Abstract

We propose and demonstrate optofluidic tunable manipulation of polystyrene microparticles based on the combination of a graded-index fiber (GIF) taper and a microcavity. The tunability on the manipulation length is experimentally explored by changing the balance between the optical force and the microfluidic flow force, as well as by tuning the focus of light emitting from the GIF taper via adjusting the length of an air microcavity. By optimizing the geometric shape of the GIF taper, as well as the flow rate and laser power, a manipulation length of 177 μm is achieved, more than 4 times longer than the state-of-the-art optical fiber tweezers. This method has advantages of high flexibility, ease of fabrication and use, integration with microfluidics and has the potential for optofluidic sensing applications.

© 2015 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  4. J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
    [Crossref] [PubMed]
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  7. G. Brambilla, G. S. Murugan, J. S. Wilkinson, and D. J. Richardson, “Optical manipulation of microspheres along a subwavelength optical wire,” Opt. Lett. 32(20), 3041–3043 (2007).
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    [Crossref]
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    [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  16. Y. Gong, A. Y. Ye, Y. Wu, Y. J. Rao, Y. Yao, and S. Xiao, “Graded-index fiber tip optical tweezers: Numerical simulation and trapping experiment,” Opt. Express 21(13), 16181–16190 (2013).
    [Crossref] [PubMed]
  17. G. Brambilla and F. Xu, “Adiabatic submicrometric tapers for optical tweezers,” Electron. Lett. 43(4), 204–205 (2007).
    [Crossref]
  18. H. Xin and B. Li, “Optical orientation and shifting of a single multiwalled carbon nanotube,” Light: Sci. Appl. 3(9), e205 (2014).
    [Crossref]
  19. H. Xin, X. Li, and B. Li, “Massive photothermal trapping and migration of particles by a tapered optical fiber,” Opt. Express 19(18), 17065–17074 (2011).
    [Crossref] [PubMed]
  20. M. Righini, G. Volpe, C. Girard, D. Petrov, and R. Quidant, “Surface plasmon optical tweezers: Tunable optical manipulation in the femtonewton range,” Phys. Rev. Lett. 100(18), 186804 (2008).
    [Crossref] [PubMed]
  21. B. Hester, G. K. Campbell, C. López-Mariscal, C. L. Filgueira, R. Huschka, N. J. Halas, and K. Helmerson, “Tunable optical tweezers for wavelength-dependent measurements,” Rev. Sci. Instrum. 83(4), 043114 (2012).
    [Crossref] [PubMed]
  22. M. Gecevičius, R. Drevinskas, M. Beresna, and P. G. Kazansky, “Single beam optical vortex tweezers with tunable orbital angular momentum,” Appl. Phys. Lett. 104(23), 231110 (2014).
    [Crossref]
  23. H. Xin, Y. Li, L. Li, R. Xu, and B. Li, “Optofluidic manipulation of Escherichia coli in a microfluidic channel using an abruptly tapered optical fiber,” Appl. Phys. Lett. 103(3), 033703 (2013).
    [Crossref]

2014 (4)

J. Berthelot, S. S. Aćimović, M. L. Juan, M. P. Kreuzer, J. Renger, and R. Quidant, “Three-dimensional manipulation with scanning near-field optical nanotweezers,” Nat. Nanotechnol. 9(4), 295–299 (2014).
[Crossref] [PubMed]

Y. Gong, W. Huang, Q. F. Liu, Y. Wu, Y. Rao, G. D. Peng, J. Lang, and K. Zhang, “Graded-index optical fiber tweezers with long manipulation length,” Opt. Express 22(21), 25267–25276 (2014).
[Crossref] [PubMed]

H. Xin and B. Li, “Optical orientation and shifting of a single multiwalled carbon nanotube,” Light: Sci. Appl. 3(9), e205 (2014).
[Crossref]

M. Gecevičius, R. Drevinskas, M. Beresna, and P. G. Kazansky, “Single beam optical vortex tweezers with tunable orbital angular momentum,” Appl. Phys. Lett. 104(23), 231110 (2014).
[Crossref]

2013 (5)

H. Xin, Y. Li, L. Li, R. Xu, and B. Li, “Optofluidic manipulation of Escherichia coli in a microfluidic channel using an abruptly tapered optical fiber,” Appl. Phys. Lett. 103(3), 033703 (2013).
[Crossref]

C. Liberale, G. Cojoc, F. Bragheri, P. Minzioni, G. Perozziello, R. La Rocca, L. Ferrara, V. Rajamanickam, E. Di Fabrizio, and I. Cristiani, “Integrated microfluidic device for single-cell trapping and spectroscopy,” Sci. Rep. 3, 1258–1263 (2013).
[PubMed]

Y. Liu, F. Stief, and M. Yu, “Subwavelength optical trapping with a fiber-based surface plasmonic lens,” Opt. Lett. 38(5), 721–723 (2013).
[Crossref] [PubMed]

Y. Gong, A. Y. Ye, Y. Wu, Y. J. Rao, Y. Yao, and S. Xiao, “Graded-index fiber tip optical tweezers: Numerical simulation and trapping experiment,” Opt. Express 21(13), 16181–16190 (2013).
[Crossref] [PubMed]

Y. Zhang and B. Li, “Particle sorting using a subwavelength optical fiber,” Laser Photon. Rev. 7(2), 289–296 (2013).
[Crossref]

2012 (1)

B. Hester, G. K. Campbell, C. López-Mariscal, C. L. Filgueira, R. Huschka, N. J. Halas, and K. Helmerson, “Tunable optical tweezers for wavelength-dependent measurements,” Rev. Sci. Instrum. 83(4), 043114 (2012).
[Crossref] [PubMed]

2011 (1)

2009 (2)

Y. X. Liu and M. Yu, “Investigation of inclined dual-fiber optical tweezers for 3D manipulation and force sensing,” Opt. Express 17(16), 13624–13638 (2009).
[Crossref] [PubMed]

A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, “Optical manipulation of nanoparticles and biomolecules in sub-wavelength slot waveguides,” Nature 457(7225), 71–75 (2009).
[Crossref] [PubMed]

2008 (2)

L. Yuan, Z. Liu, J. Yang, and C. Guan, “Twin-core fiber optical tweezers,” Opt. Express 16(7), 4559–4566 (2008).
[Crossref] [PubMed]

M. Righini, G. Volpe, C. Girard, D. Petrov, and R. Quidant, “Surface plasmon optical tweezers: Tunable optical manipulation in the femtonewton range,” Phys. Rev. Lett. 100(18), 186804 (2008).
[Crossref] [PubMed]

2007 (3)

G. Brambilla and F. Xu, “Adiabatic submicrometric tapers for optical tweezers,” Electron. Lett. 43(4), 204–205 (2007).
[Crossref]

G. Brambilla, G. S. Murugan, J. S. Wilkinson, and D. J. Richardson, “Optical manipulation of microspheres along a subwavelength optical wire,” Opt. Lett. 32(20), 3041–3043 (2007).
[Crossref] [PubMed]

C. Liberale, P. Minzioni, F. Bragheri, F. De Angelis, E. Di Fabrizio, and I. Cristiani, “Miniaturized all-fibre probe for three-dimensional optical trapping and manipulation,” Nat. Photonics 1(12), 723–727 (2007).
[Crossref]

2006 (2)

2005 (1)

P. Y. Chiou, A. T. Ohta, and M. C. Wu, “Massively parallel manipulation of single cells and microparticles using optical images,” Nature 436(7049), 370–372 (2005).
[Crossref] [PubMed]

2003 (1)

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

2001 (1)

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
[Crossref] [PubMed]

Acimovic, S. S.

J. Berthelot, S. S. Aćimović, M. L. Juan, M. P. Kreuzer, J. Renger, and R. Quidant, “Three-dimensional manipulation with scanning near-field optical nanotweezers,” Nat. Nanotechnol. 9(4), 295–299 (2014).
[Crossref] [PubMed]

Ananthakrishnan, R.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
[Crossref] [PubMed]

Beresna, M.

M. Gecevičius, R. Drevinskas, M. Beresna, and P. G. Kazansky, “Single beam optical vortex tweezers with tunable orbital angular momentum,” Appl. Phys. Lett. 104(23), 231110 (2014).
[Crossref]

Berthelot, J.

J. Berthelot, S. S. Aćimović, M. L. Juan, M. P. Kreuzer, J. Renger, and R. Quidant, “Three-dimensional manipulation with scanning near-field optical nanotweezers,” Nat. Nanotechnol. 9(4), 295–299 (2014).
[Crossref] [PubMed]

Bragheri, F.

C. Liberale, G. Cojoc, F. Bragheri, P. Minzioni, G. Perozziello, R. La Rocca, L. Ferrara, V. Rajamanickam, E. Di Fabrizio, and I. Cristiani, “Integrated microfluidic device for single-cell trapping and spectroscopy,” Sci. Rep. 3, 1258–1263 (2013).
[PubMed]

C. Liberale, P. Minzioni, F. Bragheri, F. De Angelis, E. Di Fabrizio, and I. Cristiani, “Miniaturized all-fibre probe for three-dimensional optical trapping and manipulation,” Nat. Photonics 1(12), 723–727 (2007).
[Crossref]

Brambilla, G.

Campbell, G. K.

B. Hester, G. K. Campbell, C. López-Mariscal, C. L. Filgueira, R. Huschka, N. J. Halas, and K. Helmerson, “Tunable optical tweezers for wavelength-dependent measurements,” Rev. Sci. Instrum. 83(4), 043114 (2012).
[Crossref] [PubMed]

Chiou, P. Y.

P. Y. Chiou, A. T. Ohta, and M. C. Wu, “Massively parallel manipulation of single cells and microparticles using optical images,” Nature 436(7049), 370–372 (2005).
[Crossref] [PubMed]

Cojoc, G.

C. Liberale, G. Cojoc, F. Bragheri, P. Minzioni, G. Perozziello, R. La Rocca, L. Ferrara, V. Rajamanickam, E. Di Fabrizio, and I. Cristiani, “Integrated microfluidic device for single-cell trapping and spectroscopy,” Sci. Rep. 3, 1258–1263 (2013).
[PubMed]

Cristiani, I.

C. Liberale, G. Cojoc, F. Bragheri, P. Minzioni, G. Perozziello, R. La Rocca, L. Ferrara, V. Rajamanickam, E. Di Fabrizio, and I. Cristiani, “Integrated microfluidic device for single-cell trapping and spectroscopy,” Sci. Rep. 3, 1258–1263 (2013).
[PubMed]

C. Liberale, P. Minzioni, F. Bragheri, F. De Angelis, E. Di Fabrizio, and I. Cristiani, “Miniaturized all-fibre probe for three-dimensional optical trapping and manipulation,” Nat. Photonics 1(12), 723–727 (2007).
[Crossref]

Cunningham, C. C.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
[Crossref] [PubMed]

De Angelis, F.

C. Liberale, P. Minzioni, F. Bragheri, F. De Angelis, E. Di Fabrizio, and I. Cristiani, “Miniaturized all-fibre probe for three-dimensional optical trapping and manipulation,” Nat. Photonics 1(12), 723–727 (2007).
[Crossref]

Dholakia, K.

Di Fabrizio, E.

C. Liberale, G. Cojoc, F. Bragheri, P. Minzioni, G. Perozziello, R. La Rocca, L. Ferrara, V. Rajamanickam, E. Di Fabrizio, and I. Cristiani, “Integrated microfluidic device for single-cell trapping and spectroscopy,” Sci. Rep. 3, 1258–1263 (2013).
[PubMed]

C. Liberale, P. Minzioni, F. Bragheri, F. De Angelis, E. Di Fabrizio, and I. Cristiani, “Miniaturized all-fibre probe for three-dimensional optical trapping and manipulation,” Nat. Photonics 1(12), 723–727 (2007).
[Crossref]

Drevinskas, R.

M. Gecevičius, R. Drevinskas, M. Beresna, and P. G. Kazansky, “Single beam optical vortex tweezers with tunable orbital angular momentum,” Appl. Phys. Lett. 104(23), 231110 (2014).
[Crossref]

Erickson, D.

A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, “Optical manipulation of nanoparticles and biomolecules in sub-wavelength slot waveguides,” Nature 457(7225), 71–75 (2009).
[Crossref] [PubMed]

Ferrara, L.

C. Liberale, G. Cojoc, F. Bragheri, P. Minzioni, G. Perozziello, R. La Rocca, L. Ferrara, V. Rajamanickam, E. Di Fabrizio, and I. Cristiani, “Integrated microfluidic device for single-cell trapping and spectroscopy,” Sci. Rep. 3, 1258–1263 (2013).
[PubMed]

Filgueira, C. L.

B. Hester, G. K. Campbell, C. López-Mariscal, C. L. Filgueira, R. Huschka, N. J. Halas, and K. Helmerson, “Tunable optical tweezers for wavelength-dependent measurements,” Rev. Sci. Instrum. 83(4), 043114 (2012).
[Crossref] [PubMed]

Garcés-Chávez, V.

Gecevicius, M.

M. Gecevičius, R. Drevinskas, M. Beresna, and P. G. Kazansky, “Single beam optical vortex tweezers with tunable orbital angular momentum,” Appl. Phys. Lett. 104(23), 231110 (2014).
[Crossref]

Girard, C.

M. Righini, G. Volpe, C. Girard, D. Petrov, and R. Quidant, “Surface plasmon optical tweezers: Tunable optical manipulation in the femtonewton range,” Phys. Rev. Lett. 100(18), 186804 (2008).
[Crossref] [PubMed]

Gong, Y.

Grier, D. G.

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

Guan, C.

Guck, J.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
[Crossref] [PubMed]

Guo, C.

Halas, N. J.

B. Hester, G. K. Campbell, C. López-Mariscal, C. L. Filgueira, R. Huschka, N. J. Halas, and K. Helmerson, “Tunable optical tweezers for wavelength-dependent measurements,” Rev. Sci. Instrum. 83(4), 043114 (2012).
[Crossref] [PubMed]

Helmerson, K.

B. Hester, G. K. Campbell, C. López-Mariscal, C. L. Filgueira, R. Huschka, N. J. Halas, and K. Helmerson, “Tunable optical tweezers for wavelength-dependent measurements,” Rev. Sci. Instrum. 83(4), 043114 (2012).
[Crossref] [PubMed]

Herrington, C. S.

Hester, B.

B. Hester, G. K. Campbell, C. López-Mariscal, C. L. Filgueira, R. Huschka, N. J. Halas, and K. Helmerson, “Tunable optical tweezers for wavelength-dependent measurements,” Rev. Sci. Instrum. 83(4), 043114 (2012).
[Crossref] [PubMed]

Huang, W.

Huschka, R.

B. Hester, G. K. Campbell, C. López-Mariscal, C. L. Filgueira, R. Huschka, N. J. Halas, and K. Helmerson, “Tunable optical tweezers for wavelength-dependent measurements,” Rev. Sci. Instrum. 83(4), 043114 (2012).
[Crossref] [PubMed]

Jess, P. R. T.

Juan, M. L.

J. Berthelot, S. S. Aćimović, M. L. Juan, M. P. Kreuzer, J. Renger, and R. Quidant, “Three-dimensional manipulation with scanning near-field optical nanotweezers,” Nat. Nanotechnol. 9(4), 295–299 (2014).
[Crossref] [PubMed]

Käs, J.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
[Crossref] [PubMed]

Kazansky, P. G.

M. Gecevičius, R. Drevinskas, M. Beresna, and P. G. Kazansky, “Single beam optical vortex tweezers with tunable orbital angular momentum,” Appl. Phys. Lett. 104(23), 231110 (2014).
[Crossref]

Klug, M.

A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, “Optical manipulation of nanoparticles and biomolecules in sub-wavelength slot waveguides,” Nature 457(7225), 71–75 (2009).
[Crossref] [PubMed]

Kreuzer, M. P.

J. Berthelot, S. S. Aćimović, M. L. Juan, M. P. Kreuzer, J. Renger, and R. Quidant, “Three-dimensional manipulation with scanning near-field optical nanotweezers,” Nat. Nanotechnol. 9(4), 295–299 (2014).
[Crossref] [PubMed]

La Rocca, R.

C. Liberale, G. Cojoc, F. Bragheri, P. Minzioni, G. Perozziello, R. La Rocca, L. Ferrara, V. Rajamanickam, E. Di Fabrizio, and I. Cristiani, “Integrated microfluidic device for single-cell trapping and spectroscopy,” Sci. Rep. 3, 1258–1263 (2013).
[PubMed]

Lang, J.

Li, B.

H. Xin and B. Li, “Optical orientation and shifting of a single multiwalled carbon nanotube,” Light: Sci. Appl. 3(9), e205 (2014).
[Crossref]

H. Xin, Y. Li, L. Li, R. Xu, and B. Li, “Optofluidic manipulation of Escherichia coli in a microfluidic channel using an abruptly tapered optical fiber,” Appl. Phys. Lett. 103(3), 033703 (2013).
[Crossref]

Y. Zhang and B. Li, “Particle sorting using a subwavelength optical fiber,” Laser Photon. Rev. 7(2), 289–296 (2013).
[Crossref]

H. Xin, X. Li, and B. Li, “Massive photothermal trapping and migration of particles by a tapered optical fiber,” Opt. Express 19(18), 17065–17074 (2011).
[Crossref] [PubMed]

Li, L.

H. Xin, Y. Li, L. Li, R. Xu, and B. Li, “Optofluidic manipulation of Escherichia coli in a microfluidic channel using an abruptly tapered optical fiber,” Appl. Phys. Lett. 103(3), 033703 (2013).
[Crossref]

Li, X.

Li, Y.

H. Xin, Y. Li, L. Li, R. Xu, and B. Li, “Optofluidic manipulation of Escherichia coli in a microfluidic channel using an abruptly tapered optical fiber,” Appl. Phys. Lett. 103(3), 033703 (2013).
[Crossref]

Liberale, C.

C. Liberale, G. Cojoc, F. Bragheri, P. Minzioni, G. Perozziello, R. La Rocca, L. Ferrara, V. Rajamanickam, E. Di Fabrizio, and I. Cristiani, “Integrated microfluidic device for single-cell trapping and spectroscopy,” Sci. Rep. 3, 1258–1263 (2013).
[PubMed]

C. Liberale, P. Minzioni, F. Bragheri, F. De Angelis, E. Di Fabrizio, and I. Cristiani, “Miniaturized all-fibre probe for three-dimensional optical trapping and manipulation,” Nat. Photonics 1(12), 723–727 (2007).
[Crossref]

Lipson, M.

A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, “Optical manipulation of nanoparticles and biomolecules in sub-wavelength slot waveguides,” Nature 457(7225), 71–75 (2009).
[Crossref] [PubMed]

Liu, Q. F.

Liu, Y.

Liu, Y. X.

Liu, Z.

López-Mariscal, C.

B. Hester, G. K. Campbell, C. López-Mariscal, C. L. Filgueira, R. Huschka, N. J. Halas, and K. Helmerson, “Tunable optical tweezers for wavelength-dependent measurements,” Rev. Sci. Instrum. 83(4), 043114 (2012).
[Crossref] [PubMed]

Mahmood, H.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
[Crossref] [PubMed]

Mazilu, M.

Minzioni, P.

C. Liberale, G. Cojoc, F. Bragheri, P. Minzioni, G. Perozziello, R. La Rocca, L. Ferrara, V. Rajamanickam, E. Di Fabrizio, and I. Cristiani, “Integrated microfluidic device for single-cell trapping and spectroscopy,” Sci. Rep. 3, 1258–1263 (2013).
[PubMed]

C. Liberale, P. Minzioni, F. Bragheri, F. De Angelis, E. Di Fabrizio, and I. Cristiani, “Miniaturized all-fibre probe for three-dimensional optical trapping and manipulation,” Nat. Photonics 1(12), 723–727 (2007).
[Crossref]

Moon, T. J.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
[Crossref] [PubMed]

Moore, S. D.

A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, “Optical manipulation of nanoparticles and biomolecules in sub-wavelength slot waveguides,” Nature 457(7225), 71–75 (2009).
[Crossref] [PubMed]

Murugan, G. S.

Ohta, A. T.

P. Y. Chiou, A. T. Ohta, and M. C. Wu, “Massively parallel manipulation of single cells and microparticles using optical images,” Nature 436(7049), 370–372 (2005).
[Crossref] [PubMed]

Paterson, L.

Peng, G. D.

Perozziello, G.

C. Liberale, G. Cojoc, F. Bragheri, P. Minzioni, G. Perozziello, R. La Rocca, L. Ferrara, V. Rajamanickam, E. Di Fabrizio, and I. Cristiani, “Integrated microfluidic device for single-cell trapping and spectroscopy,” Sci. Rep. 3, 1258–1263 (2013).
[PubMed]

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[PubMed]

Supplementary Material (2)

» Media 1: MOV (9118 KB)     
» Media 2: MOV (5978 KB)     

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Figures (7)

Fig. 1
Fig. 1 Schematic experimental setup for optofluidic tunable manipulation of PS microparticle.
Fig. 2
Fig. 2 Principles of optofluidic tunable manipulation of PS microparticle, (a) by tuning the force balance and (b) by tuning the light focusing effect.
Fig. 3
Fig. 3 Microscopic images of (a) the GIF taper and (b) optofluidic tunable manipulation of a 7.0μm-diameter PS microparticle. (Media 1) 9.8 Mb.
Fig. 4
Fig. 4 Manipulation length versus flow rate at different laser powers.
Fig. 5
Fig. 5 Microscopic images of (a) a sharp GIF taper, and optofluidic tunable manipulation with Lm of (b) 36.2 μm and (c) 177.0 μm.
Fig. 6
Fig. 6 Microscopic images of optofluidic tunable manipulation of PS microparticle. (Media 2) 5.9 Mb.
Fig. 7
Fig. 7 (a) Typical reflective spectrum of the microcavity for determining the cavity length and (b) manipulation length as a function of the cavity length at laser power of 43.5 mW and flow rate of 450 nL/min.

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