Abstract

A modified T-matrix method is presented to compute the scattered fields of various realistically shaped particles; then the radiation forces on the particles can be calculated via the Maxwell stress tensor integral. Numerical results of transverse trapping efficiencies of a focused Gaussian beam on ellipsoidal and spherical particles with the same volume are compared, which show that the shape and orientation of particles affect the maximal transverse trapping force and the displacement corresponding to the maximum. The effect of the polarization direction of the incident beam on the transverse trapping forces is also revealed.

© 2007 Optical Society of America

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  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] [PubMed]
  2. Y. Harada and T. Asakura, "Radiation forces on a dielectric sphere in the Rayleigh scattering regime," Opt. Commun. 124, 529-541 (1996).
    [CrossRef]
  3. J. P. Barton and D. R. Alexander, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989).
    [CrossRef]
  4. J. S. Kim and S. S. Lee, "Scattering of laser beams and the optical potential well for a homogeneous sphere," J. Opt. Soc. Am. 73, 303-312 (1983).
    [CrossRef]
  5. G. Gouesbet, B. Maheu, and G. Grehan, "Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation," J. Opt. Soc. Am. A 5, 1427-1442 (1988).
    [CrossRef]
  6. V. V. Kotlyar and A. G. Nalimov, "Analytical expression for radiation forces on a dielectric cylinder illuminated by a cylindrical Gaussian beam," Opt. Express 14, 6136-6321 (2006).
    [CrossRef]
  7. R. C. Gauthier, "Theoretical investigation of the optical trapping force and torque on cylindrical micro-objects," J. Opt. Soc. Am. B 14, 3323-3333 (1997).
    [CrossRef]
  8. T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, "Calculation and optical measurement of laser trapping forces on non-spherical particles," J. Quant. Spectrosc. Radiat. Transf. 70, 627-637 (2001).
    [CrossRef]
  9. T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Computational modeling of optical tweezers," Proc. SPIE 5514, 514-523 (2004).
    [CrossRef]
  10. S. Bayoudh, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Orientation of biological cells using plane-polarized Gaussian beam optical tweezers," J. Mod. Opt. 50, 1581-1590 (2003).
    [CrossRef]
  11. 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]
  12. 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]
  13. W. Singer, T. A. Nieminen, U. J. Gibson, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Orientation of optically trapped nonspherical birefringent particles," Phys. Rev. E 73, 021911 (2006).
    [CrossRef]
  14. S. H. Simpson and S. Hanna, "Optical trapping of spheroidal particles in Gaussian beams," J. Opt. Soc. Am. A 24, 430-443 (2007).
    [CrossRef]
  15. P. C. Waterman, "Matrix formulation of electromagnetic scattering," Proc. IEEE 53, 805-812 (1965).
    [CrossRef]
  16. P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
    [CrossRef]
  17. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002).
  18. P. C. Waterman, "Symmetry, unitary, and geometry in electromagnetic scattering," Phys. Rev. D 3, 825-839 (1971).
    [CrossRef]
  19. M. I. Mishchenko, "Light scattering by randomly oriented axially symmetric particles," J. Opt. Soc. Am. A 8, 871-882 (1991).
    [CrossRef]
  20. T. Wriedt, "Using the T-matrix method for light scattering computations by non-axisymmetric particles: superellipsoids and realistically shaped particles," Part. Part. Syst. Charact. 19, 256-268 (2002).
    [CrossRef]
  21. E. E. M. Khaled, S. C. Hill, and P. W. Barber, "Scattered and internal intensity of a sphere illuminated with a Gaussian beam," IEEE Trans. Antennas Propag. 41, 295-303 (1993).
    [CrossRef]
  22. A. Doicu and T. Wriedt, "Plane wave spectrum of electromagnetic beams," Opt. Commun. 136, 114-124 (1997).
    [CrossRef]
  23. D. Ganic, X. Gan, and M. Gu, "Exact radiation trapping force calculation based on vectorial diffraction theory," Opt. Express 12, 2670-2675 (2004).
    [CrossRef] [PubMed]
  24. J. A. Lock, "Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. II. On-axis trapping force," Appl. Opt. 43, 2545-2554 (2004).
    [CrossRef] [PubMed]
  25. A. Rohrbach and E. H. K. Stelzer, "Optical trapping of dielectric particles in arbitrary fields," J. Opt. Soc. Am. A 18, 839-853 (2001).
    [CrossRef]

2007

2006

W. Singer, T. A. Nieminen, U. J. Gibson, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Orientation of optically trapped nonspherical birefringent particles," Phys. Rev. E 73, 021911 (2006).
[CrossRef]

V. V. Kotlyar and A. G. Nalimov, "Analytical expression for radiation forces on a dielectric cylinder illuminated by a cylindrical Gaussian beam," Opt. Express 14, 6136-6321 (2006).
[CrossRef]

2004

2003

S. Bayoudh, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Orientation of biological cells using plane-polarized Gaussian beam optical tweezers," J. Mod. Opt. 50, 1581-1590 (2003).
[CrossRef]

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]

2002

T. Wriedt, "Using the T-matrix method for light scattering computations by non-axisymmetric particles: superellipsoids and realistically shaped particles," Part. Part. Syst. Charact. 19, 256-268 (2002).
[CrossRef]

2001

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, "Calculation and optical measurement of laser trapping forces on non-spherical particles," J. Quant. Spectrosc. Radiat. Transf. 70, 627-637 (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]

A. Rohrbach and E. H. K. Stelzer, "Optical trapping of dielectric particles in arbitrary fields," J. Opt. Soc. Am. A 18, 839-853 (2001).
[CrossRef]

1997

1996

Y. Harada and T. Asakura, "Radiation forces on a dielectric sphere in the Rayleigh scattering regime," Opt. Commun. 124, 529-541 (1996).
[CrossRef]

1993

E. E. M. Khaled, S. C. Hill, and P. W. Barber, "Scattered and internal intensity of a sphere illuminated with a Gaussian beam," IEEE Trans. Antennas Propag. 41, 295-303 (1993).
[CrossRef]

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

1989

J. P. Barton and D. R. Alexander, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989).
[CrossRef]

1988

1983

1971

P. C. Waterman, "Symmetry, unitary, and geometry in electromagnetic scattering," Phys. Rev. D 3, 825-839 (1971).
[CrossRef]

1965

P. C. Waterman, "Matrix formulation of electromagnetic scattering," Proc. IEEE 53, 805-812 (1965).
[CrossRef]

Alexander, D. R.

J. P. Barton and D. R. Alexander, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989).
[CrossRef]

Asakura, T.

Y. Harada and T. Asakura, "Radiation forces on a dielectric sphere in the Rayleigh scattering regime," Opt. Commun. 124, 529-541 (1996).
[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] [PubMed]

Barber, P. W.

E. E. M. Khaled, S. C. Hill, and P. W. Barber, "Scattered and internal intensity of a sphere illuminated with a Gaussian beam," IEEE Trans. Antennas Propag. 41, 295-303 (1993).
[CrossRef]

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
[CrossRef]

Barton, J. P.

J. P. Barton and D. R. Alexander, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989).
[CrossRef]

Bayoudh, S.

S. Bayoudh, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Orientation of biological cells using plane-polarized Gaussian beam optical tweezers," J. Mod. Opt. 50, 1581-1590 (2003).
[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]

Doicu, A.

A. Doicu and T. Wriedt, "Plane wave spectrum of electromagnetic beams," Opt. Commun. 136, 114-124 (1997).
[CrossRef]

Gan, X.

Ganic, D.

Gauthier, R. C.

Gibson, U. J.

W. Singer, T. A. Nieminen, U. J. Gibson, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Orientation of optically trapped nonspherical birefringent particles," Phys. Rev. E 73, 021911 (2006).
[CrossRef]

Gouesbet, G.

Grehan, G.

Gu, M.

Hanna, S.

Harada, Y.

Y. Harada and T. Asakura, "Radiation forces on a dielectric sphere in the Rayleigh scattering regime," Opt. Commun. 124, 529-541 (1996).
[CrossRef]

Heckenberg, N. R.

W. Singer, T. A. Nieminen, U. J. Gibson, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Orientation of optically trapped nonspherical birefringent particles," Phys. Rev. E 73, 021911 (2006).
[CrossRef]

T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Computational modeling of optical tweezers," Proc. SPIE 5514, 514-523 (2004).
[CrossRef]

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]

S. Bayoudh, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Orientation of biological cells using plane-polarized Gaussian beam optical tweezers," J. Mod. Opt. 50, 1581-1590 (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]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, "Calculation and optical measurement of laser trapping forces on non-spherical particles," J. Quant. Spectrosc. Radiat. Transf. 70, 627-637 (2001).
[CrossRef]

Hill, S. C.

E. E. M. Khaled, S. C. Hill, and P. W. Barber, "Scattered and internal intensity of a sphere illuminated with a Gaussian beam," IEEE Trans. Antennas Propag. 41, 295-303 (1993).
[CrossRef]

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
[CrossRef]

Khaled, E. E. M.

E. E. M. Khaled, S. C. Hill, and P. W. Barber, "Scattered and internal intensity of a sphere illuminated with a Gaussian beam," IEEE Trans. Antennas Propag. 41, 295-303 (1993).
[CrossRef]

Kim, J. S.

Kotlyar, V. V.

V. V. Kotlyar and A. G. Nalimov, "Analytical expression for radiation forces on a dielectric cylinder illuminated by a cylindrical Gaussian beam," Opt. Express 14, 6136-6321 (2006).
[CrossRef]

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002).

Lee, S. S.

Lock, J. A.

Maheu, B.

Mishchenko, M. I.

M. I. Mishchenko, "Light scattering by randomly oriented axially symmetric particles," J. Opt. Soc. Am. A 8, 871-882 (1991).
[CrossRef]

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002).

Nalimov, A. G.

V. V. Kotlyar and A. G. Nalimov, "Analytical expression for radiation forces on a dielectric cylinder illuminated by a cylindrical Gaussian beam," Opt. Express 14, 6136-6321 (2006).
[CrossRef]

Nieminen, T. A.

W. Singer, T. A. Nieminen, U. J. Gibson, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Orientation of optically trapped nonspherical birefringent particles," Phys. Rev. E 73, 021911 (2006).
[CrossRef]

T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Computational modeling of optical tweezers," Proc. SPIE 5514, 514-523 (2004).
[CrossRef]

S. Bayoudh, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Orientation of biological cells using plane-polarized Gaussian beam optical tweezers," J. Mod. Opt. 50, 1581-1590 (2003).
[CrossRef]

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]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, "Calculation and optical measurement of laser trapping forces on non-spherical particles," J. Quant. Spectrosc. Radiat. Transf. 70, 627-637 (2001).
[CrossRef]

Rohrbach, A.

Rubinsztein-Dunlop, H.

W. Singer, T. A. Nieminen, U. J. Gibson, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Orientation of optically trapped nonspherical birefringent particles," Phys. Rev. E 73, 021911 (2006).
[CrossRef]

T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Computational modeling of optical tweezers," Proc. SPIE 5514, 514-523 (2004).
[CrossRef]

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]

S. Bayoudh, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Orientation of biological cells using plane-polarized Gaussian beam optical tweezers," J. Mod. Opt. 50, 1581-1590 (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]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, "Calculation and optical measurement of laser trapping forces on non-spherical particles," J. Quant. Spectrosc. Radiat. Transf. 70, 627-637 (2001).
[CrossRef]

Simpson, S. H.

Singer, W.

W. Singer, T. A. Nieminen, U. J. Gibson, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Orientation of optically trapped nonspherical birefringent particles," Phys. Rev. E 73, 021911 (2006).
[CrossRef]

Stelzer, E. H. K.

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002).

Waterman, P. C.

P. C. Waterman, "Symmetry, unitary, and geometry in electromagnetic scattering," Phys. Rev. D 3, 825-839 (1971).
[CrossRef]

P. C. Waterman, "Matrix formulation of electromagnetic scattering," Proc. IEEE 53, 805-812 (1965).
[CrossRef]

Wriedt, T.

T. Wriedt, "Using the T-matrix method for light scattering computations by non-axisymmetric particles: superellipsoids and realistically shaped particles," Part. Part. Syst. Charact. 19, 256-268 (2002).
[CrossRef]

A. Doicu and T. Wriedt, "Plane wave spectrum of electromagnetic beams," Opt. Commun. 136, 114-124 (1997).
[CrossRef]

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]

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]

IEEE Trans. Antennas Propag.

E. E. M. Khaled, S. C. Hill, and P. W. Barber, "Scattered and internal intensity of a sphere illuminated with a Gaussian beam," IEEE Trans. Antennas Propag. 41, 295-303 (1993).
[CrossRef]

J. Appl. Phys.

J. P. Barton and D. R. Alexander, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989).
[CrossRef]

J. Mod. Opt.

S. Bayoudh, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Orientation of biological cells using plane-polarized Gaussian beam optical tweezers," J. Mod. Opt. 50, 1581-1590 (2003).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Quant. Spectrosc. Radiat. Transf.

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, "Calculation and optical measurement of laser trapping forces on non-spherical particles," J. Quant. Spectrosc. Radiat. Transf. 70, 627-637 (2001).
[CrossRef]

Opt. Commun.

Y. Harada and T. Asakura, "Radiation forces on a dielectric sphere in the Rayleigh scattering regime," Opt. Commun. 124, 529-541 (1996).
[CrossRef]

A. Doicu and T. Wriedt, "Plane wave spectrum of electromagnetic beams," Opt. Commun. 136, 114-124 (1997).
[CrossRef]

Opt. Express

D. Ganic, X. Gan, and M. Gu, "Exact radiation trapping force calculation based on vectorial diffraction theory," Opt. Express 12, 2670-2675 (2004).
[CrossRef] [PubMed]

V. V. Kotlyar and A. G. Nalimov, "Analytical expression for radiation forces on a dielectric cylinder illuminated by a cylindrical Gaussian beam," Opt. Express 14, 6136-6321 (2006).
[CrossRef]

Part. Part. Syst. Charact.

T. Wriedt, "Using the T-matrix method for light scattering computations by non-axisymmetric particles: superellipsoids and realistically shaped particles," Part. Part. Syst. Charact. 19, 256-268 (2002).
[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. D

P. C. Waterman, "Symmetry, unitary, and geometry in electromagnetic scattering," Phys. Rev. D 3, 825-839 (1971).
[CrossRef]

Phys. Rev. E

W. Singer, T. A. Nieminen, U. J. Gibson, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Orientation of optically trapped nonspherical birefringent particles," Phys. Rev. E 73, 021911 (2006).
[CrossRef]

Proc. IEEE

P. C. Waterman, "Matrix formulation of electromagnetic scattering," Proc. IEEE 53, 805-812 (1965).
[CrossRef]

Proc. SPIE

T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Computational modeling of optical tweezers," Proc. SPIE 5514, 514-523 (2004).
[CrossRef]

Other

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
[CrossRef]

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002).

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Figures (7)

Fig. 1
Fig. 1

Differential scattering cross section (DSCS) of the ellipsoid particle ( a = 0.5 μ m , b = 0.7 μ m , c = 1.0 μ m , n 2 = 1.58 ) in vacuum illuminated by a plane wave with λ = 1.064 μ m .

Fig. 2
Fig. 2

Geometries of the ellipsoidal particle with parameters a = 0.766 μ m , b = 0.574 μ m , c = 0.957 μ m and the spherical particle with radius r = 0.75 μ m .

Fig. 3
Fig. 3

Axial trapping efficiencies of the sphere of r = 0.75 μ m and ellipsoid with parameters a = 0.766 μ m , b = 0.574 μ m , c = 0.957 μ m , n 2 = 1.58 , trapped in water ( n 1 = 1.33 ) .

Fig. 4
Fig. 4

Transverse trapping efficiencies of the sphere ( r = 0.75 μ m ) and ellipsoid ( a = 0.766 μ m , b = 0.574 μ m , c = 0.957 μ m ). Q x represents x-axis trapping efficiencies (filled symbols), and Q y represents y-axis trapping efficiencies (open symbols).

Fig. 5
Fig. 5

Transverse trapping efficiencies in the x and y axes for different sizes of spherical particles: (a) r = 1.5 μ m , (b) r = 0.1 μ m . The polarization direction of the beam is in the x axis.

Fig. 6
Fig. 6

Transverse trapping efficiencies of the sphere ( r = 0.75 μ m ) and ellipsoid ( a = 0.957 μ m , b = 0.574 μ m , c = 0.766 μ m ). Q x represents x-axis trapping efficiencies (filled symbols), and Q y represents y-axis trapping efficiencies (open symbols).

Fig. 7
Fig. 7

Transverse trapping efficiencies of the sphere ( r = 0.75 μ m ) and ellipsoid ( a = 0.766 μ m , b = 0.957 μ m , c = 0.574 μ m ). Q x represents x-axis trapping efficiencies (filled symbols), and Q y represents y represents y-axis trapping efficiencies (open symbols).

Equations (22)

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

E i ( r ) = l = 1 m = l l [ c l m M l m 1 ( k r ) + d l m N l m 1 ( k r ) ] ,
E s ( r ) = l = 1 m = l l [ e l m M l m 3 ( k r ) + f l m N l m 3 ( k r ) ] ,
M l m 1 , 3 ( k r ) = γ l m z l 1 , 3 ( k r ) C l m ( ϑ , ϕ ) ,
N l m 1 , 3 ( k r ) = γ l m { l ( l + 1 ) k r z l 1 , 3 ( k r ) P l m ( ϑ , ϕ ) + 1 k r d d ( k r ) [ ( k r ) z l 1 , 3 ( k r ) ] B l m ( ϑ , ϕ ) } ,
B l m ( ϑ , ϕ ) = r [ P l m ( ϑ ) e i m ϕ ] ,
C l m ( ϑ , ϕ ) = × [ r P l m ( ϑ ) e i m ϕ ] ,
P l m ( ϑ , ϕ ) = r r P l m ( ϑ ) e i m ϕ
{ E ( r ) 0 } = E i ( r ) + k × S d s ( n × E + ) g ( k R ) + i × × S d s ( n × H + ) g ( k R ) , r { outside S inside S } ,
e l m = l = 1 m = l l [ T l m l m 11 c l m + T l m l m 12 d l m ] ,
f l m = l = 1 m = l l [ T l m l m 21 c l m + T l m l m 22 d l m ] .
S f d s i f ( v i , c ) area [ v i , 1 , v i , 2 , v i , 3 ] ,
F = S d s n T ¯ ,
T ¯ = 1 2 Re [ ϵ 0 E ( r ) E * ( r ) + μ 0 H ( r ) H * ( r ) 1 2 ( ϵ 0 E ( r ) 2 + μ 0 H ( r ) 2 ) I ¯ ] ,
F = r 2 4 π d Ω n T ¯ ,
F = ϵ 0 r 2 4 Re 4 π d Ω n [ E s 2 + E i ( E s ) * + E s ( E i ) * ] μ 0 r 2 4 Re 4 π d Ω n [ H s 2 + H i ( H s ) * + H s ( H i ) * ] = 1 2 c 0 r 2 Re 4 π d Ω { [ E i × ( H s ) * + E s × ( H i ) * ] + E s × ( H s ) * } = r 2 4 π d Ω ( S exi c 0 + S sca c 0 ) ,
F z ϵ 0 E 0 2 = 1 k 2 l = 1 m = l l Im ( 1 ( l + 1 ) l ( l + 2 ) ( l m + 1 ) ( l + m + 1 ) ( 2 l + 1 ) ( 2 l + 3 ) ( p l + 1 , m * p l , m + q l + 1 , m * q l , m a l + 1 , m * a l , m b l + 1 , m * b l , m ) + m l ( l + 1 ) i ( q l , m p l , m * b l , m a l , m * ) ) ,
F x + i F y ϵ 0 E 0 2 = 1 2 k 2 l = 1 m = l l ( C 1 l ( l + 1 ) ( l + m + 1 ) ( l m ) ( p l , m q l , m + 1 * + q l , m p l , m + 1 * a l , m b l , m + 1 * b l , m a l , m + 1 * ) + C 2 l ( l + 1 ) l ( l + 2 ) ( l + m + 1 ) ( l + m + 2 ) ( 2 l + 1 ) ( 2 l + 3 ) ( p l , m p l , 1 , m + 1 * + q l , m q l + 1 , m + 1 * a l , m a l + 1 , m + 1 * b l , m b l + 1 , m + 1 * ) + C 3 l ( l + 1 ) l ( l + 2 ) ( l m + 1 ) ( l m + 2 ) ( 2 l + 1 ) ( 2 l + 3 ) ( p l + 1 , m 1 p l , m * + q l + 1 , m 1 q l , m * a l + 1 , m 1 a l , m * b l , 1 , m 1 b l , m * ) ) ,
C 1 = { 1 m 0 1 m < 0 } , C 2 = { i m 0 i m < 0 } ,
C 3 = { i m 1 i m < 1 } ,
a l , m = c l , m 2 , b l , m = d l , m 2 ,
p l , m = a l , m + e l , m , q l , m = b l , m + f l , m .
Q i = F i ( n 1 P c 0 ) , i = x , y , z ,

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