Abstract

Using the theory of electromagnetic scattering of a uniaxial anisotropic sphere, we derive the analytical expressions of the radiation forces exerted on a uniaxial anisotropic sphere by an off-axis incident Gaussian beam. The beam’s propagation direction is parallel to the primary optical axis of the anisotropic sphere. The effects of the permittivity tensor elements εt and εz on the axial radiation forces are numerically analyzed in detail. The two transverse components of radiation forces exerted on a uniaxial anisotropic sphere, which is distinct from that exerted on an isotropic sphere due to the two eigen waves in the uniaxial anisotropic sphere, are numerically studied as well. The characteristics of the axial and transverse radiation forces are discussed for different radii of the sphere, beam waist width, and distances from the sphere center to the beam center of an off-axis Gaussian beam. The theoretical predictions of radiation forces exerted on a uniaxial anisotropic sphere are hoped to provide effective ways to achieve the improvement of optical tweezers as well as the capture, suspension, and high-precision delivery of anisotropic particles.

© 2011 OSA

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  31. A. Doicu and T. Wriedt, “Computation of the beam-shape coefficients in the generalized Lorenz-Mie theory by using the translational addition theorem for spherical vector wave functions,” Appl. Opt. 36(13), 2971–2978 (1997).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]

2011

2010

2009

2008

2007

F. L. Mao, Q. R. Xing, K. Wang, L. Y. Lang, L. Chai, and Q. Y. Wang, “Calculation of axial optical forces exerted on medium-sized particles by optical trap,” Opt. Laser Technol. 39(1), 34–39 (2007).
[CrossRef]

F. Xu, K. F. Ren, G. Gouesbet, X. S. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026613 (2007).
[CrossRef] [PubMed]

C. W. Qiu, S. Zouhdi, and A. Razek, “Modified Spherical Wave Functions With Anisotropy Ratio: Application to the Analysis of Scattering by Multilayered Anisotropic Shells,” IEEE Trans. Antenn. Propag. 55(12), 3515–3523 (2007).
[CrossRef]

2006

2004

2002

Y. K. Nahmias and D. J. Odde, “Analysis of Radiation Forces in Laser Trapping and Laser-Guided Direct Writing Applications,” IEEE J. Quantum Electron. 38(2), 131–141 (2002).
[CrossRef]

1998

1997

1996

K. F. Ren, G. Gréhan, and G. Gouesbet, “Prediction of reverse radiation pressure by generalized Lorenz-Mie theory,” Appl. Opt. 35(15), 2702–2710 (1996).
[CrossRef] [PubMed]

T. Wohland, A. Rosin, and E. H. K. Stelzer, “Theoretical determination of the influence of the polarization on forces exerted by optical tweezers,” Optik (Stuttg.) 102, 181–190 (1996).

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 (1996).
[CrossRef]

1995

G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Grehan, and G. Gouesbet, ““Trapping and levitation of a dielectric sphere with off-centred Gaussian beams. II. GLMT analysis,” Pure Appl. Opt. J. Euro. Opt. Soc. Part A 4(5), 571–585 (1995).
[CrossRef]

R. C. Gauthier and S. Wallace, “Optical levitation of spheres: analytical development and numerical computations of force equations,” J. Opt. Soc. Am. B 12(9), 1680–1687 (1995).
[CrossRef]

1994

J. A. Lock and G. Gouesbet, “Rigorous justification fo the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory.I. On-axis beams,” J. Opt. Soc. Am. A 11(9), 2503–2515 (1994).
[CrossRef]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory and associated resonance effects,” Opt. Commun. 108(4-6), 343–354 (1994).
[CrossRef]

1991

T. C. B. Schut, G. Hesselink, B. G. de Grooth, and J. Greve, “Experimental and theoretical investigations on the validity of the geometrical optics model for calculating the stability of optical traps,” Cytometry 12(6), 479–485 (1991).
[CrossRef] [PubMed]

1990

1989

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4954–4962 (1989).
[CrossRef]

1988

1986

1980

A. Ashkin, “Applications of laser radiation pressure,” Science 210(4474), 1081–1088 (1980).
[CrossRef] [PubMed]

1970

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970).
[CrossRef]

Alexander, D. R.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4954–4962 (1989).
[CrossRef]

Angelova, M. I.

G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Grehan, and G. Gouesbet, ““Trapping and levitation of a dielectric sphere with off-centred Gaussian beams. II. GLMT analysis,” Pure Appl. Opt. J. Euro. Opt. Soc. Part A 4(5), 571–585 (1995).
[CrossRef]

Asakura, T.

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 (1996).
[CrossRef]

Ashkin, A.

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(5), 288–290 (1986).
[CrossRef] [PubMed]

A. Ashkin, “Applications of laser radiation pressure,” Science 210(4474), 1081–1088 (1980).
[CrossRef] [PubMed]

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970).
[CrossRef]

Barton, J. P.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4954–4962 (1989).
[CrossRef]

Bjorkholm, J. E.

Blakely, J. T.

Braat, J. J. M.

Cai, X. S.

F. Xu, K. F. Ren, G. Gouesbet, X. S. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026613 (2007).
[CrossRef] [PubMed]

Chai, L.

F. L. Mao, Q. R. Xing, K. Wang, L. Y. Lang, L. Chai, and Q. Y. Wang, “Calculation of axial optical forces exerted on medium-sized particles by optical trap,” Opt. Laser Technol. 39(1), 34–39 (2007).
[CrossRef]

Chu, S.

de Boer, D. K.

de Grooth, B. G.

T. C. B. Schut, G. Hesselink, B. G. de Grooth, and J. Greve, “Experimental and theoretical investigations on the validity of the geometrical optics model for calculating the stability of optical traps,” Cytometry 12(6), 479–485 (1991).
[CrossRef] [PubMed]

Doicu, A.

Dziedzic, J. M.

Gao, B. Z.

Gauthier, R. C.

Geng, Y. L.

Y. L. Geng, X. B. Wu, and L. W. Li, “Mie scattering by a uniaxial anisotropic sphere,” physical review E 70, 056609 (2004).
[CrossRef]

Gordon, R.

Gouesbet, G.

F. Xu, K. F. Ren, G. Gouesbet, X. S. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026613 (2007).
[CrossRef] [PubMed]

H. Polaert, G. Gréhan, and G. Gouesbet, “Improved standard beams with application to reverse radiation pressure,” Appl. Opt. 37(12), 2435–2440 (1998).
[CrossRef] [PubMed]

Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Gréhan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt. 36(21), 5188–5198 (1997).
[CrossRef] [PubMed]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Prediction of reverse radiation pressure by generalized Lorenz-Mie theory,” Appl. Opt. 35(15), 2702–2710 (1996).
[CrossRef] [PubMed]

G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Grehan, and G. Gouesbet, ““Trapping and levitation of a dielectric sphere with off-centred Gaussian beams. II. GLMT analysis,” Pure Appl. Opt. J. Euro. Opt. Soc. Part A 4(5), 571–585 (1995).
[CrossRef]

J. A. Lock and G. Gouesbet, “Rigorous justification fo the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory.I. On-axis beams,” J. Opt. Soc. Am. A 11(9), 2503–2515 (1994).
[CrossRef]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory and associated resonance effects,” Opt. Commun. 108(4-6), 343–354 (1994).
[CrossRef]

G. Gouesbet, G. Grehan, and B. Maheu, “Computations of the gn coefficients in the geneeralized Lorenz-Mie theory using three different methods,” Appl. Opt. 27(23), 4874–4883 (1988).
[CrossRef] [PubMed]

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(9), 1427–1443 (1988).
[CrossRef]

Gousbet, G.

Grehan, G.

Gréhan, G.

F. Xu, K. F. Ren, G. Gouesbet, X. S. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026613 (2007).
[CrossRef] [PubMed]

H. Polaert, G. Gréhan, and G. Gouesbet, “Improved standard beams with application to reverse radiation pressure,” Appl. Opt. 37(12), 2435–2440 (1998).
[CrossRef] [PubMed]

Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Gréhan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt. 36(21), 5188–5198 (1997).
[CrossRef] [PubMed]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Prediction of reverse radiation pressure by generalized Lorenz-Mie theory,” Appl. Opt. 35(15), 2702–2710 (1996).
[CrossRef] [PubMed]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory and associated resonance effects,” Opt. Commun. 108(4-6), 343–354 (1994).
[CrossRef]

Greve, J.

T. C. B. Schut, G. Hesselink, B. G. de Grooth, and J. Greve, “Experimental and theoretical investigations on the validity of the geometrical optics model for calculating the stability of optical traps,” Cytometry 12(6), 479–485 (1991).
[CrossRef] [PubMed]

Guo, L. X.

Harada, Y.

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 (1996).
[CrossRef]

Hesselink, G.

T. C. B. Schut, G. Hesselink, B. G. de Grooth, and J. Greve, “Experimental and theoretical investigations on the validity of the geometrical optics model for calculating the stability of optical traps,” Cytometry 12(6), 479–485 (1991).
[CrossRef] [PubMed]

Kawano, M.

Lang, L. Y.

F. L. Mao, Q. R. Xing, K. Wang, L. Y. Lang, L. Chai, and Q. Y. Wang, “Calculation of axial optical forces exerted on medium-sized particles by optical trap,” Opt. Laser Technol. 39(1), 34–39 (2007).
[CrossRef]

Li, H. Y.

Li, L. W.

Y. L. Geng, X. B. Wu, and L. W. Li, “Mie scattering by a uniaxial anisotropic sphere,” physical review E 70, 056609 (2004).
[CrossRef]

Li, Z. J.

Lock, J. A.

Maheu, B.

Mao, F. L.

F. L. Mao, Q. R. Xing, K. Wang, L. Y. Lang, L. Chai, and Q. Y. Wang, “Calculation of axial optical forces exerted on medium-sized particles by optical trap,” Opt. Laser Technol. 39(1), 34–39 (2007).
[CrossRef]

Martinot-Lagarde, G.

G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Grehan, and G. Gouesbet, ““Trapping and levitation of a dielectric sphere with off-centred Gaussian beams. II. GLMT analysis,” Pure Appl. Opt. J. Euro. Opt. Soc. Part A 4(5), 571–585 (1995).
[CrossRef]

Nahmias, Y. K.

Y. K. Nahmias, B. Z. Gao, and D. J. Odde, “Dimensionless parameters for the design of optical traps and laser guidance systems,” Appl. Opt. 43(20), 3999–4006 (2004).
[CrossRef] [PubMed]

Y. K. Nahmias and D. J. Odde, “Analysis of Radiation Forces in Laser Trapping and Laser-Guided Direct Writing Applications,” IEEE J. Quantum Electron. 38(2), 131–141 (2002).
[CrossRef]

Nemoto, S.

Nevière, M.

Odde, D. J.

Y. K. Nahmias, B. Z. Gao, and D. J. Odde, “Dimensionless parameters for the design of optical traps and laser guidance systems,” Appl. Opt. 43(20), 3999–4006 (2004).
[CrossRef] [PubMed]

Y. K. Nahmias and D. J. Odde, “Analysis of Radiation Forces in Laser Trapping and Laser-Guided Direct Writing Applications,” IEEE J. Quantum Electron. 38(2), 131–141 (2002).
[CrossRef]

Peng, Y.

Polaert, H.

Popov, E.

Pouligny, B.

G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Grehan, and G. Gouesbet, ““Trapping and levitation of a dielectric sphere with off-centred Gaussian beams. II. GLMT analysis,” Pure Appl. Opt. J. Euro. Opt. Soc. Part A 4(5), 571–585 (1995).
[CrossRef]

Qiu, C. W.

C. W. Qiu, S. Zouhdi, and A. Razek, “Modified Spherical Wave Functions With Anisotropy Ratio: Application to the Analysis of Scattering by Multilayered Anisotropic Shells,” IEEE Trans. Antenn. Propag. 55(12), 3515–3523 (2007).
[CrossRef]

Razek, A.

C. W. Qiu, S. Zouhdi, and A. Razek, “Modified Spherical Wave Functions With Anisotropy Ratio: Application to the Analysis of Scattering by Multilayered Anisotropic Shells,” IEEE Trans. Antenn. Propag. 55(12), 3515–3523 (2007).
[CrossRef]

Ren, K. F.

F. Xu, K. F. Ren, G. Gouesbet, X. S. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026613 (2007).
[CrossRef] [PubMed]

Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Gréhan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt. 36(21), 5188–5198 (1997).
[CrossRef] [PubMed]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Prediction of reverse radiation pressure by generalized Lorenz-Mie theory,” Appl. Opt. 35(15), 2702–2710 (1996).
[CrossRef] [PubMed]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory and associated resonance effects,” Opt. Commun. 108(4-6), 343–354 (1994).
[CrossRef]

Rosin, A.

T. Wohland, A. Rosin, and E. H. K. Stelzer, “Theoretical determination of the influence of the polarization on forces exerted by optical tweezers,” Optik (Stuttg.) 102, 181–190 (1996).

Schaub, S. A.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4954–4962 (1989).
[CrossRef]

Schut, T. C. B.

T. C. B. Schut, G. Hesselink, B. G. de Grooth, and J. Greve, “Experimental and theoretical investigations on the validity of the geometrical optics model for calculating the stability of optical traps,” Cytometry 12(6), 479–485 (1991).
[CrossRef] [PubMed]

Sinton, D.

Sluijter, M.

Stelzer, E. H. K.

T. Wohland, A. Rosin, and E. H. K. Stelzer, “Theoretical determination of the influence of the polarization on forces exerted by optical tweezers,” Optik (Stuttg.) 102, 181–190 (1996).

Stout, B.

Togo, H.

Wallace, S.

Wang, K.

F. L. Mao, Q. R. Xing, K. Wang, L. Y. Lang, L. Chai, and Q. Y. Wang, “Calculation of axial optical forces exerted on medium-sized particles by optical trap,” Opt. Laser Technol. 39(1), 34–39 (2007).
[CrossRef]

Wang, Q. Y.

F. L. Mao, Q. R. Xing, K. Wang, L. Y. Lang, L. Chai, and Q. Y. Wang, “Calculation of axial optical forces exerted on medium-sized particles by optical trap,” Opt. Laser Technol. 39(1), 34–39 (2007).
[CrossRef]

Wohland, T.

T. Wohland, A. Rosin, and E. H. K. Stelzer, “Theoretical determination of the influence of the polarization on forces exerted by optical tweezers,” Optik (Stuttg.) 102, 181–190 (1996).

Wriedt, T.

Wu, X. B.

Y. L. Geng, X. B. Wu, and L. W. Li, “Mie scattering by a uniaxial anisotropic sphere,” physical review E 70, 056609 (2004).
[CrossRef]

Wu, Z. S.

Xing, Q. R.

F. L. Mao, Q. R. Xing, K. Wang, L. Y. Lang, L. Chai, and Q. Y. Wang, “Calculation of axial optical forces exerted on medium-sized particles by optical trap,” Opt. Laser Technol. 39(1), 34–39 (2007).
[CrossRef]

Xu, F.

F. Xu, K. F. Ren, G. Gouesbet, X. S. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026613 (2007).
[CrossRef] [PubMed]

Yuan, Q. K.

Zouhdi, S.

C. W. Qiu, S. Zouhdi, and A. Razek, “Modified Spherical Wave Functions With Anisotropy Ratio: Application to the Analysis of Scattering by Multilayered Anisotropic Shells,” IEEE Trans. Antenn. Propag. 55(12), 3515–3523 (2007).
[CrossRef]

Appl. Opt.

A. Doicu and T. Wriedt, “Computation of the beam-shape coefficients in the generalized Lorenz-Mie theory by using the translational addition theorem for spherical vector wave functions,” Appl. Opt. 36(13), 2971–2978 (1997).
[CrossRef] [PubMed]

S. Nemoto and H. Togo, “Axial force acting on a dielectric sphere in a focused laser beam,” Appl. Opt. 37(27), 6386–6394 (1998).
[CrossRef] [PubMed]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Prediction of reverse radiation pressure by generalized Lorenz-Mie theory,” Appl. Opt. 35(15), 2702–2710 (1996).
[CrossRef] [PubMed]

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Figures (11)

Fig. 1
Fig. 1

Uniaxial anisotropic sphere illuminated by an off-axis Gaussian beam.

Fig. 2
Fig. 2

Comparison of the axial radiation force from the theory when the anisotropic sphere is reduced to an isotropic sphere (solid curve) with the experimental results (the curves are denoted by Static measurement and Dynamic measurement) and Ray Optics results in reference [5].( w 0 = 1.8 μ m , ε t = ε z = 2.3716 ε z , μ t = μ z = μ 0 , x 0 = y 0 = 0 , λ = 0.488 μ m , a = 3.75 μ m , n 0 = 1.33 )

Fig. 3
Fig. 3

Variation of the axial radiation force F z with d for different beam waist widths. ( a = 2.0 μ m , ε t = 2.0 ε 0 , ε z = 2.4 ε 0 , μ t = 2.0 μ 0 , μ z = 2.8 μ 0 , x 0 = y 0 = 0 , λ = 0.6328 μ m , n 0 = 1.0 .)

Fig. 4
Fig. 4

Variation of the axial radiation force F z with d for different sphere radii.( w 0 = 0.5 μ m , ε t = 2.0 ε 0 , ε z = 2.4 ε 0 , μ t = 2.0 μ 0 , μ z = 2.8 μ 0 , x 0 = y 0 = 0 , λ = 0.6328 μ m , n 0 = 1.0 .)

Fig. 7
Fig. 7

Variation of the axial radiation force F z with dfor different ε z .( ε t = 2.0 ε 0 , μ t = μ z = μ 0 , x 0 = y 0 = 0 , λ = 0.6328 μ m , w 0 = 0.4 μ m , a = 2.0 μ m , n 0 = 1.33 .)

Fig. 5
Fig. 5

Variation of the axial radiation force F z with dfor different ε z .( ε t = 1.4 ε 0 , μ t = μ z = μ 0 , x 0 = y 0 = 0 , λ = 0.6328 μ m , w 0 = 0.4 μ m , a = 2.0 μ m , n 0 = 1.33 .)

Fig. 8
Fig. 8

Variation of the axial radiation force F z with dfor different ε z .( ε t = 4.0 ε 0 , μ t = μ z = μ 0 , x 0 = y 0 = 0 , λ = 0.6328 μ m , w 0 = 0.4 μ m , a = 2.0 μ m , n 0 = 1.33 .).

Fig. 6
Fig. 6

Variation of the axial radiation force F z with dfor different ε z .( ε t = 1.6 ε 0 , μ t = μ z = μ 0 , x 0 = y 0 = 0 , λ = 0.6328 μ m , w 0 = 0.4 μ m , a = 2.0 μ m , n 0 = 1.33 .)

Fig. 9
Fig. 9

Variation of the transverse radiation force F x ( F y ) with x 0 ( y 0 ) for different sphere radii. ( ε t = 2.0 ε 0 , ε z = 2.4 ε 0 , μ t = μ z = μ 0 , z 0 = 1 μ m , y 0 = 0 ( x 0 = 0 ) , λ = 0.6328 μ m , w 0 = 0.4 μ m , n 0 = 1.33 )

Fig. 10
Fig. 10

Variation of the transverse radiation force F x with x 0 for different z 0 .( w 0 = 0.4 μ m , ε t = 2.0 ε 0 , ε z = 2.4 ε 0 , μ t = μ z = μ 0 , y 0 = 0 , λ = 0.6328 μ m , a = 1.0 μ m , n 0 = 1.33 .)

Fig. 11
Fig. 11

Variation of the transverse radiation force F x with x 0 for different w 0. ( z 0 = 1 μ m , ε t = 2.0 ε 0 , ε z = 2.4 ε 0 , μ t = μ z = μ 0 , y 0 = 0 , λ = 0.6328 μ m , a = 1.0 μ m , n 0 = 1.33 .)

Equations (31)

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E i = E 0 n = 1 m = n n [ a m n i M m n ( 1 ) ( r , k 0 ) + b m n i N m n ( 1 ) ( r , k 0 ) ] , H i = E 0 k 0 i ω μ 0 n = 1 m = n n [ a m n i N m n ( 1 ) ( r , k 0 ) + b m n i M m n ( 1 ) ( r , k 0 ) ] ,
M m n ( l ) ( k r , θ , ϕ ) = z n ( l ) ( k r ) [ i m P n m ( cos θ ) sin θ e i m ϕ θ ^ d P n m ( cos θ ) d θ e i m ϕ ϕ ^ ] , N m n ( l ) ( k r , θ , ϕ ) = n ( n + 1 ) z n ( l ) ( k r ) k r P n m ( cos θ ) e i m ϕ r ^ + 1 k r d ( r z n ( l ) ( k r ) ) d r [ d P n m ( cos θ ) d θ θ ^ + i m P n m ( cos θ ) sin θ ϕ ^ ] e i m ϕ ,
[ a m n i b m n i ] = C n m ( 1 ) m 1 K n m ψ ¯ 0 0 e i k 0 z 0 1 2 [ e i ( m 1 ) φ 0 J m 1 ( 2 Q ¯ ρ 0 ρ n w 0 2 ) e i ( m + 1 ) φ 0 J m + 1 ( 2 Q ¯ ρ 0 ρ n w 0 2 ) ] ,
C n m = { i n 1 2 n + 1 n ( n + 1 ) , m 0 ( 1 ) | m | ( n + | m | ) ! ( n | m | ) ! i n 1 2 n + 1 n ( n + 1 ) , m < 0 ,     K n m = { ( i ) | m | i ( n + 1 / 2 ) | m | 1 ,                       m 0 n ( n + 1 ) n + 1 / 2 ,                                                                                                                     m = 0 ,
ψ ¯ 0 0 = i Q ¯ exp ( i Q ¯ ρ 0 2 w 0 2 ) exp ( i Q ¯ ( n + 1 / 2 ) 2 k 0 2 w 0 2 ) ,
ρ n = ( n + 1 / 2 ) / k 0 , Q ¯ = ( i 2 z 0 / l ) 1 , ρ 0 = x 0 2 + y 0 2 , φ 0 = arctan ( x 0 / y 0 ) .
E s = E 0 n = 1 m = n n [ A m n s M m n ( 3 ) ( r , k 0 ) + B m n s N m n ( 3 ) ( r , k 0 ) ] , H s = E 0 k 0 i ω μ 0 n = 1 m = n n [ A m n s N m n ( 3 ) ( r , k 0 ) + B m n s M m n ( 3 ) ( r , k 0 ) ] .
ε ¯ ¯ = [ ε t 0 0 0 ε t 0 0 0 ε z ] ,              μ ¯ ¯ = [ μ t 0 0 0 μ t 0 0 0 μ z ] .
E I ( r ) = E 0 q = 1 2 n = 1 m = n n n = 1 2 π G m n q 0 π [ A m n q e M m n ( 1 ) ( r , k q )                                                                   + B m n q e N m n ( 1 ) ( r , k q ) + C m n q e L m n ( 1 ) ( r , k q ) ] p n m ( cos θ k ) k q 2 sin θ k d θ k ,
H I ( r ) = E 0 q = 1 2 n = 1 m = n n n = 1 2 π G m n q 0 π [ A m n q h M m n ( 1 ) ( r , k q )                                                                         + B m n q h N m n ( 1 ) ( r , k q ) + C m n q h L m n ( 1 ) ( r , k q ) ] p n m ( cos θ k ) k q 2 sin θ k d θ k ,
k 1 2 = ω 2 ε t μ t μ z μ t sin 2 θ k + μ z cos 2 θ k ,       k 2 2 = ω 2 ε t ε z μ t ε t sin 2 θ k + ε z cos 2 θ k .
A m n s = 1 h n ( 1 ) ( k 0 a ) [ n = 1 2 π G m n q q = 1 2 0 π A m n q e j n ( k q a ) P n m ( cos θ k ) k q 2 sin θ k d θ k a m n i j n ( k 0 a ) ] ,
B m n s = 1 h n ( 1 ) ( k 0 a ) [ i ω μ 0 k 0 q = 1 2 n = 1 2 π G m n q 0 π A m n q h j n ( k q a ) P n m ( cos θ k ) k q 2 sin θ k d θ k b m n i j n ( k 0 a ) ] ,
q = 1 2 n = 1 2 π G m n q 0 π U m n q P n m ( cos θ k ) k q 2 sin θ k d θ k = a m n i i ( k 0 a ) 2 ,
q = 1 2 n = 1 2 π G m n q 0 π V m n q P n m ( cos θ k ) k q 2 sin θ k d θ k = b m n i i ( k 0 a ) 2 ,
U m n q = { A m n q e 1 k 0 r d d r [ r h n ( 1 ) ( k 0 r ) ] j n ( k q r ) i ω μ 0 k 0 [ B m n q h 1 k q r d d r [ r j n ( k q r ) ] + C m n q h j n ( k q r ) r ] h n ( 1 ) ( k 0 r ) } r = a ,
V m n q = { i ω μ 0 k 0 A m n q h 1 k 0 r d d r [ r h n ( 1 ) ( k 0 r ) ] j n ( k q r ) [ B m n q e 1 k q r d d r [ r j n ( k q r ) ] + C m n q e j n ( k q r ) r ] h n ( 1 ) ( k 0 r ) } r = a .
F = 1 2 Re 0 2 π 0 π [ ε 0 E r E + μ 0 H r H 1 2 ( ε 0 E 2 + μ 0 H 2 ) r ^ ] d S ,
x = r sin θ cos ϕ , y = r sin θ sin ϕ , z = r cos θ .
F x = 1 2 Re 0 2 π 0 π { 1 2 [ ε 0 ( E r E r * E θ E θ * E ϕ E ϕ * ) + μ 0 ( H r H r * H θ H θ * H ϕ H ϕ * ) ] r ^         + ( ε 0 E r E θ * + μ 0 H r H θ * ) θ ^ + ( ε 0 E r E ϕ * + μ 0 H r H ϕ * ) ϕ ^ } r 2 sin 2 θ cos ϕ d θ d ϕ | r > a ,
F y = 1 2 Re 0 2 π 0 π { 1 2 [ ε 0 ( E r E r * E θ E θ * E ϕ E ϕ * ) + μ 0 ( H r H r * H θ H θ * H ϕ H ϕ * ) ] r ^         + ( ε 0 E r E θ * + μ 0 H r H θ * ) θ ^ + ( ε 0 E r E ϕ * + μ 0 H r H ϕ * ) ϕ ^ } r 2 sin 2 θ sin ϕ d θ d ϕ | r > a ,
F z = 1 2 Re 0 2 π 0 π { 1 2 [ ε 0 ( E r E r * E θ E θ * E ϕ E ϕ * ) + μ 0 ( H r H r * H θ H θ * H ϕ H ϕ * ) ] r ^         + ( ε 0 E r E θ * + μ 0 H r H θ * ) θ ^ + ( ε 0 E r E ϕ * + μ 0 H r H ϕ * ) ϕ ^ } r 2 sin θ cos θ d θ d ϕ | r > a ,
F x + i F y = n 0 P 0 π c k 0 2 w 0 2 n = 1 m = n n [ ( n m ) ( n + m + 1 ) N m n 1 N m + 1 n 1 ( a m n i B m + 1 n S *                               + b m n i A m + 1 n S * + B m n S a m + 1 n i * + A m n S b m + 1 n i * + 2 A m n S B m + 1 n S * + 2 B m n S A m + 1 n S * )                               i ( n m 1 ) ( n m ) ( 2 n 1 ) ( 2 n + 1 ) ( n 1 ) ( n + 1 ) N m n 1 N m + 1 n 1 1 ( a m n i A m + 1 n 1 S *                             + b m n i B m + 1 n 1 S * + A m n S a m + 1 n 1 i * + B m n S b m + 1 n 1 i * + 2 A m n S A m + 1 n 1 S * + 2 B m n S B m + 1 n 1 S * )                             i ( n + m + 1 ) ( n + m + 2 ) ( 2 n + 1 ) ( 2 n + 3 ) n ( n + 2 ) N m n 1 N m + 1 n + 1 1 ( a m n i A m + 1 n + 1 S * +                             b m n i B m + 1 n + 1 S * + A m n S a m + 1 n + 1 i * + B m n S b m + 1 n + 1 i * + 2 A m n S A m + 1 n + 1 S * + 2 B m n S B m + 1 n + 1 S * ) ] ,
F z = 2 n 0 P 0 π c k 0 2 w 0 2 Re n = 1 m = n n [ i n ( n + 2 ) ( n m + 1 ) ( n + m + 1 ) ( 2 n + 1 ) ( 2 n + 3 ) N m n 1 N m n + 1 1 ( a m n + 1 i A m n S * +                                                     A m n + 1 S a m n i * + b m n + 1 i B m n S * + B m n + 1 S b m n i * + 2 A m n + 1 S A m n S * + 2 B m n + 1 S B m n S * )                                                       m N m n 2 ( a m n i B m n S * + b m n i A m n S * + 2 A m n S B m n S * ) ] .
N m n = ( 2 n + 1 ) ( n m ) ! 4 π ( n + m ) !         ( m = 0 , ± 1 , , ± n ) .
0 2 π e i ( m m ' ) ϕ d ϕ 0 π [ m ' d P n m ( cos θ ) d θ P n ' m ' ( cos θ ) sin θ + m P n m ( cos θ ) sin θ d P n ' m ' ( cos θ ) d θ ] sin 2 θ e i ϕ d θ = ( n m ) ( n + m + 1 ) N m n 1 N m ' n ' 1 δ m + 1 , m ' δ n , n '
0 2 π e i ( m ' m ) ϕ d ϕ 0 π [ m P n m ( cos θ ) sin θ d P n ' m ' ( cos θ ) d θ + m ' d P n m ( cos θ ) d θ P n ' m ' ( cos θ ) sin θ ] sin 2 θ e i ϕ d θ = ( n ' m ' ) ( n ' + m ' + 1 ) N m n 1 N m ' n ' 1 δ m ' + 1 , m δ n , n '
0 2 π e i ( m m ' ) ϕ d ϕ 0 π [ d P n m ( cos θ ) d θ d P n ' m ' ( cos θ ) d θ + m m ' P n m ( cos θ ) sin θ P n ' m ' ( cos θ ) sin θ ] sin 2 θ e i ϕ d θ = N m n 1 N m ' n ' 1 [ ( n + m + 1 ) ( n + m + 2 ) ( 2 n + 1 ) ( 2 n + 3 ) n ( n + 2 ) δ n + 1 , n ' ( n ' m ) ( n ' m + 1 ) ( 2 n ' + 1 ) ( 2 n ' + 3 ) n ' ( n ' + 2 ) δ n , n ' + 1 ] δ m + 1 , m '
0 2 π e i ( m ' m ) ϕ d ϕ 0 π [ d P n m ( cos θ ) d θ d P n ' m ' ( cos θ ) d θ + m m ' P n m ( cos θ ) sin θ P n ' m ' ( cos θ ) sin θ ] sin 2 θ e i ϕ d θ = N m n 1 N m ' n ' 1 [ ( n ' + m ' + 1 ) ( n ' + m ' + 2 ) ( 2 n ' + 1 ) ( 2 n ' + 3 ) n ' ( n ' + 2 ) δ n ' + 1 , n ( n m ' ) ( n m ' + 1 ) ( 2 n + 1 ) ( 2 n + 3 ) n ( n + 2 ) δ n ' , n + 1 ] δ m ' + 1 , m
0 2 π e i ( m m ' ) ϕ d ϕ 0 π [ m ' d P n m ( cos θ ) d θ P n ' m ' ( cos θ ) sin θ + m P n m ( cos θ ) sin θ d P n ' m ' ( cos θ ) d θ ] sin θ cos θ d θ = m N m n 1 N m ' n ' 1 δ m , m ' δ n , n '
0 2 π e i ( m m ' ) ϕ d ϕ 0 π [ d P n m ( cos θ ) d θ d P n ' m ' ( cos θ ) d θ + m m ' P n m ( cos θ ) sin θ P n ' m ' ( cos θ ) sin θ ] sin θ cos θ d θ = N m n 1 N m ' n ' 1 ( n ' m + 1 ) ( n ' + m + 1 ) ( 2 n ' + 1 ) ( 2 n ' + 3 ) n ' ( n ' + 2 ) δ m , m ' δ n , n ' + 1                 + N m n 1 N m ' n ' 1 ( n m + 1 ) ( n + m + 1 ) ( 2 n + 1 ) ( 2 n + 3 ) n ( n + 2 ) δ m , m ' δ n + 1 , n '

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