Abstract

We derive the analytical expression for the propagation of elegant Hermite-cosine-Gaussian (EHcosG) beams through a paraxial ABCD optical system and use it to study the radiation forces produced by highly focused this kind of beams acting on a Rayleigh dielectric sphere. Owing to the characteristics of focused EHcosG beams our analysis shows that it can be expected to simultaneously trap and manipulate dielectric spheres with low-refractive index at the focus point, and spheres with high-refractive index nearby the focus point. Finally, we discuss the stability conditions for effective trapping and manipulating the particle.

© 2012 OSA

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    [CrossRef] [PubMed]
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  9. M. Dienerowitz, M. Mazilu, P. J. Reece, T. F. Krauss, and K. Dholakia, “Optical vortex trap for resonant confinement of metal nanoparticles,” Opt. Express 16(7), 4991–4999 (2008).
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    [CrossRef] [PubMed]
  14. M. Bhattacharya and P. Meystre, “Using a Laguerre-Gaussian beam to trap and cool the rotational motion of a mirror,” Phys. Rev. Lett. 99(15), 153603 (2007).
    [CrossRef] [PubMed]
  15. C. L. Zhao, L. G. Wang, and X. H. Lu, “Radiation forces on a dielectric sphere produced by highly focused hollow Gaussian beams,” Phys. Lett. A 363(5-6), 502–506 (2007).
    [CrossRef]
  16. H. Little, C. T. A. Brown, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical guiding of microscopic particles in femtosecond and continuous wave Bessel light beams,” Opt. Express 12(11), 2560–2565 (2004).
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    [CrossRef]
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    [CrossRef]
  33. D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236(4-6), 225–235 (2004).
    [CrossRef]
  34. Z. Mei and D. Zhao, “Propagation of Laguerre-Gaussian and elegant Laguerre-Gaussian beams in apertured fractional Hankel transform systems,” J. Opt. Soc. Am. A 21(12), 2375–2381 (2004).
    [CrossRef] [PubMed]
  35. H. Mao and D. Zhao, “Different models for a hard-aperture function and corresponding approximate analytical propagation equations of a Gaussian beam through an apertured optical system,” J. Opt. Soc. Am. A 22(4), 647–653 (2005).
    [CrossRef] [PubMed]
  36. D. Deng, “Propagation of elegant Hermite cosine Gaussian laser beams,” Opt. Commun. 259(2), 409–414 (2006).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2011 (1)

2010 (1)

Y. J. Zhang, B. F. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[CrossRef]

2008 (1)

2007 (4)

M. Bhattacharya and P. Meystre, “Using a Laguerre-Gaussian beam to trap and cool the rotational motion of a mirror,” Phys. Rev. Lett. 99(15), 153603 (2007).
[CrossRef] [PubMed]

C. L. Zhao, L. G. Wang, and X. H. Lu, “Radiation forces on a dielectric sphere produced by highly focused hollow Gaussian beams,” Phys. Lett. A 363(5-6), 502–506 (2007).
[CrossRef]

L. G. Wang and C. L. Zhao, “Dynamic radiation force of a pulsed gaussian beam acting on rayleigh dielectric sphere,” Opt. Express 15(17), 10615–10621 (2007).
[CrossRef] [PubMed]

L. G. Wang, C. L. Zhao, L. Q. Wang, X. H. Lu, and S. Y. Zhu, “Effect of spatial coherence on radiation forces acting on a Rayleigh dielectric sphere,” Opt. Lett. 32(11), 1393–1395 (2007).
[CrossRef] [PubMed]

2006 (2)

D. Deng, “Propagation of elegant Hermite cosine Gaussian laser beams,” Opt. Commun. 259(2), 409–414 (2006).
[CrossRef]

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(5), 058301 (2006).
[CrossRef] [PubMed]

2005 (4)

2004 (8)

2003 (1)

D. Zhao, H. Mao, W. Zhang, and S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224(1-3), 5–12 (2003).
[CrossRef]

2000 (2)

B. Lü and H. Ma, “A comparative study of elegant and standard Hermite-Gaussian beams,” Opt. Commun. 174(1-4), 99–104 (2000).
[CrossRef]

A. Ashkin, “History of optical trapping and manipulation of small-neutral particle, atoms, and molecules,” IEEE J. Sel. Top. Quantum Electron. 6(6), 841–856 (2000).
[CrossRef]

1999 (1)

K. Okamoto and S. Kawata, “Radiation force exerted on subwavelength particles near a nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 (1999).
[CrossRef]

1998 (2)

1997 (1)

K. Taguchi, H. Ueno, T. Hiramatsu, and M. Ikeda, “Optical trapping of dielectric particle and biological cell using optical fibre,” Electron. Lett. 33(5), 413–414 (1997).
[CrossRef]

1996 (1)

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]

1986 (1)

1978 (1)

A. Ashkin, “Trapping of atoms by resonance radiation pressure,” Phys. Rev. Lett. 40(12), 729–732 (1978).
[CrossRef]

1973 (2)

1970 (2)

S. A. Collins., “Lens-system diffraction integral written in terms of Matrix Optics,” J. Opt. Soc. Am. 60(9), 1168–1177 (1970).
[CrossRef]

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

Ambardekar, A. A.

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, “History of optical trapping and manipulation of small-neutral particle, atoms, and molecules,” IEEE J. Sel. Top. Quantum Electron. 6(6), 841–856 (2000).
[CrossRef]

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, “Trapping of atoms by resonance radiation pressure,” Phys. Rev. Lett. 40(12), 729–732 (1978).
[CrossRef]

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

Baykal, Y.

Bhattacharya, M.

M. Bhattacharya and P. Meystre, “Using a Laguerre-Gaussian beam to trap and cool the rotational motion of a mirror,” Phys. Rev. Lett. 99(15), 153603 (2007).
[CrossRef] [PubMed]

Bjorkholm, J. E.

Block, S. M.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004).
[CrossRef] [PubMed]

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(5), 058301 (2006).
[CrossRef] [PubMed]

Brown, C. T. A.

Casperson, L. W.

Chen, C. H.

Chu, S.

Chuu, C. S.

Collins, S. A.

Deng, D.

D. Deng, “Propagation of elegant Hermite cosine Gaussian laser beams,” Opt. Commun. 259(2), 409–414 (2006).
[CrossRef]

Deng, J. L.

Dholakia, K.

Dienerowitz, M.

Ding, B. F.

Y. J. Zhang, B. F. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[CrossRef]

Dziedzic, J. M.

Eyyuboglu, H. T.

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(5), 058301 (2006).
[CrossRef] [PubMed]

Garcés-Chávez, V.

V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85(18), 4001–4003 (2004).
[CrossRef]

H. Little, C. T. A. Brown, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical guiding of microscopic particles in femtosecond and continuous wave Bessel light beams,” Opt. Express 12(11), 2560–2565 (2004).
[CrossRef] [PubMed]

Gordon, J. P.

J. P. Gordon, “Radiation forces and momenta in dielectric media,” Phys. Rev. A 8(1), 14–21 (1973).
[CrossRef]

Hanssen, J. L.

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]

Hiramatsu, T.

K. Taguchi, H. Ueno, T. Hiramatsu, and M. Ikeda, “Optical trapping of dielectric particle and biological cell using optical fibre,” Electron. Lett. 33(5), 413–414 (1997).
[CrossRef]

Hsieh, W. F.

Huang, K. K.

Ikeda, M.

K. Taguchi, H. Ueno, T. Hiramatsu, and M. Ikeda, “Optical trapping of dielectric particle and biological cell using optical fibre,” Electron. Lett. 33(5), 413–414 (1997).
[CrossRef]

Jiang, Y. F.

Jing, F.

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236(4-6), 225–235 (2004).
[CrossRef]

Kawata, S.

K. Okamoto and S. Kawata, “Radiation force exerted on subwavelength particles near a nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 (1999).
[CrossRef]

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(5), 058301 (2006).
[CrossRef] [PubMed]

Krauss, T. F.

Li, Y. Q.

Little, H.

Liu, H.

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236(4-6), 225–235 (2004).
[CrossRef]

Lu, X. H.

Lü, B.

B. Lü and H. Ma, “A comparative study of elegant and standard Hermite-Gaussian beams,” Opt. Commun. 174(1-4), 99–104 (2000).
[CrossRef]

Ma, H.

B. Lü and H. Ma, “A comparative study of elegant and standard Hermite-Gaussian beams,” Opt. Commun. 174(1-4), 99–104 (2000).
[CrossRef]

Mao, H.

H. Mao and D. Zhao, “Different models for a hard-aperture function and corresponding approximate analytical propagation equations of a Gaussian beam through an apertured optical system,” J. Opt. Soc. Am. A 22(4), 647–653 (2005).
[CrossRef] [PubMed]

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236(4-6), 225–235 (2004).
[CrossRef]

D. Zhao, H. Mao, W. Zhang, and S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224(1-3), 5–12 (2003).
[CrossRef]

Mazilu, M.

McGloin, D.

V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85(18), 4001–4003 (2004).
[CrossRef]

Mei, Z.

Melville, H.

V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85(18), 4001–4003 (2004).
[CrossRef]

Meyrath, T. P.

Meystre, P.

M. Bhattacharya and P. Meystre, “Using a Laguerre-Gaussian beam to trap and cool the rotational motion of a mirror,” Phys. Rev. Lett. 99(15), 153603 (2007).
[CrossRef] [PubMed]

Neuman, K. C.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004).
[CrossRef] [PubMed]

Okamoto, K.

K. Okamoto and S. Kawata, “Radiation force exerted on subwavelength particles near a nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 (1999).
[CrossRef]

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(5), 058301 (2006).
[CrossRef] [PubMed]

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(5), 058301 (2006).
[CrossRef] [PubMed]

Raizen, M. G.

Reece, P. J.

Roskey, D.

V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85(18), 4001–4003 (2004).
[CrossRef]

Schreck, F.

Sibbett, W.

Siegman, A. E.

Summers, M. D.

V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85(18), 4001–4003 (2004).
[CrossRef]

Suyama, T.

Y. J. Zhang, B. F. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[CrossRef]

Taguchi, K.

K. Taguchi, H. Ueno, T. Hiramatsu, and M. Ikeda, “Optical trapping of dielectric particle and biological cell using optical fibre,” Electron. Lett. 33(5), 413–414 (1997).
[CrossRef]

Tai, P. T.

Tovar, A. A.

Ueno, H.

K. Taguchi, H. Ueno, T. Hiramatsu, and M. Ikeda, “Optical trapping of dielectric particle and biological cell using optical fibre,” Electron. Lett. 33(5), 413–414 (1997).
[CrossRef]

Wang, L. G.

Wang, L. Q.

Wang, S.

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236(4-6), 225–235 (2004).
[CrossRef]

D. Zhao, H. Mao, W. Zhang, and S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224(1-3), 5–12 (2003).
[CrossRef]

Wang, Y. Z.

Wei, Q.

Wei, X.

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236(4-6), 225–235 (2004).
[CrossRef]

Wright, E. M.

V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85(18), 4001–4003 (2004).
[CrossRef]

Zhan, Q. W.

Zhang, W.

D. Zhao, H. Mao, W. Zhang, and S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224(1-3), 5–12 (2003).
[CrossRef]

Zhang, Y. J.

Y. J. Zhang, B. F. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[CrossRef]

Zhao, C. L.

Zhao, D.

H. Mao and D. Zhao, “Different models for a hard-aperture function and corresponding approximate analytical propagation equations of a Gaussian beam through an apertured optical system,” J. Opt. Soc. Am. A 22(4), 647–653 (2005).
[CrossRef] [PubMed]

Z. Mei and D. Zhao, “Propagation of Laguerre-Gaussian and elegant Laguerre-Gaussian beams in apertured fractional Hankel transform systems,” J. Opt. Soc. Am. A 21(12), 2375–2381 (2004).
[CrossRef] [PubMed]

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236(4-6), 225–235 (2004).
[CrossRef]

D. Zhao, H. Mao, W. Zhang, and S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224(1-3), 5–12 (2003).
[CrossRef]

Zhu, S. Y.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85(18), 4001–4003 (2004).
[CrossRef]

Electron. Lett. (1)

K. Taguchi, H. Ueno, T. Hiramatsu, and M. Ikeda, “Optical trapping of dielectric particle and biological cell using optical fibre,” Electron. Lett. 33(5), 413–414 (1997).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

A. Ashkin, “History of optical trapping and manipulation of small-neutral particle, atoms, and molecules,” IEEE J. Sel. Top. Quantum Electron. 6(6), 841–856 (2000).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

Opt. Commun. (5)

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]

B. Lü and H. Ma, “A comparative study of elegant and standard Hermite-Gaussian beams,” Opt. Commun. 174(1-4), 99–104 (2000).
[CrossRef]

D. Zhao, H. Mao, W. Zhang, and S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224(1-3), 5–12 (2003).
[CrossRef]

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236(4-6), 225–235 (2004).
[CrossRef]

D. Deng, “Propagation of elegant Hermite cosine Gaussian laser beams,” Opt. Commun. 259(2), 409–414 (2006).
[CrossRef]

Opt. Express (8)

L. G. Wang and C. L. Zhao, “Dynamic radiation force of a pulsed gaussian beam acting on rayleigh dielectric sphere,” Opt. Express 15(17), 10615–10621 (2007).
[CrossRef] [PubMed]

M. Dienerowitz, M. Mazilu, P. J. Reece, T. F. Krauss, and K. Dholakia, “Optical vortex trap for resonant confinement of metal nanoparticles,” Opt. Express 16(7), 4991–4999 (2008).
[CrossRef] [PubMed]

Y. F. Jiang, K. K. Huang, and X. H. Lu, “Radiation force of highly focused Lorentz-Gauss beams on a Rayleigh particle,” Opt. Express 19(10), 9708–9713 (2011).
[CrossRef] [PubMed]

T. P. Meyrath, F. Schreck, J. L. Hanssen, C. S. Chuu, and M. G. Raizen, “A high frequency optical trap for atoms using Hermite-Gaussian beams,” Opt. Express 13(8), 2843–2851 (2005).
[CrossRef] [PubMed]

J. L. Deng, Q. Wei, Y. Z. Wang, and Y. Q. Li, “Numerical modeling of optical levitation and trapping of the “stuck” particles with a pulsed optical tweezers,” Opt. Express 13(10), 3673–3680 (2005).
[CrossRef] [PubMed]

H. Little, C. T. A. Brown, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical guiding of microscopic particles in femtosecond and continuous wave Bessel light beams,” Opt. Express 12(11), 2560–2565 (2004).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

Opt. Lett. (3)

Phys. Lett. A (1)

C. L. Zhao, L. G. Wang, and X. H. Lu, “Radiation forces on a dielectric sphere produced by highly focused hollow Gaussian beams,” Phys. Lett. A 363(5-6), 502–506 (2007).
[CrossRef]

Phys. Rev. A (2)

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[CrossRef]

Y. J. Zhang, B. F. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

(a), (b) Intensity distribution of an EHcosG beam in the x direction at the z=0 plane (a) for various values of the order of the Hermite polynomial p (Here we assume q=p ) with Ω=1 m 1 , (b) for various values of the cosine part associated parameter Ω with p=2 . The other simulation parameter is selected as w 0 =1.0mm . (c) Schematic of an unapertured thin lens optical system, and in our case s=200 mm and f=2 mm are selected.

Fig. 2
Fig. 2

Intensity distributions of the EHcosG beams near the focal plane at different distances: (a) z 1 =0 , (b) z 1 =1.415μm , (c) z 1 =2.83μm , and (d) z 1 =5.00μm . The other parameters are λ=1064 nm, w 0 =1.0 mm, p=2 , Ω=2 m−1, s=200 mm, f=2 mm, n 1 =1.592 , and the input power of the beams is 100 mW.

Fig. 3
Fig. 3

(a)-(c) The transverse gradient force produced by highly focused EHcosG beams at different position z 1 : (a) z 1 =0 , (b) z 1 =1.415μm , (c) z 1 =2.83μm ; (d)-(f) The longitudinal gradient force produced by highly focused EHcosG beams at different transverse position x : (d) x=0 , (e) x=0.509μm and (f) x=1.18μm . Solid curves for the particles with n 1 =1.592 , dashed curves for the particles with n 1 =1.0 .

Fig. 4
Fig. 4

(a)-(d) The scattering force produced by highly focused EHcosG beams at different distance z 1 : (a) z 1 =0 , (b) z 1 =1.415μm , (c) z 1 =2.83μm and (d) z 1 =5.00μm . Solid curves for the particles with n 1 =1.592 , dashed curves for the particles with n 1 =1.0 .

Fig. 5
Fig. 5

Comparison of F grad,x m (solid black curve), F grad,z m (dashed red curve), F scat m (dotted blue curve), F b (dotted-dashed green curve) and F g (dotted-dashed Brown curve) with different particles’ radius a , while the other parameters are w 0 =1 mm, p=2 , Ω=2 m−1, f=2 mm, s=200 mm, n 1 =1.592 , n 2 =1.332 . F grad,x m , F grad,z m and F scat m occur at (0.325μm,0.509μm,0), (0,0,0.679μm), (0.509μm,0.509μm,0), respectively.

Equations (18)

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E pq ( x 1 , y 1 ;z=0)= G 0 H p ( x 1 w 0 )cos(Ω x 1 ) H q ( y 1 w 0 )cos(Ω y 1 )exp( x 1 2 + y 1 2 w 0 2 ),
E pq ( x 1 , y 1 ;z=0)= G 0 H p ( x 1 w 0 ) 1 2 [exp(iΩ x 1 )+exp(iΩ x 1 )] × H q ( y 1 w 0 ) 1 2 [exp(iΩ y 1 )+exp(iΩ y 1 )]exp( x 1 2 + y 1 2 w 0 2 ).
E pq (x,y;z)= i λB E pq ( x 1 , y 1 ;z=0) ×exp{ ik 2B [A( x 1 2 + y 1 2 )2(x x 1 +y y 1 )+D( x 2 +y ) 2 ] }d x 1 d y 1 ,
H n (αx)exp[ (xy) 2 ] dx= π 1/2 (1 α 2 ) n/2 H n ( αy 1 α 2 ),
E pq (x,y;z) = i G 0 λB exp[ ik 2B D( x 2 +y ) 2 ]A(x)B(y),
A(x) = 1 2 α exp( β 1 2 4α ) π 1/2 (1 1 α w 0 2 ) p/2 H p ( β 1 2α w 0 1 1 α w 0 2 ) + 1 2 α exp( β 2 2 4α ) π 1/2 (1 1 α w 0 2 ) p/2 H p ( β 2 2α w 0 1 1 α w 0 2 ),
B(y) = 1 2 α exp( β ' 1 2 4α ) π 1/2 (1 1 α w 0 2 ) q/2 H q ( β ' 1 2α w 0 1 1 α w 0 2 ) + 1 2 α exp( β ' 2 2 4α ) π 1/2 (1 1 α w 0 2 ) q/2 H q ( β ' 2 2α w 0 1 1 α w 0 2 ),
α= 1 w 0 2 + ikA 2B ,
β 1 = ikx B +iΩ,
β 2 = ikx B iΩ,
β ' 1 = iky B +iΩ,
β ' 2 = iky B iΩ.
[ A B C D ]=[ 1 z 1 +f 0 1 ][ 1 0 1/f 1 ][ 1 s 0 1 ]=[ z 1 /f z 1 /fs+f+ z 1 1/f 1s/f ],
F scat (r,z)= z ^ n 2 c C pr I(r,z),
I(r,z)= z ^ n 2 ε 0 c 2 | E(r,z) | 2 .
C pr = C scat = 8 3 π (ka) 4 a 2 ( m 2 1 m 2 +2 ) 2 ,
F grad (r,z)= 2π n 2 a 3 c ( m 2 1 m 2 +2 )I(r,z).
R thermal =exp( U max / k B T)<<1,

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