Abstract

Abstract

We investigate the dynamic evolution of the radiation forces produced by the pulsed Gaussian beams acting on a Rayleigh dielectric sphere. We derive the analytical expressions for the scattering force and all components of the ponderomotive force induced by the pulsed Gaussian beams. Our analysis shows that the radiation force, for both the transverse and longitudinal components, can be greatly enhanced as the pulse duration decreases. It is further found that for the pulse with long pulse duration, it can be used for the stable trapping and manipulating the particle, while for the pulse with short pulse duration it may be used for guiding and moving the small dielectric particle. Finally we discuss the stability conditions of the effective manipulating the particle by the pulsed beam.

© 2007 Optical Society of America

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  1. A. Ashkin, "Acceleration and trapping of particles by radiation pressure," Phys. Rev. Lett. 24, 156-159 (1970).
    [CrossRef]
  2. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, S. Chu, "Observation of a single-beam gradient force optical trap for dielectric particles," Opt. Lett. 11, 288-290 (1986).
    [CrossRef] [PubMed]
  3. A. Ashkin, in Methods in cell Biology, M. P. Sheetz, ed., (Academic Press, San Diego, 1998), 55, pp.1-27.
    [CrossRef]
  4. A. J. Hunt, F. Gittes, and J. Howard, "The force exerted by a single kinesin molecule against a viscous load," Biophys. J. 67, 766-781 (1994).
    [CrossRef] [PubMed]
  5. J. Dai and M. P. Sheetz, "Mechanical properties of neuronal growth cone membrane by tether formation with laser optical tweezers," Biophys. J. 68, 988-996 (1995).
    [CrossRef] [PubMed]
  6. M. D. Wang, M. J. Schnitzer, H. Yin, R. Landick, J. Gelles, S. M. Block, "Force and velocity measured for single molecules of RNA polymerase," Science 282, 902-907 (1998).
    [CrossRef] [PubMed]
  7. A. D. Metha, M. Rief, J. A. Spudich, D. A. Smith, R. M. Simmons, "Single-molecule biomechanics with optical methods," Science. 283, 1689-1695 (1999).
    [CrossRef]
  8. A. Ashkin, "Trapping of Atoms by resonance radiation pressure," Phys. Rev. Lett. 40, 729-732 (1978).
    [CrossRef]
  9. S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, "Experimental observation of optically trapped atoms," Phys. Rev. Lett. 57, 314-317 (1986).
    [CrossRef] [PubMed]
  10. A. A. Ambardekar and Y. Q. Li, "Optical levitation and manipulation of stuck particles with pulsed optical tweezers," Opt. Lett. 30, 1797-1799 (2005).
    [CrossRef] [PubMed]
  11. 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. Expresss. 13, 3673-3680 (2006).
    [CrossRef]
  12. H. Little, C. T. A. Brown, W. Sibbett, and K. Dholakia, "Optical guiding of microscopic particles in femtosecond and continuous wave Bessel light beams," Opt. Express. 11, 2560-2565 (2004).
    [CrossRef]
  13. B. Agate, C. T. A. Brown, W. Sibbett, and K. Dholakia, "Femtosecond optical tweezers for in-situ control of two-photon fluorescence," Opt. Express. 12, 3011-3017 (2004).
    [CrossRef] [PubMed]
  14. J. P. Gordon, "Radiation forces and momenta in dielectric media," Phys. Rev. A. 8, 14-21 (1973).
    [CrossRef]
  15. I. Brevik, "Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor," Phys. Rep. 52, 133-201 (1979).
    [CrossRef]
  16. H. M. Lai, C. K. Ng, and K. Young, "Radiation force on an object and momentum of light in a liquid dielectric," Phys. Rev. A 30, 1060-1066 (1984).
    [CrossRef]
  17. Y. Harada and T. Asakura, "Radiation forces on a dielectric sphere in the Rayleigh scatting regime," Opt. Commun. 124, 529-541 (1996).
    [CrossRef]
  18. L. G. Wang, C. L. Zhao, L. Q. Wang, X. H. Lu, S. and Y. Zhu, "Effect of spatial coherence on radiation forces acting on a Rayleigh dielectric sphere," Opt. Lett. 32, 1393-1395 (2007).
    [CrossRef] [PubMed]
  19. C. L. Zhao, L. G. Wang, X. and H. Lu, "Radiation forces on a dielectric sphere produced by highly focused hollow Gaussian beams," Phys. Lett. A. 363, 502-506 (2006).
    [CrossRef]
  20. R. Gussgard, T. Lindmo, and I. Brevik, "Calculation of the trapping force in a strongly focused laser beam," J. Opt. Soc. Am. B 9, 1922-1930 (1992).
    [CrossRef]

2007

2006

C. L. Zhao, L. G. Wang, X. and H. Lu, "Radiation forces on a dielectric sphere produced by highly focused hollow Gaussian beams," Phys. Lett. A. 363, 502-506 (2006).
[CrossRef]

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. Expresss. 13, 3673-3680 (2006).
[CrossRef]

2005

2004

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

B. Agate, C. T. A. Brown, W. Sibbett, and K. Dholakia, "Femtosecond optical tweezers for in-situ control of two-photon fluorescence," Opt. Express. 12, 3011-3017 (2004).
[CrossRef] [PubMed]

1999

A. D. Metha, M. Rief, J. A. Spudich, D. A. Smith, R. M. Simmons, "Single-molecule biomechanics with optical methods," Science. 283, 1689-1695 (1999).
[CrossRef]

1998

M. D. Wang, M. J. Schnitzer, H. Yin, R. Landick, J. Gelles, S. M. Block, "Force and velocity measured for single molecules of RNA polymerase," Science 282, 902-907 (1998).
[CrossRef] [PubMed]

1996

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

1995

J. Dai and M. P. Sheetz, "Mechanical properties of neuronal growth cone membrane by tether formation with laser optical tweezers," Biophys. J. 68, 988-996 (1995).
[CrossRef] [PubMed]

1994

A. J. Hunt, F. Gittes, and J. Howard, "The force exerted by a single kinesin molecule against a viscous load," Biophys. J. 67, 766-781 (1994).
[CrossRef] [PubMed]

1992

1986

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, S. Chu, "Observation of a single-beam gradient force optical trap for dielectric particles," Opt. Lett. 11, 288-290 (1986).
[CrossRef] [PubMed]

S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, "Experimental observation of optically trapped atoms," Phys. Rev. Lett. 57, 314-317 (1986).
[CrossRef] [PubMed]

1984

H. M. Lai, C. K. Ng, and K. Young, "Radiation force on an object and momentum of light in a liquid dielectric," Phys. Rev. A 30, 1060-1066 (1984).
[CrossRef]

1979

I. Brevik, "Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor," Phys. Rep. 52, 133-201 (1979).
[CrossRef]

1978

A. Ashkin, "Trapping of Atoms by resonance radiation pressure," Phys. Rev. Lett. 40, 729-732 (1978).
[CrossRef]

1973

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

1970

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

Agate, B.

B. Agate, C. T. A. Brown, W. Sibbett, and K. Dholakia, "Femtosecond optical tweezers for in-situ control of two-photon fluorescence," Opt. Express. 12, 3011-3017 (2004).
[CrossRef] [PubMed]

Ambardekar, A. A.

Asakura, T.

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

Ashkin, A.

S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, "Experimental observation of optically trapped atoms," Phys. Rev. Lett. 57, 314-317 (1986).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, S. Chu, "Observation of a single-beam gradient force optical trap for dielectric particles," Opt. Lett. 11, 288-290 (1986).
[CrossRef] [PubMed]

A. Ashkin, "Trapping of Atoms by resonance radiation pressure," Phys. Rev. Lett. 40, 729-732 (1978).
[CrossRef]

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

Bjorkholm, J. E.

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, S. Chu, "Observation of a single-beam gradient force optical trap for dielectric particles," Opt. Lett. 11, 288-290 (1986).
[CrossRef] [PubMed]

S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, "Experimental observation of optically trapped atoms," Phys. Rev. Lett. 57, 314-317 (1986).
[CrossRef] [PubMed]

Block, S. M.

M. D. Wang, M. J. Schnitzer, H. Yin, R. Landick, J. Gelles, S. M. Block, "Force and velocity measured for single molecules of RNA polymerase," Science 282, 902-907 (1998).
[CrossRef] [PubMed]

Brevik, I.

R. Gussgard, T. Lindmo, and I. Brevik, "Calculation of the trapping force in a strongly focused laser beam," J. Opt. Soc. Am. B 9, 1922-1930 (1992).
[CrossRef]

I. Brevik, "Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor," Phys. Rep. 52, 133-201 (1979).
[CrossRef]

Brown, C. T. A.

B. Agate, C. T. A. Brown, W. Sibbett, and K. Dholakia, "Femtosecond optical tweezers for in-situ control of two-photon fluorescence," Opt. Express. 12, 3011-3017 (2004).
[CrossRef] [PubMed]

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

Cable, A.

S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, "Experimental observation of optically trapped atoms," Phys. Rev. Lett. 57, 314-317 (1986).
[CrossRef] [PubMed]

Chu, S.

S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, "Experimental observation of optically trapped atoms," Phys. Rev. Lett. 57, 314-317 (1986).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, S. Chu, "Observation of a single-beam gradient force optical trap for dielectric particles," Opt. Lett. 11, 288-290 (1986).
[CrossRef] [PubMed]

Dai, J.

J. Dai and M. P. Sheetz, "Mechanical properties of neuronal growth cone membrane by tether formation with laser optical tweezers," Biophys. J. 68, 988-996 (1995).
[CrossRef] [PubMed]

Deng, J. L.

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. Expresss. 13, 3673-3680 (2006).
[CrossRef]

Dholakia, K.

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

B. Agate, C. T. A. Brown, W. Sibbett, and K. Dholakia, "Femtosecond optical tweezers for in-situ control of two-photon fluorescence," Opt. Express. 12, 3011-3017 (2004).
[CrossRef] [PubMed]

Dziedzic, J. M.

Gelles, J.

M. D. Wang, M. J. Schnitzer, H. Yin, R. Landick, J. Gelles, S. M. Block, "Force and velocity measured for single molecules of RNA polymerase," Science 282, 902-907 (1998).
[CrossRef] [PubMed]

Gittes, F.

A. J. Hunt, F. Gittes, and J. Howard, "The force exerted by a single kinesin molecule against a viscous load," Biophys. J. 67, 766-781 (1994).
[CrossRef] [PubMed]

Gordon, J. P.

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

Gussgard, R.

Harada, Y.

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

Howard, J.

A. J. Hunt, F. Gittes, and J. Howard, "The force exerted by a single kinesin molecule against a viscous load," Biophys. J. 67, 766-781 (1994).
[CrossRef] [PubMed]

Hunt, A. J.

A. J. Hunt, F. Gittes, and J. Howard, "The force exerted by a single kinesin molecule against a viscous load," Biophys. J. 67, 766-781 (1994).
[CrossRef] [PubMed]

Lai, H. M.

H. M. Lai, C. K. Ng, and K. Young, "Radiation force on an object and momentum of light in a liquid dielectric," Phys. Rev. A 30, 1060-1066 (1984).
[CrossRef]

Landick, R.

M. D. Wang, M. J. Schnitzer, H. Yin, R. Landick, J. Gelles, S. M. Block, "Force and velocity measured for single molecules of RNA polymerase," Science 282, 902-907 (1998).
[CrossRef] [PubMed]

Li, Y. Q.

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. Expresss. 13, 3673-3680 (2006).
[CrossRef]

A. A. Ambardekar and Y. Q. Li, "Optical levitation and manipulation of stuck particles with pulsed optical tweezers," Opt. Lett. 30, 1797-1799 (2005).
[CrossRef] [PubMed]

Lindmo, T.

Little, H.

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

Lu, X. H.

Metha, A. D.

A. D. Metha, M. Rief, J. A. Spudich, D. A. Smith, R. M. Simmons, "Single-molecule biomechanics with optical methods," Science. 283, 1689-1695 (1999).
[CrossRef]

Ng, C. K.

H. M. Lai, C. K. Ng, and K. Young, "Radiation force on an object and momentum of light in a liquid dielectric," Phys. Rev. A 30, 1060-1066 (1984).
[CrossRef]

Rief, M.

A. D. Metha, M. Rief, J. A. Spudich, D. A. Smith, R. M. Simmons, "Single-molecule biomechanics with optical methods," Science. 283, 1689-1695 (1999).
[CrossRef]

Schnitzer, M. J.

M. D. Wang, M. J. Schnitzer, H. Yin, R. Landick, J. Gelles, S. M. Block, "Force and velocity measured for single molecules of RNA polymerase," Science 282, 902-907 (1998).
[CrossRef] [PubMed]

Sheetz, M. P.

J. Dai and M. P. Sheetz, "Mechanical properties of neuronal growth cone membrane by tether formation with laser optical tweezers," Biophys. J. 68, 988-996 (1995).
[CrossRef] [PubMed]

Sibbett, W.

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

B. Agate, C. T. A. Brown, W. Sibbett, and K. Dholakia, "Femtosecond optical tweezers for in-situ control of two-photon fluorescence," Opt. Express. 12, 3011-3017 (2004).
[CrossRef] [PubMed]

Simmons, R. M.

A. D. Metha, M. Rief, J. A. Spudich, D. A. Smith, R. M. Simmons, "Single-molecule biomechanics with optical methods," Science. 283, 1689-1695 (1999).
[CrossRef]

Smith, D. A.

A. D. Metha, M. Rief, J. A. Spudich, D. A. Smith, R. M. Simmons, "Single-molecule biomechanics with optical methods," Science. 283, 1689-1695 (1999).
[CrossRef]

Spudich, J. A.

A. D. Metha, M. Rief, J. A. Spudich, D. A. Smith, R. M. Simmons, "Single-molecule biomechanics with optical methods," Science. 283, 1689-1695 (1999).
[CrossRef]

Wang, L. G.

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

C. L. Zhao, L. G. Wang, X. and H. Lu, "Radiation forces on a dielectric sphere produced by highly focused hollow Gaussian beams," Phys. Lett. A. 363, 502-506 (2006).
[CrossRef]

Wang, L. Q.

Wang, M. D.

M. D. Wang, M. J. Schnitzer, H. Yin, R. Landick, J. Gelles, S. M. Block, "Force and velocity measured for single molecules of RNA polymerase," Science 282, 902-907 (1998).
[CrossRef] [PubMed]

Wang, Y. Z.

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. Expresss. 13, 3673-3680 (2006).
[CrossRef]

Wei, Q.

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. Expresss. 13, 3673-3680 (2006).
[CrossRef]

Yin, H.

M. D. Wang, M. J. Schnitzer, H. Yin, R. Landick, J. Gelles, S. M. Block, "Force and velocity measured for single molecules of RNA polymerase," Science 282, 902-907 (1998).
[CrossRef] [PubMed]

Young, K.

H. M. Lai, C. K. Ng, and K. Young, "Radiation force on an object and momentum of light in a liquid dielectric," Phys. Rev. A 30, 1060-1066 (1984).
[CrossRef]

Zhao, C. L.

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

C. L. Zhao, L. G. Wang, X. and H. Lu, "Radiation forces on a dielectric sphere produced by highly focused hollow Gaussian beams," Phys. Lett. A. 363, 502-506 (2006).
[CrossRef]

Biophys. J.

A. J. Hunt, F. Gittes, and J. Howard, "The force exerted by a single kinesin molecule against a viscous load," Biophys. J. 67, 766-781 (1994).
[CrossRef] [PubMed]

J. Dai and M. P. Sheetz, "Mechanical properties of neuronal growth cone membrane by tether formation with laser optical tweezers," Biophys. J. 68, 988-996 (1995).
[CrossRef] [PubMed]

J. Opt. Soc. Am. B

Opt. Commun.

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

Opt. Express.

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

B. Agate, C. T. A. Brown, W. Sibbett, and K. Dholakia, "Femtosecond optical tweezers for in-situ control of two-photon fluorescence," Opt. Express. 12, 3011-3017 (2004).
[CrossRef] [PubMed]

Opt. Expresss.

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. Expresss. 13, 3673-3680 (2006).
[CrossRef]

Opt. Lett.

Phys. Lett. A.

C. L. Zhao, L. G. Wang, X. and H. Lu, "Radiation forces on a dielectric sphere produced by highly focused hollow Gaussian beams," Phys. Lett. A. 363, 502-506 (2006).
[CrossRef]

Phys. Rep.

I. Brevik, "Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor," Phys. Rep. 52, 133-201 (1979).
[CrossRef]

Phys. Rev. A

H. M. Lai, C. K. Ng, and K. Young, "Radiation force on an object and momentum of light in a liquid dielectric," Phys. Rev. A 30, 1060-1066 (1984).
[CrossRef]

Phys. Rev. A.

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

Phys. Rev. Lett.

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

A. Ashkin, "Trapping of Atoms by resonance radiation pressure," Phys. Rev. Lett. 40, 729-732 (1978).
[CrossRef]

S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, "Experimental observation of optically trapped atoms," Phys. Rev. Lett. 57, 314-317 (1986).
[CrossRef] [PubMed]

Science

M. D. Wang, M. J. Schnitzer, H. Yin, R. Landick, J. Gelles, S. M. Block, "Force and velocity measured for single molecules of RNA polymerase," Science 282, 902-907 (1998).
[CrossRef] [PubMed]

Science.

A. D. Metha, M. Rief, J. A. Spudich, D. A. Smith, R. M. Simmons, "Single-molecule biomechanics with optical methods," Science. 283, 1689-1695 (1999).
[CrossRef]

Other

A. Ashkin, in Methods in cell Biology, M. P. Sheetz, ed., (Academic Press, San Diego, 1998), 55, pp.1-27.
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

Schematics of a pulse laser radiating on a particle.

Fig. 2.
Fig. 2.

The temporal evolutions of (a–c) the transverse and (e–f) the longitudinal radiation forces for the pulses with different durations: τ=1 ps for (a) and (e), τ=0.1 ps for (b) and (f), τ=0.01 ps for (c) and (g). (d) and (h) show the dependence of both the maximal transverse and longitudinal radiation forces on the duration τ, respectively.

Fig. 3.
Fig. 3.

The dynamic distributions of (a–c) the longitudinal and (d–e) the transverse radiation forces for the pulses with duration τ=1 ps at different times: t=-0.5τ for (a) and (d), t=0 for (b) and (e), and t=0.5τ for (c) and (f).

Fig. 4.
Fig. 4.

The dynamic distribution of (a–e) the longitudinal and (f–j) transverse radiation forces for the pulse with τ=0.1 ps at different times: t=-0.2τ for (a) and (f), t=-0.1τ for (b) and (g), t=0 for (c) and (h), t=0.1τ for (d) and (i), and t=0.2τ for (e) and (j).

Fig. 5.
Fig. 5.

The dynamic distribution of (a–c) the longitudinal and (d–f) transverse radiation forces for the pulse with τ=0.01 ps at different times: t=-0.2τ for (a) and (d), t=0 for (b) and (e), and t=0.2τ for (c) and (f).

Equations (9)

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

E ( ρ , z , t ) = x ̂ E ( ρ , z , t )
= x ̂ i E 0 i + 2 z k w 0 2 exp [ i ω 0 t i k z i 2 k z ρ 2 ( k w 0 2 ) 2 + 4 z 2 ( k w 0 ) 2 ρ 2 ( k w 0 2 ) 2 + 4 z 2 ] exp [ ( t z c ) 2 τ 2 ] ,
H ( ρ , z , , t ) y ̂ n 2 ε 0 c E ( ρ , z , t ) ,
I ( ρ , z , t ) < S ( ρ , z , t ) > T = z ̂ I ( ρ , z , t ) = z ̂ P 1 + 4 z ˜ 2 exp [ 2 ρ ˜ 2 1 + 4 z ˜ 2 ] exp [ 2 ( t ˜ z ˜ k w 0 2 c τ ) 2 ] ,
F p ( ρ , z , t ) = [ p ( ρ , z , t ) · ] E ( ρ , z , t ) + [ t p ( ρ , z , t ) ] × B ( ρ , z , t ) = F grad + F t ,
F grad , ρ = ρ ̂ 2 β I ( ρ , z , t ) ρ ˜ [ c n 2 ε 0 w 0 ( 1 + 4 z ˜ 2 ) ] ,
F grad , z = z ̂ 2 β I ( ρ , z , t ) n 2 ε 0 c k w 0 2 [ z ˜ k 2 w 0 4 c 2 τ 2 k t ˜ w 0 2 c τ + 2 z ˜ ( 1 + 4 z ˜ 2 2 ρ ˜ 2 ) ( 1 + 4 z ˜ 2 ) 2 ] ,
F t = z ̂ 8 μ 0 β I ( ρ , z , t ) t ˜ τ + z ̂ 8 z ˜ μ 0 β I ( ρ , z , t ) k w 0 2 ( c τ 2 ) ,
F scat ( ρ , z , t ) = C pr < S ( ρ , z , t ) > T ( c n 2 ) = z ̂ ( n 2 c ) C pr I ( ρ , z , t ) ,

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