Focusing of a Femtosecond Vortex Light Pulse Through a High Numerical Aperture Objective

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 OCIS codes: (320.7120) Ultrafast phenomena; (260.6042) Singular Optics; (050.1960) Diffraction Theory. 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). 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). 3. D. an der Brügge, and A. Pukhov, “Ultrashort focused electromagnetic pulses,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(1), 016603 (2009). 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). 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). 6. L. E. Helseth, “Strongly focused polarized light pulse,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(4), 047602 (2005). 7. M. S. Soskin, and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001). 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). 9. Q. Zhan, “Properties of circularly polarized vortex beams,” Opt. Lett. 31(7), 867–869 (2006). 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). 11. N. Bokor, and N. Davidson, “A three dimensional dark focal spot uniformly surrounded by light,” Opt. Commun. 279(2), 229–234 (2007). 12. T. Grosjean, and D. Courjon, “Smallest focal spots,” Opt. Commun. 272(2), 314–319 (2007). 13. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). 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). 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). 16. B. Chen, and J. Pu, “Tight focusing of elliptically polarized vortex beams,” Appl. Opt. 48(7), 1288–1294 (2009). 17. Z. Bor, and Z. L. Horvath, “Distortion of femtosecond pulses in lenses. Wave optical description,” Opt. Commun. 94(4), 249–258 (1992). 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). 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). #121146 $15.00 USD Received 9 Dec 2009; revised 20 Feb 2010; accepted 17 Mar 2010; published 10 May 2010 (C) 2010 OSA 10 May 2010 / Vol. 18, No. 10 / OPTICS EXPRESS 10822 21. W. D. St John, “Cylinder gauge measurement using a position sensitive detector,” Appl. Opt. 46(30), 7469–7474 (2007). 22. L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999). 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).


Introduction
In some optical measurements, it is often desired to achieve high temporal and spatial resolution [1][2][3].Femtosecond light pulses are often employed to increase the temporal resolution, and thus they have been extensively studied [3][4][5][6].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 [7][8][9].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 [12][13][14][15][16].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.

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 where 0

E and 0
V are the constant amplitude and the beam size, ( ) D is the Bessel function of the first kind in which D is the Bessel parameter.exp( ) imI 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].Z is its central angular frequency.
The focusing objective is assumed to obey sine condition sin r f T [18], where f is the focal length of the high NA objective, and T is the numerical-aperture angle, shown in Fig. 1.
Then we obtain the pupil apodization function of a single spectral component as [6] 2 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 [18][19][20]  Z is the wave vector related to the angular frequency of the pulse, and c is the light velocity in vacuum.max T is the maximum numerical angle of the objective.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] 0 ( , , , ) ( , , , ) exp( ) , ( , , ) Then we get the total intensity near the focus as follows , , ) .
The velocity of the light pulse along the z-axis can be expressed by the formula ( ) ( ) / v t dz t dt (7) where ( ) z 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] , where sin k N T , Z is the angular frequency of the incident pulse, V is the helicity of the light beam related to the polarization state of the light pulse, m is the topological charge of the light pulse.Obviously, when the incident light pulse is x -polarized, V equals to zero.
Therefore, the total OAM of the vortex light pulse can be written as [7] 0 ( , , , ) , where 0 H 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.

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 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).The other parameters are chosen to be the same as in Fig. 2.
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.0 z 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 .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.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( 1 m ), the velocity of a pulse with a smaller pulse duration is larger than that of a pulse with a larger pulse duration.).The other parameters are chosen to be the same as in Fig. 2.
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.

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.

Fig. 2 .
Fig. 2. Contour plots of the intensity distributions in the focal plane.(a) The total intensity I ; (b) the x -component x I ; (c) the y -component y I ; (d) the z -component z I .The other parameters

Fig. 3 .
Fig. 3. Phase distributions in the focal plane.(a) the x -component; (b) the y -component; (c) the z -component.The other parameters are chosen to be the same as inFig.2.

Fig. 4 .Fig. 5 .
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. 6 .
Fig. 6.Pulse velocity distribution with (a) different topological charges and (b) different pulse duration( 1 m).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.