Abstract

The radiation force of highly focused partially coherent and radially polarized vortex beams on a Rayleigh particle is theoretically studied. The dependence of the radiation force on coherence lengths, beam widths, topological charges of incident vortex beams, and numerical apertures of an objective is analyzed. The transverse scattering force is also investigated. It is found that the azimuthal scattering force can produce torques with respect to the optical axis if the optical tweezers are constructed by the vortex beams carrying orbit angular momentum. The direction of the torque depends on the sign of the topological charge of vortex beams, and the magnitude of the torque increases with the increase of the value of the topological charge. A Rayleigh particle can revolve around the optical axis driven by the vortex beams.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  7. Y. Tadir, W. H. Wright, O. Vafa, T. Ord, R. H. Asch, and M. W. Berns, “Micromanipulation of sperm by a laser generated optical trap,” Fertil. Steril. 52, 870–873 (1989).
  8. N. Ponelies, J. Scheef, A. Harim, G. Leitz, and K. O. Greulich, “Laser micromanipulations for biotechnology and genome research,” J. Biotech. 35, 109–120 (1994).
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    [CrossRef]
  10. D. Li, J. Pu, and X. Wang, “Radiation forces on a dielectric medium plate induced by a Gaussian beam,” Opt. Commun. 285, 1680–1683 (2012).
    [CrossRef]
  11. Z. Zhang, J. Pu, and X. Wang, “Focusing of partially coherent Bessel–Gaussian beams through a high numerical-aperture objective,” Opt. Lett. 33, 49–51 (2008).
    [CrossRef]
  12. J. Arlt, K. Dholakia, J. Soneson, and E. M. Wright, “Optical dipole traps and atomic waveguides based on Bessel light beams,” Phys. Rev. A 63, 063602 (2001).
    [CrossRef]
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    [CrossRef]
  14. Q. Zhan, “Radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams,” J. Opt. A 5, 229–232 (2003).
    [CrossRef]
  15. S. Yan and B. Yao, “Radiation forces of a highly focused radially polarized beam on spherical particles,” Phys. Rev. A 76, 053836 (2007).
    [CrossRef]
  16. E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
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  18. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009).
  19. S. Yan and B. Yao, “Accurate description of a radially polarized Gaussian beam,” Phys. Rev. A 77, 023827 (2008).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  26. I. S. Gradysteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 2000).

2012 (3)

2011 (1)

2009 (1)

2008 (2)

2007 (4)

L. Wang, C. Zhao, L. Wang, X. Lu, and S. Y. Zhu, “Effect of spatial coherence on radiation forces acting on a Rayleigh dielectric sphere,” Opt. Lett. 32, 1393–1395 (2007).
[CrossRef]

S. Yan and B. Yao, “Radiation forces of a highly focused radially polarized beam on spherical particles,” Phys. Rev. A 76, 053836 (2007).
[CrossRef]

Y. Zhao, J. Scott Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef]

L. Rao and J. Pu, “Focusing of partially coherent vortex by an aperture lens,” Chin. Phys. Lett. 24, 1252–1255 (2007).
[CrossRef]

2006 (1)

L. Oroszi, P. Galajda, H. Kirei, S. Bottka, and P. Ormos, “Direct measurement of torque in an optical trap and its application to double-strand DNA,” Phys. Rev. Lett. 97, 058301 (2006).
[CrossRef]

2005 (1)

2003 (2)

Q. Zhan, “Radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams,” J. Opt. A 5, 229–232 (2003).
[CrossRef]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
[CrossRef]

2001 (1)

J. Arlt, K. Dholakia, J. Soneson, and E. M. Wright, “Optical dipole traps and atomic waveguides based on Bessel light beams,” Phys. Rev. A 63, 063602 (2001).
[CrossRef]

1996 (1)

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

1994 (2)

K. Svoboda and S. M. Block, “Optical trapping of metallic Rayleigh particles,” Opt. Lett. 19, 930–932 (1994).
[CrossRef]

N. Ponelies, J. Scheef, A. Harim, G. Leitz, and K. O. Greulich, “Laser micromanipulations for biotechnology and genome research,” J. Biotech. 35, 109–120 (1994).

1989 (1)

Y. Tadir, W. H. Wright, O. Vafa, T. Ord, R. H. Asch, and M. W. Berns, “Micromanipulation of sperm by a laser generated optical trap,” Fertil. Steril. 52, 870–873 (1989).

1986 (1)

1982 (1)

1959 (1)

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).

Ambardekar, A. A.

Arlt, J.

J. Arlt, K. Dholakia, J. Soneson, and E. M. Wright, “Optical dipole traps and atomic waveguides based on Bessel light beams,” Phys. Rev. A 63, 063602 (2001).
[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]

Asch, R. H.

Y. Tadir, W. H. Wright, O. Vafa, T. Ord, R. H. Asch, and M. W. Berns, “Micromanipulation of sperm by a laser generated optical trap,” Fertil. Steril. 52, 870–873 (1989).

Ashkin, A.

Berns, M. W.

Y. Tadir, W. H. Wright, O. Vafa, T. Ord, R. H. Asch, and M. W. Berns, “Micromanipulation of sperm by a laser generated optical trap,” Fertil. Steril. 52, 870–873 (1989).

Bjorkholm, J. E.

Block, S. M.

Bottka, S.

L. Oroszi, P. Galajda, H. Kirei, S. Bottka, and P. Ormos, “Direct measurement of torque in an optical trap and its application to double-strand DNA,” Phys. Rev. Lett. 97, 058301 (2006).
[CrossRef]

Brzobohaty, O.

Chen, X.

Chiu, D. T.

Y. Zhao, J. Scott Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef]

Chu, S.

Dholakia, K.

J. Arlt, K. Dholakia, J. Soneson, and E. M. Wright, “Optical dipole traps and atomic waveguides based on Bessel light beams,” Phys. Rev. A 63, 063602 (2001).
[CrossRef]

Dziedzic, J. M.

Galajda, P.

L. Oroszi, P. Galajda, H. Kirei, S. Bottka, and P. Ormos, “Direct measurement of torque in an optical trap and its application to double-strand DNA,” Phys. Rev. Lett. 97, 058301 (2006).
[CrossRef]

Gradysteyn, I. S.

I. S. Gradysteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 2000).

Greulich, K. O.

N. Ponelies, J. Scheef, A. Harim, G. Leitz, and K. O. Greulich, “Laser micromanipulations for biotechnology and genome research,” J. Biotech. 35, 109–120 (1994).

Gu, M.

M. Gu, Advanced Optical Imaging Theory (Springer-Verlag,1999).

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]

Harim, A.

N. Ponelies, J. Scheef, A. Harim, G. Leitz, and K. O. Greulich, “Laser micromanipulations for biotechnology and genome research,” J. Biotech. 35, 109–120 (1994).

Hoang, T. X.

Huang, K.

Jakl, P.

Jeffries, G. D. M.

Y. Zhao, J. Scott Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef]

Jiang, Y.

Kirei, H.

L. Oroszi, P. Galajda, H. Kirei, S. Bottka, and P. Ormos, “Direct measurement of torque in an optical trap and its application to double-strand DNA,” Phys. Rev. Lett. 97, 058301 (2006).
[CrossRef]

Leitz, G.

N. Ponelies, J. Scheef, A. Harim, G. Leitz, and K. O. Greulich, “Laser micromanipulations for biotechnology and genome research,” J. Biotech. 35, 109–120 (1994).

Li, D.

D. Li, J. Pu, and X. Wang, “Radiation forces on a dielectric medium plate induced by a Gaussian beam,” Opt. Commun. 285, 1680–1683 (2012).
[CrossRef]

Li, Y. Q.

Lu, X.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995) Chap. 6.

McGloin, D.

Y. Zhao, J. Scott Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef]

Ord, T.

Y. Tadir, W. H. Wright, O. Vafa, T. Ord, R. H. Asch, and M. W. Berns, “Micromanipulation of sperm by a laser generated optical trap,” Fertil. Steril. 52, 870–873 (1989).

Ormos, P.

L. Oroszi, P. Galajda, H. Kirei, S. Bottka, and P. Ormos, “Direct measurement of torque in an optical trap and its application to double-strand DNA,” Phys. Rev. Lett. 97, 058301 (2006).
[CrossRef]

Oroszi, L.

L. Oroszi, P. Galajda, H. Kirei, S. Bottka, and P. Ormos, “Direct measurement of torque in an optical trap and its application to double-strand DNA,” Phys. Rev. Lett. 97, 058301 (2006).
[CrossRef]

Ponelies, N.

N. Ponelies, J. Scheef, A. Harim, G. Leitz, and K. O. Greulich, “Laser micromanipulations for biotechnology and genome research,” J. Biotech. 35, 109–120 (1994).

Pu, J.

D. Li, J. Pu, and X. Wang, “Radiation forces on a dielectric medium plate induced by a Gaussian beam,” Opt. Commun. 285, 1680–1683 (2012).
[CrossRef]

Z. Zhang, J. Pu, and X. Wang, “Focusing of partially coherent Bessel–Gaussian beams through a high numerical-aperture objective,” Opt. Lett. 33, 49–51 (2008).
[CrossRef]

L. Rao and J. Pu, “Focusing of partially coherent vortex by an aperture lens,” Chin. Phys. Lett. 24, 1252–1255 (2007).
[CrossRef]

Rao, L.

L. Rao and J. Pu, “Focusing of partially coherent vortex by an aperture lens,” Chin. Phys. Lett. 24, 1252–1255 (2007).
[CrossRef]

Ryzhik, I. M.

I. S. Gradysteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 2000).

Scheef, J.

N. Ponelies, J. Scheef, A. Harim, G. Leitz, and K. O. Greulich, “Laser micromanipulations for biotechnology and genome research,” J. Biotech. 35, 109–120 (1994).

Scott Edgar, J.

Y. Zhao, J. Scott Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef]

Sheppard, C. J. R.

Siler, M.

Soneson, J.

J. Arlt, K. Dholakia, J. Soneson, and E. M. Wright, “Optical dipole traps and atomic waveguides based on Bessel light beams,” Phys. Rev. A 63, 063602 (2001).
[CrossRef]

Svoboda, K.

Tadir, Y.

Y. Tadir, W. H. Wright, O. Vafa, T. Ord, R. H. Asch, and M. W. Berns, “Micromanipulation of sperm by a laser generated optical trap,” Fertil. Steril. 52, 870–873 (1989).

Vafa, O.

Y. Tadir, W. H. Wright, O. Vafa, T. Ord, R. H. Asch, and M. W. Berns, “Micromanipulation of sperm by a laser generated optical trap,” Fertil. Steril. 52, 870–873 (1989).

Wang, L.

Wang, X.

D. Li, J. Pu, and X. Wang, “Radiation forces on a dielectric medium plate induced by a Gaussian beam,” Opt. Commun. 285, 1680–1683 (2012).
[CrossRef]

Z. Zhang, J. Pu, and X. Wang, “Focusing of partially coherent Bessel–Gaussian beams through a high numerical-aperture objective,” Opt. Lett. 33, 49–51 (2008).
[CrossRef]

Wolf, E.

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
[CrossRef]

E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
[CrossRef]

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995) Chap. 6.

Wright, E. M.

J. Arlt, K. Dholakia, J. Soneson, and E. M. Wright, “Optical dipole traps and atomic waveguides based on Bessel light beams,” Phys. Rev. A 63, 063602 (2001).
[CrossRef]

Wright, W. H.

Y. Tadir, W. H. Wright, O. Vafa, T. Ord, R. H. Asch, and M. W. Berns, “Micromanipulation of sperm by a laser generated optical trap,” Fertil. Steril. 52, 870–873 (1989).

Yan, S.

S. Yan and B. Yao, “Accurate description of a radially polarized Gaussian beam,” Phys. Rev. A 77, 023827 (2008).
[CrossRef]

S. Yan and B. Yao, “Radiation forces of a highly focused radially polarized beam on spherical particles,” Phys. Rev. A 76, 053836 (2007).
[CrossRef]

Yao, B.

S. Yan and B. Yao, “Accurate description of a radially polarized Gaussian beam,” Phys. Rev. A 77, 023827 (2008).
[CrossRef]

S. Yan and B. Yao, “Radiation forces of a highly focused radially polarized beam on spherical particles,” Phys. Rev. A 76, 053836 (2007).
[CrossRef]

Zemanek, P.

Zhan, Q.

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009).

Q. Zhan, “Radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams,” J. Opt. A 5, 229–232 (2003).
[CrossRef]

Zhang, Z.

Zhao, C.

Zhao, Y.

Y. Zhao, J. Scott Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef]

Zhu, S. Y.

Adv. Opt. Photon. (1)

Chin. Phys. Lett. (1)

L. Rao and J. Pu, “Focusing of partially coherent vortex by an aperture lens,” Chin. Phys. Lett. 24, 1252–1255 (2007).
[CrossRef]

Fertil. Steril. (1)

Y. Tadir, W. H. Wright, O. Vafa, T. Ord, R. H. Asch, and M. W. Berns, “Micromanipulation of sperm by a laser generated optical trap,” Fertil. Steril. 52, 870–873 (1989).

J. Biotech. (1)

N. Ponelies, J. Scheef, A. Harim, G. Leitz, and K. O. Greulich, “Laser micromanipulations for biotechnology and genome research,” J. Biotech. 35, 109–120 (1994).

J. Opt. A (1)

Q. Zhan, “Radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams,” J. Opt. A 5, 229–232 (2003).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Commun. (2)

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

D. Li, J. Pu, and X. Wang, “Radiation forces on a dielectric medium plate induced by a Gaussian beam,” Opt. Commun. 285, 1680–1683 (2012).
[CrossRef]

Opt. Express (2)

Opt. Lett. (5)

Phys. Lett. A (1)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
[CrossRef]

Phys. Rev. A (3)

J. Arlt, K. Dholakia, J. Soneson, and E. M. Wright, “Optical dipole traps and atomic waveguides based on Bessel light beams,” Phys. Rev. A 63, 063602 (2001).
[CrossRef]

S. Yan and B. Yao, “Accurate description of a radially polarized Gaussian beam,” Phys. Rev. A 77, 023827 (2008).
[CrossRef]

S. Yan and B. Yao, “Radiation forces of a highly focused radially polarized beam on spherical particles,” Phys. Rev. A 76, 053836 (2007).
[CrossRef]

Phys. Rev. Lett. (2)

L. Oroszi, P. Galajda, H. Kirei, S. Bottka, and P. Ormos, “Direct measurement of torque in an optical trap and its application to double-strand DNA,” Phys. Rev. Lett. 97, 058301 (2006).
[CrossRef]

Y. Zhao, J. Scott Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).

Other (3)

M. Gu, Advanced Optical Imaging Theory (Springer-Verlag,1999).

I. S. Gradysteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 2000).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995) Chap. 6.

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Figures (6)

Fig. 1.
Fig. 1.

Scheme of a tight focusing system.

Fig. 2.
Fig. 2.

Radiation forces produced by highly focused partially coherent and radially polarized vortex beams. (a) The transverse gradient force in the focal plane, (b) the longitudinal gradient forces on the z-axis, (c) the radial scattering force at different z-planes, and (d) the longitudinal scattering force on the z-axis. The parameters for calculations are P=4W, λ0=1064nm, f=1cm, a=30nm, np=1.592, nm=1.332, NA=0.96, w0=1cm, Lc=1cm, and m=0.

Fig. 3.
Fig. 3.

Transverse gradient force distribution produced by highly focused partially coherent and radially polarized vortex beams with different Lc (a), different w0 (b), different NA (c), and different m with Lc=5cm (d). The other parameters are the same as those in Fig. 2.

Fig. 4.
Fig. 4.

Longitudinal radiation force distribution produced by highly focused partially coherent and radially polarized vortex beams with different Lc: (a) longitudinal gradient force and (b) longitudinal scattering force. The other parameters are the same as those in Fig. 2.

Fig. 5.
Fig. 5.

Influence of varying topological charges of the vortex beam with Lc=5cm on the azimuthal scattering forces: (a) m>0 and (b) m<0. The other parameters are the same as those in Fig. 2.

Fig. 6.
Fig. 6.

Influence of varying topological charges of the vortex beam with Lc=5cm on the z-component of torques. (a) m>0 and (b) m<0. The other parameters are the same as those in Fig. 2.

Equations (27)

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

E(r,φ,z)=[ErEφEz]=ikf2π0α02πcosθsinθexp(ikzcosθ)exp[ikrsinθcos(ϕφ)]A(θ,ϕ)[cosθcos(ϕφ)cosθsin(ϕφ)sinθ]dϕdθ,
Hr=1ik(1rEzφEφz),Hφ=1ik(ErzEzr),
H(r,φ,z)=[HrHφ]=ikf2π0α02πcosθsinθexp(ikzcosθ)exp[ikrsinθcos(ϕφ)]×A(θ,ϕ)cos2θ[sin(ϕφ)cos(ϕφ)]dϕdθ.
Wjk(r1,r2,z)=Ej*(r1,φ1,z)Ek(r2,φ2,z)Mjk(r1,r2,z)=Ej*(r1,φ1,z)Hk(r2,φ2,z),j,k=x,y,z,
I(r,z)=Wxx(r,r,z)+Wyy(r,r,z)+Wzz(r,r,z),
Sj(r,z)=εjklRe[Mkl(r,r,z)Mlk(r,r,z)],j,k,l=x,y,z.
Ex(r,φ,z)=Er(r,φ,z)cosφEφ(r,φ,z)sinφ,Ey(r,φ,z)=Er(r,φ,z)sinφ+Eφ(r,φ,z)cosφ.
I(r,φ,z)=Wrr(r,r,z)+Wφφ(r,r,z)+Wzz(r,r,z),
Sj(r,z)=εjklRe[Mkl(r,r,z)Mlk(r,r,z)],
Wjk(r1,r2,z)=Ej*(r1,φ1,z)Ek(r2,φ2,z),Mjk(r1,r2,z)=Ej*(r1,φ1,z)Hk(r2,φ2,z),j,k,l=r,φ,z.
Fscat(r,φ,z)=nmcχS(r,φ,z)Fgrad(r,φ,z)=2πnmcηI(r,φ,z),
A(ρ,ϕ)=E0(2ρw0)|m|exp(ρ2w02)exp(imϕ)exp(iβ),
A(θ,ϕ)=E0(2fsinθw0)|m|exp((fsinθ)2w02)exp(imϕ)exp(iβ),
A(θ1,θ2,ϕ1,ϕ2)=|E0|2(2f2sinθ1sinθ2w02)|m|exp[(fsinθ1)2+(fsinθ2)2w02]×exp[im(ϕ2ϕ1)]exp{[(fsinθ1)2+(fsinθ2)22f2sinθ1sinθ2cos(ϕ1ϕ2)]Lc2},
P=SI0(ρ,ϕ)ds,
|E0|2=2P|m|![1e2i=0|m|2|m|i(|m|i)!]πw02,
exp[ixcos(ϕφ)]=n=inJn(x)exp[in(ϕφ)],
02πexp[imϕ1+ycos(ϕ1ϕ2)]dϕ1=2πexp(imϕ2)Im(y),
02πexp(imϕ)dϕ=2πifm=0=0ifm=0,
Wrr(r1,r2,φ1,φ2,z)=(kf2)2n=0α0αC*(θ1,z)C(θ2,z)Jn(kr1sinθ1){In+m1(2f2sinθ1sinθ2Lc2)exp[i(n1)(φ1φ2)][Jn(kr2sinθ2)Jn2(kr2sinθ2)]+In+m+1(2f2sinθ1sinθ2Lc2)exp[i(n+1)(φ1φ2)][Jn(kr2sinθ2)Jn+2(kr2sinθ2)]}cosθ1cosθ2dθ1dθ2,
Wφφ(r1,r2,φ1,φ2,z)=(kf2)2n=0α0αC*(θ1,z)C(θ2,z)Jn(kr1sinθ1){In+m1(2f2sinθ1sinθ2Lc2)exp[i(n1)(φ1φ2)]×[Jn(kr2sinθ2)+Jn2(kr2sinθ2)]+In+m+1(2f2sinθ1sinθ2Lc2)exp[i(n+1)(φ1φ2)]×[Jn(kr2sinθ2)+Jn+2(kr2sinθ2)]}cosθ1cosθ2dθ1dθ2,
Wzz(r1,r2,φ1,φ2,z)=(kf)2n=0α0αC*(θ1,z)C(θ2,z)sinθ1sinθ2Jn(kr1sinθ1)×Jn(kr2sinθ2)In+m(2f2sinθ1sinθ2Lc2)exp[in(φ1φ2)]dθ1dθ2,
Mrφ(r1,r2,φ1,φ2,z)=(kf2)2n=0α0αC*(θ1,z)C(θ2,z)Jn(kr1sinθ1){In+m1(2f2sinθ1sinθ2Lc2)exp[i(n1)(φ1φ2)]×[Jn(kr2sinθ2)Jn2(kr2sinθ2)]+In+m+1(2f2sinθ1sinθ2Lc2)exp[i(n+1)(φ1φ2)][Jn(kr2sinθ2)Jn+2(kr2sinθ2)]}cosθ1cos2θ2dθ1dθ2,
Mzφ(r1,r2,φ1,φ2,z)=i(kf)22n=0α0αC*(θ1,z)C(θ2,z)sinθ1cos2θ2In+m(2f2sinθ1sinθ2Lc2)Jn(kr1sinθ1)[Jn+1(kr2sinθ2)Jn1(kr2sinθ2)]exp[in(φ1φ2)]dθ1dθ2,
Mφr(r1,r2,φ1,φ2,z)=(kf2)2n=0α0αC*(θ1,z)C(θ2,z)Jn(kr1sinθ1){In+m1(2f2sinθ1sinθ2Lc2)exp[i(n1)(φ1φ2)][Jn(kr2sinθ2)+Jn2(kr2sinθ2)]+In+m+1(2f2sinθ1sinθ2Lc2)exp[i(n+1)(φ1φ2)][Jn(kr2sinθ2)+Jn+2(kr2sinθ2)]}cosθ1cos2θ2dθ1dθ2,
Mzr(r1,r2,φ1,φ2,z)=(kf)22n=0α0αC*(θ1,z)C(θ2,z)sinθ1cos2θ2In+m(2f2sinθ1sinθ2Lc2)Jn(kr1sinθ1)[Jn+1(kr2sinθ2)+Jn1(kr2sinθ2)]exp[in(φ1φ2)]dθ1dθ2,
C(θl,z)=E0sinθlcosθlexp(ikzcosθl)×(2fsinθlw0)|m|exp[(1w02+1Lc2)f2sin2θl],l=1,2.

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