Abstract

The axial and transverse optical forces exerted by Gaussian beams on an arbitrarily oriented and homogeneous spheroid are calculated and studied within the framework of the Mie theory. The results are applied to study the behavior of the forces in a counterpropagating optical trap. We calculate the trapping efficiencies for a wide range of physical parameters, including the beam waist separation distance, the equivalent spheroid radius, the spheroid eccentricity, and the refractive index ratio between the particle and the surrounding medium.

© 2009 Optical Society of America

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  1. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156-159 (1970).
    [CrossRef]
  2. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of s single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288-290 (1986).
    [CrossRef] [PubMed]
  3. 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] [PubMed]
  4. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810-816 (2003).
    [CrossRef] [PubMed]
  5. K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787-2804 (2004).
    [CrossRef]
  6. K. Dholakia, P. Reece, and M. Gu, “Optical micromanipulation,” Chem. Soc. Rev. 37, 42-55 (2008).
    [CrossRef] [PubMed]
  7. H. A. Pohl, Dielectrophoresis: The Behavior of Neutral Matter in Non-Uniform Electric Fields (Cambridge Univ. Press, 1978).
  8. N. G. Green and H. Morgan, “Dielectrophoretic investigations of sub-micrometre latex spheres,” J. Phys. D 30, 2626-2633 (1997).
    [CrossRef]
  9. G. Roosen and C. Imbert, “Optical levitation by means of two horizontal laser beams: a theoretical and experimental study,” Phys. Lett. A 59, 6-8 (1976).
    [CrossRef]
  10. E. Sidick, S. D. Collins, and A. Knoesen, “Trapping forces in a multiple-beam fiber-optic trap,” Appl. Opt. 36, 6423-6433 (1997).
    [CrossRef]
  11. A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical application and measurement of torque on microparticles of isotropic nonabsorbing material,” Phys. Rev. A 68, 033802 (2003).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  15. F. Xu, K. Ren, G. Gouesbet, X. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E 75, 026613 (2007).
    [CrossRef]
  16. F. Xu, K. Ren, G. Gouesbet, G. Gréhan, and X. Cai, “Generalized Lorenz-Mie theory for an arbitrarily oriented, located, and shaped beam scattered by a homogeneous spheroid,” J. Opt. Soc. Am. A 24, 119-131 (2007).
    [CrossRef]
  17. F. Xu, J. A. Lock, G. Gouesbet, and C. Tropea, “Radiation torque exerted on a spheroid: analytical solution,” Phys. Rev. E 78, 013843 (2008).
  18. Y. Han and Z. Wu, “Scattering of a spheroidal particle illuminated by a Gaussian beam,” Appl. Opt. 40, 2501-2509 (2001).
    [CrossRef]
  19. S. H. Simpson and S. Hanna, “Optical trapping of spheroidal particles in Gaussian beams,” J. Opt. Soc. Am. A 24, 430-443 (2007).
    [CrossRef]
  20. R. W. Going, B. L. Conover, and M. J. Escuti, “Electrostatic force and torque escription of generalized spheroidal particles in optical landscapes,” Proc. SPIE 7038, 703826 (2008).
    [CrossRef]
  21. T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, “Numerical modelling of optical trapping,” Comput. Phys. Commun. 142, 468-471 (2001).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  24. D. Rudd, C. López-Mariscal, M. Summers, A. Shahvisi, J. C. Gutiérrez-Vega, and D. McGloin, “Fiber based optical trapping of aerosols,” Opt. Express 16, 14550-14560 (2008).
    [CrossRef] [PubMed]
  25. M. Guillon, O. Moine, and B. Stout, “Longitudinal optical binding of high optical contrast microdroplets in air,” Phys. Rev. Lett. 96, 143902 (2006).
    [CrossRef] [PubMed]
  26. M. Guillon, O. Moine, and B. Stout, “Erratum,” Phys. Rev. Lett. 99, 079901 (2007).
    [CrossRef]
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    [CrossRef] [PubMed]
  28. J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81, 767-784 (2001).
    [CrossRef] [PubMed]

2008

K. Dholakia, P. Reece, and M. Gu, “Optical micromanipulation,” Chem. Soc. Rev. 37, 42-55 (2008).
[CrossRef] [PubMed]

F. Xu, J. A. Lock, G. Gouesbet, and C. Tropea, “Radiation torque exerted on a spheroid: analytical solution,” Phys. Rev. E 78, 013843 (2008).

R. W. Going, B. L. Conover, and M. J. Escuti, “Electrostatic force and torque escription of generalized spheroidal particles in optical landscapes,” Proc. SPIE 7038, 703826 (2008).
[CrossRef]

D. Rudd, C. López-Mariscal, M. Summers, A. Shahvisi, J. C. Gutiérrez-Vega, and D. McGloin, “Fiber based optical trapping of aerosols,” Opt. Express 16, 14550-14560 (2008).
[CrossRef] [PubMed]

2007

M. Guillon, O. Moine, and B. Stout, “Erratum,” Phys. Rev. Lett. 99, 079901 (2007).
[CrossRef]

S. H. Simpson and S. Hanna, “Optical trapping of spheroidal particles in Gaussian beams,” J. Opt. Soc. Am. A 24, 430-443 (2007).
[CrossRef]

F. Xu, K. Ren, G. Gouesbet, X. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E 75, 026613 (2007).
[CrossRef]

F. Xu, K. Ren, G. Gouesbet, G. Gréhan, and X. Cai, “Generalized Lorenz-Mie theory for an arbitrarily oriented, located, and shaped beam scattered by a homogeneous spheroid,” J. Opt. Soc. Am. A 24, 119-131 (2007).
[CrossRef]

2006

2004

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787-2804 (2004).
[CrossRef]

2003

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

A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical application and measurement of torque on microparticles of isotropic nonabsorbing material,” Phys. Rev. A 68, 033802 (2003).
[CrossRef]

Y. Han, G. Gréhan, and G. Gouesbet, “Generalized Lorenz-Mie theory for a spheroidal particle with off-axis Gaussian-beam illumination,” Appl. Opt. 42, 6621-6629 (2003).
[CrossRef] [PubMed]

2001

Y. Han and Z. Wu, “Scattering of a spheroidal particle illuminated by a Gaussian beam,” Appl. Opt. 40, 2501-2509 (2001).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, “Numerical modelling of optical trapping,” Comput. Phys. Commun. 142, 468-471 (2001).
[CrossRef]

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81, 767-784 (2001).
[CrossRef] [PubMed]

1997

E. Sidick, S. D. Collins, and A. Knoesen, “Trapping forces in a multiple-beam fiber-optic trap,” Appl. Opt. 36, 6423-6433 (1997).
[CrossRef]

N. G. Green and H. Morgan, “Dielectrophoretic investigations of sub-micrometre latex spheres,” J. Phys. D 30, 2626-2633 (1997).
[CrossRef]

1993

1992

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] [PubMed]

1991

1986

1979

1976

G. Roosen and C. Imbert, “Optical levitation by means of two horizontal laser beams: a theoretical and experimental study,” Phys. Lett. A 59, 6-8 (1976).
[CrossRef]

1975

1970

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

Ananthakrishnan, R.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81, 767-784 (2001).
[CrossRef] [PubMed]

Asano, S.

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] [PubMed]

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of s single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288-290 (1986).
[CrossRef] [PubMed]

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

Bishop, A. I.

A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical application and measurement of torque on microparticles of isotropic nonabsorbing material,” Phys. Rev. A 68, 033802 (2003).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, “Numerical modelling of optical trapping,” Comput. Phys. Commun. 142, 468-471 (2001).
[CrossRef]

Bjorkholm, J. E.

Block, S. M.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787-2804 (2004).
[CrossRef]

Cai, X.

Chu, S.

Collins, S. D.

Conover, B. L.

R. W. Going, B. L. Conover, and M. J. Escuti, “Electrostatic force and torque escription of generalized spheroidal particles in optical landscapes,” Proc. SPIE 7038, 703826 (2008).
[CrossRef]

Constable, A.

Cunningham, C. C.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81, 767-784 (2001).
[CrossRef] [PubMed]

Dholakia, K.

K. Dholakia, P. Reece, and M. Gu, “Optical micromanipulation,” Chem. Soc. Rev. 37, 42-55 (2008).
[CrossRef] [PubMed]

Dziedzic, J. M.

Escuti, M. J.

R. W. Going, B. L. Conover, and M. J. Escuti, “Electrostatic force and torque escription of generalized spheroidal particles in optical landscapes,” Proc. SPIE 7038, 703826 (2008).
[CrossRef]

Fang Ren, K.

Going, R. W.

R. W. Going, B. L. Conover, and M. J. Escuti, “Electrostatic force and torque escription of generalized spheroidal particles in optical landscapes,” Proc. SPIE 7038, 703826 (2008).
[CrossRef]

Gouesbet, G.

F. Xu, J. A. Lock, G. Gouesbet, and C. Tropea, “Radiation torque exerted on a spheroid: analytical solution,” Phys. Rev. E 78, 013843 (2008).

F. Xu, K. Ren, G. Gouesbet, X. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E 75, 026613 (2007).
[CrossRef]

F. Xu, K. Ren, G. Gouesbet, G. Gréhan, and X. Cai, “Generalized Lorenz-Mie theory for an arbitrarily oriented, located, and shaped beam scattered by a homogeneous spheroid,” J. Opt. Soc. Am. A 24, 119-131 (2007).
[CrossRef]

Y. Han, G. Gréhan, and G. Gouesbet, “Generalized Lorenz-Mie theory for a spheroidal particle with off-axis Gaussian-beam illumination,” Appl. Opt. 42, 6621-6629 (2003).
[CrossRef] [PubMed]

Green, N. G.

N. G. Green and H. Morgan, “Dielectrophoretic investigations of sub-micrometre latex spheres,” J. Phys. D 30, 2626-2633 (1997).
[CrossRef]

Gréhan, G.

Grier, D. G.

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

Gu, M.

K. Dholakia, P. Reece, and M. Gu, “Optical micromanipulation,” Chem. Soc. Rev. 37, 42-55 (2008).
[CrossRef] [PubMed]

Guck, J.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81, 767-784 (2001).
[CrossRef] [PubMed]

Guillon, M.

M. Guillon, O. Moine, and B. Stout, “Erratum,” Phys. Rev. Lett. 99, 079901 (2007).
[CrossRef]

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

Gutiérrez-Vega, J. C.

Han, Y.

Hanna, S.

Heckenberg, N. R.

A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical application and measurement of torque on microparticles of isotropic nonabsorbing material,” Phys. Rev. A 68, 033802 (2003).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, “Numerical modelling of optical trapping,” Comput. Phys. Commun. 142, 468-471 (2001).
[CrossRef]

Hovenac, E. A.

Imbert, C.

G. Roosen and C. Imbert, “Optical levitation by means of two horizontal laser beams: a theoretical and experimental study,” Phys. Lett. A 59, 6-8 (1976).
[CrossRef]

Käs, J.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81, 767-784 (2001).
[CrossRef] [PubMed]

Kim, J.

Knoesen, A.

Lock, J. A.

F. Xu, J. A. Lock, G. Gouesbet, and C. Tropea, “Radiation torque exerted on a spheroid: analytical solution,” Phys. Rev. E 78, 013843 (2008).

López-Mariscal, C.

Mahmood, H.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81, 767-784 (2001).
[CrossRef] [PubMed]

McGloin, D.

Mervis, J.

Moine, O.

M. Guillon, O. Moine, and B. Stout, “Erratum,” Phys. Rev. Lett. 99, 079901 (2007).
[CrossRef]

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

Moon, T. J.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81, 767-784 (2001).
[CrossRef] [PubMed]

Morgan, H.

N. G. Green and H. Morgan, “Dielectrophoretic investigations of sub-micrometre latex spheres,” J. Phys. D 30, 2626-2633 (1997).
[CrossRef]

Neuman, K. C.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787-2804 (2004).
[CrossRef]

Nieminen, T. A.

A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical application and measurement of torque on microparticles of isotropic nonabsorbing material,” Phys. Rev. A 68, 033802 (2003).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, “Numerical modelling of optical trapping,” Comput. Phys. Commun. 142, 468-471 (2001).
[CrossRef]

Pohl, H. A.

H. A. Pohl, Dielectrophoresis: The Behavior of Neutral Matter in Non-Uniform Electric Fields (Cambridge Univ. Press, 1978).

Prentiss, M.

Reece, P.

K. Dholakia, P. Reece, and M. Gu, “Optical micromanipulation,” Chem. Soc. Rev. 37, 42-55 (2008).
[CrossRef] [PubMed]

Ren, K.

F. Xu, K. Ren, G. Gouesbet, X. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E 75, 026613 (2007).
[CrossRef]

F. Xu, K. Ren, G. Gouesbet, G. Gréhan, and X. Cai, “Generalized Lorenz-Mie theory for an arbitrarily oriented, located, and shaped beam scattered by a homogeneous spheroid,” J. Opt. Soc. Am. A 24, 119-131 (2007).
[CrossRef]

Roosen, G.

G. Roosen and C. Imbert, “Optical levitation by means of two horizontal laser beams: a theoretical and experimental study,” Phys. Lett. A 59, 6-8 (1976).
[CrossRef]

Rubinsztein-Dunlop, H.

A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical application and measurement of torque on microparticles of isotropic nonabsorbing material,” Phys. Rev. A 68, 033802 (2003).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, “Numerical modelling of optical trapping,” Comput. Phys. Commun. 142, 468-471 (2001).
[CrossRef]

Rudd, D.

Shahvisi, A.

Shen, J.

Sidick, E.

Simpson, S. H.

Stout, B.

M. Guillon, O. Moine, and B. Stout, “Erratum,” Phys. Rev. Lett. 99, 079901 (2007).
[CrossRef]

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

Summers, M.

Tropea, C.

F. Xu, J. A. Lock, G. Gouesbet, and C. Tropea, “Radiation torque exerted on a spheroid: analytical solution,” Phys. Rev. E 78, 013843 (2008).

Wu, Z.

Xu, F.

F. Xu, J. A. Lock, G. Gouesbet, and C. Tropea, “Radiation torque exerted on a spheroid: analytical solution,” Phys. Rev. E 78, 013843 (2008).

F. Xu, K. Ren, G. Gouesbet, X. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E 75, 026613 (2007).
[CrossRef]

F. Xu, K. Ren, G. Gouesbet, G. Gréhan, and X. Cai, “Generalized Lorenz-Mie theory for an arbitrarily oriented, located, and shaped beam scattered by a homogeneous spheroid,” J. Opt. Soc. Am. A 24, 119-131 (2007).
[CrossRef]

F. Xu, K. Fang Ren, X. Cai, and J. Shen, “Extension of geometrical-optics approximation to on-axis Gaussian beam scattering. II. By a spheroidal particle with end-on incidence,” Appl. Opt. 45, 5000-5009 (2006).
[CrossRef] [PubMed]

Yamamoto, G.

Zarinetchi, F.

Appl. Opt.

Biophys. J.

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] [PubMed]

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81, 767-784 (2001).
[CrossRef] [PubMed]

Chem. Soc. Rev.

K. Dholakia, P. Reece, and M. Gu, “Optical micromanipulation,” Chem. Soc. Rev. 37, 42-55 (2008).
[CrossRef] [PubMed]

Comput. Phys. Commun.

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, “Numerical modelling of optical trapping,” Comput. Phys. Commun. 142, 468-471 (2001).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. D

N. G. Green and H. Morgan, “Dielectrophoretic investigations of sub-micrometre latex spheres,” J. Phys. D 30, 2626-2633 (1997).
[CrossRef]

Nature

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

Opt. Express

Opt. Lett.

Phys. Lett. A

G. Roosen and C. Imbert, “Optical levitation by means of two horizontal laser beams: a theoretical and experimental study,” Phys. Lett. A 59, 6-8 (1976).
[CrossRef]

Phys. Rev. A

A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical application and measurement of torque on microparticles of isotropic nonabsorbing material,” Phys. Rev. A 68, 033802 (2003).
[CrossRef]

Phys. Rev. E

F. Xu, K. Ren, G. Gouesbet, X. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E 75, 026613 (2007).
[CrossRef]

F. Xu, J. A. Lock, G. Gouesbet, and C. Tropea, “Radiation torque exerted on a spheroid: analytical solution,” Phys. Rev. E 78, 013843 (2008).

Phys. Rev. Lett.

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

M. Guillon, O. Moine, and B. Stout, “Erratum,” Phys. Rev. Lett. 99, 079901 (2007).
[CrossRef]

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

Proc. SPIE

R. W. Going, B. L. Conover, and M. J. Escuti, “Electrostatic force and torque escription of generalized spheroidal particles in optical landscapes,” Proc. SPIE 7038, 703826 (2008).
[CrossRef]

Rev. Sci. Instrum.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787-2804 (2004).
[CrossRef]

Other

H. A. Pohl, Dielectrophoresis: The Behavior of Neutral Matter in Non-Uniform Electric Fields (Cambridge Univ. Press, 1978).

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Figures (11)

Fig. 1
Fig. 1

Geometry and axes definitions for the spheroid.

Fig. 2
Fig. 2

Geometry for calculating the force due to the scattering of a single incident ray of power P by a dielectric spheroid, showing the reflected ray P R 0 and infinite set of refracted rays P T 0 ( j = 1 m 1 R j ) T m .

Fig. 3
Fig. 3

(a), (b) Rotational trajectories in a meridional plane without and with total internal reflection. (c), (d) Librational trajectories without and with total internal reflection.

Fig. 4
Fig. 4

Plane of incidence of the input ray (a) equal to a meridional plane and (b) orthogonal to the meridional plane.

Fig. 5
Fig. 5

Trapping efficiencies q ( s ) ( η , θ 0 ) and q ( g ) ( η , θ 0 ) corresponding to the situation depicted in Fig. 4a for spheroids of eccentricities e = 0 , 0.3 , 0.6 and refractive index ratio N = 1.2 . Note that each subplot has its own vertical scale.

Fig. 6
Fig. 6

Scattering and gradient trapping efficiencies corresponding to the situation depicted in Fig. 4b for spheroids of eccentricities e = 0.3 and e = 0.6 and refractive index ratio N = 1.2 . Note that each subplot has its own vertical scale.

Fig. 7
Fig. 7

Geometry of counterpropagating dual Gaussian beams illuminating a spheroid.

Fig. 8
Fig. 8

Variation of the transverse trapping efficiency Q x as a function of the transverse offset D with (a) the beam waist separation S, (b) the spheroid size R 0 , and (c) the effective index of refraction N as a parameter. Parameters not shown in the legends are S = 30 , R 0 = 1 , and N = 1.2 . Also, the spheroid has an eccentricity e = 0.7 .

Fig. 9
Fig. 9

Variation of the transverse trapping efficiency Q x as a function of the angle β between the z axis of the spheroid coordinate system and the axis of propagation of the laser beam for different values of the eccentricity of the spheroid. Parameters not shown in the legends are S = 15 , R 0 = 1 , and N = 1.2 .

Fig. 10
Fig. 10

Variation of the transverse trapping efficiency Q x as a function of the eccentricity of the spheroid. Parameters not shown in the legends are S = 15 , R 0 = 1 , and N = 1.2 .

Fig. 11
Fig. 11

Variation of the axial trapping efficiency Q z as a function of the axial offset Z 0 with (a) the beam waist separation S, (b) the spheroid size R 0 , and (c) the effective index of refraction N as a parameter. Parameters not shown in the legends are S = 30 , R 0 = 1 , and N = 1.2 . Also, the spheroid has an eccentricity e = 0.7 .

Equations (39)

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r = x x ̂ + y y ̂ + z z ̂ .
r cm = x cm x ̂ cm + y cm y ̂ cm + z cm z ̂ cm ,
r sp = x sp x ̂ sp + y sp y ̂ sp + z sp z ̂ sp
C sp = ( C x sp C y sp C z sp ) = ( b sin η cos φ b sin η sin φ a cos η ) ,
C cm = ( C x cm C y cm C z cm ) = T ( α , β ) ( C x sp C y sp C z sp ) ,
T ( α , β ) = ( cos α cos β sin α cos α sin β sin α cos β cos α sin α sin β sin β 0 cos β )
C = C cm + ( 0 0 z 0 ) = T ( α , β ) ( b sin η cos φ b sin η sin φ a cos η ) + ( 0 0 z 0 ) .
n ̂ sp = ( C x sp b 2 ) x ̂ sp + ( C y sp b 2 ) y ̂ sp + ( C z sp a 2 ) z ̂ sp ( C x sp b 2 ) 2 + ( C y sp b 2 ) 2 + ( C z sp a 2 ) 2 .
n ̂ = T ( α , β ) n ̂ sp ,
d P = I ( r ) d S ,
d F = n 0 c d P = n 0 c I ( r ) d S ,
f = n 0 P c ı ̂ 0 ( n 0 P 0 c r ̂ 0 + m = 1 n 0 P m c t ̂ m ) ,
r ̂ 0 = ı ̂ 0 + ( 2 cos θ 0 ) n ̂ 0 ,
t ̂ 0 = ı ̂ 1 = 1 N ı ̂ 0 ( 1 sin 2 θ 0 N 2 cos θ 0 N ) n ̂ 0 ,
r ̂ m = ı ̂ m + 1 = ı ̂ m ( 2 cos θ m ) n ̂ m ,
t ̂ m = N ı ̂ m + ( 1 N 2 sin 2 θ m N cos θ m ) n ̂ m ,
P 0 = P R 0 ,
P m = P T 0 ( j = 1 m 1 R j ) T m , m 1 ,
f = ( n 0 P c ) q ,
q ı ̂ 0 R 0 r ̂ 0 T 0 m = 1 T m ( j = 1 m 1 R j ) t ̂ m
R m = { ( R m TE + R m TM ) 2 θ m θ c 1 θ m > θ c } ,
T m = { 1 R m θ m θ c 0 θ m > θ c } ,
R m TE = | N ( ı ̂ m n ̂ m ) ( t ̂ m n ̂ m ) N ( ı ̂ m n ̂ m ) + ( t ̂ m n ̂ m ) | 2 ,
R m TM = | ( ı ̂ m n ̂ m ) N ( t ̂ m n ̂ m ) ( ı ̂ m n ̂ m ) + N ( t ̂ m n ̂ m ) | 2
f ( s ) = n 0 P c q ( s ) = n 0 P c ( q ı ̂ 0 ) ı ̂ 0 .
u ̂ n ̂ 0 × ı ̂ 0 | n ̂ 0 × ı ̂ 0 | ,
f ( g ) = n 0 P c q ( g ) = n 0 P c ( q u ̂ ) u ̂ .
u ̂ ( n ̂ 0 × ı ̂ 0 ) × ı ̂ 0 | ( n ̂ 0 × ı ̂ 0 ) × ı ̂ 0 | ,
f ( g ) = n 0 P c q ( g ) = n 0 P c ( q u ̂ ) u ̂ .
F = n 0 c d P j f j = n 0 c j P j q j ,
F = ( n 0 P T c ) Q = ( n 0 P T c ) j P j P T q j .
P j = I ( r ) cos θ j Δ S = 2 P T π w 2 exp ( 2 r 2 w 2 ) cos θ j Δ S ,
ı ̂ = r r ̂ + R ( z ) z ̂ r 2 + R 2 ( z ) ,
Q = j I ( r j ) cos θ j P T q j
= 2 π j exp ( 2 r j 2 w j 2 ) w j 2 cos θ j Δ S j q j ,
Q x = Q x ̂ , Q y = Q y ̂ , Q z = Q z ̂ .
F z = F z 1 F z 2 = n 0 c ( P 1 T Q z 1 P 2 T Q z 2 ) ,
F x y = F x y 1 + F x y 2 = n 0 c ( P 1 T Q x y 1 + P 2 T Q x y 2 ) ,
Z 0 = Z 0 S 2

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