Abstract

We present simultaneous separation of polydisperse particles driven by an optical gradient force in the absence of microfluidic flow. The separation mechanism involves particle-size dependence of the potential landscape generated by a one-dimensional asymmetric optical stripe pattern. The outcome is that the particles align in different stacks according to their sizes. The dynamics of Brownian particles inside the optical potential landscapes are investigated theoretically and experimentally for various optical intensities and particle sizes. By introducing sequential changes in the optical profile, we also show that this technique allows semipassive arrangement of particles in arbitrary configurations.

© 2009 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  32. B. Sun, Y. Roichman, and D. G. Grier, “Theory of holographic optical trapping,” Opt. Express 16, 15765-15776 (2008).
    [CrossRef] [PubMed]
  33. R. L. Eriksen, V. R. Daria, and J. Glückstad, “Fully dynamic multiple-beam optical tweezers,” Opt. Express 10, 597-602(2002).
    [PubMed]
  34. C. A. Alonzo, P. J. Rodrigo, and J. Glückstad, “Photon-efficient grey level image projection by the generalized phase contrast method,” New J. Phys. 9, 132 (2007).
    [CrossRef]

2008

2007

G. Milne, D. Rhodes, M. MacDonald, and K. Dholakia, “Fractionation of polydisperse colloid with acousto-optically generated potential energy landscapes,” Opt. Lett. 32, 1144-1146(2007).
[CrossRef] [PubMed]

C. A. Alonzo, P. J. Rodrigo, and J. Glückstad, “Photon-efficient grey level image projection by the generalized phase contrast method,” New J. Phys. 9, 132 (2007).
[CrossRef]

Y. Y. Sun, X.-C. Yuan, L. S. Ong, J. Bu, S. W. Zhu, and R. Liu, “Large-scale optical traps on a chip for optical sorting,” Appl. Phys. Lett. 90, 031107 (2007).
[CrossRef]

2006

I. Ricárdez-Vargas, P. Rodríguez-Montero, R. Ramos-García, and K. Volke-Sepúlveda, “Modulated optical sieve for sorting of polydisperse microparticles,” Appl. Phys. Lett. 88, 121116 (2006).
[CrossRef]

T. Čižmár, M. Šiler, M. Šerý, P. Zemánek, V. Garcés-Chávez, and K. Dholakia, “Optical sorting and detection of submicrometer objects in a motional standing wave,” Phys. Rev. B 74, 035105 (2006).
[CrossRef]

2005

L. Paterson, E. Papagiakoumou, G. Milne, V. Garcés-Chávez, S. A. Tatarkova, W. Sibbet, F. J. Gunn-Moore, P. E. Bryant, A. C. Riches, and K. Dholakia, “Light-induced cell separation in a tailored optical landscape,” Appl. Phys. Lett. 87, 123901(2005).
[CrossRef]

A. M. Lacasta, J. M. Sancho, A. H. Romero, and K. Lindenberg, “Sorting on periodic surfaces,” Phys. Rev. Lett. 94, 160601(2005).
[CrossRef] [PubMed]

2004

2003

R. L. Eriksen, P. J. Rodrigo, V. R. Daria, and J. Glückstad, “Spatial light modulator-controlled alignment and spinning of birefringent particles optically trapped in an array,” Appl. Opt. 42, 5107-5111 (2003).
[CrossRef] [PubMed]

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

V. R. Daria, R. L. Eriksen, and J. Glückstad, “Dynamic optical manipulation of colloidal systems using a spatial light modulator,” J. Mod. Opt. 50, 1601-1614 (2003).

S. J. Hart and A. V. Terray, “Refractive-index-driven separation of colloidal polymer particles using optical chromatography,” Appl. Phys. Lett. 83, 5316-5318 (2003).
[CrossRef]

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421-424(2003).
[CrossRef] [PubMed]

2002

2001

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbet, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912-914 (2001).
[CrossRef] [PubMed]

2000

P. A. Maia Neto and H. M. Nussenzveig, “Theory of optical tweezers,” Europhys. Lett. 50, 702-708 (2000).
[CrossRef]

1994

L. P. Faucheux and A. J. Libchaber, “Confined Brownian motion,” Phys. Rev. E 49, 5158-5163 (1994).
[CrossRef]

1992

1989

M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, “Optical binding,” Phys. Rev. Lett. 63, 1233-1236 (1989).
[CrossRef] [PubMed]

1970

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

1940

H. A. Kramers, “Brownian motion in a field of force and the diffusion model of chemical reactions,” Physica 7, 284-304 (1940).
[CrossRef]

Alonzo, C. A.

C. A. Alonzo, P. J. Rodrigo, and J. Glückstad, “Photon-efficient grey level image projection by the generalized phase contrast method,” New J. Phys. 9, 132 (2007).
[CrossRef]

Arlt, J.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbet, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Ashihara, S.

Y. Hayashi, S. Ashihara, T. Shimura, and K. Kuroda, “Particle sorting using optically induced asymmetric double-well potential,” Opt. Commun. 281, 3792-3798 (2008).
[CrossRef]

Ashkin, A.

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

Brenner, H.

J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics (Prentice-Hall, 1965).

Brevik, I.

Bryant, P. E.

L. Paterson, E. Papagiakoumou, G. Milne, V. Garcés-Chávez, S. A. Tatarkova, W. Sibbet, F. J. Gunn-Moore, P. E. Bryant, A. C. Riches, and K. Dholakia, “Light-induced cell separation in a tailored optical landscape,” Appl. Phys. Lett. 87, 123901(2005).
[CrossRef]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbet, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Bu, J.

Y. Y. Sun, X.-C. Yuan, L. S. Ong, J. Bu, S. W. Zhu, and R. Liu, “Large-scale optical traps on a chip for optical sorting,” Appl. Phys. Lett. 90, 031107 (2007).
[CrossRef]

Burns, M. M.

M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, “Optical binding,” Phys. Rev. Lett. 63, 1233-1236 (1989).
[CrossRef] [PubMed]

Chávez-Cerda, S.

Cižmár, T.

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

T. Čižmár, M. Šiler, M. Šerý, P. Zemánek, V. Garcés-Chávez, and K. Dholakia, “Optical sorting and detection of submicrometer objects in a motional standing wave,” Phys. Rev. B 74, 035105 (2006).
[CrossRef]

Courtial, J.

Curtis, J. E.

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

Daria, V. R.

Dholakia, K.

R. F. Marchington, M. Mazilu, S. Kuriakose, V. Garcés-Chávez, P. J. Reece, T. F. Krauss, M. Gu, and K. Dholakia, “Optical deflection and sorting of microparticles in a near-field optical geometry,” Opt. Express 16, 3712-3726 (2008).
[CrossRef] [PubMed]

G. Milne, D. Rhodes, M. MacDonald, and K. Dholakia, “Fractionation of polydisperse colloid with acousto-optically generated potential energy landscapes,” Opt. Lett. 32, 1144-1146(2007).
[CrossRef] [PubMed]

T. Čižmár, M. Šiler, M. Šerý, P. Zemánek, V. Garcés-Chávez, and K. Dholakia, “Optical sorting and detection of submicrometer objects in a motional standing wave,” Phys. Rev. B 74, 035105 (2006).
[CrossRef]

L. Paterson, E. Papagiakoumou, G. Milne, V. Garcés-Chávez, S. A. Tatarkova, W. Sibbet, F. J. Gunn-Moore, P. E. Bryant, A. C. Riches, and K. Dholakia, “Light-induced cell separation in a tailored optical landscape,” Appl. Phys. Lett. 87, 123901(2005).
[CrossRef]

K. Volke-Sepúlveda, S. Chávez-Cerda, V. Garcés-Chávez, and K. Dholakia, “Three-dimensional optical forces and transfer of orbital angular momentum from multiringed light beams to spherical microparticles,” J. Opt. Soc. Am. B 21, 1749-1757(2004).
[CrossRef]

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421-424(2003).
[CrossRef] [PubMed]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbet, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Eriksen, R. L.

Faucheux, L. P.

L. P. Faucheux and A. J. Libchaber, “Confined Brownian motion,” Phys. Rev. E 49, 5158-5163 (1994).
[CrossRef]

Fournier, J.-M.

M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, “Optical binding,” Phys. Rev. Lett. 63, 1233-1236 (1989).
[CrossRef] [PubMed]

Garcés-Chávez, V.

R. F. Marchington, M. Mazilu, S. Kuriakose, V. Garcés-Chávez, P. J. Reece, T. F. Krauss, M. Gu, and K. Dholakia, “Optical deflection and sorting of microparticles in a near-field optical geometry,” Opt. Express 16, 3712-3726 (2008).
[CrossRef] [PubMed]

T. Čižmár, M. Šiler, M. Šerý, P. Zemánek, V. Garcés-Chávez, and K. Dholakia, “Optical sorting and detection of submicrometer objects in a motional standing wave,” Phys. Rev. B 74, 035105 (2006).
[CrossRef]

L. Paterson, E. Papagiakoumou, G. Milne, V. Garcés-Chávez, S. A. Tatarkova, W. Sibbet, F. J. Gunn-Moore, P. E. Bryant, A. C. Riches, and K. Dholakia, “Light-induced cell separation in a tailored optical landscape,” Appl. Phys. Lett. 87, 123901(2005).
[CrossRef]

K. Volke-Sepúlveda, S. Chávez-Cerda, V. Garcés-Chávez, and K. Dholakia, “Three-dimensional optical forces and transfer of orbital angular momentum from multiringed light beams to spherical microparticles,” J. Opt. Soc. Am. B 21, 1749-1757(2004).
[CrossRef]

Gardiner, C. W.

C. W. Gardiner, Handbook of Stochastic Methods, 3rd ed. (Springer, 2004).

Gibson, G.

Glückstad, J.

C. A. Alonzo, P. J. Rodrigo, and J. Glückstad, “Photon-efficient grey level image projection by the generalized phase contrast method,” New J. Phys. 9, 132 (2007).
[CrossRef]

R. L. Eriksen, P. J. Rodrigo, V. R. Daria, and J. Glückstad, “Spatial light modulator-controlled alignment and spinning of birefringent particles optically trapped in an array,” Appl. Opt. 42, 5107-5111 (2003).
[CrossRef] [PubMed]

V. R. Daria, R. L. Eriksen, and J. Glückstad, “Dynamic optical manipulation of colloidal systems using a spatial light modulator,” J. Mod. Opt. 50, 1601-1614 (2003).

R. L. Eriksen, V. R. Daria, and J. Glückstad, “Fully dynamic multiple-beam optical tweezers,” Opt. Express 10, 597-602(2002).
[PubMed]

Golovchenko, J. A.

M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, “Optical binding,” Phys. Rev. Lett. 63, 1233-1236 (1989).
[CrossRef] [PubMed]

Grier, D. G.

B. Sun, Y. Roichman, and D. G. Grier, “Theory of holographic optical trapping,” Opt. Express 16, 15765-15776 (2008).
[CrossRef] [PubMed]

K. Ladavac and D. G. Grier, “Microoptomechanical pumps assembled and driven by holographic optical vortex arrays,” Opt. Express 12, 1144-1149 (2004).
[CrossRef] [PubMed]

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

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

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

Gu, M.

Gunn-Moore, F. J.

L. Paterson, E. Papagiakoumou, G. Milne, V. Garcés-Chávez, S. A. Tatarkova, W. Sibbet, F. J. Gunn-Moore, P. E. Bryant, A. C. Riches, and K. Dholakia, “Light-induced cell separation in a tailored optical landscape,” Appl. Phys. Lett. 87, 123901(2005).
[CrossRef]

Gussgard, R.

Happel, J.

J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics (Prentice-Hall, 1965).

Hart, S. J.

S. J. Hart and A. V. Terray, “Refractive-index-driven separation of colloidal polymer particles using optical chromatography,” Appl. Phys. Lett. 83, 5316-5318 (2003).
[CrossRef]

Hayashi, Y.

Y. Hayashi, S. Ashihara, T. Shimura, and K. Kuroda, “Particle sorting using optically induced asymmetric double-well potential,” Opt. Commun. 281, 3792-3798 (2008).
[CrossRef]

Jákl, P.

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

Jonáš, A.

Jordan, P.

Kasza, K.

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

Ketterson, J. B.

Koss, B. A.

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

Kramers, H. A.

H. A. Kramers, “Brownian motion in a field of force and the diffusion model of chemical reactions,” Physica 7, 284-304 (1940).
[CrossRef]

Krauss, T. F.

Kuriakose, S.

Kuroda, K.

Y. Hayashi, S. Ashihara, T. Shimura, and K. Kuroda, “Particle sorting using optically induced asymmetric double-well potential,” Opt. Commun. 281, 3792-3798 (2008).
[CrossRef]

Lacasta, A. M.

A. M. Lacasta, J. M. Sancho, A. H. Romero, and K. Lindenberg, “Sorting on periodic surfaces,” Phys. Rev. Lett. 94, 160601(2005).
[CrossRef] [PubMed]

Laczik, Z. J.

Ladavac, K.

K. Ladavac and D. G. Grier, “Microoptomechanical pumps assembled and driven by holographic optical vortex arrays,” Opt. Express 12, 1144-1149 (2004).
[CrossRef] [PubMed]

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

Leach, J.

Li, Z.

Libchaber, A. J.

L. P. Faucheux and A. J. Libchaber, “Confined Brownian motion,” Phys. Rev. E 49, 5158-5163 (1994).
[CrossRef]

Lindenberg, K.

A. M. Lacasta, J. M. Sancho, A. H. Romero, and K. Lindenberg, “Sorting on periodic surfaces,” Phys. Rev. Lett. 94, 160601(2005).
[CrossRef] [PubMed]

Lindmo, T.

Liška, M.

Liu, R.

Y. Y. Sun, X.-C. Yuan, L. S. Ong, J. Bu, S. W. Zhu, and R. Liu, “Large-scale optical traps on a chip for optical sorting,” Appl. Phys. Lett. 90, 031107 (2007).
[CrossRef]

Luan, L.

MacDonald, M.

MacDonald, M. P.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421-424(2003).
[CrossRef] [PubMed]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbet, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Maia Neto, P. A.

P. A. Maia Neto and H. M. Nussenzveig, “Theory of optical tweezers,” Europhys. Lett. 50, 702-708 (2000).
[CrossRef]

Marchington, R. F.

Mazilu, M.

Milne, G.

G. Milne, D. Rhodes, M. MacDonald, and K. Dholakia, “Fractionation of polydisperse colloid with acousto-optically generated potential energy landscapes,” Opt. Lett. 32, 1144-1146(2007).
[CrossRef] [PubMed]

L. Paterson, E. Papagiakoumou, G. Milne, V. Garcés-Chávez, S. A. Tatarkova, W. Sibbet, F. J. Gunn-Moore, P. E. Bryant, A. C. Riches, and K. Dholakia, “Light-induced cell separation in a tailored optical landscape,” Appl. Phys. Lett. 87, 123901(2005).
[CrossRef]

Mu, W.

Nussenzveig, H. M.

P. A. Maia Neto and H. M. Nussenzveig, “Theory of optical tweezers,” Europhys. Lett. 50, 702-708 (2000).
[CrossRef]

Ong, L. S.

Y. Y. Sun, X.-C. Yuan, L. S. Ong, J. Bu, S. W. Zhu, and R. Liu, “Large-scale optical traps on a chip for optical sorting,” Appl. Phys. Lett. 90, 031107 (2007).
[CrossRef]

Padgett, M. J.

Papagiakoumou, E.

L. Paterson, E. Papagiakoumou, G. Milne, V. Garcés-Chávez, S. A. Tatarkova, W. Sibbet, F. J. Gunn-Moore, P. E. Bryant, A. C. Riches, and K. Dholakia, “Light-induced cell separation in a tailored optical landscape,” Appl. Phys. Lett. 87, 123901(2005).
[CrossRef]

Paterson, L.

L. Paterson, E. Papagiakoumou, G. Milne, V. Garcés-Chávez, S. A. Tatarkova, W. Sibbet, F. J. Gunn-Moore, P. E. Bryant, A. C. Riches, and K. Dholakia, “Light-induced cell separation in a tailored optical landscape,” Appl. Phys. Lett. 87, 123901(2005).
[CrossRef]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbet, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Ramos-García, R.

I. Ricárdez-Vargas, P. Rodríguez-Montero, R. Ramos-García, and K. Volke-Sepúlveda, “Modulated optical sieve for sorting of polydisperse microparticles,” Appl. Phys. Lett. 88, 121116 (2006).
[CrossRef]

Reece, P. J.

Rhodes, D.

Ricárdez-Vargas, I.

I. Ricárdez-Vargas, P. Rodríguez-Montero, R. Ramos-García, and K. Volke-Sepúlveda, “Modulated optical sieve for sorting of polydisperse microparticles,” Appl. Phys. Lett. 88, 121116 (2006).
[CrossRef]

Riches, A. C.

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Supplementary Material (1)

» Media 1: MOV (8209 KB)     

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Figures (8)

Fig. 1
Fig. 1

(a) Intensity distribution of an asymmetric nonperiodic optical stripe pattern. Peak positions from left to right are indicated as x 1 , x 2 , and x 3 , respectively. (b) Optical potential energy versus transverse position x for spheres of 0.49, 1.03, and 1.61 μm (solid, dashed, and dotted curves, respectively) radii. Values for the 1.03 and 1.61 μm spheres are scaled by 1 / 8 and 1 / 25 , respectively, and offset by 15 and 29 k B T , respectively.

Fig. 2
Fig. 2

Experimental setup. The computer-controlled LCP and the polarizing beam splitter (PBS2) spatially modulate the optical intensity distribution. The telescope system composed of lenses L3, L4, and L5 and a microscope objective images the optical pattern at the LCP on top of the sample chamber with a magnification factor of 1 / 360 . Inset 1, image of the optical pattern at LCP; inset 2, corresponding average cross-sectional intensity distribution along the x axis.

Fig. 3
Fig. 3

Typical particle trajectories of three separate trials for polystyrene spheres of 0.49 (three trials from the left), 1.03 (middle three trials), and 1.61 μm (three trials from the right) radii. Each trial was monitored for 10 s with a sampling interval of 33 ms . The tics on the bottom axis represent t = 0 for each trial.

Fig. 4
Fig. 4

Probability of a sphere being in a particular potential well at an elapsed time: (a)  t = 1.13 s , (b)  2.0 s , (c)  5.0 s , (d)  9.5 s . The bars filled with vertical, diagonal, and horizontal stripes correspond to polystyrene spheres of a = 0.49, 1.03, and 1.61 μm radii, respectively.

Fig. 5
Fig. 5

Temporal evolutions of (a)  P S , 1 ( t ) , (b)  P M , 2 ( t ) , (c)  P L , 3 ( t ) , (d)  S ( t ) at various intensity levels I 1 , I 2 , I 3 , and I 4 that correspond to I 0 = 3.8 , 4.9, 6.9, and 10.4 kW / cm 2 , respectively. Radii of small, medium, and large spheres are 0.49, 1.03, and 1.61 μm , respectively. Markers and curves (dashed, dashed–dotted, solid, and dotted) represent measured and calculated values. Inset in (c) shows the fitted result of P L , 3 ( t ) with α = 1.4 .

Fig. 6
Fig. 6

(a) State of particle distribution as a function of sphere radius at various intensity levels I 1 , I 2 , I 3 , I 4 , and I 5 ( I 0 = 0.8 , 1.3, 8.9, 13.4, and 22.3 kW / cm 2 , respectively). The operating time is set at t = 4.0 s . Regions S, M, and L bounded by dashed gray lines correspond to small, medium, and large sphere regions for which the numbers of potential energy wells are 3, 2, and 1, respectively. (b) Intensity dependence of size resolutions Δ a 1 and Δ a 2 , i.e., width of the dotted gray lines in (a) at t = 4.0 s . (c) Time dependence of size resolutions Δ a 1 and Δ a 2 at I 0 = 18.0 kW / cm 2 .

Fig. 7
Fig. 7

Frame excerpts from video recording showing separation and arrangement of three types of polystyrene spheres (Media 1). Optical profiles used are shown in the insets along with the circles indicating the relative positions of the spheres (solid, dotted, and dashed circles represent 0.49, 1.03, and 1.61 μm radius spheres, respectively): (a) pairs of 0.49, 1.03, and 1.61 μm spheres move to the dashed light pattern; (b) a stack of spheres is exposed to the asymmetric nonperiodic optical stripe pattern; (c)  1.03 μm and 1.61 μm spheres reach the middle optical peak; (d)  1.61 μm spheres move further to the bottom optical peak and separation is complete; (e) asymmetric nonperiodic optical stripe pattern is stretched vertically to increase the separation between spheres of different sizes; (f) spheres move to the central area as the optical line pattern shortens with time; (g) all spheres are located at the central area; (h) spheres are aligned equally by outward moving optical spots.

Fig. 8
Fig. 8

Schematics showing calculated regions and barrier types for (a)  P S , 1 ( t ) , (b),(c)  P M , 2 ( t ) , (d)  P L , 3 ( t ) calculations. Solid curves represent optical potential landscapes for (a) small, (b),(c) medium, (d) large spheres. The thick vertical lines represent reflecting barriers, and absorbing regions are filled with diagonal stripes. The calculated regions x a x x b are depicted as thick horizontal lines. All the spheres are assumed to be at the asymmetric nonperiodic optical stripe pattern’s lowest intensity peak x = x 1 at t = 0 .

Equations (3)

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I ( x , y ) = I 0 { sin 2 π λ 1 ( x b 1 ) + c 1 ( x b 1 ) + d 1 ( b 1 x < b 2 ) sin 2 π λ 2 ( x b 2 ) + c 2 ( x b 2 ) + d 2 ( b 2 x < b 3 ) sin 2 π λ 3 ( x b 3 ) + c 3 ( x b 3 ) + d 3 ( b 3 x < b 4 ) .
G ( x , t ) t = U ( x ) γ G ( x , t ) x + D 2 G ( x , t ) x 2 .
( P S , 1 ( t ) η 1 ) ( P M , 2 ( t ) η 2 ) ( P L , 3 ( t ) η 3 )

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