Abstract

Motivated by the recent optical trapping experiments using ultra-short pulsed lasers [Opt. Express 18, 7554 (2010); Appl. Opt. 48, G33 (2009)], in this paper we have re-investigated the trapping effects of the pulsed radiation force (PRF), which is induced by a pulsed Gaussian beam acting on a Rayleigh dielectric sphere. Based on our previous model [Opt. Express 15, 10615 (2007)], we have considered the effects arisen from both the transverse and axial PRFs, which lead to the different behaviors of both velocities and displacements of a Rayleigh particle within a pulse duration. Our analysis shows that, for the small-sized Rayleigh particles, when the pulse has the large pulse duration, it might provide the three-dimensional optical trapping; and when the pulse has the short pulse duration, it only provides the two-dimensional optical trapping with the axial movement along the pulse propagation. When the particle is in the vacuum or in the situation with the very weak Brownian motion, the particle can always be trapped stably due to the particle’s cumulative momentum transferred from the pulse, and only in this case the trapping effect is independent of pulse duration. Finally, we have predicted that for the large-sized Rayleigh particles, the pulse beam can only provide the two-dimensional optical trap (optical guiding). Our results provide the important information about the trapping mechanism of pulsed tweezers.

© 2011 OSA

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  28. H. Misawa, M. Koshioka, K. Sasaki, N. Kitamura, and H. Masuhara, “Three-dimensional optical trapping and laser ablation of a single polymer latex particle in water,” J. Appl. Phys. 70(7), 3829–3836 (1991).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2011 (1)

2010 (2)

J. C. Shane, M. Mazilu, W. M. Lee, and K. Dholakia, “Effect of pulse temporal shape on optical trapping and impulse transfer using ultrashort pulsed lasers,” Opt. Express 18(7), 7554–7568 (2010).
[CrossRef] [PubMed]

T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, “Measurement of the instantaneous velocity of a Brownian particle,” Science 328(5986), 1673–1675 (2010).
[CrossRef] [PubMed]

2009 (1)

2008 (2)

L. Jauffred, A. C. Richardson, and L. B. Oddershede, “Three-dimensional optical control of individual quantum dots,” Nano Lett. 8(10), 3376–3380 (2008).
[CrossRef] [PubMed]

J. Shane, M. Mazilu, W. M. Lee, and K. Dholakia, ““Optical trapping using ultashort 12.9fs pulses,” Optical Trapping and Optical Micromanipulation V,” Proc. SPIE 7038, 70380Y, 70380Y–11 (2008).
[CrossRef]

2007 (5)

L. Pan, A. Ishikawa, and N. Tamai, “Detection of optical trapping of CdTe quantum dots by two-photon-induced luminescence,” Phys. Rev. B 75, 161305 (2007).
[CrossRef]

Y. Deng, J. Bechhoefer, and N. R. Forde, “Brownian motion in a modulated optical trap,” J. Opt. A, Pure Appl. Opt. 9(8), S256–S263 (2007).
[CrossRef]

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]

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]

L. G. Wang and C. L. Zhao, “Dynamic radiation force of a pulsed gaussian beam acting on rayleigh dielectric sphere,” Opt. Express 15(17), 10615–10621 (2007).
[CrossRef] [PubMed]

2005 (4)

J. L. Deng, Q. Wei, Y. Z. Wang, and Y. Q. Li, “Numerical modeling of optical levitation and trapping of the “stuck” particles with a pulsed optical tweezers,” Opt. Express 13(10), 3673–3680 (2005).
[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]

K. Berg-Sørensen and H. Flyvbjerg, “The color of thermal noise in classical Brownian motion: a feasibility study of direct experimental observation,” N. J. Phys. 7, 38 (2005).
[CrossRef]

B. Lukić, S. Jeney, C. Tischer, A. J. Kulik, L. Forró, and E.-L. Florin, “Direct observation of nondiffusive motion of a Brownian particle,” Phys. Rev. Lett. 95(16), 160601 (2005).
[CrossRef] [PubMed]

2004 (3)

2002 (1)

P. T. Korda, M. B. Taylor, and D. G. Grier, “Kinetically locked-in colloidal transport in an array of optical tweezers,” Phys. Rev. Lett. 89(12), 128301 (2002).
[CrossRef] [PubMed]

2001 (1)

J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197(4-6), 239–245 (2001).
[CrossRef]

1999 (1)

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283(5408), 1689–1695 (1999).
[CrossRef] [PubMed]

1998 (1)

M. D. Wang, M. J. Schnitzer, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Force and velocity measured for single molecules of RNA polymerase,” Science 282(5390), 902–907 (1998).
[CrossRef] [PubMed]

1995 (1)

J. Dai and M. P. Sheetz, “Mechanical properties of neuronal growth cone membranes studied by tether formation with laser optical tweezers,” Biophys. J. 68(3), 988–996 (1995).
[CrossRef] [PubMed]

1994 (1)

A. J. Hunt, F. Gittes, and J. Howard, “The force exerted by a single kinesin molecule against a viscous load,” Biophys. J. 67(2), 766–781 (1994).
[CrossRef] [PubMed]

1991 (1)

H. Misawa, M. Koshioka, K. Sasaki, N. Kitamura, and H. Masuhara, “Three-dimensional optical trapping and laser ablation of a single polymer latex particle in water,” J. Appl. Phys. 70(7), 3829–3836 (1991).
[CrossRef]

1987 (1)

A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235(4795), 1517–1520 (1987).
[CrossRef] [PubMed]

1986 (2)

S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, “Experimental observation of optically trapped atoms,” Phys. Rev. Lett. 57(3), 314–317 (1986).
[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]

1978 (1)

A. Ashkin, “Trapping of Atoms by Resonance Radiation Pressure,” Phys. Rev. Lett. 40(12), 729–732 (1978).
[CrossRef]

1975 (1)

E. J. Hinch, “Application of the Langevin equation to fluid suspensions,” J. Fluid Mech. 72(03), 499–511 (1975).
[CrossRef]

1970 (1)

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970).
[CrossRef]

Agate, B.

Ambardekar, A. A.

Arlt, J.

J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197(4-6), 239–245 (2001).
[CrossRef]

Ashkin, A.

A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235(4795), 1517–1520 (1987).
[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]

S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, “Experimental observation of optically trapped atoms,” Phys. Rev. Lett. 57(3), 314–317 (1986).
[CrossRef] [PubMed]

A. Ashkin, “Trapping of Atoms by Resonance Radiation Pressure,” Phys. Rev. Lett. 40(12), 729–732 (1978).
[CrossRef]

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970).
[CrossRef]

Bechhoefer, J.

Y. Deng, J. Bechhoefer, and N. R. Forde, “Brownian motion in a modulated optical trap,” J. Opt. A, Pure Appl. Opt. 9(8), S256–S263 (2007).
[CrossRef]

Berg-Sørensen, K.

K. Berg-Sørensen and H. Flyvbjerg, “The color of thermal noise in classical Brownian motion: a feasibility study of direct experimental observation,” N. J. Phys. 7, 38 (2005).
[CrossRef]

Bjorkholm, J. E.

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]

S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, “Experimental observation of optically trapped atoms,” Phys. Rev. Lett. 57(3), 314–317 (1986).
[CrossRef] [PubMed]

Block, S. M.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004).
[CrossRef] [PubMed]

M. D. Wang, M. J. Schnitzer, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Force and velocity measured for single molecules of RNA polymerase,” Science 282(5390), 902–907 (1998).
[CrossRef] [PubMed]

Brown, C. T. A.

Cable, A.

S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, “Experimental observation of optically trapped atoms,” Phys. Rev. Lett. 57(3), 314–317 (1986).
[CrossRef] [PubMed]

Chu, S.

S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, “Experimental observation of optically trapped atoms,” Phys. Rev. Lett. 57(3), 314–317 (1986).
[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]

Dai, J.

J. Dai and M. P. Sheetz, “Mechanical properties of neuronal growth cone membranes studied by tether formation with laser optical tweezers,” Biophys. J. 68(3), 988–996 (1995).
[CrossRef] [PubMed]

De, A. K.

Deng, J. L.

Deng, Y.

Y. Deng, J. Bechhoefer, and N. R. Forde, “Brownian motion in a modulated optical trap,” J. Opt. A, Pure Appl. Opt. 9(8), S256–S263 (2007).
[CrossRef]

Dholakia, K.

Dutta, A.

Dziedzic, J. M.

Florin, E.-L.

B. Lukić, S. Jeney, C. Tischer, A. J. Kulik, L. Forró, and E.-L. Florin, “Direct observation of nondiffusive motion of a Brownian particle,” Phys. Rev. Lett. 95(16), 160601 (2005).
[CrossRef] [PubMed]

Flyvbjerg, H.

K. Berg-Sørensen and H. Flyvbjerg, “The color of thermal noise in classical Brownian motion: a feasibility study of direct experimental observation,” N. J. Phys. 7, 38 (2005).
[CrossRef]

Forde, N. R.

Y. Deng, J. Bechhoefer, and N. R. Forde, “Brownian motion in a modulated optical trap,” J. Opt. A, Pure Appl. Opt. 9(8), S256–S263 (2007).
[CrossRef]

Forró, L.

B. Lukić, S. Jeney, C. Tischer, A. J. Kulik, L. Forró, and E.-L. Florin, “Direct observation of nondiffusive motion of a Brownian particle,” Phys. Rev. Lett. 95(16), 160601 (2005).
[CrossRef] [PubMed]

Garcés-Chávez, V.

Gelles, J.

M. D. Wang, M. J. Schnitzer, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Force and velocity measured for single molecules of RNA polymerase,” Science 282(5390), 902–907 (1998).
[CrossRef] [PubMed]

Gittes, F.

A. J. Hunt, F. Gittes, and J. Howard, “The force exerted by a single kinesin molecule against a viscous load,” Biophys. J. 67(2), 766–781 (1994).
[CrossRef] [PubMed]

Goswami, D.

Grier, D. G.

P. T. Korda, M. B. Taylor, and D. G. Grier, “Kinetically locked-in colloidal transport in an array of optical tweezers,” Phys. Rev. Lett. 89(12), 128301 (2002).
[CrossRef] [PubMed]

Hinch, E. J.

E. J. Hinch, “Application of the Langevin equation to fluid suspensions,” J. Fluid Mech. 72(03), 499–511 (1975).
[CrossRef]

Howard, J.

A. J. Hunt, F. Gittes, and J. Howard, “The force exerted by a single kinesin molecule against a viscous load,” Biophys. J. 67(2), 766–781 (1994).
[CrossRef] [PubMed]

Hunt, A. J.

A. J. Hunt, F. Gittes, and J. Howard, “The force exerted by a single kinesin molecule against a viscous load,” Biophys. J. 67(2), 766–781 (1994).
[CrossRef] [PubMed]

Ishikawa, A.

L. Pan, A. Ishikawa, and N. Tamai, “Detection of optical trapping of CdTe quantum dots by two-photon-induced luminescence,” Phys. Rev. B 75, 161305 (2007).
[CrossRef]

Jauffred, L.

L. Jauffred, A. C. Richardson, and L. B. Oddershede, “Three-dimensional optical control of individual quantum dots,” Nano Lett. 8(10), 3376–3380 (2008).
[CrossRef] [PubMed]

Jeney, S.

B. Lukić, S. Jeney, C. Tischer, A. J. Kulik, L. Forró, and E.-L. Florin, “Direct observation of nondiffusive motion of a Brownian particle,” Phys. Rev. Lett. 95(16), 160601 (2005).
[CrossRef] [PubMed]

Kheifets, S.

T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, “Measurement of the instantaneous velocity of a Brownian particle,” Science 328(5986), 1673–1675 (2010).
[CrossRef] [PubMed]

Kitamura, N.

H. Misawa, M. Koshioka, K. Sasaki, N. Kitamura, and H. Masuhara, “Three-dimensional optical trapping and laser ablation of a single polymer latex particle in water,” J. Appl. Phys. 70(7), 3829–3836 (1991).
[CrossRef]

Korda, P. T.

P. T. Korda, M. B. Taylor, and D. G. Grier, “Kinetically locked-in colloidal transport in an array of optical tweezers,” Phys. Rev. Lett. 89(12), 128301 (2002).
[CrossRef] [PubMed]

Koshioka, M.

H. Misawa, M. Koshioka, K. Sasaki, N. Kitamura, and H. Masuhara, “Three-dimensional optical trapping and laser ablation of a single polymer latex particle in water,” J. Appl. Phys. 70(7), 3829–3836 (1991).
[CrossRef]

Kulik, A. J.

B. Lukić, S. Jeney, C. Tischer, A. J. Kulik, L. Forró, and E.-L. Florin, “Direct observation of nondiffusive motion of a Brownian particle,” Phys. Rev. Lett. 95(16), 160601 (2005).
[CrossRef] [PubMed]

Landick, R.

M. D. Wang, M. J. Schnitzer, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Force and velocity measured for single molecules of RNA polymerase,” Science 282(5390), 902–907 (1998).
[CrossRef] [PubMed]

Lee, W. M.

J. C. Shane, M. Mazilu, W. M. Lee, and K. Dholakia, “Effect of pulse temporal shape on optical trapping and impulse transfer using ultrashort pulsed lasers,” Opt. Express 18(7), 7554–7568 (2010).
[CrossRef] [PubMed]

J. Shane, M. Mazilu, W. M. Lee, and K. Dholakia, ““Optical trapping using ultashort 12.9fs pulses,” Optical Trapping and Optical Micromanipulation V,” Proc. SPIE 7038, 70380Y, 70380Y–11 (2008).
[CrossRef]

Li, H.

Li, J.

Li, T.

T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, “Measurement of the instantaneous velocity of a Brownian particle,” Science 328(5986), 1673–1675 (2010).
[CrossRef] [PubMed]

Li, Y. Q.

Little, H.

Lu, X. H.

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]

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]

Lukic, B.

B. Lukić, S. Jeney, C. Tischer, A. J. Kulik, L. Forró, and E.-L. Florin, “Direct observation of nondiffusive motion of a Brownian particle,” Phys. Rev. Lett. 95(16), 160601 (2005).
[CrossRef] [PubMed]

Masuhara, H.

H. Misawa, M. Koshioka, K. Sasaki, N. Kitamura, and H. Masuhara, “Three-dimensional optical trapping and laser ablation of a single polymer latex particle in water,” J. Appl. Phys. 70(7), 3829–3836 (1991).
[CrossRef]

Mazilu, M.

J. C. Shane, M. Mazilu, W. M. Lee, and K. Dholakia, “Effect of pulse temporal shape on optical trapping and impulse transfer using ultrashort pulsed lasers,” Opt. Express 18(7), 7554–7568 (2010).
[CrossRef] [PubMed]

J. Shane, M. Mazilu, W. M. Lee, and K. Dholakia, ““Optical trapping using ultashort 12.9fs pulses,” Optical Trapping and Optical Micromanipulation V,” Proc. SPIE 7038, 70380Y, 70380Y–11 (2008).
[CrossRef]

Medellin, D.

T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, “Measurement of the instantaneous velocity of a Brownian particle,” Science 328(5986), 1673–1675 (2010).
[CrossRef] [PubMed]

Mehta, A. D.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283(5408), 1689–1695 (1999).
[CrossRef] [PubMed]

Misawa, H.

H. Misawa, M. Koshioka, K. Sasaki, N. Kitamura, and H. Masuhara, “Three-dimensional optical trapping and laser ablation of a single polymer latex particle in water,” J. Appl. Phys. 70(7), 3829–3836 (1991).
[CrossRef]

Neuman, K. C.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004).
[CrossRef] [PubMed]

Oddershede, L. B.

L. Jauffred, A. C. Richardson, and L. B. Oddershede, “Three-dimensional optical control of individual quantum dots,” Nano Lett. 8(10), 3376–3380 (2008).
[CrossRef] [PubMed]

Pan, L.

L. Pan, A. Ishikawa, and N. Tamai, “Detection of optical trapping of CdTe quantum dots by two-photon-induced luminescence,” Phys. Rev. B 75, 161305 (2007).
[CrossRef]

Qiang, L.

Raizen, M. G.

T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, “Measurement of the instantaneous velocity of a Brownian particle,” Science 328(5986), 1673–1675 (2010).
[CrossRef] [PubMed]

Richardson, A. C.

L. Jauffred, A. C. Richardson, and L. B. Oddershede, “Three-dimensional optical control of individual quantum dots,” Nano Lett. 8(10), 3376–3380 (2008).
[CrossRef] [PubMed]

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Sasaki, K.

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J. Shane, M. Mazilu, W. M. Lee, and K. Dholakia, ““Optical trapping using ultashort 12.9fs pulses,” Optical Trapping and Optical Micromanipulation V,” Proc. SPIE 7038, 70380Y, 70380Y–11 (2008).
[CrossRef]

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Sheetz, M. P.

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Simmons, R. M.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283(5408), 1689–1695 (1999).
[CrossRef] [PubMed]

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A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283(5408), 1689–1695 (1999).
[CrossRef] [PubMed]

Spudich, J. A.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283(5408), 1689–1695 (1999).
[CrossRef] [PubMed]

Tamai, N.

L. Pan, A. Ishikawa, and N. Tamai, “Detection of optical trapping of CdTe quantum dots by two-photon-induced luminescence,” Phys. Rev. B 75, 161305 (2007).
[CrossRef]

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P. T. Korda, M. B. Taylor, and D. G. Grier, “Kinetically locked-in colloidal transport in an array of optical tweezers,” Phys. Rev. Lett. 89(12), 128301 (2002).
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B. Lukić, S. Jeney, C. Tischer, A. J. Kulik, L. Forró, and E.-L. Florin, “Direct observation of nondiffusive motion of a Brownian particle,” Phys. Rev. Lett. 95(16), 160601 (2005).
[CrossRef] [PubMed]

Wang, L. G.

Wang, L. Q.

Wang, M. D.

M. D. Wang, M. J. Schnitzer, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Force and velocity measured for single molecules of RNA polymerase,” Science 282(5390), 902–907 (1998).
[CrossRef] [PubMed]

Wang, Y. Z.

Wei, Q.

Yin, H.

M. D. Wang, M. J. Schnitzer, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Force and velocity measured for single molecules of RNA polymerase,” Science 282(5390), 902–907 (1998).
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Zhao, C. L.

Zhu, S. Y.

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L. Jauffred, A. C. Richardson, and L. B. Oddershede, “Three-dimensional optical control of individual quantum dots,” Nano Lett. 8(10), 3376–3380 (2008).
[CrossRef] [PubMed]

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[CrossRef]

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P. T. Korda, M. B. Taylor, and D. G. Grier, “Kinetically locked-in colloidal transport in an array of optical tweezers,” Phys. Rev. Lett. 89(12), 128301 (2002).
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[CrossRef] [PubMed]

Proc. SPIE (1)

J. Shane, M. Mazilu, W. M. Lee, and K. Dholakia, ““Optical trapping using ultashort 12.9fs pulses,” Optical Trapping and Optical Micromanipulation V,” Proc. SPIE 7038, 70380Y, 70380Y–11 (2008).
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[CrossRef] [PubMed]

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Figures (8)

Fig. 1
Fig. 1

Schematic for a Gaussian-shaped pulse passing through the focusing region.

Fig. 2
Fig. 2

Typical evolutions of transverse (dashed line) and axial (solid line) PRFs for pulses with different durations: (a) τ = 1ps, (b) τ = 0.1ps, and (c) τ = 0.01ps. Other parameters are λ 0 = 0.514 μm, w 0 = 1 μm, a = 5 nm, U = 0.1 μJ, and m = n 1 / n 2 = 1.592 / 1.332 (for example, the small glass bead and water). The particle is located at the position ( ρ ˜ , z ˜ ) = (0.2, 0.5).

Fig. 3
Fig. 3

Time evolutions of the transverse and axial components for (a-b) the velocities v t r a n s and v a x i a l , and (c-d) the displacements s t r a n s and s a x i a l , of the particle under the action of the different pulses. Dashed lines are forτ = 1ps, solid lines for τ = 0.1ps, dot-dashed lines for τ = 0.03ps, and short-dashed lines for τ = 0.01ps. In (b) and (d), the color arrows denote the ends of the pulses for τ = 0.01ps and τ = 0.03ps (i. e., the PRF nearly disappears). Other parameters are the same as in Fig. 2.

Fig. 4
Fig. 4

Final distribution of the particle’s velocity for the different pulses( τ = 1 , 0 .1 , a n d 0 .01 ps) at time t ˜ = 5 . Other parameters are the same as in Fig. 2.

Fig. 5
Fig. 5

Dynamic distributions of the velocity (a-c) and displacement (d-f) of the particle under the action of the pulse with τ = 1 ps at different times: t ˜ = 1 for (a) and (d), t ˜ = 0 for (b) and (e), and t ˜ = 1 for (c) and (f). Other parameters are the same as in Fig. 2.

Fig. 6
Fig. 6

Dynamic distributions of the velocity (a-d) and displacement (e-h) of the particle under the action of the PRF for the pulse with τ = 0.01ps at different times: t ˜ = 1 for (a) and (e), t ˜ = 0 for (b) and (f), t ˜ = 1 for (c) and (g), and t ˜ = 2 for (d) and (h). Other parameters are the same as in Fig. 2.

Fig. 7
Fig. 7

Dynamic distributions of the velocity (a-c) and displacement (d-f) of the particle under the action of the PRF for the pulse with τ = 1ps at different times: t ˜ = 1 for (a) and (d), t ˜ = 0 for (b) and (e), and t ˜ = 1 for (c) and (f). Other parameters are the same as in Fig. 2 except for a = 50nm.

Fig. 8
Fig. 8

Dynamic distributions of the velocity (a-d) and displacement (e-h) of the particle under the action of the PRF for the pulse with τ = 0.01ps at different times: t ˜ = 1 for (a) and (e), t ˜ = 0 for (b) and (f), t ˜ = 1 for (c) and (g), and t ˜ = 2 for (d) and (h). Other parameters are the same as in Fig. 2 except for a = 50nm.

Equations (14)

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E ( ρ , z , t ) = e x E ( ρ , z , t ) = e x ( i Z R E 0 i Z R + z ) exp [ i ω 0 t i k z k Z R ρ 2 + i k z ρ 2 2 ( z 2 + Z R 2 ) ] exp [ ( t z / c ) 2 τ 2 ] ,
F t r a n s = F g r a d , ρ = ρ ^ 2 β I ( ρ ˜ , z ˜ , t ˜ ) ρ ˜ / [ c n 2 ε 0 w 0 ( 1 + z ˜ 2 ) ] ,
F a x i a l = F g r a d , z + F t + F s c a t = z ^ β I ( ρ ˜ , z ˜ , t ˜ ) n 2 ε 0 c Z R [ 2 Z R 2 z ˜ c 2 τ 2 2 Z R t ˜ c τ + z ˜ ( 1 + z ˜ 2 2 ρ ˜ 2 ) ( 1 + z ˜ 2 ) 2 ] z ^ 8 μ 0 β I ( ρ ˜ , z ˜ , t ˜ ) t ˜ / τ + z ^ 8 z ˜ μ 0 β I ( ρ ˜ , z ˜ , t ˜ ) Z R / ( c τ 2 ) + z ^ ( n 2 / c ) C p r I ( ρ ˜ , z ˜ , t ˜ ) ,
I ( ρ ˜ , z ˜ , t ˜ ) = P 1 + z ˜ 2 exp [ 2 ρ ˜ 2 1 + z ˜ 2 ] exp [ 2 ( t ˜ z ˜ Z R c τ ) 2 ] ,
v t r a n s ( ρ ˜ , z ˜ , t ˜ ) = 2 β P ρ ˜ c n 2 ε 0 w 0 ( 1 + z ˜ 2 ) 2 exp ( 2 ρ ˜ 2 1 + z ˜ 2 ) Φ 1 ( z ˜ , t ˜ ) ,
s t r a n s ( ρ ˜ , z ˜ , t ˜ ) = 2 β P ρ ˜ c n 2 ε 0 w 0 ( 1 + z ˜ 2 ) 2 exp ( 2 ρ ˜ 2 1 + z ˜ 2 ) Φ 2 ( z ˜ , t ˜ ) ,
v a x i a l ( ρ ˜ , z ˜ , t ˜ ) = Y 1 ( ρ ˜ , z ˜ ) Φ 1 ( z ˜ , t ˜ ) + Y 2 ( ρ ˜ , z ˜ ) Φ 3 ( z ˜ , t ˜ ) ,
s a x i a l ( ρ ˜ , z ˜ , t ˜ ) = Y 1 ( ρ ˜ , z ˜ ) Φ 2 ( z ˜ , t ˜ ) + Y 2 ( ρ ˜ , z ˜ ) Φ 4 ( z ˜ , t ˜ ) ,
Y 1 ( ρ ˜ , z ˜ ) = P ( 1 + z ˜ 2 ) M p exp [ 2 ρ ˜ 2 1 + z ˜ 2 ] { 8 z ˜ μ 0 β Z R c τ 2 + n 2 C p r c β M p n 2 ε 0 c Z R [ 2 z ˜ Z R 2 c 2 τ 2 + z ˜ ( 1 + z ˜ 2 2 ρ ˜ 2 ) ( 1 + z ˜ 2 ) 2 ] } ,
Y 2 ( ρ ˜ , z ˜ ) = P ( 1 + z ˜ 2 ) M p exp [ 2 ρ ˜ 2 1 + z ˜ 2 ] ( 2 β n 2 ε 0 c 2 τ 8 μ 0 β τ ) ,
Φ 1 ( z ˜ , t ˜ ) = 2 π τ 4 { 1 + E r f [ 2 ( t ˜ z ˜ Z R c τ ) ] } ,
Φ 2 ( z ˜ , t ˜ ) = τ 2 4 exp [ 2 ( t ˜ z ˜ Z R c τ ) 2 ] + τ ( t ˜ z ˜ Z R c τ ) Φ 1 ( z ˜ , t ˜ ) ,
Φ 3 ( z ˜ , t ˜ ) = τ 4 exp [ 2 ( t ˜ z ˜ Z R c τ ) 2 ] + z ˜ Z R c τ Φ 1 ( z ˜ , t ˜ ) ,
Φ 4 ( z ˜ , t ˜ ) = z ˜ Z R c τ Φ 2 ( z ˜ , t ˜ ) τ 4 Φ 1 ( z ˜ , t ˜ ) .

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