Abstract

A comprehensive dynamics analysis of microsphere has been presented in a dual-beam fiber-optic trap with transverse offset. As the offset distance between two counterpropagating beams increases, the motion type of the microsphere starts with capture, then spiral motion, then orbital rotation, and ends with escape. We analyze the transformation process and mechanism of the four motion types based on ray optics approximation. Dynamic simulations show that the existence of critical offset distances at which different motion types transform. The result is an important step toward explaining physical phenomena in a dual-beam fiber-optic trap with transverse offset, and is generally applicable to achieving controllable motions of microspheres in integrated systems, such as microfluidic systems and lab-on-a-chip systems.

© 2016 Optical Society of America

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References

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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  22. T. Kolb, S. Albert, M. Haug, and G. Whyte, “Dynamically reconfigurable fibre optical spanner,” Lab Chip 14(6), 1186–1190 (2014).
    [Crossref] [PubMed]
  23. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61(2), 569–582 (1992).
    [Crossref] [PubMed]
  24. E. Sidick, S. D. Collins, and A. Knoesen, “Trapping forces in a multiple-beam fiber-optic trap,” Appl. Opt. 36(25), 6423–6433 (1997).
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    [Crossref]

2016 (2)

M. Li, S. Yan, B. Yao, Y. Liang, M. Lei, and Y. Yang, “Optically induced rotation of Rayleigh particles by vortex beams with different states of polarization,” Phys. Lett. A 380(1-2), 311–315 (2016).
[Crossref]

G. Xiao, K. Yang, H. Luo, X. Chen, and W. Xiong, “Orbital rotation of trapped particle in a transversely misaligned dual-fiber optical trap,” IEEE Photonics J. 8(1), 1–8 (2016).
[Crossref]

2014 (1)

T. Kolb, S. Albert, M. Haug, and G. Whyte, “Dynamically reconfigurable fibre optical spanner,” Lab Chip 14(6), 1186–1190 (2014).
[Crossref] [PubMed]

2013 (2)

Y. Arita, M. Mazilu, and K. Dholakia, “Laser-induced rotation and cooling of a trapped microgyroscope in vacuum,” Nat. Commun. 4, 2374 (2013).
[Crossref] [PubMed]

M. C. Zhong, X. B. Wei, J. H. Zhou, Z. Q. Wang, and Y. M. Li, “Trapping red blood cells in living animals using optical tweezers,” Nat. Commun. 4, 1768 (2013).
[Crossref] [PubMed]

2012 (4)

M. Koch and A. Rohrbach, “Object-adapted optical trapping and shapetracking of energy-switching helical bacteria,” Nat. Photonics 6(10), 680–686 (2012).
[Crossref]

N. Watanabe and K. Taguchi, “Theoretical study of optical vibration and circulation of a microsphere,” Key Eng. Mater. 516, 563–568 (2012).
[Crossref]

N. Bellini, F. Bragheri, I. Cristiani, J. Guck, R. Osellame, and G. Whyte, “Validation and perspectives of a femtosecond laser fabricated monolithic optical stretcher,” Biomed. Opt. Express 3(10), 2658–2668 (2012).
[Crossref] [PubMed]

B. J. Black and S. K. Mohanty, “Fiber-optic spanner,” Opt. Lett. 37(24), 5030–5032 (2012).
[Crossref] [PubMed]

2011 (1)

K. Wang, E. Schonbrun, P. Steinvurzel, and K. B. Crozier, “Trapping and rotating nanoparticles using a plasmonic nano-tweezer with an integrated heat sink,” Nat. Commun. 2, 469 (2011).
[Crossref] [PubMed]

2010 (2)

2009 (1)

S. Zwick, T. Haist, Y. Miyamoto, L. He, M. Warber, A. Hermerschmidt, and W. Osten, “Holographic twin traps,” J. Opt. A, Pure Appl. Opt. 11(3), 034011 (2009).
[Crossref]

2008 (3)

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77(1), 205–228 (2008).
[Crossref] [PubMed]

J. T. Blakely, R. Gordon, and D. Sinton, “Flow-dependent optofluidic particle trapping and circulation,” Lab Chip 8(8), 1350–1356 (2008).
[Crossref] [PubMed]

V. Karásek, T. Cizmár, O. Brzobohatý, P. Zemánek, V. Garcés-Chávez, and K. Dholakia, “Long-range one-dimensional longitudinal optical binding,” Phys. Rev. Lett. 101(14), 143601 (2008).
[Crossref] [PubMed]

2004 (2)

K. Ladavac and D. G. Grier, “Sorting by periodic potential energy landscapes: optical fractionation,” Phys. Rev. E. 70(1), 010901 (2004).
[Crossref] [PubMed]

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

2000 (1)

K. Taguchi, K. Atsuta, T. Nakata, and M. Ikeda, “Levitation of a microscopic object using plural optical fibers,” Opt. Commun. 176(1), 43–47 (2000).
[Crossref]

1997 (2)

1993 (1)

1992 (1)

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61(2), 569–582 (1992).
[Crossref] [PubMed]

1987 (1)

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

1970 (1)

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

Albert, S.

T. Kolb, S. Albert, M. Haug, and G. Whyte, “Dynamically reconfigurable fibre optical spanner,” Lab Chip 14(6), 1186–1190 (2014).
[Crossref] [PubMed]

Allen, L.

Arita, Y.

Y. Arita, M. Mazilu, and K. Dholakia, “Laser-induced rotation and cooling of a trapped microgyroscope in vacuum,” Nat. Commun. 4, 2374 (2013).
[Crossref] [PubMed]

Ashkin, A.

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61(2), 569–582 (1992).
[Crossref] [PubMed]

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

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

Atsuta, K.

K. Taguchi, K. Atsuta, T. Nakata, and M. Ikeda, “Levitation of a microscopic object using plural optical fibers,” Opt. Commun. 176(1), 43–47 (2000).
[Crossref]

Bañas, A.

Bellini, N.

Berghoff, K.

Black, B. J.

Blakely, J. T.

J. T. Blakely, R. Gordon, and D. Sinton, “Flow-dependent optofluidic particle trapping and circulation,” Lab Chip 8(8), 1350–1356 (2008).
[Crossref] [PubMed]

Block, S. M.

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

Bragheri, F.

Brzobohatý, O.

V. Karásek, T. Cizmár, O. Brzobohatý, P. Zemánek, V. Garcés-Chávez, and K. Dholakia, “Long-range one-dimensional longitudinal optical binding,” Phys. Rev. Lett. 101(14), 143601 (2008).
[Crossref] [PubMed]

Bustamante, C.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77(1), 205–228 (2008).
[Crossref] [PubMed]

Chemla, Y. R.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77(1), 205–228 (2008).
[Crossref] [PubMed]

Chen, X.

G. Xiao, K. Yang, H. Luo, X. Chen, and W. Xiong, “Orbital rotation of trapped particle in a transversely misaligned dual-fiber optical trap,” IEEE Photonics J. 8(1), 1–8 (2016).
[Crossref]

Cizmár, T.

V. Karásek, T. Cizmár, O. Brzobohatý, P. Zemánek, V. Garcés-Chávez, and K. Dholakia, “Long-range one-dimensional longitudinal optical binding,” Phys. Rev. Lett. 101(14), 143601 (2008).
[Crossref] [PubMed]

Collins, S. D.

Constable, A.

Cristiani, I.

Crozier, K. B.

K. Wang, E. Schonbrun, P. Steinvurzel, and K. B. Crozier, “Trapping and rotating nanoparticles using a plasmonic nano-tweezer with an integrated heat sink,” Nat. Commun. 2, 469 (2011).
[Crossref] [PubMed]

Denz, C.

Dholakia, K.

Y. Arita, M. Mazilu, and K. Dholakia, “Laser-induced rotation and cooling of a trapped microgyroscope in vacuum,” Nat. Commun. 4, 2374 (2013).
[Crossref] [PubMed]

V. Karásek, T. Cizmár, O. Brzobohatý, P. Zemánek, V. Garcés-Chávez, and K. Dholakia, “Long-range one-dimensional longitudinal optical binding,” Phys. Rev. Lett. 101(14), 143601 (2008).
[Crossref] [PubMed]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22(1), 52–54 (1997).
[Crossref] [PubMed]

Dziedzic, J. M.

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

Garcés-Chávez, V.

V. Karásek, T. Cizmár, O. Brzobohatý, P. Zemánek, V. Garcés-Chávez, and K. Dholakia, “Long-range one-dimensional longitudinal optical binding,” Phys. Rev. Lett. 101(14), 143601 (2008).
[Crossref] [PubMed]

Glückstad, J.

Gordon, R.

J. T. Blakely, R. Gordon, and D. Sinton, “Flow-dependent optofluidic particle trapping and circulation,” Lab Chip 8(8), 1350–1356 (2008).
[Crossref] [PubMed]

Grier, D. G.

K. Ladavac and D. G. Grier, “Sorting by periodic potential energy landscapes: optical fractionation,” Phys. Rev. E. 70(1), 010901 (2004).
[Crossref] [PubMed]

Guck, J.

Haist, T.

S. Zwick, T. Haist, Y. Miyamoto, L. He, M. Warber, A. Hermerschmidt, and W. Osten, “Holographic twin traps,” J. Opt. A, Pure Appl. Opt. 11(3), 034011 (2009).
[Crossref]

Haug, M.

T. Kolb, S. Albert, M. Haug, and G. Whyte, “Dynamically reconfigurable fibre optical spanner,” Lab Chip 14(6), 1186–1190 (2014).
[Crossref] [PubMed]

He, L.

S. Zwick, T. Haist, Y. Miyamoto, L. He, M. Warber, A. Hermerschmidt, and W. Osten, “Holographic twin traps,” J. Opt. A, Pure Appl. Opt. 11(3), 034011 (2009).
[Crossref]

Hermerschmidt, A.

S. Zwick, T. Haist, Y. Miyamoto, L. He, M. Warber, A. Hermerschmidt, and W. Osten, “Holographic twin traps,” J. Opt. A, Pure Appl. Opt. 11(3), 034011 (2009).
[Crossref]

Ikeda, M.

K. Taguchi, K. Atsuta, T. Nakata, and M. Ikeda, “Levitation of a microscopic object using plural optical fibers,” Opt. Commun. 176(1), 43–47 (2000).
[Crossref]

Karásek, V.

V. Karásek, T. Cizmár, O. Brzobohatý, P. Zemánek, V. Garcés-Chávez, and K. Dholakia, “Long-range one-dimensional longitudinal optical binding,” Phys. Rev. Lett. 101(14), 143601 (2008).
[Crossref] [PubMed]

Kim, J.

Knoesen, A.

Koch, M.

M. Koch and A. Rohrbach, “Object-adapted optical trapping and shapetracking of energy-switching helical bacteria,” Nat. Photonics 6(10), 680–686 (2012).
[Crossref]

Kolb, T.

T. Kolb, S. Albert, M. Haug, and G. Whyte, “Dynamically reconfigurable fibre optical spanner,” Lab Chip 14(6), 1186–1190 (2014).
[Crossref] [PubMed]

Ladavac, K.

K. Ladavac and D. G. Grier, “Sorting by periodic potential energy landscapes: optical fractionation,” Phys. Rev. E. 70(1), 010901 (2004).
[Crossref] [PubMed]

Lei, M.

M. Li, S. Yan, B. Yao, Y. Liang, M. Lei, and Y. Yang, “Optically induced rotation of Rayleigh particles by vortex beams with different states of polarization,” Phys. Lett. A 380(1-2), 311–315 (2016).
[Crossref]

Li, M.

M. Li, S. Yan, B. Yao, Y. Liang, M. Lei, and Y. Yang, “Optically induced rotation of Rayleigh particles by vortex beams with different states of polarization,” Phys. Lett. A 380(1-2), 311–315 (2016).
[Crossref]

Li, Y. M.

M. C. Zhong, X. B. Wei, J. H. Zhou, Z. Q. Wang, and Y. M. Li, “Trapping red blood cells in living animals using optical tweezers,” Nat. Commun. 4, 1768 (2013).
[Crossref] [PubMed]

Liang, Y.

M. Li, S. Yan, B. Yao, Y. Liang, M. Lei, and Y. Yang, “Optically induced rotation of Rayleigh particles by vortex beams with different states of polarization,” Phys. Lett. A 380(1-2), 311–315 (2016).
[Crossref]

Luo, H.

G. Xiao, K. Yang, H. Luo, X. Chen, and W. Xiong, “Orbital rotation of trapped particle in a transversely misaligned dual-fiber optical trap,” IEEE Photonics J. 8(1), 1–8 (2016).
[Crossref]

Mazilu, M.

Y. Arita, M. Mazilu, and K. Dholakia, “Laser-induced rotation and cooling of a trapped microgyroscope in vacuum,” Nat. Commun. 4, 2374 (2013).
[Crossref] [PubMed]

Mervis, J.

Miyamoto, Y.

S. Zwick, T. Haist, Y. Miyamoto, L. He, M. Warber, A. Hermerschmidt, and W. Osten, “Holographic twin traps,” J. Opt. A, Pure Appl. Opt. 11(3), 034011 (2009).
[Crossref]

Moffitt, J. R.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77(1), 205–228 (2008).
[Crossref] [PubMed]

Mohanty, S. K.

Nakata, T.

K. Taguchi, K. Atsuta, T. Nakata, and M. Ikeda, “Levitation of a microscopic object using plural optical fibers,” Opt. Commun. 176(1), 43–47 (2000).
[Crossref]

Neuman, K. C.

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

Osellame, R.

Osten, W.

S. Zwick, T. Haist, Y. Miyamoto, L. He, M. Warber, A. Hermerschmidt, and W. Osten, “Holographic twin traps,” J. Opt. A, Pure Appl. Opt. 11(3), 034011 (2009).
[Crossref]

Padgett, M. J.

Palima, D.

Prentiss, M.

Rohrbach, A.

M. Koch and A. Rohrbach, “Object-adapted optical trapping and shapetracking of energy-switching helical bacteria,” Nat. Photonics 6(10), 680–686 (2012).
[Crossref]

Schonbrun, E.

K. Wang, E. Schonbrun, P. Steinvurzel, and K. B. Crozier, “Trapping and rotating nanoparticles using a plasmonic nano-tweezer with an integrated heat sink,” Nat. Commun. 2, 469 (2011).
[Crossref] [PubMed]

Sidick, E.

Simpson, N. B.

Sinton, D.

J. T. Blakely, R. Gordon, and D. Sinton, “Flow-dependent optofluidic particle trapping and circulation,” Lab Chip 8(8), 1350–1356 (2008).
[Crossref] [PubMed]

Smith, S. B.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77(1), 205–228 (2008).
[Crossref] [PubMed]

Steinvurzel, P.

K. Wang, E. Schonbrun, P. Steinvurzel, and K. B. Crozier, “Trapping and rotating nanoparticles using a plasmonic nano-tweezer with an integrated heat sink,” Nat. Commun. 2, 469 (2011).
[Crossref] [PubMed]

Taguchi, K.

N. Watanabe and K. Taguchi, “Theoretical study of optical vibration and circulation of a microsphere,” Key Eng. Mater. 516, 563–568 (2012).
[Crossref]

K. Taguchi, K. Atsuta, T. Nakata, and M. Ikeda, “Levitation of a microscopic object using plural optical fibers,” Opt. Commun. 176(1), 43–47 (2000).
[Crossref]

Tauro, S.

Wang, K.

K. Wang, E. Schonbrun, P. Steinvurzel, and K. B. Crozier, “Trapping and rotating nanoparticles using a plasmonic nano-tweezer with an integrated heat sink,” Nat. Commun. 2, 469 (2011).
[Crossref] [PubMed]

Wang, Z. Q.

M. C. Zhong, X. B. Wei, J. H. Zhou, Z. Q. Wang, and Y. M. Li, “Trapping red blood cells in living animals using optical tweezers,” Nat. Commun. 4, 1768 (2013).
[Crossref] [PubMed]

Warber, M.

S. Zwick, T. Haist, Y. Miyamoto, L. He, M. Warber, A. Hermerschmidt, and W. Osten, “Holographic twin traps,” J. Opt. A, Pure Appl. Opt. 11(3), 034011 (2009).
[Crossref]

Watanabe, N.

N. Watanabe and K. Taguchi, “Theoretical study of optical vibration and circulation of a microsphere,” Key Eng. Mater. 516, 563–568 (2012).
[Crossref]

Wei, X. B.

M. C. Zhong, X. B. Wei, J. H. Zhou, Z. Q. Wang, and Y. M. Li, “Trapping red blood cells in living animals using optical tweezers,” Nat. Commun. 4, 1768 (2013).
[Crossref] [PubMed]

Whyte, G.

Woerdemann, M.

Xiao, G.

G. Xiao, K. Yang, H. Luo, X. Chen, and W. Xiong, “Orbital rotation of trapped particle in a transversely misaligned dual-fiber optical trap,” IEEE Photonics J. 8(1), 1–8 (2016).
[Crossref]

Xiong, W.

G. Xiao, K. Yang, H. Luo, X. Chen, and W. Xiong, “Orbital rotation of trapped particle in a transversely misaligned dual-fiber optical trap,” IEEE Photonics J. 8(1), 1–8 (2016).
[Crossref]

Yan, S.

M. Li, S. Yan, B. Yao, Y. Liang, M. Lei, and Y. Yang, “Optically induced rotation of Rayleigh particles by vortex beams with different states of polarization,” Phys. Lett. A 380(1-2), 311–315 (2016).
[Crossref]

Yang, K.

G. Xiao, K. Yang, H. Luo, X. Chen, and W. Xiong, “Orbital rotation of trapped particle in a transversely misaligned dual-fiber optical trap,” IEEE Photonics J. 8(1), 1–8 (2016).
[Crossref]

Yang, Y.

M. Li, S. Yan, B. Yao, Y. Liang, M. Lei, and Y. Yang, “Optically induced rotation of Rayleigh particles by vortex beams with different states of polarization,” Phys. Lett. A 380(1-2), 311–315 (2016).
[Crossref]

Yao, B.

M. Li, S. Yan, B. Yao, Y. Liang, M. Lei, and Y. Yang, “Optically induced rotation of Rayleigh particles by vortex beams with different states of polarization,” Phys. Lett. A 380(1-2), 311–315 (2016).
[Crossref]

Zarinetchi, F.

Zemánek, P.

V. Karásek, T. Cizmár, O. Brzobohatý, P. Zemánek, V. Garcés-Chávez, and K. Dholakia, “Long-range one-dimensional longitudinal optical binding,” Phys. Rev. Lett. 101(14), 143601 (2008).
[Crossref] [PubMed]

Zhong, M. C.

M. C. Zhong, X. B. Wei, J. H. Zhou, Z. Q. Wang, and Y. M. Li, “Trapping red blood cells in living animals using optical tweezers,” Nat. Commun. 4, 1768 (2013).
[Crossref] [PubMed]

Zhou, J. H.

M. C. Zhong, X. B. Wei, J. H. Zhou, Z. Q. Wang, and Y. M. Li, “Trapping red blood cells in living animals using optical tweezers,” Nat. Commun. 4, 1768 (2013).
[Crossref] [PubMed]

Zwick, S.

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Figures (8)

Fig. 1
Fig. 1

Schematic of dual-beam fiber-optic trap with transverse offset. d: offset distance. S: beam waist separation distance.

Fig. 2
Fig. 2

(a) Simulation results of the trapping forces exerted on the microsphere when D = 0. The colors and directions of arrows respectively represent the magnitudes and directions of trapping forces. The red solid curve denotes the dynamic trajectory of the microsphere. (b) Schematic showing the motion type of the microsphere when two fibers are perfectly collinear.

Fig. 3
Fig. 3

(a)The dynamic trajectory of microsphere when D = 2.58. (b) Schematic of the spiral motion of the microsphere.

Fig. 4
Fig. 4

(a) Axial trapping force Fz versus axial position (b) Transverse trapping force Fx versus transverse position for varying D.

Fig. 5
Fig. 5

(a) The dynamic trajectory of the microsphere when D = 2.8. (b) Schematic of the orbital rotation of the microsphere.

Fig. 6
Fig. 6

(a) Orbital rotation perimeter and (b) orbital rotation rate versus transverse offset for varying microsphere radius r0. The triangles represent the critical values D1. The squares represent the critical values D2

Fig. 7
Fig. 7

The transition of the motion type from orbital rotation to escape with the increasing of D. (a) D = 3.0 (b) D = 3.194 (c) D = 3.2. Red solid lines: motion trajectories of the microsphere. Black solid lines with solid circles: stable transverse equilibrium points. Black solid lines with hollow triangles: unstable transverse equilibrium points. The symbol “+” and “-” denote the direction of the transverse force Fx.

Fig. 8
Fig. 8

(a) The dynamic trajectory of the microsphere when D = 3.2. (b) The microsphere escapes when D>D2.

Equations (8)

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F z1 ( z 1 , d 1 )= n 1 P 1 c × 2 r 0 2 π 0 π dφ 0 θ max dφ sin2θ exp(2 r 2 / ω 2 ) ω 2 R c ×{ q s R z + q g tanγ ×[ R z R c ( R z + r 0 cosθ) acosγ ]},
F x1 ( z 1 , d 1 )= n 1 P 1 c × 2 r 0 2 π 0 π dφ 0 θ max dφ sin2θ exp(2 r 2 / ω 2 ) ω 2 R c ×{ q s ( r 0 sinθcosφ d 1 )+ q g tanγ ×[ r 0 sinθcosφ d 1 (1 R c acosγ )]},
q s =1+Rcos2 α i T 2 cos(2 α i 2 α r )+Rcos2 α i 1+ R 2 +2Rcos2 α r , q g =Rsin2 α i + T 2 sin(2 α i 2 α r )+Rsin2 α i 1+ R 2 +2Rcos2 α r ,
R= 1 2 [ sin ( α i α r ) 2 sin ( α i + α r ) 2 + tan ( α i α r ) 2 tan ( α i + α r ) 2 ], T=1R, R c =( z 1 rcosθ)[1+ ( π n 1 ω 0 2 λ 0 ) 2 ], R z = ( R c 2 r 2 ) 1/2 , α i = 1 2 r 0 R c { [ d 1 2 + ( R z + r 0 cosθ) 2 ] 2 r 0 2 R c 2 }, sin α r = n 1 n 2 sin α i , r 2 = d 1 2 + ( r 0 sinθ) 2 2 d 1 r 0 sinθcosφ, a 2 = d 1 2 + ( r 0 cosθ+ R z ) 2 , θ max = cos 1 ( r 0 z 1 ), ω 2 = ω 0 2 [1+ ( z p λ 0 π n 1 ω 0 2 ) 2 ], γ= sin 1 ( r 0 sinθ R c ).
F z = F z1 ( z 1 , d 1 )+ F z2 ( z 2 , d 2 ), F x = F x1 ( z 1 , d 1 )+ F x2 ( z 2 , d 2 ),
F v =6π r 0 vη,
m r ¨ ( t )= F x (r) x ^ + F z (r) z ^ + F v (t),
v( t+Δt )=v(t)+ r ¨ (t)Δt, r( t+Δt )=r(t)+v(t)Δt.

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