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.

©2010 Optical Society of America

1. Introduction

In some optical measurements, it is often desired to achieve high temporal and spatial resolution [13]. Femtosecond light pulses are often employed to increase the temporal resolution, and thus they have been extensively studied [36]. Recently, with an increasing number of new applications, the vortex beams have generated great research interest, which lead to a new branch of singular optics in modern optics [79]. Vortices in femtosecond pulse are of great use in topological spectroscopy. Spatially controlled light with vortices can be used for characterization of topological properties of materials [10]. Therefore, it is important to study on the femtosecond vortex light pulse. On the other hand, the three-dimensional spatial resolution can be increased by tight focusing system [1,11,12]. We notice that the focusing of laser beams through a high numerical aperture (NA) objective will achieve tighter focal spots which can be used in applications such as microscopy, lithography, optical data storage, optical trapping and plasma physics [1216]. However, to the best of our knowledge, there are no papers studying the focusing of femtosecond vortex light pulses through a high NA objective. This paper is devoted to study on the tight focusing properties of the femtosecond vortex light pulses in the focal field.

2. Theory

In this paper, we use the x-polarized Bessel-Gaussian femtosecond light pulse as the vortex model. The electric field of such a pulse can be expressed as

E(r,ϕ,t)=E0Jm(αr)exp(r2/σ02)exp(imϕ)A(t),
where E0 and σ0 are the constant amplitude and the beam size, Jm(αr) is the Bessel function of the first kind in which α is the Bessel parameter. exp(imϕ)is the vortex phase factor and m is the corresponding topological charge. A(t)is the temporal pulse shape which we assume to have Gaussian profile, i.e [17].
A(t)=exp[(agt/T)2]exp(iω0t),
where ag=(2ln2)1/2, T is the pulse duration, and ω0 is its central angular frequency.

The focusing objective is assumed to obey sine condition r=fsinθ [18], where f is the focal length of the high NA objective, and θ is the numerical-aperture angle, shown in Fig. 1 . Then we obtain the pupil apodization function of a single spectral component as [6]

S(θ,ϕ,ω)=12πE(r,ϕ,t)exp(iωt)dt     =T2agE0Jm(αfsinθ)exp[(fsinθ)2σ02]exp(imϕ)exp[T2(ωω0)2/4ag2]
We use the above field distribution as illumination. Then the electric field of a single spectral component in the focal region when the light pulse is focused by a high NA objective is given by the Richards-Wolf vectorial diffraction method as [1820]
E(r,φ,z,ω)=ikf2π0θmax02πS(θ,ϕ,ω)exp[ikrsinθcos(ϕφ)+ikzcosθ]                                           ×sinθcosθ[(cos2ϕcosθ+sin2ϕ)excosϕsinϕ(cosθ1)eysinθcosϕez]dϕdθ
wherek=ω/cis the wave vector related to the angular frequency of the pulse, and c is the light velocity in vacuum.θmaxis the maximum numerical angle of the objective.

 figure: Fig. 1

Fig. 1 Tight focusing system.

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By a Fourier-transformation, the electric fields of the femtosecond pulse in the vicinity of the focal spot can be calculated by the superposition of each spectral component as [6]

Ej(r,φ,z,t)=0Ej(r,φ,z,ω)exp(iωt) dω,(j=x,y,z)
Then we get the total intensity near the focus as follows

I(r,φ,z,t)=j=x,y,zIj(r,φ,z,t)=j=x,y,z|Ej(r,φ,z,t)|2.

The velocity of the light pulse along the z-axis can be expressed by the formula

v(t)=dz(t)/dt,
wherez(t)is the average longitudinal propagation distance defined by the beam centroid [21].

Finally, the orbital angular momentum (OAM) of the vortex light pulse is also investigated. Since the vortex light pulse is focused by a high NA objective, the OAM should be analyzed under the nonparaxial condition suggested by [7,22]

JzW=(m+σ)ω+σω0kdκ[|E(κ)|2κ/(k2κ2)]0kdκ[|E(κ)|2(2k2κ2)/κ(k2κ2)],
where κ=ksinθ, ω is the angular frequency of the incident pulse, σ is the helicity of the light beam related to the polarization state of the light pulse, mis the topological charge of the light pulse. Obviously, when the incident light pulse is x-polarized, σ equals to zero. Therefore, the total OAM of the vortex light pulse can be written as [7]
Lz=ε0mωI(r,φ,z,t)rdrdφ,
where ε0 is the permittivity constant in vacuum. It is shown that the OAM distribution is related to the intensity distribution on the transverse plane.

Based on the above derived equations, we will investigate the focusing properties of the femtosecond light pulse through a high NA objective in the following by some numerical calculations.

3. Results and discussions

We show in Fig. 2 the total intensity distribution and its x, y and z components in the focal plane when the femtosecond light pulse is tightly focused. We found that there is a tiny dark core with non-zero central intensity in the total intensity distribution. It is shown that the intensity has three components in the focal region which means that the x -polarized Bessel-Gaussian femtosecond light pulse is depolarized when it is focused by a high NA objective. Moreover, the maximum intensity of the x, y and z components are calculated to be 94.5%, 0.2% and 10.6% of the total intensity respectively, indicating that the y -component contributes least to the total intensity. This means that the x component plays a dominant role in shaping the total intensity. The central intensity of the z component leads to the non-zero central intensity of the overall intensity [23].

 figure: Fig. 2

Fig. 2 Contour plots of the intensity distributions in the focal plane. (a) The total intensityI; (b) thex-componentIx; (c) they-componentIy; (d) thez-componentIz. 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.

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Figure 3 shows the phase distributions of the x, y and z components with central angular frequency. It is shown that the phase distributions present screw wave-front. From Fig. 3(a), it can be seen that there exists a screw phase distribution corresponding to the x-component intensity distribution in Fig. 2(b). And in Fig. 2(d), we can find that there are two dark regions in the center, leading to the two screw wave-fronts in the phase distributions in Fig. 3(c).

 figure: 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.

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The propagation evolution of the femtosecond vortex light pulse focused by a high NA objective is illustrated in Fig. 4 (Media 1). It is shown that there is a dark region with non-zero central intensity along the z-axis, i.e., in the longitudinal direction, corresponding to the dark core with non-zero central intensity in Fig. 2. Moreover, we notice that the light pulse propagates faster when it is far enough away from the focal plane (i.e. z=0 plane) and slows down near the focus. The more detailed velocity variation is given in Fig. 5 . We evaluate the pulse speed by the average longitudinal propagation distance of the beam centroid, i.e., the velocity along the z-axis, which is described in Eq. (7) and also defined in detail in Reference [21]. The pulse velocity is normalized to the velocity of light in vacuum c=3×108m/s. It is seen that the total velocity and the three component velocity all slow down to about half the velocity in vacuum at the focus. That is because in tight focusing process, the light pulse velocity c is projected onto the z-axis as velocity v(t), shown in the inset in Fig. 1. This result is important since it offers a new technique to control the motion state of photons and enables research on controlling the interaction of light and materials.

 figure: 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.

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 figure: 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.

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Then we compare the total velocity variation with certain parameters in Fig. 6 . Figure 6(a) shows the comparison between non-vortex light pulse and vortex light pulse. We found that the femtosecond vortex light pulse generally propagates slower than the non-vortex light pulse. And the larger the topological charge is, the smaller the propagation speed is, indicating that the slow light phenomenon is more pronounced in femtosecond vortex light pulse. That is because the vortex light pulse carries OAM, which means that there is an azimuthal component of the linear momentum density at all points within the pulse. And this will reduce the linear momentum comparing with a non-vortex light pulse, or a lower-charge vortex, leading to a speed reduction of the light pulse. The influence of pulse duration T on the longitudinal velocity is illustrated in Fig. 6(b). It is shown that for the same topological charge(m=1), the velocity of a pulse with a smaller pulse duration is larger than that of a pulse with a larger pulse duration.

 figure: 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.

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As is known to all, the vortex beam carries OAM, and so does the vortex light pulse. We then present the dependence of normalized OAM distribution on the numerical aperture of the focusing objective (NA) and the pulse duration of the femtosecond vortex light pulse (T) as the vortex light pulse is focused by a high NA objective in Fig. 7 . It is seen that the OAM increases near the focus and reaches peak on the focal plane. That is because the femtosecond vortex light pulse is focused into a tight spot with intense intensity when the light pulse is focused by a high NA objective and the OAM coincides with the intensity distribution. Moreover, from Fig. 7(a), we can see that the larger the NA of the focusing objective is, the larger is the OAM. And from Fig. 7(b), it is obvious that the OAM increases with the increment of the pulse duration T.

 figure: Fig. 7

Fig. 7 Dependence of OAM distribution on (a)NA and (b)T. The other parameters are the same as in Fig. 2.

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4. Conclusions

In conclusion, we have studied the focusing properties of a femtosecond vortex light pulse by a high NA objective based on Richards-Wolf vectorial diffraction theory. We found that the propagation velocity slows down near the focus and that non-vortex light pulse propagates faster than vortex light pulse. We also found that a smaller pulse duration will lead to a higher propagation velocity. Moreover, the OAM increases as the NA of the focusing objective increases or as the pulse duration T increases. The results obtained in this paper might be useful in applications of femtosecond vortex pulses, such as optical tweezers, etc.

Acknowledgements

This research is supported by National Natural Science Foundation of China (Grants No. 60977068), Natural Science Foundation of Fujian Province(Grants No. A0810012), and the Open Research Fund of State Key Laboratory of Transient Optics and Photonics, Chinese Academy of Sciences (Grants No. SKLST 200912). O. Korotkova’s research is funded by the AFOSR (Grant FA 95500810102).

References and links

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]  

References

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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]

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]

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]

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)

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]

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

2005 (2)

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

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]

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

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]

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)

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

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]

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)

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

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]

Opt. Express (2)

Opt. Lett. (3)

Phys. Rev. A (2)

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]

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]

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

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

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]

Phys. Rev. Lett. (4)

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]

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]

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)

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

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

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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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