Abstract

We investigate the focusing properties of a femtosecond vortex light pulse focused by a high numerical aperture objective. By using the Richards-Wolf vectorial diffraction method, the intensity distribution, the velocity variation and the orbital angular momentum near the focus are studied in great detail. We have discovered that the femtosecond vortex light pulse can travel at various speeds, that is, slower or faster than light with a tight focusing system. Moreover, we have found that the numerical aperture of the focusing objective and the duration of the vortex light pulse will influence the orbital angular momentum distribution in the focused field.

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  1. T. Brixner, F. J. García de Abajo, J. Schneider, and W. Pfeiffer, “Nanoscopic ultrafast space-time-resolved spectroscopy,” Phys. Rev. Lett. 95(9), 093901 (2005).
    [CrossRef] [PubMed]
  2. Z. Bor, Z. Gogolak, and G. Szabo, “Femtosecond-resolution pulse-front distortion measurement by time-of-flight interferometry,” Opt. Lett. 14(16), 862–864 (1989).
    [CrossRef] [PubMed]
  3. D. an der Brügge and A. Pukhov, “Ultrashort focused electromagnetic pulses,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(1), 016603 (2009).
    [CrossRef] [PubMed]
  4. K. M. Romallosa, J. Bantang, and C. Saloma, “Three-dimensional light distribution near the focus of a tightly focused beam of few-cycle optical pulses,” Phys. Rev. A 68(3), 033812 (2003).
    [CrossRef]
  5. M. Kempe, U. Stamm, and B. Wilhelmi, “Spatial and temporal transformation of femtosecond laser pulses by lenses with annular aperture,” Opt. Commun. 89(2-4), 119–125 (1992).
    [CrossRef]
  6. L. E. Helseth, “Strongly focused polarized light pulse,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(4), 047602 (2005).
    [CrossRef] [PubMed]
  7. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
    [CrossRef]
  8. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997).
    [CrossRef]
  9. Q. Zhan, “Properties of circularly polarized vortex beams,” Opt. Lett. 31(7), 867–869 (2006).
    [CrossRef] [PubMed]
  10. Y. Tokizane, K. Oka, and R. Morita, “Supercontinuum optical vortex pulse generation without spatial or topological-charge dispersion,” Opt. Express 17(17), 14517–14525 (2009).
    [CrossRef] [PubMed]
  11. N. Bokor and N. Davidson, “A three dimensional dark focal spot uniformly surrounded by light,” Opt. Commun. 279(2), 229–234 (2007).
    [CrossRef]
  12. T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272(2), 314–319 (2007).
    [CrossRef]
  13. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
    [CrossRef] [PubMed]
  14. E. S. Efimenko, A. V. Kim, and M. Quiroga-Teixeiro, “Ionization-induced small-scaled plasma structures in tightly focused ultrashort laser pulses,” Phys. Rev. Lett. 102(1), 015002 (2009).
    [CrossRef] [PubMed]
  15. T. T. Xi, X. Lu, and J. Zhang, “Interaction of light filaments generated by femtosecond laser pulses in air,” Phys. Rev. Lett. 96(2), 025003 (2006).
    [CrossRef] [PubMed]
  16. B. Chen and J. Pu, “Tight focusing of elliptically polarized vortex beams,” Appl. Opt. 48(7), 1288–1294 (2009).
    [CrossRef] [PubMed]
  17. Z. Bor and Z. L. Horvath, “Distortion of femtosecond pulses in lenses. Wave optical description,” Opt. Commun. 94(4), 249–258 (1992).
    [CrossRef]
  18. M. Gu, Advanced optical imaging theory (Springer, Heidelberg, 1999).
  19. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Ser. A 253(1274), 358–379 (1959).
    [CrossRef]
  20. Z. Zhang, J. Pu, and X. Wang, “Focusing of partially coherent Bessel-Gaussian beams through a high-numerical-aperture objective,” Opt. Lett. 33(1), 49–51 (2008).
    [CrossRef]
  21. W. D. St John, “Cylinder gauge measurement using a position sensitive detector,” Appl. Opt. 46(30), 7469–7474 (2007).
    [CrossRef] [PubMed]
  22. L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
    [CrossRef]
  23. D. Ganic, X. Gan, and M. Gu, “Focusing of doughnut laser beams by a high numerical-aperture objective in free space,” Opt. Express 11(21), 2747–2752 (2003).
    [CrossRef] [PubMed]

2009 (4)

D. an der Brügge and A. Pukhov, “Ultrashort focused electromagnetic pulses,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(1), 016603 (2009).
[CrossRef] [PubMed]

E. S. Efimenko, A. V. Kim, and M. Quiroga-Teixeiro, “Ionization-induced small-scaled plasma structures in tightly focused ultrashort laser pulses,” Phys. Rev. Lett. 102(1), 015002 (2009).
[CrossRef] [PubMed]

B. Chen and J. Pu, “Tight focusing of elliptically polarized vortex beams,” Appl. Opt. 48(7), 1288–1294 (2009).
[CrossRef] [PubMed]

Y. Tokizane, K. Oka, and R. Morita, “Supercontinuum optical vortex pulse generation without spatial or topological-charge dispersion,” Opt. Express 17(17), 14517–14525 (2009).
[CrossRef] [PubMed]

2008 (1)

2007 (3)

W. D. St John, “Cylinder gauge measurement using a position sensitive detector,” Appl. Opt. 46(30), 7469–7474 (2007).
[CrossRef] [PubMed]

N. Bokor and N. Davidson, “A three dimensional dark focal spot uniformly surrounded by light,” Opt. Commun. 279(2), 229–234 (2007).
[CrossRef]

T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272(2), 314–319 (2007).
[CrossRef]

2006 (2)

Q. Zhan, “Properties of circularly polarized vortex beams,” Opt. Lett. 31(7), 867–869 (2006).
[CrossRef] [PubMed]

T. T. Xi, X. Lu, and J. Zhang, “Interaction of light filaments generated by femtosecond laser pulses in air,” Phys. Rev. Lett. 96(2), 025003 (2006).
[CrossRef] [PubMed]

2005 (2)

T. Brixner, F. J. García de Abajo, J. Schneider, and W. Pfeiffer, “Nanoscopic ultrafast space-time-resolved spectroscopy,” Phys. Rev. Lett. 95(9), 093901 (2005).
[CrossRef] [PubMed]

L. E. Helseth, “Strongly focused polarized light pulse,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(4), 047602 (2005).
[CrossRef] [PubMed]

2003 (3)

K. M. Romallosa, J. Bantang, and C. Saloma, “Three-dimensional light distribution near the focus of a tightly focused beam of few-cycle optical pulses,” Phys. Rev. A 68(3), 033812 (2003).
[CrossRef]

D. Ganic, X. Gan, and M. Gu, “Focusing of doughnut laser beams by a high numerical-aperture objective in free space,” Opt. Express 11(21), 2747–2752 (2003).
[CrossRef] [PubMed]

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

2001 (1)

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

1999 (1)

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[CrossRef]

1997 (1)

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997).
[CrossRef]

1992 (2)

M. Kempe, U. Stamm, and B. Wilhelmi, “Spatial and temporal transformation of femtosecond laser pulses by lenses with annular aperture,” Opt. Commun. 89(2-4), 119–125 (1992).
[CrossRef]

Z. Bor and Z. L. Horvath, “Distortion of femtosecond pulses in lenses. Wave optical description,” Opt. Commun. 94(4), 249–258 (1992).
[CrossRef]

1989 (1)

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Ser. A 253(1274), 358–379 (1959).
[CrossRef]

Allen, L.

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[CrossRef]

an der Brügge, D.

D. an der Brügge and A. Pukhov, “Ultrashort focused electromagnetic pulses,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(1), 016603 (2009).
[CrossRef] [PubMed]

Babiker, M.

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[CrossRef]

Bantang, J.

K. M. Romallosa, J. Bantang, and C. Saloma, “Three-dimensional light distribution near the focus of a tightly focused beam of few-cycle optical pulses,” Phys. Rev. A 68(3), 033812 (2003).
[CrossRef]

Bokor, N.

N. Bokor and N. Davidson, “A three dimensional dark focal spot uniformly surrounded by light,” Opt. Commun. 279(2), 229–234 (2007).
[CrossRef]

Bor, Z.

Z. Bor and Z. L. Horvath, “Distortion of femtosecond pulses in lenses. Wave optical description,” Opt. Commun. 94(4), 249–258 (1992).
[CrossRef]

Z. Bor, Z. Gogolak, and G. Szabo, “Femtosecond-resolution pulse-front distortion measurement by time-of-flight interferometry,” Opt. Lett. 14(16), 862–864 (1989).
[CrossRef] [PubMed]

Brixner, T.

T. Brixner, F. J. García de Abajo, J. Schneider, and W. Pfeiffer, “Nanoscopic ultrafast space-time-resolved spectroscopy,” Phys. Rev. Lett. 95(9), 093901 (2005).
[CrossRef] [PubMed]

Chen, B.

Courjon, D.

T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272(2), 314–319 (2007).
[CrossRef]

Davidson, N.

N. Bokor and N. Davidson, “A three dimensional dark focal spot uniformly surrounded by light,” Opt. Commun. 279(2), 229–234 (2007).
[CrossRef]

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

Efimenko, E. S.

E. S. Efimenko, A. V. Kim, and M. Quiroga-Teixeiro, “Ionization-induced small-scaled plasma structures in tightly focused ultrashort laser pulses,” Phys. Rev. Lett. 102(1), 015002 (2009).
[CrossRef] [PubMed]

Gan, X.

Ganic, D.

García de Abajo, F. J.

T. Brixner, F. J. García de Abajo, J. Schneider, and W. Pfeiffer, “Nanoscopic ultrafast space-time-resolved spectroscopy,” Phys. Rev. Lett. 95(9), 093901 (2005).
[CrossRef] [PubMed]

Gogolak, Z.

Gorshkov, V. N.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997).
[CrossRef]

Grosjean, T.

T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272(2), 314–319 (2007).
[CrossRef]

Gu, M.

Heckenberg, N. R.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997).
[CrossRef]

Helseth, L. E.

L. E. Helseth, “Strongly focused polarized light pulse,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(4), 047602 (2005).
[CrossRef] [PubMed]

Horvath, Z. L.

Z. Bor and Z. L. Horvath, “Distortion of femtosecond pulses in lenses. Wave optical description,” Opt. Commun. 94(4), 249–258 (1992).
[CrossRef]

Kempe, M.

M. Kempe, U. Stamm, and B. Wilhelmi, “Spatial and temporal transformation of femtosecond laser pulses by lenses with annular aperture,” Opt. Commun. 89(2-4), 119–125 (1992).
[CrossRef]

Kim, A. V.

E. S. Efimenko, A. V. Kim, and M. Quiroga-Teixeiro, “Ionization-induced small-scaled plasma structures in tightly focused ultrashort laser pulses,” Phys. Rev. Lett. 102(1), 015002 (2009).
[CrossRef] [PubMed]

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

Lu, X.

T. T. Xi, X. Lu, and J. Zhang, “Interaction of light filaments generated by femtosecond laser pulses in air,” Phys. Rev. Lett. 96(2), 025003 (2006).
[CrossRef] [PubMed]

Malos, J. T.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997).
[CrossRef]

Morita, R.

Oka, K.

Padgett, M. J.

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[CrossRef]

Pfeiffer, W.

T. Brixner, F. J. García de Abajo, J. Schneider, and W. Pfeiffer, “Nanoscopic ultrafast space-time-resolved spectroscopy,” Phys. Rev. Lett. 95(9), 093901 (2005).
[CrossRef] [PubMed]

Pu, J.

Pukhov, A.

D. an der Brügge and A. Pukhov, “Ultrashort focused electromagnetic pulses,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(1), 016603 (2009).
[CrossRef] [PubMed]

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

Quiroga-Teixeiro, M.

E. S. Efimenko, A. V. Kim, and M. Quiroga-Teixeiro, “Ionization-induced small-scaled plasma structures in tightly focused ultrashort laser pulses,” Phys. Rev. Lett. 102(1), 015002 (2009).
[CrossRef] [PubMed]

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Ser. A 253(1274), 358–379 (1959).
[CrossRef]

Romallosa, K. M.

K. M. Romallosa, J. Bantang, and C. Saloma, “Three-dimensional light distribution near the focus of a tightly focused beam of few-cycle optical pulses,” Phys. Rev. A 68(3), 033812 (2003).
[CrossRef]

Saloma, C.

K. M. Romallosa, J. Bantang, and C. Saloma, “Three-dimensional light distribution near the focus of a tightly focused beam of few-cycle optical pulses,” Phys. Rev. A 68(3), 033812 (2003).
[CrossRef]

Schneider, J.

T. Brixner, F. J. García de Abajo, J. Schneider, and W. Pfeiffer, “Nanoscopic ultrafast space-time-resolved spectroscopy,” Phys. Rev. Lett. 95(9), 093901 (2005).
[CrossRef] [PubMed]

Soskin, M. S.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997).
[CrossRef]

St John, W. D.

Stamm, U.

M. Kempe, U. Stamm, and B. Wilhelmi, “Spatial and temporal transformation of femtosecond laser pulses by lenses with annular aperture,” Opt. Commun. 89(2-4), 119–125 (1992).
[CrossRef]

Szabo, G.

Tokizane, Y.

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997).
[CrossRef]

Wang, X.

Wilhelmi, B.

M. Kempe, U. Stamm, and B. Wilhelmi, “Spatial and temporal transformation of femtosecond laser pulses by lenses with annular aperture,” Opt. Commun. 89(2-4), 119–125 (1992).
[CrossRef]

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Ser. A 253(1274), 358–379 (1959).
[CrossRef]

Xi, T. T.

T. T. Xi, X. Lu, and J. Zhang, “Interaction of light filaments generated by femtosecond laser pulses in air,” Phys. Rev. Lett. 96(2), 025003 (2006).
[CrossRef] [PubMed]

Zhan, Q.

Zhang, J.

T. T. Xi, X. Lu, and J. Zhang, “Interaction of light filaments generated by femtosecond laser pulses in air,” Phys. Rev. Lett. 96(2), 025003 (2006).
[CrossRef] [PubMed]

Zhang, Z.

Appl. Opt. (2)

Opt. Commun. (4)

M. Kempe, U. Stamm, and B. Wilhelmi, “Spatial and temporal transformation of femtosecond laser pulses by lenses with annular aperture,” Opt. Commun. 89(2-4), 119–125 (1992).
[CrossRef]

N. Bokor and N. Davidson, “A three dimensional dark focal spot uniformly surrounded by light,” Opt. Commun. 279(2), 229–234 (2007).
[CrossRef]

T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272(2), 314–319 (2007).
[CrossRef]

Z. Bor and Z. L. Horvath, “Distortion of femtosecond pulses in lenses. Wave optical description,” Opt. Commun. 94(4), 249–258 (1992).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Phys. Rev. A (2)

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997).
[CrossRef]

K. M. Romallosa, J. Bantang, and C. Saloma, “Three-dimensional light distribution near the focus of a tightly focused beam of few-cycle optical pulses,” Phys. Rev. A 68(3), 033812 (2003).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (2)

D. an der Brügge and A. Pukhov, “Ultrashort focused electromagnetic pulses,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(1), 016603 (2009).
[CrossRef] [PubMed]

L. E. Helseth, “Strongly focused polarized light pulse,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(4), 047602 (2005).
[CrossRef] [PubMed]

Phys. Rev. Lett. (4)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

E. S. Efimenko, A. V. Kim, and M. Quiroga-Teixeiro, “Ionization-induced small-scaled plasma structures in tightly focused ultrashort laser pulses,” Phys. Rev. Lett. 102(1), 015002 (2009).
[CrossRef] [PubMed]

T. T. Xi, X. Lu, and J. Zhang, “Interaction of light filaments generated by femtosecond laser pulses in air,” Phys. Rev. Lett. 96(2), 025003 (2006).
[CrossRef] [PubMed]

T. Brixner, F. J. García de Abajo, J. Schneider, and W. Pfeiffer, “Nanoscopic ultrafast space-time-resolved spectroscopy,” Phys. Rev. Lett. 95(9), 093901 (2005).
[CrossRef] [PubMed]

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

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Ser. A 253(1274), 358–379 (1959).
[CrossRef]

Prog. Opt. (2)

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

Other (1)

M. Gu, Advanced optical imaging theory (Springer, Heidelberg, 1999).

Supplementary Material (1)

» Media 1: MPG (596 KB)     

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Figures (7)

Fig. 1
Fig. 1

Tight focusing system.

Fig. 2
Fig. 2

Contour plots of the intensity distributions in the focal plane. (a) The total intensityI; (b) thex-component Ix ; (c) they-component Iy ; (d) thez-component Iz . The other parameters are chosen as ω0=7.57×1015s1 , T=5fs , α=0.5mm1 , f=1cm , σ0=2cm , m=1 , NA=0.9 , t=0fs , E0=1 .

Fig. 3
Fig. 3

Phase distributions in the focal plane. (a) thex-component; (b) they-component; (c) thez-component. The other parameters are chosen to be the same as in Fig. 2.

Fig. 4
Fig. 4

(Media 1) The propagation evolution of the femtosecond vortex light pulse. The other parameters are chosen to be the same as in Fig. 2.

Fig. 5
Fig. 5

Pulse velocity distribution and its x, y and z component velocity distributions near the focus. The parameters are chosen to be the same as in Fig. 2.

Fig. 6
Fig. 6

Pulse velocity distribution with (a) different topological charges and (b) different pulse duration( m=1 ). The other parameters are chosen to be the same as in Fig. 2.

Fig. 7
Fig. 7

Dependence of OAM distribution on (a)NA and (b)T. The other parameters are the same as in Fig. 2.

Equations (9)

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

E(r,ϕ,t)=E0Jm(αr)exp(r2/σ02)exp(imϕ)A(t),
A(t)=exp[(agt/T)2]exp(iω0t),
S(θ,ϕ,ω)=12πE(r,ϕ,t)exp(iωt)dt     =T2agE0Jm(αfsinθ)exp[(fsinθ)2σ02]exp(imϕ)exp[T2(ωω0)2/4ag2]
E(r,φ,z,ω)=ikf2π0θmax02πS(θ,ϕ,ω)exp[ikrsinθcos(ϕφ)+ikzcosθ]                                           ×sinθcosθ[(cos2ϕcosθ+sin2ϕ)excosϕsinϕ(cosθ1)eysinθcosϕez]dϕdθ
Ej(r,φ,z,t)=0Ej(r,φ,z,ω)exp(iωt) dω,(j=x,y,z)
I(r,φ,z,t)=j=x,y,zIj(r,φ,z,t)=j=x,y,z|Ej(r,φ,z,t)|2.
v(t)=dz(t)/dt,
JzW=(m+σ)ω+σω0kdκ[|E(κ)|2κ/(k2κ2)]0kdκ[|E(κ)|2(2k2κ2)/κ(k2κ2)],
Lz=ε0mωI(r,φ,z,t)rdrdφ,

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