Abstract

The axial and transverse optical forces and torques exerted by circularly ringed beams on an arbitrarily oriented and homogeneous spheroid are calculated and studied within the framework of the geometrical optics regime. The results are applied to study the behavior of the forces in a counter-propagating optical trap. We calculate the trapping efficiencies and torques for several values 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.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  27. 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]

2009

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]

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

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

2006

F. Xu, K. F. 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]

M. Guillon, O. Moine, and B. Stout, “Longitudinal optical binding of high optical contrast microdroplets in air,” Phys. Rev. Lett. 96, 143902 (2006); M. Guillon, O. Moine, and B. Stout, erratum, Phys. Rev. 99, 079901 (2007).
[CrossRef] [PubMed]

M. Guillon, O. Moine, and B. Stout, “Longitudinal optical binding of high optical contrast microdroplets in air,” Phys. Rev. Lett. 96, 143902 (2006); M. Guillon, O. Moine, and B. Stout, erratum, Phys. Rev. 99, 079901 (2007).
[CrossRef] [PubMed]

C. López-Mariscal, J. C. Gutiérrez-Vega, G. Milne and K. Dholakia, “Orbital angular momentum transfer in helical Mathieu beams,” Opt. Express 14, 4182–4187 (2006).
[CrossRef] [PubMed]

2004

2003

2002

V. Garcés-Chávez, K. Volke-Sepulveda, S. Chavez-Cerda, W. Sibbett, and K. Dholakia, “Orbital angular momentum transfer to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

K. P. Volke-Sepulveda, V. Garcés-Chávez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclassical Opt. 4, S82–S789 (2002).
[CrossRef]

2001

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]

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]

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

J. P. Barton, “Internal, near-surface, and scattered electromagnetic fields for a layered spheroid with arbitrary illumination,” Appl. Opt. 40, 3598–3607 (2001).
[CrossRef]

2000

L. Allen, and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000).
[CrossRef]

1999

M. Mansuripur, “Geometric-optical rays, Poynting’s vector and field momenta,” Opt. Photonics News 10(3), 53–56 (1999).
[CrossRef]

1997

1995

1993

1991

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

Allen, L.

L. Allen, and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000).
[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]

Arlt, J.

K. P. Volke-Sepulveda, V. Garcés-Chávez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclassical Opt. 4, S82–S789 (2002).
[CrossRef]

Asano, S.

Barton, J. P.

Bishop, A. I.

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]

Cai, X.

Chavez-Cerda, S.

K. P. Volke-Sepulveda, V. Garcés-Chávez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclassical Opt. 4, S82–S789 (2002).
[CrossRef]

V. Garcés-Chávez, K. Volke-Sepulveda, S. Chavez-Cerda, W. Sibbett, and K. Dholakia, “Orbital angular momentum transfer to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

Chávez-Cerda, 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.

C. López-Mariscal, J. C. Gutiérrez-Vega, G. Milne and K. Dholakia, “Orbital angular momentum transfer in helical Mathieu beams,” Opt. Express 14, 4182–4187 (2006).
[CrossRef] [PubMed]

K. Volke-Sepúlveda, S. Chávez-Cerda, V. Garcés-Chávez, and K. Dholakia, “Three-dimensional optical forces and transfer of orbital angular momentum from multiringed light beams to spherical microparticles,” J. Opt. Soc. Am. B 21, 1749–1757 (2004).
[CrossRef]

V. Garcés-Chávez, K. Volke-Sepulveda, S. Chavez-Cerda, W. Sibbett, and K. Dholakia, “Orbital angular momentum transfer to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

K. P. Volke-Sepulveda, V. Garcés-Chávez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclassical Opt. 4, S82–S789 (2002).
[CrossRef]

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]

Garcés-Chávez, V.

K. Volke-Sepúlveda, S. Chávez-Cerda, V. Garcés-Chávez, and K. Dholakia, “Three-dimensional optical forces and transfer of orbital angular momentum from multiringed light beams to spherical microparticles,” J. Opt. Soc. Am. B 21, 1749–1757 (2004).
[CrossRef]

V. Garcés-Chávez, K. Volke-Sepulveda, S. Chavez-Cerda, W. Sibbett, and K. Dholakia, “Orbital angular momentum transfer to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

K. P. Volke-Sepulveda, V. Garcés-Chávez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclassical Opt. 4, S82–S789 (2002).
[CrossRef]

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]

Gréhan, G.

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, “Longitudinal optical binding of high optical contrast microdroplets in air,” Phys. Rev. Lett. 96, 143902 (2006); M. Guillon, O. Moine, and B. Stout, erratum, Phys. Rev. 99, 079901 (2007).
[CrossRef] [PubMed]

M. Guillon, O. Moine, and B. Stout, “Longitudinal optical binding of high optical contrast microdroplets in air,” Phys. Rev. Lett. 96, 143902 (2006); M. Guillon, O. Moine, and B. Stout, erratum, Phys. Rev. 99, 079901 (2007).
[CrossRef] [PubMed]

Gutiérrez-Vega, J. C.

Han, Y.

Hanna, S.

Heckenberg, N. R.

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]

Mansuripur, M.

M. Mansuripur, “Geometric-optical rays, Poynting’s vector and field momenta,” Opt. Photonics News 10(3), 53–56 (1999).
[CrossRef]

McGloin, D.

Mervis, J.

Milne, G.

Moine, O.

M. Guillon, O. Moine, and B. Stout, “Longitudinal optical binding of high optical contrast microdroplets in air,” Phys. Rev. Lett. 96, 143902 (2006); M. Guillon, O. Moine, and B. Stout, erratum, Phys. Rev. 99, 079901 (2007).
[CrossRef] [PubMed]

M. Guillon, O. Moine, and B. Stout, “Longitudinal optical binding of high optical contrast microdroplets in air,” Phys. Rev. Lett. 96, 143902 (2006); M. Guillon, O. Moine, and B. Stout, erratum, Phys. Rev. 99, 079901 (2007).
[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]

Nieminen, T. A.

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]

Padgett, M. J.

L. Allen, and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000).
[CrossRef]

Prentiss, M.

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]

Ren, K. F.

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.

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.

Sibbett, W.

V. Garcés-Chávez, K. Volke-Sepulveda, S. Chavez-Cerda, W. Sibbett, and K. Dholakia, “Orbital angular momentum transfer to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

Sidick, E.

Simpson, S. H.

Sosa-Martínez, H.

Stout, B.

M. Guillon, O. Moine, and B. Stout, “Longitudinal optical binding of high optical contrast microdroplets in air,” Phys. Rev. Lett. 96, 143902 (2006); M. Guillon, O. Moine, and B. Stout, erratum, Phys. Rev. 99, 079901 (2007).
[CrossRef] [PubMed]

M. Guillon, O. Moine, and B. Stout, “Longitudinal optical binding of high optical contrast microdroplets in air,” Phys. Rev. Lett. 96, 143902 (2006); M. Guillon, O. Moine, and B. Stout, erratum, Phys. Rev. 99, 079901 (2007).
[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).

Volke-Sepulveda, K.

V. Garcés-Chávez, K. Volke-Sepulveda, S. Chavez-Cerda, W. Sibbett, and K. Dholakia, “Orbital angular momentum transfer to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

Volke-Sepulveda, K. P.

K. P. Volke-Sepulveda, V. Garcés-Chávez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclassical Opt. 4, S82–S789 (2002).
[CrossRef]

Volke-Sepúlveda, K.

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, 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. 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. F. 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.

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]

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. B: Quantum Semiclassical Opt.

K. P. Volke-Sepulveda, V. Garcés-Chávez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclassical Opt. 4, S82–S789 (2002).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

L. Allen, and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Photonics News

M. Mansuripur, “Geometric-optical rays, Poynting’s vector and field momenta,” Opt. Photonics News 10(3), 53–56 (1999).
[CrossRef]

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

V. Garcés-Chávez, K. Volke-Sepulveda, S. Chavez-Cerda, W. Sibbett, and K. Dholakia, “Orbital angular momentum transfer to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

Phys. Rev. E

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]

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); M. Guillon, O. Moine, and B. Stout, erratum, Phys. Rev. 99, 079901 (2007).
[CrossRef] [PubMed]

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]

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Figures (9)

Fig. 1
Fig. 1

Parameter definitions and geometry of the problem.

Fig. 2
Fig. 2

(a) Radial variation of the intensity of several LG beams with different mode numbers ( m , l ) as a function of the normalized radius D = r w 0 . (b) (c) (d) Respectively, the Q r . Q z , and Q θ trapping efficiencies of a spherical particle immersed in the LG beams.

Fig. 3
Fig. 3

Trapping efficiencies and normalized torques exerted by a Gaussian beam on (a) a spheroid of eccentricity 0.6. The curves in subplots (b), (c) and (d) correspond to different orientations β of the spheroid (see Fig. 1) with α = 0 .

Fig. 4
Fig. 4

Trapping efficiencies and normalized torques exerted by a LG 1 , 0 beam on (a) a spheroid of eccentricity 0.6 for different orientations β (b), (c), (d) of the spheroid (see Fig. 1) with α = 0 .

Fig. 5
Fig. 5

Trapping efficiencies and normalized torques exerted by a LG 1 , 0 beam on (a) a spheroid with different orientations [ α , β ] (b), (c), (d), (e) in units of π 4 .

Fig. 6
Fig. 6

Trapping efficiencies and normalized torques exerted by a LG 1 , 0 annular beam on (a) a spheroid of eccentricity 0.6 for different orientations β (b), (c), (d), (e) of the spheroid (see Fig. 1) with α = 0 .

Fig. 7
Fig. 7

Trapping efficiencies and normalized torques exerted by a LG 1 , 0 annular beam on (a) a spheroid of eccentricity 0.6 with different orientations [ α , β ] (b), (c), (d), (e) in units of π 4 .

Fig. 8
Fig. 8

Trapping efficiencies and normalized torques exerted by two perfectly aligned counter-propagating beams LG 1 , 0 on (a) a spheroid with different orientations β and α = 0 (b), (c), (d), (e).

Fig. 9
Fig. 9

Trapping efficiencies and normalized torques exerted by two misaligned counter-propagating beams LG 1 , 0 on (a) a spheroid with different orientations (b), (c), (d) β and α = 0 . Subplots (e) and (f) show the cases of constructive and destructive superposition of the helical wavefronts, respectively.

Equations (13)

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C = H ( α , β ) ( b sin η cos φ b sin η sin φ a cos η ) + ( d 0 z 0 ) ,
H ( 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 )
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 .
T = ( n 0 P T r 0 c ) τ = ( n 0 P T r 0 c ) j P j P T τ ¯ j ,
LG m , l ( r , θ , z ) = 2 m ! P T π ( m + l ) ! exp ( i l θ ) w ( z ) × [ 2 r w ( z ) ] l L m l ( 2 r 2 w ( z ) 2 ) exp [ r 2 w 2 ( z ) ] × exp i [ k z + k r 2 2 R ( z ) ( 2 m + l + 1 ) ψ GS ( z ) ] x ̂ ,
P j = I ( r j ) cos θ j Δ S j ,
= 2 m ! P T π ( m + l ) ! w j 2 ( 2 r j w j ) 2 l × [ L m l ( 2 r j 2 w j 2 ) ] 2 exp ( 2 r j 2 w j 2 ) cos θ j Δ S j ,
ı ̂ = r 2 r ̂ + ( l R k ) θ ̂ + r R z ̂ r 4 + l 2 R 2 k 2 + r 2 R 2 .
Q = j I ( r j ) cos θ j P T q j ,
= 2 m ! π ( m + l ) ! j ( 2 r j w j ) 2 l w j 2 × [ L m l ( 2 r j 2 w j 2 ) ] 2 exp ( 2 r 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 ̂ ,
τ x = τ x ̂ , τ y = τ y ̂ , τ z = τ z ̂ .

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