. Loke, T. A. Nieminen, S. J. Parkin, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "FDFD/T-matrix hybrid method," J. Quant. Spectrosc. Radiat. Transfer 106, 274-284 (2007).
[CrossRef]

Pereda, J. A.

J. A. Pereda, A. Vegas, and A. Prieto, "An improved compact 2D fullwave FDFD method for general guided wave structures," Microwave Opt. Technol. Lett. 38, 331-335 (2003).
[CrossRef]

Prieto, A.

J. A. Pereda, A. Vegas, and A. Prieto, "An improved compact 2D fullwave FDFD method for general guided wave structures," Microwave Opt. Technol. Lett. 38, 331-335 (2003).
[CrossRef]

Roichman, Y.

Rubinsztein-Dunlop, H.

V. L. Y. Loke, T. A. Nieminen, S. J. Parkin, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "FDFD/T-matrix hybrid method," J. Quant. Spectrosc. Radiat. Transfer 106, 274-284 (2007).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg and A. I. Bishop, "Numerical modelling of optical trapping," Comput. Phys. Commun. 142, 468-471 (2001).
[CrossRef]

Schmitz, C. H. J.

Sheets, S. A.

E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets and D. G. Grier, "Computer-generated holographic optical tweezer arrays," Rev. Sci. Instrum. 72, 1810-1816 (2001).
[CrossRef]

Simpson, S. H.

S. H. Simpson and S. Hanna, "Optical trapping of spheroidal particles in Gaussian beams," J. Opt. Soc. Am. A 24, 430-443 (2007).
[CrossRef]

S. H. Simpson, D. C. Benito and S. Hanna, "Polarization-induced torque in optical traps," Phys. Rev. A 76, 408 (2007).
[CrossRef]

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

S. H. Simpson and S. Hanna, "Scanning near-field optical microscopy of metallic features," Opt. Commun. 256, 476-488 (2005).
[CrossRef]

S. H. Simpson and S. Hanna, "Analysis of the effects arising from the near-field optical microscopy of homogeneous dielectric slabs," Opt. Commun. 196, 17-31 (2001).
[CrossRef]

Spalding, G. C.

E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets and D. G. Grier, "Computer-generated holographic optical tweezer arrays," Rev. Sci. Instrum. 72, 1810-1816 (2001).
[CrossRef]

Spatz, J. P.

Stout, B.

Tjin, S. C.

Travis, L. D.

M. I. Mishchenko, L. D. Travis and D.W. Mackowski, "T-matrix computations of light scattering by nonspherical particles: A review," J. Quant. Spectrosc. Radiat. Transfer 55, 535-575 (1996).
[CrossRef]

Uhrig, K.

Vegas, A.

J. A. Pereda, A. Vegas, and A. Prieto, "An improved compact 2D fullwave FDFD method for general guided wave structures," Microwave Opt. Technol. Lett. 38, 331-335 (2003).
[CrossRef]

Voshchinnikov, N. V.

N. V. Voshchinnikov and V. G. Farafonov, "Optical properties of spheroidal particles," Astrophys. Space Sci. 204, 19-86 (1993).
[CrossRef]

White, D. A.

D. A. White, "Vector finite element modeling of optical tweezers," Comput. Phys. Commun. 128, 558-564 (2000).
[CrossRef]

Yee, K. S.

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas Propag. Mag. 14, 302-307 (1966).
[CrossRef]

Yu, G. H.

Yuan, X. C.

Zakharian, A. R.

Zhang, D. W.

Astrophys. Space Sci.

N. V. Voshchinnikov and V. G. Farafonov, "Optical properties of spheroidal particles," Astrophys. Space Sci. 204, 19-86 (1993).
[CrossRef]

Comput. Phys. Commun.

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg and A. I. Bishop, "Numerical modelling of optical trapping," Comput. Phys. Commun. 142, 468-471 (2001).
[CrossRef]

D. A. White, "Vector finite element modeling of optical tweezers," Comput. Phys. Commun. 128, 558-564 (2000).
[CrossRef]

IEEE Trans. Antennas Propag. Mag.

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas Propag. Mag. 14, 302-307 (1966).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Quant. Spectrosc. Radiat. Transfer

V. L. Y. Loke, T. A. Nieminen, S. J. Parkin, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "FDFD/T-matrix hybrid method," J. Quant. Spectrosc. Radiat. Transfer 106, 274-284 (2007).
[CrossRef]

M. I. Mishchenko, L. D. Travis and D.W. Mackowski, "T-matrix computations of light scattering by nonspherical particles: A review," J. Quant. Spectrosc. Radiat. Transfer 55, 535-575 (1996).
[CrossRef]

Microwave Opt. Technol. Lett.

J. A. Pereda, A. Vegas, and A. Prieto, "An improved compact 2D fullwave FDFD method for general guided wave structures," Microwave Opt. Technol. Lett. 38, 331-335 (2003).
[CrossRef]

Opt. Commun.

S. H. Simpson and S. Hanna, "Scanning near-field optical microscopy of metallic features," Opt. Commun. 256, 476-488 (2005).
[CrossRef]

J. E. Curtis, B. A. Koss and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002).
[CrossRef]

Opt. Express

Opt. Lett.

Optics Commun.

S. H. Simpson and S. Hanna, "Analysis of the effects arising from the near-field optical microscopy of homogeneous dielectric slabs," Opt. Commun. 196, 17-31 (2001).
[CrossRef]

Phys. Rev. A

S. H. Simpson, D. C. Benito and S. Hanna, "Polarization-induced torque in optical traps," Phys. Rev. A 76, 408 (2007).
[CrossRef]

Phys. Rev. E

M. K. Liu, N. Ji, Z. F. Lin, and S. T. Chui, "Radiation torque on a birefringent sphere caused by an electromagnetic wave," Phys. Rev. E 72, 610 (2005).
[CrossRef]

Phys. Rev. Lett.

A. Ashkin, "Acceleration and trapping of particles by radiation pressure," Phys. Rev. Lett. 24, 156-159 (1970).
[CrossRef]

Rev. Sci. Instrum.

E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets and D. G. Grier, "Computer-generated holographic optical tweezer arrays," Rev. Sci. Instrum. 72, 1810-1816 (2001).
[CrossRef]

Other

A. Taflove and S. C. Hagness, Computational Electrodynamics The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, Inc, Norwood, MA, 2005).

S. A. Schelkunoff, Electromagnetic Waves (Van Nostrand, New York, 1943).

P. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields, vol. 12 of International Series of Monographs in Electromagnetic Waves, 1st ed. (Pergamon Press, London, 1966).

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (John Wiley & Sons, New York, 1985).

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Figures (9)

Fig. 1.
Fig. 1.

Side views of a Gaussian beam produced using (a) the VSWF expansion and (b) the paraxial approximation. The dashed line indicates the source plane and the shaded region indicates the position of a uniaxial perfectly matched layer (UPML), used to eliminate reflections from the simulation boundaries [18].

Fig. 2.
Fig. 2.

(a) The force on a 1 µm diameter silica sphere lying on the axis of a linearly polarized Gaussian beam, simulated with different FDTD cell sizes. Positive distances indicate positions closer to the source. (b) The force on a 1µm diameter silica sphere positioned 300 nm below the trapping point in a linearly polarized Gaussian beam calculated for several source area sizes. The cell side dimension was kept at λ/40 nm throughout.

Fig. 3.
Fig. 3.

The excess, optically induced forces (Fx , Fy and Fz ) arising between two 1 µm diameter silica spheres held at their trapping positions in two parallel Gaussian traps. The sphere separation is defined as the minimum gap between the two spherical surfaces.

Fig. 4.
Fig. 4.

Side view showing the relative intensity of three Gaussian beams trapping a 3µm long silica rod with radius 45 nm. The dashed line indicates the source. The intensity variation is shown as false colours varying from red (most intense) to purple (least intense). The vertical scale is stretched compared with the horizontal scale.

Fig. 5.
Fig. 5.

The restoring force on a cylinder trapped by three parallel traps, as it is moved parallel to the x-axis from the central position seen in Fig. 4. The cylinder is 3µm in length, and has a radius of 45 nm. The traps are 1.14µm apart.

Fig. 6.
Fig. 6.

Torque on a dielectric cylinder as it is rotated about the y-axis, in varying numbers of parallel traps.

Fig. 7.
Fig. 7.

(a) The torque on a dielectric cylinder held in several parallel traps, as it is rotated about the beam axis. (b) Diagram showing the view along the beam axis for a cylinder in three beams. (c) Similar diagram for five beams.

Fig. 8.
Fig. 8.

The torque exerted by a circularly polarized Gaussian beam on oblate spheroids with aspect ratios, δ, in the range 0.2-0.5 and a flat coin shape with thickness 150 nm.

Fig. 9.
Fig. 9.

(a) The torque on a 1µm diameter sphere of calcite trapped in a 10mW linearly polarized beam. Both calculated results (points) and a sin2ϕ fit (solid line) are shown. (b) The torque about the beam axis acting on different sizes of calcite sphere in a circularly polarized Gaussian beam.

Tables (4)

Tables Icon

Table 1. The parameters used to define the beam and particle characteristics.

Tables Icon

Table 2. Sample timings for FDTD simulations of a 1µm diameter silica particle in a Gaussian trap, using various lattice sizes. Each simulation consists of 2800 iterations i.e. 14 optical time periods, running on 16 processor cores.

Tables Icon

Table 3. The spring constants in three perpendicular directions for a 1µm silica sphere in water trapped by a 10mW linearly polarized laser beam with a Gaussian intensity profile.

Tables Icon

Table 4. Effective spring constants for displacements parallel to the coordinate axes defined in Fig. 4, for a silica cylinder trapped in multiple Gaussian beams.

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

J inc = n ̂ × H inc and M inc = n ̂ × E inc
× H = D t + J inc sin ( ω t + θ ( x , y , z + Δ 2 ) + θ ( 0 , 0 , 0 ) )
× E = B t + M inc sin ( ω t + θ ( x , y , z ) + θ ( 0 , 0 , 0 ) )
E inc = n = 1 m = n n a nm Rg M nm ( k r , θ , ϕ ) + b nm Rg N nm ( k r , θ , ϕ )
H inc = j ε μ n = 1 m = n n b nm Rg M nm ( k r , θ , ϕ ) + a nm Rg N nm ( k r , θ , ϕ )
a nm = δ m , 1 g 5 , n and b nm = δ m , 1 m g 5 , n
g 5 , n = exp [ s 2 ( n 1 ) ( n + 2 ) ] { 1 + ( n 1 ) ( n + 2 ) s 4 [ 3 ( n 1 ) ( n + 2 ) s 2 ]
+ ( n 1 ) 2 ( n + 2 ) 2 s 8 [ 10 5 ( n 1 ) ( n + 2 ) s 2 + 0.5 ( n 1 ) 2 ( n + 2 ) 2 s 4 ] }
F = S T ˜ ( r , t ) · d S
Γ = S r d S · [ T ˜ ( r , t ) × r ̂ ]
T ˜ ij ( r , t ) = ε ε 0 E i ( r , t ) E j ( r , t ) + μ μ 0 H i ( r , t ) H j ( r , t ) 1 2 ( ε ε 0 E ( r , t ) 2 + μ μ 0 H ( r , t ) 2 ) δ ij
E = κ D
E x ( i , j , k ) = κ xx D x ( i , j , k ) +
κ xy [ D y ( i , j 1 , k ) + D y ( i + 1 , j 1 , k ) + D y ( i , j , k ) + D y ( i + 1 , j , k ) ] 4 +
κ xz [ D z ( i , j , k 1 ) + D z ( i + 1 , j , k 1 ) + D z ( i , j , k ) + D z ( i + 1 , j , k ) ] 4

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