Abstract

In this paper, we develop a theoretical method based on ray optics to calculate the optical force and torque on a metallo-dielectric Janus particle in an optical trap made from a tightly focused Gaussian beam. The Janus particle is a 2.8 μm diameter polystyrene sphere half-coated with gold thin film several nanometers in thickness. The calculation result shows that the focused beam will push the Janus particle away from the center of the trap, and the equilibrium position of the Janus particle, where the optical force and torque are both zero, is located in a circular orbit surrounding the laser beam axis. The theoretical results are in good agreement qualitatively and quantitatively with our experimental observation. As the ray-optics model is simple in principle, user friendly in formalism, and cost effective in terms of computation resources and time compared with other usual rigorous electromagnetics approaches, the developed theoretical method can become an invaluable tool for understanding and designing ways to control the mechanical motion of complicated microscopic particles in various optical tweezers.

© 2015 Chinese Laser Press

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References

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  1. H. L. Guo and Z. Y. Li, “Optical tweezers technique and its applications,” Sci. China Phys. Astro. & Mech. 56, 2351–2360 (2013).
    [Crossref]
  2. 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, 288–290 (1986).
    [Crossref]
  3. M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys. J. 72, 1335–1346 (1997).
    [Crossref]
  4. E. Qu, H. L. Guo, C. H. Xu, Z. L. Li, M. Yuan, B. Y. Cheng, and D. Z. Zhang, “Kinetics of microtubule -AtMAP65-1 bond studied with dual-optical tweezers,” Jpn. J. Appl. Phys. 46, 7514–7518 (2007).
    [Crossref]
  5. A. K. Rai, A. Rai, A. J. Ramaiya, R. Jha, and R. Mallik, “Molecular adaptations allow dynein to generate large collective forces inside cells,” Cell 152, 172–182 (2013).
    [Crossref]
  6. J. C. Crocker and D. G. Grier, “Microscopic measurement of the pair interaction potential of charge-stabilized colloid,” Phys. Rev. Lett. 73, 352–355 (1994).
    [Crossref]
  7. G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, and D. J. Evans, “Experimental demonstration of violations of the second law of thermodynamics for small systems and short time scales,” Phys. Rev. Lett. 89, 050601 (2002).
    [Crossref]
  8. J. Köhler, R. Ghadiri, S. I. Ksouri, Q. Guo, E. L. Gurevich, and A. Ostendorf, “Generation of microfluidic flow using an optically assembled and magnetically driven microrotor,” J. Phys. D 47, 505501 (2014).
    [Crossref]
  9. G. B. Liao, P. B. Bareil, Y. L. Sheng, and A. Chiou, “One-dimensional jumping optical tweezers for optical stretching of bi-concave human red blood cells,” Opt. Express 16, 1996–2004 (2008).
    [Crossref]
  10. S. Mohanty, “Optically-actuated translational and rotational motion at the microscale for microfluidic manipulation and characterization,” Lab Chip 12, 3624–3636 (2012).
    [Crossref]
  11. L. Lin, H. L. Guo, X. L. Zhong, L. Huang, J. F. Li, L. Gan, and Z. Y. Li, “Manipulation of gold nanorods with dual-optical tweezers for surface plasmon resonance control,” Nanotechnology 23, 215302 (2012).
    [Crossref]
  12. L. Huang, H. L. Guo, K. L. Li, Y. H. Chen, B. H. Feng, and Z. Y. Li, “Three dimensional force detection of gold nanoparticles using backscattered light detection,” J. Appl. Phys. 113, 113103 (2013).
    [Crossref]
  13. M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 395, 621–635 (1998).
    [Crossref]
  14. N. J. Jenness, R. M. Erb, B. B. Yellen, and R. L. Clark, “Magnetic and optical manipulation of spherical metal-coated Janus particles,” Proc. SPIE 7762, 776227 (2010).
    [Crossref]
  15. H. Wang, S. Bhaskar, J. Lahann, and Y. G. Lee, “Optical trapping of Janus particles,” Proc. SPIE 7038, 703813 (2008).
    [Crossref]
  16. S. Nedev, S. Carretero-Palacios, P. Kühler, T. Lohmüller, A. S. Urban, L. J. E. Anderson, and J. Feldmann, “An optically controlled microscale elevator using plasmonic Janus particles,” ACS Photonics 2, 491–496 (2015).
    [Crossref]
  17. F. S. Merkt, A. Erbe, and P. Leiderer, “Capped colloids as light-mills in optical traps,” New J. Phys. 8, 216 (2006).
    [Crossref]
  18. L. Baraban, R. Streubel, D. Makarov, L. Han, D. Karnaushenko, O. G. Schmidt, and G. Cuniberti, “Fuel-free locomotion of Janus motors: magnetically induced thermophoresis,” ACS Nano 7, 1360–1367 (2013).
    [Crossref]
  19. M. Yoshida and J. Lahann, “Smart nanomaterials,” ACS Nano 2, 1101–1107 (2008).
    [Crossref]
  20. T. Nisisako, T. Torii, T. Takahashi, and Y. Takizawa, “Synthesis of monodisperse bicolored Janus particles with electrical anisotropy using a microfluidic co-flow system,” Adv. Mater. 18, 1152–1156 (2006).
    [Crossref]
  21. C. J. Behrend, J. N. Anker, B. H. McNaughton, and R. Kopelman, “Microrheology with modulated optical nanoprobes (MOONs),” J. Magn. Magn. Mater. 293, 663–670 (2005).
    [Crossref]
  22. D. A. White, “Vector finite element modeling of optical tweezers,” Comput. Phys. Comm. 128, 558–564 (2000).
    [Crossref]
  23. D. C. Benito, S. H. Simpson, and S. Hanna, “FDTD simulations of forces on particles during holographic assembly,” Opt. Express 16, 2942–2957 (2008).
    [Crossref]
  24. J. Q. Qin, X. L. Wang, D. Jia, J. Chen, Y. X. Fan, J. P. Ding, and H. T. Wang, “FDTD approach to optical forces of tightly focused vector beams on metal particles,” Opt. Express 17, 8407–8416 (2009).
    [Crossref]
  25. S. H. Simpson and S. Hanna, “Application of the discrete dipole approximation to optical trapping calculations of inhomogeneous and anisotropic particles,” Opt. Express 19, 16526–16541 (2011).
    [Crossref]
  26. L. Lin, F. Zhou, L. Huang, and Z. Y. Li, “Optical forces on arbitrary shaped particles in optical tweezers,” J. Appl. Phys. 108, 073110 (2010).
    [Crossref]
  27. F. Borghese, P. Denti, and R. Saija, “Radiation torque and force on optically trapped linear nanostructures,” Phys. Rev. Lett. 100, 163903 (2008).
    [Crossref]
  28. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
    [Crossref]
  29. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 54–59 and 628–633.

2015 (1)

S. Nedev, S. Carretero-Palacios, P. Kühler, T. Lohmüller, A. S. Urban, L. J. E. Anderson, and J. Feldmann, “An optically controlled microscale elevator using plasmonic Janus particles,” ACS Photonics 2, 491–496 (2015).
[Crossref]

2014 (1)

J. Köhler, R. Ghadiri, S. I. Ksouri, Q. Guo, E. L. Gurevich, and A. Ostendorf, “Generation of microfluidic flow using an optically assembled and magnetically driven microrotor,” J. Phys. D 47, 505501 (2014).
[Crossref]

2013 (4)

H. L. Guo and Z. Y. Li, “Optical tweezers technique and its applications,” Sci. China Phys. Astro. & Mech. 56, 2351–2360 (2013).
[Crossref]

L. Huang, H. L. Guo, K. L. Li, Y. H. Chen, B. H. Feng, and Z. Y. Li, “Three dimensional force detection of gold nanoparticles using backscattered light detection,” J. Appl. Phys. 113, 113103 (2013).
[Crossref]

L. Baraban, R. Streubel, D. Makarov, L. Han, D. Karnaushenko, O. G. Schmidt, and G. Cuniberti, “Fuel-free locomotion of Janus motors: magnetically induced thermophoresis,” ACS Nano 7, 1360–1367 (2013).
[Crossref]

A. K. Rai, A. Rai, A. J. Ramaiya, R. Jha, and R. Mallik, “Molecular adaptations allow dynein to generate large collective forces inside cells,” Cell 152, 172–182 (2013).
[Crossref]

2012 (2)

S. Mohanty, “Optically-actuated translational and rotational motion at the microscale for microfluidic manipulation and characterization,” Lab Chip 12, 3624–3636 (2012).
[Crossref]

L. Lin, H. L. Guo, X. L. Zhong, L. Huang, J. F. Li, L. Gan, and Z. Y. Li, “Manipulation of gold nanorods with dual-optical tweezers for surface plasmon resonance control,” Nanotechnology 23, 215302 (2012).
[Crossref]

2011 (1)

2010 (2)

L. Lin, F. Zhou, L. Huang, and Z. Y. Li, “Optical forces on arbitrary shaped particles in optical tweezers,” J. Appl. Phys. 108, 073110 (2010).
[Crossref]

N. J. Jenness, R. M. Erb, B. B. Yellen, and R. L. Clark, “Magnetic and optical manipulation of spherical metal-coated Janus particles,” Proc. SPIE 7762, 776227 (2010).
[Crossref]

2009 (1)

2008 (5)

G. B. Liao, P. B. Bareil, Y. L. Sheng, and A. Chiou, “One-dimensional jumping optical tweezers for optical stretching of bi-concave human red blood cells,” Opt. Express 16, 1996–2004 (2008).
[Crossref]

D. C. Benito, S. H. Simpson, and S. Hanna, “FDTD simulations of forces on particles during holographic assembly,” Opt. Express 16, 2942–2957 (2008).
[Crossref]

F. Borghese, P. Denti, and R. Saija, “Radiation torque and force on optically trapped linear nanostructures,” Phys. Rev. Lett. 100, 163903 (2008).
[Crossref]

H. Wang, S. Bhaskar, J. Lahann, and Y. G. Lee, “Optical trapping of Janus particles,” Proc. SPIE 7038, 703813 (2008).
[Crossref]

M. Yoshida and J. Lahann, “Smart nanomaterials,” ACS Nano 2, 1101–1107 (2008).
[Crossref]

2007 (1)

E. Qu, H. L. Guo, C. H. Xu, Z. L. Li, M. Yuan, B. Y. Cheng, and D. Z. Zhang, “Kinetics of microtubule -AtMAP65-1 bond studied with dual-optical tweezers,” Jpn. J. Appl. Phys. 46, 7514–7518 (2007).
[Crossref]

2006 (2)

T. Nisisako, T. Torii, T. Takahashi, and Y. Takizawa, “Synthesis of monodisperse bicolored Janus particles with electrical anisotropy using a microfluidic co-flow system,” Adv. Mater. 18, 1152–1156 (2006).
[Crossref]

F. S. Merkt, A. Erbe, and P. Leiderer, “Capped colloids as light-mills in optical traps,” New J. Phys. 8, 216 (2006).
[Crossref]

2005 (1)

C. J. Behrend, J. N. Anker, B. H. McNaughton, and R. Kopelman, “Microrheology with modulated optical nanoprobes (MOONs),” J. Magn. Magn. Mater. 293, 663–670 (2005).
[Crossref]

2002 (1)

G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, and D. J. Evans, “Experimental demonstration of violations of the second law of thermodynamics for small systems and short time scales,” Phys. Rev. Lett. 89, 050601 (2002).
[Crossref]

2000 (1)

D. A. White, “Vector finite element modeling of optical tweezers,” Comput. Phys. Comm. 128, 558–564 (2000).
[Crossref]

1998 (1)

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 395, 621–635 (1998).
[Crossref]

1997 (1)

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys. J. 72, 1335–1346 (1997).
[Crossref]

1994 (1)

J. C. Crocker and D. G. Grier, “Microscopic measurement of the pair interaction potential of charge-stabilized colloid,” Phys. Rev. Lett. 73, 352–355 (1994).
[Crossref]

1992 (1)

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

1986 (1)

Anderson, L. J. E.

S. Nedev, S. Carretero-Palacios, P. Kühler, T. Lohmüller, A. S. Urban, L. J. E. Anderson, and J. Feldmann, “An optically controlled microscale elevator using plasmonic Janus particles,” ACS Photonics 2, 491–496 (2015).
[Crossref]

Anker, J. N.

C. J. Behrend, J. N. Anker, B. H. McNaughton, and R. Kopelman, “Microrheology with modulated optical nanoprobes (MOONs),” J. Magn. Magn. Mater. 293, 663–670 (2005).
[Crossref]

Ashkin, A.

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
[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, 288–290 (1986).
[Crossref]

Baraban, L.

L. Baraban, R. Streubel, D. Makarov, L. Han, D. Karnaushenko, O. G. Schmidt, and G. Cuniberti, “Fuel-free locomotion of Janus motors: magnetically induced thermophoresis,” ACS Nano 7, 1360–1367 (2013).
[Crossref]

Bareil, P. B.

Behrend, C. J.

C. J. Behrend, J. N. Anker, B. H. McNaughton, and R. Kopelman, “Microrheology with modulated optical nanoprobes (MOONs),” J. Magn. Magn. Mater. 293, 663–670 (2005).
[Crossref]

Benito, D. C.

Bhaskar, S.

H. Wang, S. Bhaskar, J. Lahann, and Y. G. Lee, “Optical trapping of Janus particles,” Proc. SPIE 7038, 703813 (2008).
[Crossref]

Bjorkholm, J. E.

Block, S. M.

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys. J. 72, 1335–1346 (1997).
[Crossref]

Borghese, F.

F. Borghese, P. Denti, and R. Saija, “Radiation torque and force on optically trapped linear nanostructures,” Phys. Rev. Lett. 100, 163903 (2008).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 54–59 and 628–633.

Carretero-Palacios, S.

S. Nedev, S. Carretero-Palacios, P. Kühler, T. Lohmüller, A. S. Urban, L. J. E. Anderson, and J. Feldmann, “An optically controlled microscale elevator using plasmonic Janus particles,” ACS Photonics 2, 491–496 (2015).
[Crossref]

Chen, J.

Chen, Y. H.

L. Huang, H. L. Guo, K. L. Li, Y. H. Chen, B. H. Feng, and Z. Y. Li, “Three dimensional force detection of gold nanoparticles using backscattered light detection,” J. Appl. Phys. 113, 113103 (2013).
[Crossref]

Cheng, B. Y.

E. Qu, H. L. Guo, C. H. Xu, Z. L. Li, M. Yuan, B. Y. Cheng, and D. Z. Zhang, “Kinetics of microtubule -AtMAP65-1 bond studied with dual-optical tweezers,” Jpn. J. Appl. Phys. 46, 7514–7518 (2007).
[Crossref]

Chiou, A.

Chu, S.

Clark, R. L.

N. J. Jenness, R. M. Erb, B. B. Yellen, and R. L. Clark, “Magnetic and optical manipulation of spherical metal-coated Janus particles,” Proc. SPIE 7762, 776227 (2010).
[Crossref]

Crocker, J. C.

J. C. Crocker and D. G. Grier, “Microscopic measurement of the pair interaction potential of charge-stabilized colloid,” Phys. Rev. Lett. 73, 352–355 (1994).
[Crossref]

Cuniberti, G.

L. Baraban, R. Streubel, D. Makarov, L. Han, D. Karnaushenko, O. G. Schmidt, and G. Cuniberti, “Fuel-free locomotion of Janus motors: magnetically induced thermophoresis,” ACS Nano 7, 1360–1367 (2013).
[Crossref]

Denti, P.

F. Borghese, P. Denti, and R. Saija, “Radiation torque and force on optically trapped linear nanostructures,” Phys. Rev. Lett. 100, 163903 (2008).
[Crossref]

Ding, J. P.

Dziedzic, J. M.

Erb, R. M.

N. J. Jenness, R. M. Erb, B. B. Yellen, and R. L. Clark, “Magnetic and optical manipulation of spherical metal-coated Janus particles,” Proc. SPIE 7762, 776227 (2010).
[Crossref]

Erbe, A.

F. S. Merkt, A. Erbe, and P. Leiderer, “Capped colloids as light-mills in optical traps,” New J. Phys. 8, 216 (2006).
[Crossref]

Evans, D. J.

G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, and D. J. Evans, “Experimental demonstration of violations of the second law of thermodynamics for small systems and short time scales,” Phys. Rev. Lett. 89, 050601 (2002).
[Crossref]

Fan, Y. X.

Feldmann, J.

S. Nedev, S. Carretero-Palacios, P. Kühler, T. Lohmüller, A. S. Urban, L. J. E. Anderson, and J. Feldmann, “An optically controlled microscale elevator using plasmonic Janus particles,” ACS Photonics 2, 491–496 (2015).
[Crossref]

Feng, B. H.

L. Huang, H. L. Guo, K. L. Li, Y. H. Chen, B. H. Feng, and Z. Y. Li, “Three dimensional force detection of gold nanoparticles using backscattered light detection,” J. Appl. Phys. 113, 113103 (2013).
[Crossref]

Friese, M. E. J.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 395, 621–635 (1998).
[Crossref]

Gan, L.

L. Lin, H. L. Guo, X. L. Zhong, L. Huang, J. F. Li, L. Gan, and Z. Y. Li, “Manipulation of gold nanorods with dual-optical tweezers for surface plasmon resonance control,” Nanotechnology 23, 215302 (2012).
[Crossref]

Gelles, J.

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys. J. 72, 1335–1346 (1997).
[Crossref]

Ghadiri, R.

J. Köhler, R. Ghadiri, S. I. Ksouri, Q. Guo, E. L. Gurevich, and A. Ostendorf, “Generation of microfluidic flow using an optically assembled and magnetically driven microrotor,” J. Phys. D 47, 505501 (2014).
[Crossref]

Grier, D. G.

J. C. Crocker and D. G. Grier, “Microscopic measurement of the pair interaction potential of charge-stabilized colloid,” Phys. Rev. Lett. 73, 352–355 (1994).
[Crossref]

Guo, H. L.

H. L. Guo and Z. Y. Li, “Optical tweezers technique and its applications,” Sci. China Phys. Astro. & Mech. 56, 2351–2360 (2013).
[Crossref]

L. Huang, H. L. Guo, K. L. Li, Y. H. Chen, B. H. Feng, and Z. Y. Li, “Three dimensional force detection of gold nanoparticles using backscattered light detection,” J. Appl. Phys. 113, 113103 (2013).
[Crossref]

L. Lin, H. L. Guo, X. L. Zhong, L. Huang, J. F. Li, L. Gan, and Z. Y. Li, “Manipulation of gold nanorods with dual-optical tweezers for surface plasmon resonance control,” Nanotechnology 23, 215302 (2012).
[Crossref]

E. Qu, H. L. Guo, C. H. Xu, Z. L. Li, M. Yuan, B. Y. Cheng, and D. Z. Zhang, “Kinetics of microtubule -AtMAP65-1 bond studied with dual-optical tweezers,” Jpn. J. Appl. Phys. 46, 7514–7518 (2007).
[Crossref]

Guo, Q.

J. Köhler, R. Ghadiri, S. I. Ksouri, Q. Guo, E. L. Gurevich, and A. Ostendorf, “Generation of microfluidic flow using an optically assembled and magnetically driven microrotor,” J. Phys. D 47, 505501 (2014).
[Crossref]

Gurevich, E. L.

J. Köhler, R. Ghadiri, S. I. Ksouri, Q. Guo, E. L. Gurevich, and A. Ostendorf, “Generation of microfluidic flow using an optically assembled and magnetically driven microrotor,” J. Phys. D 47, 505501 (2014).
[Crossref]

Han, L.

L. Baraban, R. Streubel, D. Makarov, L. Han, D. Karnaushenko, O. G. Schmidt, and G. Cuniberti, “Fuel-free locomotion of Janus motors: magnetically induced thermophoresis,” ACS Nano 7, 1360–1367 (2013).
[Crossref]

Hanna, S.

Heckenberg, N. R.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 395, 621–635 (1998).
[Crossref]

Huang, L.

L. Huang, H. L. Guo, K. L. Li, Y. H. Chen, B. H. Feng, and Z. Y. Li, “Three dimensional force detection of gold nanoparticles using backscattered light detection,” J. Appl. Phys. 113, 113103 (2013).
[Crossref]

L. Lin, H. L. Guo, X. L. Zhong, L. Huang, J. F. Li, L. Gan, and Z. Y. Li, “Manipulation of gold nanorods with dual-optical tweezers for surface plasmon resonance control,” Nanotechnology 23, 215302 (2012).
[Crossref]

L. Lin, F. Zhou, L. Huang, and Z. Y. Li, “Optical forces on arbitrary shaped particles in optical tweezers,” J. Appl. Phys. 108, 073110 (2010).
[Crossref]

Jenness, N. J.

N. J. Jenness, R. M. Erb, B. B. Yellen, and R. L. Clark, “Magnetic and optical manipulation of spherical metal-coated Janus particles,” Proc. SPIE 7762, 776227 (2010).
[Crossref]

Jha, R.

A. K. Rai, A. Rai, A. J. Ramaiya, R. Jha, and R. Mallik, “Molecular adaptations allow dynein to generate large collective forces inside cells,” Cell 152, 172–182 (2013).
[Crossref]

Jia, D.

Karnaushenko, D.

L. Baraban, R. Streubel, D. Makarov, L. Han, D. Karnaushenko, O. G. Schmidt, and G. Cuniberti, “Fuel-free locomotion of Janus motors: magnetically induced thermophoresis,” ACS Nano 7, 1360–1367 (2013).
[Crossref]

Köhler, J.

J. Köhler, R. Ghadiri, S. I. Ksouri, Q. Guo, E. L. Gurevich, and A. Ostendorf, “Generation of microfluidic flow using an optically assembled and magnetically driven microrotor,” J. Phys. D 47, 505501 (2014).
[Crossref]

Kopelman, R.

C. J. Behrend, J. N. Anker, B. H. McNaughton, and R. Kopelman, “Microrheology with modulated optical nanoprobes (MOONs),” J. Magn. Magn. Mater. 293, 663–670 (2005).
[Crossref]

Ksouri, S. I.

J. Köhler, R. Ghadiri, S. I. Ksouri, Q. Guo, E. L. Gurevich, and A. Ostendorf, “Generation of microfluidic flow using an optically assembled and magnetically driven microrotor,” J. Phys. D 47, 505501 (2014).
[Crossref]

Kühler, P.

S. Nedev, S. Carretero-Palacios, P. Kühler, T. Lohmüller, A. S. Urban, L. J. E. Anderson, and J. Feldmann, “An optically controlled microscale elevator using plasmonic Janus particles,” ACS Photonics 2, 491–496 (2015).
[Crossref]

Lahann, J.

H. Wang, S. Bhaskar, J. Lahann, and Y. G. Lee, “Optical trapping of Janus particles,” Proc. SPIE 7038, 703813 (2008).
[Crossref]

M. Yoshida and J. Lahann, “Smart nanomaterials,” ACS Nano 2, 1101–1107 (2008).
[Crossref]

Landick, R.

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys. J. 72, 1335–1346 (1997).
[Crossref]

Lee, Y. G.

H. Wang, S. Bhaskar, J. Lahann, and Y. G. Lee, “Optical trapping of Janus particles,” Proc. SPIE 7038, 703813 (2008).
[Crossref]

Leiderer, P.

F. S. Merkt, A. Erbe, and P. Leiderer, “Capped colloids as light-mills in optical traps,” New J. Phys. 8, 216 (2006).
[Crossref]

Li, J. F.

L. Lin, H. L. Guo, X. L. Zhong, L. Huang, J. F. Li, L. Gan, and Z. Y. Li, “Manipulation of gold nanorods with dual-optical tweezers for surface plasmon resonance control,” Nanotechnology 23, 215302 (2012).
[Crossref]

Li, K. L.

L. Huang, H. L. Guo, K. L. Li, Y. H. Chen, B. H. Feng, and Z. Y. Li, “Three dimensional force detection of gold nanoparticles using backscattered light detection,” J. Appl. Phys. 113, 113103 (2013).
[Crossref]

Li, Z. L.

E. Qu, H. L. Guo, C. H. Xu, Z. L. Li, M. Yuan, B. Y. Cheng, and D. Z. Zhang, “Kinetics of microtubule -AtMAP65-1 bond studied with dual-optical tweezers,” Jpn. J. Appl. Phys. 46, 7514–7518 (2007).
[Crossref]

Li, Z. Y.

H. L. Guo and Z. Y. Li, “Optical tweezers technique and its applications,” Sci. China Phys. Astro. & Mech. 56, 2351–2360 (2013).
[Crossref]

L. Huang, H. L. Guo, K. L. Li, Y. H. Chen, B. H. Feng, and Z. Y. Li, “Three dimensional force detection of gold nanoparticles using backscattered light detection,” J. Appl. Phys. 113, 113103 (2013).
[Crossref]

L. Lin, H. L. Guo, X. L. Zhong, L. Huang, J. F. Li, L. Gan, and Z. Y. Li, “Manipulation of gold nanorods with dual-optical tweezers for surface plasmon resonance control,” Nanotechnology 23, 215302 (2012).
[Crossref]

L. Lin, F. Zhou, L. Huang, and Z. Y. Li, “Optical forces on arbitrary shaped particles in optical tweezers,” J. Appl. Phys. 108, 073110 (2010).
[Crossref]

Liao, G. B.

Lin, L.

L. Lin, H. L. Guo, X. L. Zhong, L. Huang, J. F. Li, L. Gan, and Z. Y. Li, “Manipulation of gold nanorods with dual-optical tweezers for surface plasmon resonance control,” Nanotechnology 23, 215302 (2012).
[Crossref]

L. Lin, F. Zhou, L. Huang, and Z. Y. Li, “Optical forces on arbitrary shaped particles in optical tweezers,” J. Appl. Phys. 108, 073110 (2010).
[Crossref]

Lohmüller, T.

S. Nedev, S. Carretero-Palacios, P. Kühler, T. Lohmüller, A. S. Urban, L. J. E. Anderson, and J. Feldmann, “An optically controlled microscale elevator using plasmonic Janus particles,” ACS Photonics 2, 491–496 (2015).
[Crossref]

Makarov, D.

L. Baraban, R. Streubel, D. Makarov, L. Han, D. Karnaushenko, O. G. Schmidt, and G. Cuniberti, “Fuel-free locomotion of Janus motors: magnetically induced thermophoresis,” ACS Nano 7, 1360–1367 (2013).
[Crossref]

Mallik, R.

A. K. Rai, A. Rai, A. J. Ramaiya, R. Jha, and R. Mallik, “Molecular adaptations allow dynein to generate large collective forces inside cells,” Cell 152, 172–182 (2013).
[Crossref]

McNaughton, B. H.

C. J. Behrend, J. N. Anker, B. H. McNaughton, and R. Kopelman, “Microrheology with modulated optical nanoprobes (MOONs),” J. Magn. Magn. Mater. 293, 663–670 (2005).
[Crossref]

Merkt, F. S.

F. S. Merkt, A. Erbe, and P. Leiderer, “Capped colloids as light-mills in optical traps,” New J. Phys. 8, 216 (2006).
[Crossref]

Mittag, E.

G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, and D. J. Evans, “Experimental demonstration of violations of the second law of thermodynamics for small systems and short time scales,” Phys. Rev. Lett. 89, 050601 (2002).
[Crossref]

Mohanty, S.

S. Mohanty, “Optically-actuated translational and rotational motion at the microscale for microfluidic manipulation and characterization,” Lab Chip 12, 3624–3636 (2012).
[Crossref]

Nedev, S.

S. Nedev, S. Carretero-Palacios, P. Kühler, T. Lohmüller, A. S. Urban, L. J. E. Anderson, and J. Feldmann, “An optically controlled microscale elevator using plasmonic Janus particles,” ACS Photonics 2, 491–496 (2015).
[Crossref]

Nieminen, T. A.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 395, 621–635 (1998).
[Crossref]

Nisisako, T.

T. Nisisako, T. Torii, T. Takahashi, and Y. Takizawa, “Synthesis of monodisperse bicolored Janus particles with electrical anisotropy using a microfluidic co-flow system,” Adv. Mater. 18, 1152–1156 (2006).
[Crossref]

Ostendorf, A.

J. Köhler, R. Ghadiri, S. I. Ksouri, Q. Guo, E. L. Gurevich, and A. Ostendorf, “Generation of microfluidic flow using an optically assembled and magnetically driven microrotor,” J. Phys. D 47, 505501 (2014).
[Crossref]

Qin, J. Q.

Qu, E.

E. Qu, H. L. Guo, C. H. Xu, Z. L. Li, M. Yuan, B. Y. Cheng, and D. Z. Zhang, “Kinetics of microtubule -AtMAP65-1 bond studied with dual-optical tweezers,” Jpn. J. Appl. Phys. 46, 7514–7518 (2007).
[Crossref]

Rai, A.

A. K. Rai, A. Rai, A. J. Ramaiya, R. Jha, and R. Mallik, “Molecular adaptations allow dynein to generate large collective forces inside cells,” Cell 152, 172–182 (2013).
[Crossref]

Rai, A. K.

A. K. Rai, A. Rai, A. J. Ramaiya, R. Jha, and R. Mallik, “Molecular adaptations allow dynein to generate large collective forces inside cells,” Cell 152, 172–182 (2013).
[Crossref]

Ramaiya, A. J.

A. K. Rai, A. Rai, A. J. Ramaiya, R. Jha, and R. Mallik, “Molecular adaptations allow dynein to generate large collective forces inside cells,” Cell 152, 172–182 (2013).
[Crossref]

Rubinsztein-Dunlop, H.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 395, 621–635 (1998).
[Crossref]

Saija, R.

F. Borghese, P. Denti, and R. Saija, “Radiation torque and force on optically trapped linear nanostructures,” Phys. Rev. Lett. 100, 163903 (2008).
[Crossref]

Schmidt, O. G.

L. Baraban, R. Streubel, D. Makarov, L. Han, D. Karnaushenko, O. G. Schmidt, and G. Cuniberti, “Fuel-free locomotion of Janus motors: magnetically induced thermophoresis,” ACS Nano 7, 1360–1367 (2013).
[Crossref]

Searles, D. J.

G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, and D. J. Evans, “Experimental demonstration of violations of the second law of thermodynamics for small systems and short time scales,” Phys. Rev. Lett. 89, 050601 (2002).
[Crossref]

Sevick, E. M.

G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, and D. J. Evans, “Experimental demonstration of violations of the second law of thermodynamics for small systems and short time scales,” Phys. Rev. Lett. 89, 050601 (2002).
[Crossref]

Sheng, Y. L.

Simpson, S. H.

Streubel, R.

L. Baraban, R. Streubel, D. Makarov, L. Han, D. Karnaushenko, O. G. Schmidt, and G. Cuniberti, “Fuel-free locomotion of Janus motors: magnetically induced thermophoresis,” ACS Nano 7, 1360–1367 (2013).
[Crossref]

Takahashi, T.

T. Nisisako, T. Torii, T. Takahashi, and Y. Takizawa, “Synthesis of monodisperse bicolored Janus particles with electrical anisotropy using a microfluidic co-flow system,” Adv. Mater. 18, 1152–1156 (2006).
[Crossref]

Takizawa, Y.

T. Nisisako, T. Torii, T. Takahashi, and Y. Takizawa, “Synthesis of monodisperse bicolored Janus particles with electrical anisotropy using a microfluidic co-flow system,” Adv. Mater. 18, 1152–1156 (2006).
[Crossref]

Torii, T.

T. Nisisako, T. Torii, T. Takahashi, and Y. Takizawa, “Synthesis of monodisperse bicolored Janus particles with electrical anisotropy using a microfluidic co-flow system,” Adv. Mater. 18, 1152–1156 (2006).
[Crossref]

Urban, A. S.

S. Nedev, S. Carretero-Palacios, P. Kühler, T. Lohmüller, A. S. Urban, L. J. E. Anderson, and J. Feldmann, “An optically controlled microscale elevator using plasmonic Janus particles,” ACS Photonics 2, 491–496 (2015).
[Crossref]

Wang, G. M.

G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, and D. J. Evans, “Experimental demonstration of violations of the second law of thermodynamics for small systems and short time scales,” Phys. Rev. Lett. 89, 050601 (2002).
[Crossref]

Wang, H.

H. Wang, S. Bhaskar, J. Lahann, and Y. G. Lee, “Optical trapping of Janus particles,” Proc. SPIE 7038, 703813 (2008).
[Crossref]

Wang, H. T.

Wang, M. D.

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys. J. 72, 1335–1346 (1997).
[Crossref]

Wang, X. L.

White, D. A.

D. A. White, “Vector finite element modeling of optical tweezers,” Comput. Phys. Comm. 128, 558–564 (2000).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 54–59 and 628–633.

Xu, C. H.

E. Qu, H. L. Guo, C. H. Xu, Z. L. Li, M. Yuan, B. Y. Cheng, and D. Z. Zhang, “Kinetics of microtubule -AtMAP65-1 bond studied with dual-optical tweezers,” Jpn. J. Appl. Phys. 46, 7514–7518 (2007).
[Crossref]

Yellen, B. B.

N. J. Jenness, R. M. Erb, B. B. Yellen, and R. L. Clark, “Magnetic and optical manipulation of spherical metal-coated Janus particles,” Proc. SPIE 7762, 776227 (2010).
[Crossref]

Yin, H.

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys. J. 72, 1335–1346 (1997).
[Crossref]

Yoshida, M.

M. Yoshida and J. Lahann, “Smart nanomaterials,” ACS Nano 2, 1101–1107 (2008).
[Crossref]

Yuan, M.

E. Qu, H. L. Guo, C. H. Xu, Z. L. Li, M. Yuan, B. Y. Cheng, and D. Z. Zhang, “Kinetics of microtubule -AtMAP65-1 bond studied with dual-optical tweezers,” Jpn. J. Appl. Phys. 46, 7514–7518 (2007).
[Crossref]

Zhang, D. Z.

E. Qu, H. L. Guo, C. H. Xu, Z. L. Li, M. Yuan, B. Y. Cheng, and D. Z. Zhang, “Kinetics of microtubule -AtMAP65-1 bond studied with dual-optical tweezers,” Jpn. J. Appl. Phys. 46, 7514–7518 (2007).
[Crossref]

Zhong, X. L.

L. Lin, H. L. Guo, X. L. Zhong, L. Huang, J. F. Li, L. Gan, and Z. Y. Li, “Manipulation of gold nanorods with dual-optical tweezers for surface plasmon resonance control,” Nanotechnology 23, 215302 (2012).
[Crossref]

Zhou, F.

L. Lin, F. Zhou, L. Huang, and Z. Y. Li, “Optical forces on arbitrary shaped particles in optical tweezers,” J. Appl. Phys. 108, 073110 (2010).
[Crossref]

ACS Nano (2)

L. Baraban, R. Streubel, D. Makarov, L. Han, D. Karnaushenko, O. G. Schmidt, and G. Cuniberti, “Fuel-free locomotion of Janus motors: magnetically induced thermophoresis,” ACS Nano 7, 1360–1367 (2013).
[Crossref]

M. Yoshida and J. Lahann, “Smart nanomaterials,” ACS Nano 2, 1101–1107 (2008).
[Crossref]

ACS Photonics (1)

S. Nedev, S. Carretero-Palacios, P. Kühler, T. Lohmüller, A. S. Urban, L. J. E. Anderson, and J. Feldmann, “An optically controlled microscale elevator using plasmonic Janus particles,” ACS Photonics 2, 491–496 (2015).
[Crossref]

Adv. Mater. (1)

T. Nisisako, T. Torii, T. Takahashi, and Y. Takizawa, “Synthesis of monodisperse bicolored Janus particles with electrical anisotropy using a microfluidic co-flow system,” Adv. Mater. 18, 1152–1156 (2006).
[Crossref]

Biophys. J. (2)

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys. J. 72, 1335–1346 (1997).
[Crossref]

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

Cell (1)

A. K. Rai, A. Rai, A. J. Ramaiya, R. Jha, and R. Mallik, “Molecular adaptations allow dynein to generate large collective forces inside cells,” Cell 152, 172–182 (2013).
[Crossref]

Comput. Phys. Comm. (1)

D. A. White, “Vector finite element modeling of optical tweezers,” Comput. Phys. Comm. 128, 558–564 (2000).
[Crossref]

J. Appl. Phys. (2)

L. Lin, F. Zhou, L. Huang, and Z. Y. Li, “Optical forces on arbitrary shaped particles in optical tweezers,” J. Appl. Phys. 108, 073110 (2010).
[Crossref]

L. Huang, H. L. Guo, K. L. Li, Y. H. Chen, B. H. Feng, and Z. Y. Li, “Three dimensional force detection of gold nanoparticles using backscattered light detection,” J. Appl. Phys. 113, 113103 (2013).
[Crossref]

J. Magn. Magn. Mater. (1)

C. J. Behrend, J. N. Anker, B. H. McNaughton, and R. Kopelman, “Microrheology with modulated optical nanoprobes (MOONs),” J. Magn. Magn. Mater. 293, 663–670 (2005).
[Crossref]

J. Phys. D (1)

J. Köhler, R. Ghadiri, S. I. Ksouri, Q. Guo, E. L. Gurevich, and A. Ostendorf, “Generation of microfluidic flow using an optically assembled and magnetically driven microrotor,” J. Phys. D 47, 505501 (2014).
[Crossref]

Jpn. J. Appl. Phys. (1)

E. Qu, H. L. Guo, C. H. Xu, Z. L. Li, M. Yuan, B. Y. Cheng, and D. Z. Zhang, “Kinetics of microtubule -AtMAP65-1 bond studied with dual-optical tweezers,” Jpn. J. Appl. Phys. 46, 7514–7518 (2007).
[Crossref]

Lab Chip (1)

S. Mohanty, “Optically-actuated translational and rotational motion at the microscale for microfluidic manipulation and characterization,” Lab Chip 12, 3624–3636 (2012).
[Crossref]

Nanotechnology (1)

L. Lin, H. L. Guo, X. L. Zhong, L. Huang, J. F. Li, L. Gan, and Z. Y. Li, “Manipulation of gold nanorods with dual-optical tweezers for surface plasmon resonance control,” Nanotechnology 23, 215302 (2012).
[Crossref]

Nature (1)

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 395, 621–635 (1998).
[Crossref]

New J. Phys. (1)

F. S. Merkt, A. Erbe, and P. Leiderer, “Capped colloids as light-mills in optical traps,” New J. Phys. 8, 216 (2006).
[Crossref]

Opt. Express (4)

Opt. Lett. (1)

Phys. Rev. Lett. (3)

F. Borghese, P. Denti, and R. Saija, “Radiation torque and force on optically trapped linear nanostructures,” Phys. Rev. Lett. 100, 163903 (2008).
[Crossref]

J. C. Crocker and D. G. Grier, “Microscopic measurement of the pair interaction potential of charge-stabilized colloid,” Phys. Rev. Lett. 73, 352–355 (1994).
[Crossref]

G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, and D. J. Evans, “Experimental demonstration of violations of the second law of thermodynamics for small systems and short time scales,” Phys. Rev. Lett. 89, 050601 (2002).
[Crossref]

Proc. SPIE (2)

N. J. Jenness, R. M. Erb, B. B. Yellen, and R. L. Clark, “Magnetic and optical manipulation of spherical metal-coated Janus particles,” Proc. SPIE 7762, 776227 (2010).
[Crossref]

H. Wang, S. Bhaskar, J. Lahann, and Y. G. Lee, “Optical trapping of Janus particles,” Proc. SPIE 7038, 703813 (2008).
[Crossref]

Sci. China Phys. Astro. & Mech. (1)

H. L. Guo and Z. Y. Li, “Optical tweezers technique and its applications,” Sci. China Phys. Astro. & Mech. 56, 2351–2360 (2013).
[Crossref]

Other (1)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 54–59 and 628–633.

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Figures (7)

Fig. 1.
Fig. 1. Diagram of ray-optics model used to handle trapping and motion of Janus particles in a tightly focused Gaussian laser beam. (a) 3D stereogram of the Janus particle made from a PS bead half-coated with thin gold film. The Au–PS separation plane of the particle can rotate around the axis of laser beam (z axis, denoted by the angle γ), and the orientation of its surface normal vector with respect to the laser beam axis can change (denoted as by the angle α). (b) Schematic diagram of the cross-sectional geometry of the Janus particle in the plane of x=0. The gold thin film thickness reduces gradually from the top of the PS bead (the y-axis direction) to the separation plane (the xz plane). (c) Diagram of a Gaussian laser beam tightly focused by a high-NA microscope objective lens with pupil radius rmax and illuminating the Janus particle for optical trapping and manipulation. In the ray-optics model, the laser beam is decomposed into a large amount of rays of light denoted by their directional unit vectors k⃗1,k⃗2,k⃗3,,k⃗n,. The ki ray intersects with the objective pupil at the position with polar coordinate r⃗i=(r,φ). (d) Diagram of the ray tracking for a specific ray k⃗1 within the Janus particle, where multiple events of ray reflection and refraction will take place at the positions a⃗1, a⃗2, a⃗3, etc. Momentum exchange and transfer at each event can be calculated to yield the optical force and torque, all of which will sum up to yield the net total force and torque by the laser beam upon the Janus particle.
Fig. 2.
Fig. 2. Sketch of the propagation of a ray on (a) the PS surface and (b) the Au surface. Reflection and refraction of the ray happen, and the corresponding intensity reflection and transmission coefficients can be calculated based on Fresnel’s formulae.
Fig. 3.
Fig. 3. (a) Calculated intensity reflection and transmission coefficients as a function of the incident angle θ1 on the gold thin film layer with the thickness of 0, 2, and 4 nm, respectively. (b) Calculated optical torques of laser beam upon the Janus particle as a function of the orientation angle α of the Au–PS separation plane.
Fig. 4.
Fig. 4. Comparison of the optical force and torque on a Janus particle with those on a PS particle imposed by the focused laser beam. (a) Sketch of Qx(Sx/rs). (b) Sketch of Qy(Sy/rs). (c) Sketch of Qz(Sz/rs). (d) Sketch of M(Sx/rs). (e) Sketch of M(Sy/rs). (f) Sketch of M(Sz/rs).
Fig. 5.
Fig. 5. (a) 3D stereogram of the asymmetric Janus particle with its separation plane rotating around the axis of a laser beam denoted by the angle γ. (b) Calculated equilibrium positions of the Janus particle in the xy plane as γ varies, which are located at a circle 0.107 rs in distance from the axis of the laser beam. (c) Plane graph of the Janus particle posture with respect to the laser beam axis when γ=0°,45°,90°,180°. The corresponding equilibrium positions of the particle are explicitly marked in panel (b) by colored dot signals. In each case, the equilibrium position, which is also the center of the Janus particle, when measured with respect to the laser beam axis (denoted by the red-cross signal), remains fixed with the same later offset distance. This further confirms the result of panel (b) that the center of the Janus particle will rotate around the laser beam axis in a fixed distance.
Fig. 6.
Fig. 6. Calculated distance of the equilibrium positions of the Janus particle away from the laser beam axis with variation of the thickness of gold film.
Fig. 7.
Fig. 7. Experimental results of Janus particle trapping and motion within an optical tweezers produced from a Gaussian laser beam focused by a NA=1.4 microscope objective lens. (a) Typical optical microscopy image of a Janus particle observed in experiments. The separation plane of the PS side (bright part) and gold side (dark part) particle can be clearly recognized, indicating that the Au–PS separation plane is dominantly parallel to the laser beam axis, which is normal to the paper sheet. (b) Recorded positions of the Janus particle center relative to the laser beam axis (where the optical trap center resides), which is set to locate at the origin of coordinates. The blue curve is the fitting averaged circular orbit where the center of the Janus particle resides when it rotates around the laser beam axis.

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

RE=[n1sinθ1n3cosθ3n1sinθ1+n3cosθ3]2,TE=n3cosθ3n1cosθ1[2n1sinθ1n1sinθ1+n3cosθ3]2.
RM=[n3cosθ1n1cosθ3n3cosθ1+n1cosθ3]2,TM=n1cosθ3n3cosθ1[2n3cosθ1n3cosθ1+n1cosθ3]2.
RE=|r|2=ρ122e2v2η+ρ232e2v2η+2ρ12ρ23cos(ϕ23ϕ12+2u2η)e2v2η+ρ122ρ232e2v2η+2ρ12ρ23cos(ϕ23+ϕ12+2u2η),
TE=n3cosθ3n1cosθ1|t|2=n3cosθ3n1cosθ1τ122τ232e2v2η1+ρ122ρ232e4v2η+2ρ12ρ23e2v2ηcos(ϕ23+ϕ12+2u2η).
ρ122=(n1cosθ1u2)2+v22(n1cosθ1+u2)2+v22,tanϕ12=2v2n1cosθ1u22+v22n12cos2θ1,ρ232=(n3cosθ3u2)2+v22(n3cosθ3+u2)2+v22,tanϕ23=2v2n3cosθ3u22+v22n32cos2θ3.
RM=|r|2=ρ122e2v2η+ρ232e2v2η+2ρ12ρ23cos(ϕ23ϕ12+2u2η)e2v2η+ρ122ρ232e2v2η+2ρ12ρ23cos(ϕ23+ϕ12+2u2η),
TM=n1cosθ3n3cosθ1|t|2=n1cosθ3n3cosθ1τ122τ232e2v2η1+ρ122ρ232e4v2η+2ρ12ρ23e2v2ηcos(ϕ23+ϕ12+2u2η).
ρ122=[n22(1κ22)cosθ1n1u2]2+[2n22κ2cosθ1n1v2]2[n22(1κ22)cosθ1+n1u2]2+[2n22κ2cosθ1+n1v2]2,tanϕ12=2n1n22cosθ12κ2u2(1κ22)v2n24(1+κ22)2cos2θ1n12(u22+v22),τ122=4n24(1+κ22)2cos2θ1[n22(1κ22)cosθ1+n1u2]2+[2n22κ2cosθ1+n1v2]2,ρ232=[n22(1κ22)cosθ3n3u2]2+[2n22κ2cosθ3n3v2]2[n22(1κ22)cosθ3+n3u2]2+[2n22κ2cosθ3+n3v2]2,tanϕ23=2n3n22cosθ32κ2u2(1κ22)v2n24(1+κ22)2cos2θ3n32(u22+v22),τ232=4n32(v22+u22)[n3u2+n22(1κ22)cosθ3]2+(n3v2+2n22κ2cosθ3)2.
F⃗k1=Pn1c(k⃗1k⃗1rR1k⃗2tT1T2n=3k⃗ntT1Tn(R2·R3Rn1))=Pn1cQ⃗k1.
F⃗total=i=1NWkiF⃗ki=Pn1ci=1NWkiQ⃗ki.
τ⃗k1=Pn1c[(a⃗1S⃗)×k⃗1(a⃗2S⃗)×k⃗1rR1(a⃗3S⃗)×k⃗1rk⃗2tT1T2n=3(a⃗nS⃗)×k⃗ntT1Tn(R2·R3Rn1)]=Pn1cM⃗k1,
τ⃗total=i=1NWkiτ⃗ki=Pn1ci=1NWkiM⃗ki.
P=PtotalrΔφ·Δrπrmax2
F⃗total=Ptotaln1ci=1NWkiQ⃗kirkiΔφ·Δrπrmax2,
τ⃗total=Ptotaln1ci=1NWkiM⃗kirkiΔφ·Δrπrmax2.

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