Abstract

We consider the problem of sorting, by size, spherical particles of order 100 nm radius. The scheme we analyze consists of a heterogeneous stream of spherical particles flowing at an oblique angle across an optical Gaussian mode standing wave. Sorting is achieved by the combined spatial and size dependencies of the optical force. Particles of all sizes enter the flow at a point, but exit at different locations depending on size. Exiting particles may be detected optically or separated for further processing. The scheme has the advantages of accommodating a high throughput, producing a continuous stream of continuously dispersed particles, and exhibiting excellent size resolution. We performed detailed Monte Carlo simulations of particle trajectories through the optical field under the influence of convective air flow. We also developed a method for deriving effective velocities and diffusion constants from the Fokker-Planck equation that can generate equivalent results much more quickly. With an optical wavelength of 1064 nm, polystyrene particles with radii in the neighborhood of 275 nm, for which the optical force vanishes, may be sorted with a resolution below 1 nm.

© 2016 Optical Society of America

Full Article  |  PDF Article
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References

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    [Crossref]
  47. 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, 128301 (2002).
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  48. S. Ahlawat, R. Dasgupta, R. S. Verma, V. N. Kumar, and P. K. Gupta, “Optical sorting in holographic trap arrays by tuning the inter-trap separation,” J. Opt. 14, 125501 (2012).
    [Crossref]
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    [Crossref]
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    [Crossref]

2016 (1)

P. P. Mathai, J. A. Liddle, and S. M. Stavis, “Optical tracking of nanoscale particles in microscale environments,” Appl. Phys. Rev. 3, 011105 (2016).
[Crossref] [PubMed]

2014 (2)

D. S. Dean and G. Oshanin, “Approach to asymptotically diffusive behavior for Brownian particles in periodic potentials: Extracting information from transients,” Phys. Rev. E 90, 022112 (2014).
[Crossref]

P. Jákl, A. V. Arzola, M. Šiler, L. Chvátal, K. Volke-Sepúlveda, and P. Zemánek, “Optical sorting of nonspherical and living microobjects in moving interference structures,” Opt. Express 22, 29746 (2014).
[Crossref]

2013 (3)

O. Brzobohatý, M. Šiler, J. Jezek, P. Jákl, and P. Zemánek, “Optical manipulation of aerosol droplets using a holographic dual and single beam trap,” Optics Lett. 38, 4601 (2013).
[Crossref]

N. J. Alvarez, C. Jeppesen, K. Yvind, N. A. Mortensen, and O. Hassager, “The chromatic separation of particles using optical electric fields,” Lab Chip 13, 928 (2013).
[Crossref] [PubMed]

O. Brzobohatý, V. Karásek, M. Šiler, L. Chvátal, T. Cizmár, and P. Zemánek, “Experimental demonstration of optical transport, sorting, and self-arrangement using a ‘tractor beam’,” Nat. Phot. 7, 123 (2013).
[Crossref]

2012 (2)

M. Ploschner, T. Cizmár, M. Mazilu, A. Di Falco, and K. Dholakia, “Bidirectional optical sorting of gold nanoparticles,” Nano Lett. 12, 1923 (2012).
[Crossref] [PubMed]

S. Ahlawat, R. Dasgupta, R. S. Verma, V. N. Kumar, and P. K. Gupta, “Optical sorting in holographic trap arrays by tuning the inter-trap separation,” J. Opt. 14, 125501 (2012).
[Crossref]

2011 (4)

K. H. Lee, S. B. Kim, K. S. Lee, and H. J. Sung, “Enhancement by optical force of separation in pinched flow fractionation,” Lab Chip 11354 (2011).
[Crossref]

Y. Pang and R. Gordon, “Optical trapping of 12 nm dielectric spheres using double nanoholes in a gold film,” Nano Lett. 11, 3763 (2011).
[Crossref] [PubMed]

M. Horstmann, K. Probst, and C. Fallnich, “An integrated fiber-based optical trap for single airborne particles,” Appl. Phys. B 103, 35 (2011).
[Crossref]

M. Ploschner, M. Mazilu, T. Cizmár, and K. Dholakia, “Numerical investigation of passive optical sorting of plasmon nanoparticles,” Opt. Express 19, 13922 (2011).
[Crossref] [PubMed]

2010 (4)

K. Xiao and D. G. Grier, “Multidimensional Optical Fractionation of Colloidal Particles with Holographic Verification,” Phys. Rev. Lett. 104, 028302 (2010).
[Crossref] [PubMed]

A. E. Carruthers, J. P. Reid, and A. J. Orr-Ewing, “Longitudinal optical trapping and sizing of aerosol droplets,” Optics Express 18, 14238 (2010).
[Crossref] [PubMed]

K. J. Knox, D. R. Burnham, L. I. McCann, S. L. Murphy, D. McGloin, and J. P. Reid, “Observation of bistability of trapping position in aerosol optical tweezers,” J. Opt. Soc. Am. B 27, 582 (2010).
[Crossref]

K. Xiao and D. G. Grier, “Sorting colloidal particles into multiple channels with optical forces: prismatic optical fractionation,” Phys. Rev. E 82, 051407 (2010).
[Crossref]

2009 (2)

2008 (5)

D. Rudd, C. López-Mariscal, M. Summers, A. Shahvisi, J. C. Gutiérrez-Vega, and D. McGloin, “Fiber based optical trapping of aerosols,” Opt. Express 16, 14550 (2008).
[Crossref] [PubMed]

P. Jákl, T. Cizmár, M. Šerý, and P. Zemánek, “Static optical sorting in a laser interference field,” Appl. Phys. Lett. 92, 161110 (2008).
[Crossref]

A. Jonáš and P. Zemánek, “Light at work: The use of optical forces for particle manipulation, sorting, and analysis,” Electrophoresis 29, 4813 (2008).
[Crossref]

D. McGloin, D. R. Burnham, M. D. Summers, D. Rudd, N. Dewar, and S. Anand, “Optical manipulation of airborne particles: techniques and applications,” Faraday Discuss. 137, 335 (2008).
[Crossref] [PubMed]

R. F. Marchington, M. Mazilu, S. Kuriakose, V. Garcés-Chávez, P. J. Reece, T. F. Krause, M. Gu, and K. Dholakia, “Optical deflection and sorting of microparticles in a near-field optical geometry,” Opt. Express 16, 3712 (2008).
[Crossref] [PubMed]

2007 (2)

R. L. Smith, G. C. Spalding, K. Dholakia, and M. P. MacDonald, “Colloidal sorting in dynamic optical lattices,” J. Opt. A: Pure and Appl. Opt. 9, S134 (2007).
[Crossref]

R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Colloquium: Momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys. 79, 1197 (2007).
[Crossref]

2006 (2)

K. Taji, M. Tachikawa, and K. Nagashima, “Laser trapping of ice crystals,” Appl. Phys. Lett. 88, 141111 (2006).
[Crossref]

M. Guillon, O. Moine, and B. Stout, “Longitudinal optical binding of high optical contrast microdroplets in air,” Phys. Rev. Lett. 96, 143902 (2006).
[Crossref] [PubMed]

2005 (2)

T. Cizmár, V. Garcés-chávez, K. Dholakia, and P. Zemánek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86, 174101 (2005).
[Crossref]

K. Grujic, O. G. Hellesø, J. P. Hole, and J. S. Wilkinson, “Sorting of polystyrene microspheres using a Y-branched optical waveguide,” Opt. Express 13, 1 (2005).
[Crossref] [PubMed]

2004 (2)

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787 (2004).
[Crossref]

K. Ladavac, K. Kasza, and D. G. Grier, “Sorting mesoscopic objects with periodic potential landscapes: optical fractionation,” Phys. Rev. E 70, 010901(R) (2004).
[Crossref]

2003 (3)

S. Tatarkova, W. Sibbett, and K. Dholakia, “Brownian particle in an optical potential of the washboard type,” Phys. Rev. Lett. 91, 038101 (2003).
[Crossref] [PubMed]

P. Zemánek, A. Jonáš, P. Jákl, J. Jezek, M. Šerý, and M. Liška, “Theoretical comparison of optical traps created by standing wave and single beam,” Opt. Commun. 220, 401 (2003).
[Crossref]

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421 (2003).
[Crossref] [PubMed]

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, 128301 (2002).
[Crossref] [PubMed]

2000 (2)

I. D. Nikolov and C. D. Ivanov, “Optical plastic refractive measurements in the visible and the near-infrared regions,” Appl. Opt. 39, 2067 (2000).
[Crossref]

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

1996 (1)

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529 (1996).
[Crossref]

1989 (1)

J. P. Barton, D. R. Alexander, and S. A Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594 (1989).
[Crossref]

1986 (1)

1983 (1)

B. Derrida, “Velocity and Diffusion Constant of a Periodic One-Dimensional Hoping Model,” J. Stat. Phys. 31, 433 (1983).
[Crossref]

1970 (1)

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

1943 (1)

S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Rev. Mod. Phys. 15, 1 (1943).
[Crossref]

1940 (1)

H. A. Kramers, “Brownian motion in a field of force and the diffusion model of chemical reactions,” Physica 7, 284 (1940).
[Crossref]

1909 (1)

P. Debye, “Der Lichtdruck auf Kugeln von belibigem Material, Ann. der Phyzik IV 335, 57 (1909).

Ahlawat, S.

S. Ahlawat, R. Dasgupta, R. S. Verma, V. N. Kumar, and P. K. Gupta, “Optical sorting in holographic trap arrays by tuning the inter-trap separation,” J. Opt. 14, 125501 (2012).
[Crossref]

Alexander, D. R.

J. P. Barton, D. R. Alexander, and S. A Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594 (1989).
[Crossref]

Alvarez, N. J.

N. J. Alvarez, C. Jeppesen, K. Yvind, N. A. Mortensen, and O. Hassager, “The chromatic separation of particles using optical electric fields,” Lab Chip 13, 928 (2013).
[Crossref] [PubMed]

Anand, S.

D. McGloin, D. R. Burnham, M. D. Summers, D. Rudd, N. Dewar, and S. Anand, “Optical manipulation of airborne particles: techniques and applications,” Faraday Discuss. 137, 335 (2008).
[Crossref] [PubMed]

Arzola, A. V.

Asakura, T.

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529 (1996).
[Crossref]

Ashkin, A.

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

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, 288 (1986).
[Crossref] [PubMed]

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

Barton, J. P.

J. P. Barton, D. R. Alexander, and S. A Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594 (1989).
[Crossref]

Bjorkholm, J. E.

Block, S. M.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787 (2004).
[Crossref]

Borse, G. J.

G. J. Borse, Numerical Methods with MATLAB, (PWS Publishing, 1997).

Brzobohatý, O.

O. Brzobohatý, V. Karásek, M. Šiler, L. Chvátal, T. Cizmár, and P. Zemánek, “Experimental demonstration of optical transport, sorting, and self-arrangement using a ‘tractor beam’,” Nat. Phot. 7, 123 (2013).
[Crossref]

O. Brzobohatý, M. Šiler, J. Jezek, P. Jákl, and P. Zemánek, “Optical manipulation of aerosol droplets using a holographic dual and single beam trap,” Optics Lett. 38, 4601 (2013).
[Crossref]

Burnham, D. R.

K. J. Knox, D. R. Burnham, L. I. McCann, S. L. Murphy, D. McGloin, and J. P. Reid, “Observation of bistability of trapping position in aerosol optical tweezers,” J. Opt. Soc. Am. B 27, 582 (2010).
[Crossref]

D. McGloin, D. R. Burnham, M. D. Summers, D. Rudd, N. Dewar, and S. Anand, “Optical manipulation of airborne particles: techniques and applications,” Faraday Discuss. 137, 335 (2008).
[Crossref] [PubMed]

Carruthers, A. E.

A. E. Carruthers, J. P. Reid, and A. J. Orr-Ewing, “Longitudinal optical trapping and sizing of aerosol droplets,” Optics Express 18, 14238 (2010).
[Crossref] [PubMed]

Chandrasekhar, S.

S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Rev. Mod. Phys. 15, 1 (1943).
[Crossref]

Chu, S.

Chvátal, L.

P. Jákl, A. V. Arzola, M. Šiler, L. Chvátal, K. Volke-Sepúlveda, and P. Zemánek, “Optical sorting of nonspherical and living microobjects in moving interference structures,” Opt. Express 22, 29746 (2014).
[Crossref]

O. Brzobohatý, V. Karásek, M. Šiler, L. Chvátal, T. Cizmár, and P. Zemánek, “Experimental demonstration of optical transport, sorting, and self-arrangement using a ‘tractor beam’,” Nat. Phot. 7, 123 (2013).
[Crossref]

Cizmár, T.

O. Brzobohatý, V. Karásek, M. Šiler, L. Chvátal, T. Cizmár, and P. Zemánek, “Experimental demonstration of optical transport, sorting, and self-arrangement using a ‘tractor beam’,” Nat. Phot. 7, 123 (2013).
[Crossref]

M. Ploschner, T. Cizmár, M. Mazilu, A. Di Falco, and K. Dholakia, “Bidirectional optical sorting of gold nanoparticles,” Nano Lett. 12, 1923 (2012).
[Crossref] [PubMed]

M. Ploschner, M. Mazilu, T. Cizmár, and K. Dholakia, “Numerical investigation of passive optical sorting of plasmon nanoparticles,” Opt. Express 19, 13922 (2011).
[Crossref] [PubMed]

P. Jákl, T. Cizmár, M. Šerý, and P. Zemánek, “Static optical sorting in a laser interference field,” Appl. Phys. Lett. 92, 161110 (2008).
[Crossref]

T. Cizmár, V. Garcés-chávez, K. Dholakia, and P. Zemánek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86, 174101 (2005).
[Crossref]

Curry, J. J.

Z. H. Levine and J. J. Curry, “Scattering and Gradient Forces from the Electromagnetic Stress Tensor Acting on a Dielectric Sphere,” Mathematica J. (submitted) (2016).

Z. H. Levine and J. J. Curry, (to be submitted for publication).

Dasgupta, R.

S. Ahlawat, R. Dasgupta, R. S. Verma, V. N. Kumar, and P. K. Gupta, “Optical sorting in holographic trap arrays by tuning the inter-trap separation,” J. Opt. 14, 125501 (2012).
[Crossref]

Dean, D. S.

D. S. Dean and G. Oshanin, “Approach to asymptotically diffusive behavior for Brownian particles in periodic potentials: Extracting information from transients,” Phys. Rev. E 90, 022112 (2014).
[Crossref]

Debye, P.

P. Debye, “Der Lichtdruck auf Kugeln von belibigem Material, Ann. der Phyzik IV 335, 57 (1909).

Derrida, B.

B. Derrida, “Velocity and Diffusion Constant of a Periodic One-Dimensional Hoping Model,” J. Stat. Phys. 31, 433 (1983).
[Crossref]

Dewar, N.

D. McGloin, D. R. Burnham, M. D. Summers, D. Rudd, N. Dewar, and S. Anand, “Optical manipulation of airborne particles: techniques and applications,” Faraday Discuss. 137, 335 (2008).
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M. Ploschner, T. Cizmár, M. Mazilu, A. Di Falco, and K. Dholakia, “Bidirectional optical sorting of gold nanoparticles,” Nano Lett. 12, 1923 (2012).
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R. F. Marchington, M. Mazilu, S. Kuriakose, V. Garcés-Chávez, P. J. Reece, T. F. Krause, M. Gu, and K. Dholakia, “Optical deflection and sorting of microparticles in a near-field optical geometry,” Opt. Express 16, 3712 (2008).
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S. Tatarkova, W. Sibbett, and K. Dholakia, “Brownian particle in an optical potential of the washboard type,” Phys. Rev. Lett. 91, 038101 (2003).
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M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421 (2003).
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M. Ploschner, T. Cizmár, M. Mazilu, A. Di Falco, and K. Dholakia, “Bidirectional optical sorting of gold nanoparticles,” Nano Lett. 12, 1923 (2012).
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N. A. Fuchs, The Mechanics of Aerosols, (Dover, 1989) p.27.

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K. Xiao and D. G. Grier, “Sorting colloidal particles into multiple channels with optical forces: prismatic optical fractionation,” Phys. Rev. E 82, 051407 (2010).
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S. Ahlawat, R. Dasgupta, R. S. Verma, V. N. Kumar, and P. K. Gupta, “Optical sorting in holographic trap arrays by tuning the inter-trap separation,” J. Opt. 14, 125501 (2012).
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R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Colloquium: Momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys. 79, 1197 (2007).
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N. J. Alvarez, C. Jeppesen, K. Yvind, N. A. Mortensen, and O. Hassager, “The chromatic separation of particles using optical electric fields,” Lab Chip 13, 928 (2013).
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O. Brzobohatý, M. Šiler, J. Jezek, P. Jákl, and P. Zemánek, “Optical manipulation of aerosol droplets using a holographic dual and single beam trap,” Optics Lett. 38, 4601 (2013).
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K. Ladavac, K. Kasza, and D. G. Grier, “Sorting mesoscopic objects with periodic potential landscapes: optical fractionation,” Phys. Rev. E 70, 010901(R) (2004).
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K. H. Lee, S. B. Kim, K. S. Lee, and H. J. Sung, “Enhancement by optical force of separation in pinched flow fractionation,” Lab Chip 11354 (2011).
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P. Zemánek, A. Jonáš, P. Jákl, J. Jezek, M. Šerý, and M. Liška, “Theoretical comparison of optical traps created by standing wave and single beam,” Opt. Commun. 220, 401 (2003).
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Mathai, P. P.

P. P. Mathai, J. A. Liddle, and S. M. Stavis, “Optical tracking of nanoscale particles in microscale environments,” Appl. Phys. Rev. 3, 011105 (2016).
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T. T. Perkins, “Optical traps for single molecule biophysics: a primer,” Laser Photon. Rev. 3, 203 (2009).
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M. Ploschner, T. Cizmár, M. Mazilu, A. Di Falco, and K. Dholakia, “Bidirectional optical sorting of gold nanoparticles,” Nano Lett. 12, 1923 (2012).
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M. Ploschner, M. Mazilu, T. Cizmár, and K. Dholakia, “Numerical investigation of passive optical sorting of plasmon nanoparticles,” Opt. Express 19, 13922 (2011).
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M. Horstmann, K. Probst, and C. Fallnich, “An integrated fiber-based optical trap for single airborne particles,” Appl. Phys. B 103, 35 (2011).
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Reid, J. P.

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R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Colloquium: Momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys. 79, 1197 (2007).
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D. McGloin, D. R. Burnham, M. D. Summers, D. Rudd, N. Dewar, and S. Anand, “Optical manipulation of airborne particles: techniques and applications,” Faraday Discuss. 137, 335 (2008).
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P. Jákl, T. Cizmár, M. Šerý, and P. Zemánek, “Static optical sorting in a laser interference field,” Appl. Phys. Lett. 92, 161110 (2008).
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P. Zemánek, A. Jonáš, P. Jákl, J. Jezek, M. Šerý, and M. Liška, “Theoretical comparison of optical traps created by standing wave and single beam,” Opt. Commun. 220, 401 (2003).
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Shahvisi, A.

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S. Tatarkova, W. Sibbett, and K. Dholakia, “Brownian particle in an optical potential of the washboard type,” Phys. Rev. Lett. 91, 038101 (2003).
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P. Jákl, A. V. Arzola, M. Šiler, L. Chvátal, K. Volke-Sepúlveda, and P. Zemánek, “Optical sorting of nonspherical and living microobjects in moving interference structures,” Opt. Express 22, 29746 (2014).
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O. Brzobohatý, M. Šiler, J. Jezek, P. Jákl, and P. Zemánek, “Optical manipulation of aerosol droplets using a holographic dual and single beam trap,” Optics Lett. 38, 4601 (2013).
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O. Brzobohatý, V. Karásek, M. Šiler, L. Chvátal, T. Cizmár, and P. Zemánek, “Experimental demonstration of optical transport, sorting, and self-arrangement using a ‘tractor beam’,” Nat. Phot. 7, 123 (2013).
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R. L. Smith, G. C. Spalding, K. Dholakia, and M. P. MacDonald, “Colloidal sorting in dynamic optical lattices,” J. Opt. A: Pure and Appl. Opt. 9, S134 (2007).
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R. L. Smith, G. C. Spalding, K. Dholakia, and M. P. MacDonald, “Colloidal sorting in dynamic optical lattices,” J. Opt. A: Pure and Appl. Opt. 9, S134 (2007).
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M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421 (2003).
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P. P. Mathai, J. A. Liddle, and S. M. Stavis, “Optical tracking of nanoscale particles in microscale environments,” Appl. Phys. Rev. 3, 011105 (2016).
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M. Guillon, O. Moine, and B. Stout, “Longitudinal optical binding of high optical contrast microdroplets in air,” Phys. Rev. Lett. 96, 143902 (2006).
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Summers, M. D.

D. McGloin, D. R. Burnham, M. D. Summers, D. Rudd, N. Dewar, and S. Anand, “Optical manipulation of airborne particles: techniques and applications,” Faraday Discuss. 137, 335 (2008).
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S. Tatarkova, W. Sibbett, and K. Dholakia, “Brownian particle in an optical potential of the washboard type,” Phys. Rev. Lett. 91, 038101 (2003).
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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, 128301 (2002).
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K. Xiao and D. G. Grier, “Sorting colloidal particles into multiple channels with optical forces: prismatic optical fractionation,” Phys. Rev. E 82, 051407 (2010).
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K. Xiao and D. G. Grier, “Multidimensional Optical Fractionation of Colloidal Particles with Holographic Verification,” Phys. Rev. Lett. 104, 028302 (2010).
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Yvind, K.

N. J. Alvarez, C. Jeppesen, K. Yvind, N. A. Mortensen, and O. Hassager, “The chromatic separation of particles using optical electric fields,” Lab Chip 13, 928 (2013).
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O. Brzobohatý, V. Karásek, M. Šiler, L. Chvátal, T. Cizmár, and P. Zemánek, “Experimental demonstration of optical transport, sorting, and self-arrangement using a ‘tractor beam’,” Nat. Phot. 7, 123 (2013).
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O. Brzobohatý, M. Šiler, J. Jezek, P. Jákl, and P. Zemánek, “Optical manipulation of aerosol droplets using a holographic dual and single beam trap,” Optics Lett. 38, 4601 (2013).
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P. Jákl, T. Cizmár, M. Šerý, and P. Zemánek, “Static optical sorting in a laser interference field,” Appl. Phys. Lett. 92, 161110 (2008).
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A. Jonáš and P. Zemánek, “Light at work: The use of optical forces for particle manipulation, sorting, and analysis,” Electrophoresis 29, 4813 (2008).
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T. Cizmár, V. Garcés-chávez, K. Dholakia, and P. Zemánek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86, 174101 (2005).
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P. Zemánek, A. Jonáš, P. Jákl, J. Jezek, M. Šerý, and M. Liška, “Theoretical comparison of optical traps created by standing wave and single beam,” Opt. Commun. 220, 401 (2003).
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Appl. Phys. B (1)

M. Horstmann, K. Probst, and C. Fallnich, “An integrated fiber-based optical trap for single airborne particles,” Appl. Phys. B 103, 35 (2011).
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Appl. Phys. Lett. (3)

T. Cizmár, V. Garcés-chávez, K. Dholakia, and P. Zemánek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86, 174101 (2005).
[Crossref]

P. Jákl, T. Cizmár, M. Šerý, and P. Zemánek, “Static optical sorting in a laser interference field,” Appl. Phys. Lett. 92, 161110 (2008).
[Crossref]

K. Taji, M. Tachikawa, and K. Nagashima, “Laser trapping of ice crystals,” Appl. Phys. Lett. 88, 141111 (2006).
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Appl. Phys. Rev. (1)

P. P. Mathai, J. A. Liddle, and S. M. Stavis, “Optical tracking of nanoscale particles in microscale environments,” Appl. Phys. Rev. 3, 011105 (2016).
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Electrophoresis (1)

A. Jonáš and P. Zemánek, “Light at work: The use of optical forces for particle manipulation, sorting, and analysis,” Electrophoresis 29, 4813 (2008).
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Faraday Discuss. (1)

D. McGloin, D. R. Burnham, M. D. Summers, D. Rudd, N. Dewar, and S. Anand, “Optical manipulation of airborne particles: techniques and applications,” Faraday Discuss. 137, 335 (2008).
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J. Opt. (1)

S. Ahlawat, R. Dasgupta, R. S. Verma, V. N. Kumar, and P. K. Gupta, “Optical sorting in holographic trap arrays by tuning the inter-trap separation,” J. Opt. 14, 125501 (2012).
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J. Opt. A: Pure and Appl. Opt. (1)

R. L. Smith, G. C. Spalding, K. Dholakia, and M. P. MacDonald, “Colloidal sorting in dynamic optical lattices,” J. Opt. A: Pure and Appl. Opt. 9, S134 (2007).
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K. H. Lee, S. B. Kim, K. S. Lee, and H. J. Sung, “Enhancement by optical force of separation in pinched flow fractionation,” Lab Chip 11354 (2011).
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N. J. Alvarez, C. Jeppesen, K. Yvind, N. A. Mortensen, and O. Hassager, “The chromatic separation of particles using optical electric fields,” Lab Chip 13, 928 (2013).
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Laser Photon. Rev. (1)

T. T. Perkins, “Optical traps for single molecule biophysics: a primer,” Laser Photon. Rev. 3, 203 (2009).
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Nano Lett. (2)

M. Ploschner, T. Cizmár, M. Mazilu, A. Di Falco, and K. Dholakia, “Bidirectional optical sorting of gold nanoparticles,” Nano Lett. 12, 1923 (2012).
[Crossref] [PubMed]

Y. Pang and R. Gordon, “Optical trapping of 12 nm dielectric spheres using double nanoholes in a gold film,” Nano Lett. 11, 3763 (2011).
[Crossref] [PubMed]

Nat. Phot. (1)

O. Brzobohatý, V. Karásek, M. Šiler, L. Chvátal, T. Cizmár, and P. Zemánek, “Experimental demonstration of optical transport, sorting, and self-arrangement using a ‘tractor beam’,” Nat. Phot. 7, 123 (2013).
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Nature (1)

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421 (2003).
[Crossref] [PubMed]

Opt. Commun. (2)

P. Zemánek, A. Jonáš, P. Jákl, J. Jezek, M. Šerý, and M. Liška, “Theoretical comparison of optical traps created by standing wave and single beam,” Opt. Commun. 220, 401 (2003).
[Crossref]

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529 (1996).
[Crossref]

Opt. Express (6)

Opt. Lett. (1)

Optics Express (1)

A. E. Carruthers, J. P. Reid, and A. J. Orr-Ewing, “Longitudinal optical trapping and sizing of aerosol droplets,” Optics Express 18, 14238 (2010).
[Crossref] [PubMed]

Optics Lett. (1)

O. Brzobohatý, M. Šiler, J. Jezek, P. Jákl, and P. Zemánek, “Optical manipulation of aerosol droplets using a holographic dual and single beam trap,” Optics Lett. 38, 4601 (2013).
[Crossref]

Phys. Rev. E (3)

K. Xiao and D. G. Grier, “Sorting colloidal particles into multiple channels with optical forces: prismatic optical fractionation,” Phys. Rev. E 82, 051407 (2010).
[Crossref]

K. Ladavac, K. Kasza, and D. G. Grier, “Sorting mesoscopic objects with periodic potential landscapes: optical fractionation,” Phys. Rev. E 70, 010901(R) (2004).
[Crossref]

D. S. Dean and G. Oshanin, “Approach to asymptotically diffusive behavior for Brownian particles in periodic potentials: Extracting information from transients,” Phys. Rev. E 90, 022112 (2014).
[Crossref]

Phys. Rev. Lett. (5)

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, 128301 (2002).
[Crossref] [PubMed]

S. Tatarkova, W. Sibbett, and K. Dholakia, “Brownian particle in an optical potential of the washboard type,” Phys. Rev. Lett. 91, 038101 (2003).
[Crossref] [PubMed]

M. Guillon, O. Moine, and B. Stout, “Longitudinal optical binding of high optical contrast microdroplets in air,” Phys. Rev. Lett. 96, 143902 (2006).
[Crossref] [PubMed]

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

K. Xiao and D. G. Grier, “Multidimensional Optical Fractionation of Colloidal Particles with Holographic Verification,” Phys. Rev. Lett. 104, 028302 (2010).
[Crossref] [PubMed]

Physica (1)

H. A. Kramers, “Brownian motion in a field of force and the diffusion model of chemical reactions,” Physica 7, 284 (1940).
[Crossref]

Rev. Mod. Phys. (2)

S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Rev. Mod. Phys. 15, 1 (1943).
[Crossref]

R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Colloquium: Momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys. 79, 1197 (2007).
[Crossref]

Rev. Sci. Instrum. (1)

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787 (2004).
[Crossref]

Other (7)

Nicholas A. Krall and Alvin W. Trivelpiece, Principles of Plasma Physics, (San Francisco Press, Inc., 1986) p. 295.

Z. H. Levine and J. J. Curry, “Scattering and Gradient Forces from the Electromagnetic Stress Tensor Acting on a Dielectric Sphere,” Mathematica J. (submitted) (2016).

A. Zangwill, Modern Electrodynamics, (Cambridge University, 2013) pp. 787–788.

N. A. Fuchs, The Mechanics of Aerosols, (Dover, 1989) p.27.

G. J. Borse, Numerical Methods with MATLAB, (PWS Publishing, 1997).

Z. H. Levine and J. J. Curry, (to be submitted for publication).

P. Morse, Thermal Physics2nd ed. (Benjamin, 1969) pp. 228–236.

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Figures (12)

Fig. 1
Fig. 1 The scheme for sorting particles by size consists of a narrow stream of aerosol particles flowing across a Gaussian mode standing wave at an oblique angle. The figure is not to scale; in particular, the optical fringes and particles are much finer than shown.
Fig. 2
Fig. 2 Comparison of different expressions for the optical force versus a/λ0 for λ0 = 1064 nm, np = 1.57, and k0z = π/4. The force is normalized by the factor I0πa2, so that the plotted functions give the force per unit of optical power incident on the sphere. The dashed black line is the Rayleigh expression Eq. (8). The blue curve is the 5-term expression of Eq. (7). The red curve is the actual force obtained from high-order numerical evaluation of Eq. (3).
Fig. 3
Fig. 3 The mobility for a spherical particle in air at standard temperature and pressure according to Fuchs [40] compared to the g → 0 limit.
Fig. 4
Fig. 4 The solution of the Fokker-Planck equation for a = 100 nm, Λ = 532 nm, I0 = 0.4 GW/m2, and vd = 707 μm/s. The solutions occur starting from the initial condition in blue, then in four steps of 64 μs represented sequentially by green, red, gold, and black curves.
Fig. 5
Fig. 5 Time evolution for the velocity in z, vobs, which is the average over 1/16 of the total time range, of a spatial distribution of particles of radius a = 100 nm in a standing wave with I0 = 0.4 GW/m2 and a flow velocity v0 = 0.707 mm/s. A second average is taken in the shaded interval to determine veff, the large-time limit of vobs.
Fig. 6
Fig. 6 Normalized effective velocity across interference fringes versus optical intensity I0 for different particle radii in nm (shown next to each curve).
Fig. 7
Fig. 7 Mean trajectories for particles with radii of a=40, 50, 60, 70, 80, 90, 100, and 110 nm propagating at a velocity of 1 mm/s through a standing wave with w0 = 500 μm and I0 = 2 GW/m2. The 1/e2 full width of the fundamental Gaussian mode optical field is shown in green.
Fig. 8
Fig. 8 Spatial distributions in the output plane (x = 1.5 mm) for each particle size.
Fig. 9
Fig. 9 Distribution width in the output plane versus particle size with and without an optical field.
Fig. 10
Fig. 10 Deflection of particles as a function of radius for several values of intensity I0 in GW/m2. All values of z1 are negative or zero.
Fig. 11
Fig. 11 Resolving power versus particle radius for I0 = 0.4 GW/m2. In general, the size distribution is bimodal because two particle sizes, one smaller than the zero-force size (a ≈ 274 nm) and one larger, will be deflected to the same position. The blue curve is obtained using only one mode of the distribution. The green curve includes both. The inset shows the probability that a particle observed at z1 = −29 μm has radius a, assuming a uniform distribution of input radii.
Fig. 12
Fig. 12 The escape rate from the well of a standing wave versus intensity I0. Kramers’ analytic formula valid for small escape rates is shown in blue. The escape rate found numerically from the effective fluid constant model is in red. The ratio of the two is in the inset for the same range of intensities.

Tables (4)

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Table 1 Exact values of the ciℓ coefficients in Eq. (7)

Tables Icon

Table 2 Parameters used for converged solution of Eq. (29). Small a means 40 nm ≤ a ≤ 110 nm; medium a means 120 nm ≤ a ≤ 170 nm; large a means 180 nm ≤ a ≤ 300 nm

Tables Icon

Table 3 Parameter values used in simulations

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Table 4 Ratios of results from the Monte Carlo model divided by results from the effective constants of motion model for I0 = 2.0 GW/m2

Equations (49)

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F = T ¯ d V 1 c 2 S t d V
T ¯ ( ε 0 EE + 1 μ 0 BB ) 1 2 ( ε 0 | E | 2 + 1 μ 0 | B | 2 ) I ¯
F = T ¯ d A ,
E ( x , y , z , t ) = x ^ E 0 ± cos ( ω t ± k 0 z ) ,
I ( z ) = 2 I 0 ( 1 + cos 2 k 0 z ) ,
I 0 = ε 0 μ 0 | E 0 | 2 2
F z ^ 4 π 105 c k 0 2 I 0 sin ( 2 k 0 z ) = 1 5 [ c 1 Im ( a b ) + c 2 Im ( a 1 a * b 1 b * ) + c 3 Im ( a b * ) ]
F z = 8 π k 0 a 3 I 0 c n p 2 1 n p 2 + 2 sin ( 2 k 0 z ) ,
F z = F z ( 0 ) I 0 sin 2 k 0 z e 2 ( x 2 + y 2 ) w 0 2 ,
z R = π w 0 2 λ 0 .
f ( r , v , t | r i , v i , t i ) d 3 r d 3 v
d f d t = v f + v F ( r , t ) m p f + ( d f d t ) c ,
( d f d t ) c = v ( C f ) D v 2 v 2 f ,
C = v v 0 μ m p D v = 2 k B T μ m p 2 ,
f = [ m p 2 π k B T ( 1 e t / τ ) ] 3 / 2 × exp ( m p | v v 0 ( v i v 0 ) e t / 2 τ | 2 2 k B T ( 1 e t / τ ) ) ,
τ μ m p 2
D = μ k B T
μ = 1 + A g a + Q g a e b a / g 6 π η a
F d = 6 π η a ( v v 0 ) .
d v p d t = F ( r , t ) m p + ( d v p d t ) c
d r p d t = v p
r p = + r f d 3 v d 3 r v p = + v f d 3 v d 3 r
( d v p d t ) c = Δ v Δ t .
f ( z , t ) t = D 2 f ( z , t ) z 2 μ z [ F ( z ) f ( z , t ) ] .
f ( z , t ) = 1 N Λ n = c n ( t ) e i n k z ,
1 = N Λ / 2 N Λ / 2 d z f ( z , t ) ,
f ( z , t ) = 1 N Λ ( 1 + 2 Re n = 1 c n ( t ) e i n k z ) .
F ( z ) = F 0 + F 1 sin ( 2 π z / Λ ) ,
d d t c n ( t ) = D n 2 k 2 c n ( t ) + i v 0 n k c n ( t ) μ 2 F 1 n k c n N ( t ) + μ 2 F 1 n k c n + N ( t ) .
z 1 = t 0 t 1 d t { v z , eff [ x ( t ) ] v z , 0 } = 1 v x x 0 x 1 d x { v z , eff ( x ) v z , 0 }
[ σ z ( D ) ] 2 = 2 v x x 0 x 1 d x D eff ( x ) .
σ z ( v ) = v z , eff σ t ,
σ z 2 = [ σ z ( D ) ] 2 + [ σ z ( v ) ] 2 .
σ y 2 = 2 D x 1 x 0 v x .
R a σ a ,
P s = IN σ .
P s = N σ 2 π w 0 2 P c
N 2 π w 0 2 1 R σ .
Γ π 2 w 0 1 R σ V c .
t 1 = 0 Λ d z 1 v ( z ) = μ 1 0 Λ d z [ F 0 + F 1 sin ( 2 π z Λ ) ] 1 = μ 1 Λ ( F 0 2 F 1 2 ) 1 / 2 , | F 1 | < | F 0 | .
v eff Λ t 1 = μ ( F 0 2 F 1 2 ) 1 / 2 = v 0 ( 1 F 1 2 F 0 2 ) 1 / 2 .
1 τ = μ M ω A ω C 2 π e Δ U / k T
U ( z ) M ω A 2 2 ( z z A ) 2
U ( z ) Δ U M ω C 2 2 ( z z C ) 2
U ( z ) = 4 π n 2 a 3 I 0 c n p 2 1 n p 2 + 2 cos ( 2 π z Λ ) .
Δ U = 8 π n 2 a 3 I 0 c n p 2 1 n p 2 + 2 .
M ω A 2 2 = M ω C 2 2 = M ω A ω C 2 = 8 π 3 n 2 a 3 I 0 c Λ 2 ε p ε m ε p + 2 ε m .
D eff = Λ 2 2 ( k + ( hop ) + k ( hop ) )
v eff = Λ ( k + ( hop ) k ( hop ) ) .

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