Abstract

The radiation forces and trajectories of Rayleigh dielectric particles induced by one-dimensional Airy beam were numerically analyzed. Results show that the Airy beam drags particles into the optical intensity peaks, and guides particles vertically along parabolic trajectories. Viscosity of surrounding medium significantly affects the trajectories. Random Brownian force affects the trajectories. Meanwhile, trapping potential depths and minimum trapping particle radii in different potential wells were also discussed. The trapping stability could be improved by increasing either the input peak intensity or the particle radius.

©2010 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]
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    [Crossref]
  8. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
    [Crossref] [PubMed]
  9. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
    [Crossref]
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  13. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
    [Crossref]
  14. J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9(10), 1334–1336 (2009).
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    [Crossref] [PubMed]
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    [Crossref]
  19. R. Quidant, D. Petrov, and G. Badenes, “Radiation forces on a Rayleigh dielectric sphere in a patterned optical near field,” Opt. Lett. 30(9), 1009–1011 (2005).
    [Crossref] [PubMed]
  20. A. H. J. Yang, T. Lerdsuchatawanich, and D. Erickson, “Forces and transport velocities for a particle in a slot waveguide,” Nano Lett. 9(3), 1182–1188 (2009).
    [Crossref] [PubMed]
  21. S. Abuzeid, A. A. Busnaina, and G. Ahmadi, “Wall Deposition of Aerosol particles in a Turbulent Channel Flow,” J. Aerosol Sci. 22(1), 43–62 (1991).
    [Crossref]
  22. M. D. Summers, D. R. Burnham, and D. McGloin, “Trapping solid aerosols with optical tweezers: a comparison between gas and liquid phase optical traps,” Opt. Express 16(11), 7739–7747 (2008).
    [Crossref] [PubMed]
  23. A. Neild, T. W. Ng, and W. M. S. Yii, “Optical sorting of dielectric Rayleigh spherical particles with scattering and standing waves,” Opt. Express 17(7), 5321–5329 (2009).
    [Crossref] [PubMed]

2010 (3)

2009 (4)

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9(10), 1334–1336 (2009).
[Crossref] [PubMed]

A. H. J. Yang, T. Lerdsuchatawanich, and D. Erickson, “Forces and transport velocities for a particle in a slot waveguide,” Nano Lett. 9(3), 1182–1188 (2009).
[Crossref] [PubMed]

A. Neild, T. W. Ng, and W. M. S. Yii, “Optical sorting of dielectric Rayleigh spherical particles with scattering and standing waves,” Opt. Express 17(7), 5321–5329 (2009).
[Crossref] [PubMed]

2008 (5)

2007 (3)

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91(5), 053902 (2007).
[Crossref]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

2006 (1)

P. H. Jones, E. Stride, and N. Saffari, “Trapping and manipulation of microscopic bubbles with a scanning optical tweezer,” Appl. Phys. Lett. 89(8), 081113 (2006).
[Crossref]

2005 (2)

T. Cižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86(17), 174101 (2005).
[Crossref]

R. Quidant, D. Petrov, and G. Badenes, “Radiation forces on a Rayleigh dielectric sphere in a patterned optical near field,” Opt. Lett. 30(9), 1009–1011 (2005).
[Crossref] [PubMed]

2002 (1)

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
[Crossref] [PubMed]

1996 (1)

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 (1996).
[Crossref]

1991 (1)

S. Abuzeid, A. A. Busnaina, and G. Ahmadi, “Wall Deposition of Aerosol particles in a Turbulent Channel Flow,” J. Aerosol Sci. 22(1), 43–62 (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 (1)

Aabo, T.

L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, “Efficient optical trapping and visualization of silver nanoparticles,” Nano Lett. 8(5), 1486–1491 (2008).
[Crossref] [PubMed]

Abuzeid, S.

S. Abuzeid, A. A. Busnaina, and G. Ahmadi, “Wall Deposition of Aerosol particles in a Turbulent Channel Flow,” J. Aerosol Sci. 22(1), 43–62 (1991).
[Crossref]

Agate, B.

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91(5), 053902 (2007).
[Crossref]

Ahmadi, G.

S. Abuzeid, A. A. Busnaina, and G. Ahmadi, “Wall Deposition of Aerosol particles in a Turbulent Channel Flow,” J. Aerosol Sci. 22(1), 43–62 (1991).
[Crossref]

Alfano, R. R.

Arie, A.

Asakura, T.

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 (1996).
[Crossref]

Ashkin, A.

Badenes, G.

Baumgartl, J.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9(10), 1334–1336 (2009).
[Crossref] [PubMed]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

Bendix, P. M.

L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, “Efficient optical trapping and visualization of silver nanoparticles,” Nano Lett. 8(5), 1486–1491 (2008).
[Crossref] [PubMed]

Bjorkholm, J. E.

Bosanac, L.

L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, “Efficient optical trapping and visualization of silver nanoparticles,” Nano Lett. 8(5), 1486–1491 (2008).
[Crossref] [PubMed]

Broky, J.

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

Brown, C. T. A.

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91(5), 053902 (2007).
[Crossref]

Burnham, D. R.

Busnaina, A. A.

S. Abuzeid, A. A. Busnaina, and G. Ahmadi, “Wall Deposition of Aerosol particles in a Turbulent Channel Flow,” J. Aerosol Sci. 22(1), 43–62 (1991).
[Crossref]

Chong, A.

A. Chong, W. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

Christodoulides, D. N.

A. Chong, W. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[Crossref] [PubMed]

Chu, S.

Cižmár, T.

T. Cižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86(17), 174101 (2005).
[Crossref]

Comrie, M.

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91(5), 053902 (2007).
[Crossref]

Day, D.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9(10), 1334–1336 (2009).
[Crossref] [PubMed]

Dholakia, K.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9(10), 1334–1336 (2009).
[Crossref] [PubMed]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91(5), 053902 (2007).
[Crossref]

T. Cižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86(17), 174101 (2005).
[Crossref]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
[Crossref] [PubMed]

Dogariu, A.

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

Dolev, I.

Dziedzic, J. M.

Ellenbogen, T.

Erickson, D.

A. H. J. Yang, T. Lerdsuchatawanich, and D. Erickson, “Forces and transport velocities for a particle in a slot waveguide,” Nano Lett. 9(3), 1182–1188 (2009).
[Crossref] [PubMed]

Garcés-Chávez, V.

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91(5), 053902 (2007).
[Crossref]

T. Cižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86(17), 174101 (2005).
[Crossref]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
[Crossref] [PubMed]

Gu, M.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9(10), 1334–1336 (2009).
[Crossref] [PubMed]

Gunn-Moore, F.

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91(5), 053902 (2007).
[Crossref]

Hannappel, G. M.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9(10), 1334–1336 (2009).
[Crossref] [PubMed]

Harada, Y.

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 (1996).
[Crossref]

Jones, P. H.

P. H. Jones, E. Stride, and N. Saffari, “Trapping and manipulation of microscopic bubbles with a scanning optical tweezer,” Appl. Phys. Lett. 89(8), 081113 (2006).
[Crossref]

Kolesik, M.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

Lerdsuchatawanich, T.

A. H. J. Yang, T. Lerdsuchatawanich, and D. Erickson, “Forces and transport velocities for a particle in a slot waveguide,” Nano Lett. 9(3), 1182–1188 (2009).
[Crossref] [PubMed]

Li, J. X.

Mazilu, M.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

McGloin, D.

M. D. Summers, D. R. Burnham, and D. McGloin, “Trapping solid aerosols with optical tweezers: a comparison between gas and liquid phase optical traps,” Opt. Express 16(11), 7739–7747 (2008).
[Crossref] [PubMed]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
[Crossref] [PubMed]

Melville, H.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
[Crossref] [PubMed]

Moloney, J. V.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

Neild, A.

Ng, T. W.

Oddershede, L. B.

L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, “Efficient optical trapping and visualization of silver nanoparticles,” Nano Lett. 8(5), 1486–1491 (2008).
[Crossref] [PubMed]

Petrov, D.

Polynkin, P.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

Quidant, R.

Renninger, W.

A. Chong, W. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

Saffari, N.

P. H. Jones, E. Stride, and N. Saffari, “Trapping and manipulation of microscopic bubbles with a scanning optical tweezer,” Appl. Phys. Lett. 89(8), 081113 (2006).
[Crossref]

Sibbett, W.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
[Crossref] [PubMed]

Siviloglou, G. A.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[Crossref] [PubMed]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

Stevenson, D. J.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9(10), 1334–1336 (2009).
[Crossref] [PubMed]

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91(5), 053902 (2007).
[Crossref]

Stride, E.

P. H. Jones, E. Stride, and N. Saffari, “Trapping and manipulation of microscopic bubbles with a scanning optical tweezer,” Appl. Phys. Lett. 89(8), 081113 (2006).
[Crossref]

Summers, M. D.

Sztul, H. I.

Tian, J. G.

Tsampoula, X.

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91(5), 053902 (2007).
[Crossref]

Wise, F. W.

A. Chong, W. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

Yang, A. H. J.

A. H. J. Yang, T. Lerdsuchatawanich, and D. Erickson, “Forces and transport velocities for a particle in a slot waveguide,” Nano Lett. 9(3), 1182–1188 (2009).
[Crossref] [PubMed]

Yii, W. M. S.

Zang, W. P.

Zemanek, P.

T. Cižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86(17), 174101 (2005).
[Crossref]

Appl. Phys. Lett. (3)

P. H. Jones, E. Stride, and N. Saffari, “Trapping and manipulation of microscopic bubbles with a scanning optical tweezer,” Appl. Phys. Lett. 89(8), 081113 (2006).
[Crossref]

T. Cižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86(17), 174101 (2005).
[Crossref]

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91(5), 053902 (2007).
[Crossref]

J. Aerosol Sci. (1)

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Figures (7)
Fig. 1
Fig. 1 Propagation dynamics of a finite energy Airy beam as a function of distance with (a) the z component and (b) the x component of the intensity, where λ = 1064 nm, a = 0.1, x 0 = 2 μm and I 0 = 1.4518 × 1011 W/m2, respectively.

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Fig. 2
Fig. 2 The radiation forces on a 45 nm (radius) fused silica nanoparticle at varying x positions for three different z planes. The gradient force of (a) x component and (b) z component; the scattering force of (c) x component and (d) z component.

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Fig. 3
Fig. 3 The total radiation forces on fused silicon nanoparticles at varying x positions for z = 0 plane. Plots of total forces for particles with different radii and refractive indices of (a) and (b) x component; (c) and (d) z component. The refractive index is 1.46 in (a) and (c). The radius is 50 nm in (b) and (d).

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Fig. 4
Fig. 4 The trajectories of fused silica nanoparticles in an Airy beam. Plots of trajectories for 50 nm (radius) particles at four different initial x positions (z = 0) in (a) and trajectories for two different radius particles both at the initial position of x = −3 μm (z = 0) in (b). The inset is the partial enlarged detail of the circled region.

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Fig. 5
Fig. 5 The trajectories of a 50 nm (radius) fused silica nanoparticle at the initial position of x = −11 μm (z = 0) with different viscosity of surrounding medium.

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Fig. 6
Fig. 6 Potential depths of fused silicon nanoparticles at varying x positions in an Airy beam. Plots of potential depths in (a) a 50 nm radius particle at different z planes when the input peak intensity of light is 1.4518 × 1011 W/m2, in (b) a 50 nm radius particle with different input peak intensity at z = 0 plane, in (c) three different radius particles at z = 0 plane with input peak intensity of 1.4518 × 1011 W/m2.

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Fig. 7
Fig. 7 Dependence of minimum radius of optical-field trapped particles on input optical peak intensity.

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Equations (22)

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i ∂ ϕ ∂ ξ + ∂ 2 ϕ ∂ s 2 = 0 ,
E x = − i k A 0 ( 2 k 2 x 0 2 ) { exp ( C ) [ ( 2 a 2 + 2 i a ξ − ξ 2 / 4 + 2 s + 2 k 2 x 0 2 ) A i r y ( D )       + ( 4 a + i ξ ) A i r y ' ( D ) ] } ,
E z = − i k A 0 ( 4 k 3 x 0 3 ) { i exp ( C ) [ 2 a 3 + 9 i a 2 ξ / 2 + a ( − 3 ξ 2 / 2 + 6 s − 4 k 2 x 0 2 ) − i ( 2 i + ξ 3 / 8       − 3 s ξ / 2 + ξ k 2 x 0 2 ) A i r y ( D ) + ( 6 a 2 + 4 i a ξ + 2 s − ξ 2 / 2 − 4 k 2 x 0 2 ) A i r y ' ( D ) ] } ,
B y = k A 0 ( 4 k 2 x 0 2 ) { i exp ( C ) [ ( 2 a 2 + 2 i a ξ − ξ 2 / 4 + 2 s − 4 k 2 x 0 2 ) A i r y ( D )       + ( 4 a + i ξ ) A i r y ' ( D ) ] } ,
I ⇀ = 〈 S ⇀ ( r ⇀ , t ) 〉 t = 1 2 Re [ E ⇀ × H ⇀ * ] .
I ⇀ = n ε 0 c 2 | E x | 2 e ⇀ z + n ε 0 c 2 | E x | ⋅ | E z | e ⇀ x .
I z = n ε 0 c 2 | E x | 2 ,
I x = n ε 0 c 2 | E x | ⋅ | E z |
F → g r a d ( x , z ) = ε 0 π n 2 2 R 3 ( m 2 − 1 m 2 + 2 ) ∇ | E → ⋅ E → * |                 = ε 0 π n 2 2 R 3 ( m 2 − 1 m 2 + 2 ) ∇ ( | E x | 2 + | E z | 2 ) ,
F ⇀ s c a t ( x , z ) = ( n 2 c ) C p r I ⇀ ( x , z ) ,
C p r = C s c a t = 8 3 π k 4 R 6 ( m 2 − 1 m 2 + 2 ) 2 ,
( F g r a d ) x = ε 0 π n 2 2 R 3 ( m 2 − 1 m 2 + 2 ) ∂ ∂ x ( | E x | 2 + | E z | 2 ) ,
( F g r a d ) z = ε 0 π n 2 2 R 3 ( m 2 − 1 m 2 + 2 ) ∂ ∂ z ( | E x | 2 + | E z | 2 ) ,
( F s c a t ) x = ( n 2 c ) C p r I x ( x , z ) ,
( F s c a t ) z = ( n 2 c ) C p r I z ( x , z ) .
F ⇀ D r a g = C D r a g V ⇀ .
F B n = 2 C D r a g k B T ξ .
m p d 2 x d t 2 = F x ( x , z ) + C D r a g d x d t + F B n ,
m p d 2 z d t 2 = F z ( x , z ) + C D r a g d z d t + G p + F b + F B n ,
U / ( k B T ) ≥ 10 ,
U = 2 π n 2 R 3 c ( m 2 − 1 m 2 + 2 ) [ I ( r p e a k , z ) − I ( r h o l l o w , z ) ] ,
R min = [ 10 k B T c 2 π n 2 ( m 2 + 2 m 2 − 1 ) ] 1 / 3 × [ I ( r p e a k , z ) − I ( r h o l l o w , z ) ] − 1 / 3 .

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