Dynamics of femtosecond laser filamentation in argon with non-uniform density distribution

Zhanxin Wang; Chaojin Zhang; Jiansheng Liu; Zhizhan Xu

doi:10.1364/OE.20.009558

1. Introduction

The researches on femtosecond laser filaments can be traced back to the 1990s [1,2]. Since then, lots of works have been done because of its physical interest and its potential applications [3–6]. The basic physics of filaments is generally considered as the dynamic balance between the focusing caused by Kerr effect and lens and the plasma-induced defocusing, although many controversies have recently arisen [7–13]. In most of the works on filaments, the transparent media such as fused silica, water, air, and/or argon are employed. Filamentation of ultrashort pulses in argon has been widely used to obtain few-cycle laser pulses by employing the now well-known effect of pulse self-compression [14–28]. The self-compression effect in the argon filamentation occured in two stages: When the optical Kerr effect focuses the pulse until the peak intensity is clamped, an ionization front is generated that will defocus the trail of the pulse and shortens its leading edge, which is the first stage of self-compression [14,15]. Furtherly, the pulse front can be exhausted, leaving one isolated and shortened pulse behind due to refocusing in the rear part of the pulse [17–22], that is the so-called split-isolation cycle [19–21]. Experimentally, the few-cycle pulse is extracted by termination of the filaments at proper position using a exit window [17,19–21] or a movable aperture [26,27]. In the case of movable aperture, the filament is terminated abruptly or smoothly depending on the size of aperture where a density gradient occurs. Experimental results show that a double increase in the aperture diameter can result in an approximately three-fold increase in the pulse duration [26]. This proposed the question of sensitivity of the nonlinear pulse dynamics to the gas density gradient. Recently, Diels group [29] has employed an aerodynamic window as a new diagnostic tool to experimentally investigate this question. From the point of view of numerical simulation, although A. Couairon et al. [23] have numerically shown that self-compression to the single-cycle limit can be achieved by using argon with a pressure gradient, there is still the need to improve the understanding on the dynamics in the filamentation of non-uniform argon when considering the complexity of the nonlinear propagation.

In this paper, we numerically investigated the dynamics of the femtosecond filamentation in argon with non-uniform density. Different with the traditional split-isolation cycle where the pulse front is eventually exhausted, we find that the pulse-splitting appear earlier in the non-uniform case than the uniform case because the refocusing effect is sensitive to the plasma density. During the pulse-splitting, the spatio-temporal intensity profile takes on a double $Λ$ shape.

2. Models and parameters

In our numerical experiments, a transform-limited Gaussian laser pulse with the duration of $35 fs$ (FWHM) and beam radius of $4.4 mm$ ( $1 / e^{2}$ ) was focused in a cell filled with argon by a lens with a focal length $f = 1 m .$ The length of the argon cell is $30 cm,$ which is so placed that the focus of the lens is located at the center of the cell. The input plane of our numerical simulation is just located at the front surface of the cell. The area between the lens and the cell is assumed as vacuum. Therefore, the spectra of the electric field in the input plane can be analytically expressed as

\tilde{E} (r, ω, z = 0) = \sqrt{π} T_{0} \exp (- \frac{T_{0}^{2} {(ω - ω_{0})}^{2}}{4}) \frac{1}{p q} \exp (- \frac{r^{2}}{q}) .

Where,

p = 1 / w_{0}^{2} + i k_{0} / 2 f

and

q = 1 / p + i 2 d / k_{0},

ω

is the angular frequency of the electric field,

w_{0} = 4.4 mm

is the beam radius,

T_{0} = t_{FWHM} / \sqrt{2 \ln 2}

is the pulse duration,

λ_{0} = 800 nm

is the center wavelength of the laser pulse in vacuum,

ω_{0} = 2 π / λ_{0}

is the angular frequency,

k_{0} = 2 π / λ_{0}

is the wave number,

d = 85 cm

is the distance between the lens and the front surface of the cell. In Eq. (1), the Fourier transform and its reverse are defined as

\tilde{E} (ω) = \int_{- \infty}^{+ \infty} E (t) e^{i ω t} d t

and

E (t) = (1 / 2 π) \int_{- \infty}^{+ \infty} \tilde{E} (ω) e^{- i ω t} d ω,

respectively.

The propagation of the axially symmetric laser pulse in the argon cell is described by the scalar unidirectional pulse propagation equation as [6]

\partial_{z} \tilde{E} = (\frac{i}{2 k (ω)} + i k (ω)) \tilde{E} + \frac{ω}{2 k (ω) c^{2} ε_{0}} {\tilde{F}}_{NL} .

Where

\tilde{E} \equiv \tilde{E} (r, ω, z)

is the spectra of the forward electric field,

k (ω) = n (ω) ω / c

is the frequency-dependent wave number,

{\tilde{F}}_{NL} = i ω {\tilde{P}}_{NL} - \tilde{J}

contains the Kerr nonlinear polarization

P_{NL}

and the current density

J .

The evolution of free electron density can be described as

\partial_{z} ρ = W (| E |) ρ_{nt} + σ {| E |}^{2} ρ / I_{p},

where

W (| E |)

is the ionization rate,

I_{p}

is the ionization potential, and

ρ_{nt}

is the gas density.

The parameters for argon that is used in our simulation can be described as follows. The ionization potential is $I_{p} = 15.76 eV,$ and the ionization rate is calculated by the Keldysh-Perelomov, Popov, and Terent’ev (PPT) model [5,6,30]. Here, we stress by the way that the dynamics shown in the next section is independent of the specific ionization model, which have been confirmed by using the multi-photon ionization (MPI) model. The pressure-dependent nonlinear coefficient is take as $n_{2} =1 .0 \times 10^{- 19} p {cm}^{2} / W$ in terms of the recent experimental data [31], where $p$ is the pressure ratio with respect to the standard atmospheric pressure.We have taken no account of the higher-order nonlinear coefficients than the 3th-order due to the considerable controversy [7–13]. The dispersion relation for argon is calculated in terms of the linear refractive index $n_{a t}$ at one atmospheric pressure as $k (ω) = ω \sqrt{1 + p [n_{a t}^{2} (ω) - 1]} / c,$ where the refractive index $n_{a t}$ is given in Ref [32]. Equation (2) is solved by the extended Crank-Nicolson scheme [33] which is more effective than our previously used fourth-order Runge-Kutta method [20], because the electron density need to be calculated only once in each step in the Crank-Nicolson scheme.

3. Results and discussion

In this section, we show the propagation dynamics in the gas cell filled with argon of uniform and non-uniform density, respectively. In the case of uniform density, the gas pressure is taken as 3 bar, which corresponds to the gas density of $ρ_{nt} = 8 \times 10^{19} {cm}^{- 3} .$ In the another case, the gas density distribution is assumed as Gaussian profile along the propagation axis and the peak density is the same as it in the uniform case, i.e., $ρ_{nt} = 8 \times 10^{19} \times exp(- z^{2} / w_{z}^{2}) {cm}^{- 3},$ where z is the distance from the center of argon cell, $w_{z} = 5 cm$ is the half width ( $1 / e^{2}$ ) of gas density distribution.

Figures 1(a) , 1(b) show the peak intensity and peak plasma density of the laser pulse in the argon cell along the propagation axis z in the case of uniform and non-uniform density, respectively. The position of z = 0 is located in the center of the cell and is also the focal plane of the lens. In the filamentation region, the intensity is clamped to ~106 TW/cm², and the corresponding plasma density is ~10¹⁷ cm⁻³. The calculated intensity is much higher than that in some previous works where it is 50-60 TW/cm² [16,17,20], because the Keldysh-PPT model gives a smaller ionization rate than the MPI model in the high-field region [5]. Comparing Fig. 1(a) with Fig. 1(b), we can see that, although the input laser field at the entrance of the argon cell is completely the same, there is a substantial difference in the filament length between the two cases. In the uniform case, the filament can sustain relatively long distance, whereas the intensity is almost not clamped in the non-uniform case. This can be understood from the calculated fluence profile as shown in Figs. 1(c), 1(d). The pulse energy of 1 mJ corresponds to ~10 P_cr in the uniform case, therefore it is enough to sustain several focusing-defocusing cycles as seen from Fig. 1(c). By contrast, the critical self-focusing power P_cr gradually changes along the propagation due to the change of nonlinear coefficient $n_{2}$ in the non-uniform case, and as a whole, the value of P_cr is larger. Therefore, the pulse just sustains two focusing-defocusing cycles, and the defocusing is immediately followed by a refocusing near the nonlinear focus point as seen from Fig. 1(d).

Fig. 1 (a,b) Peak intensity (solid curves) and peak electron density (dashed curves), (c,d) fluence distribution, and (e,f) on-axis temporal profile as a function of propagation distance in argon cell. (a,c,e) Uniform density distribution, (b,d,f) Non-uniform density distribution.

Download Full Size | PDF

Figures 1(e), 1(f) show the evolution of on-axis temporal intensity profile along the propagation axis. Firstly, we discuss the dynamics in the uniform density case as shown in Fig. 1(e). The rear of the pulse is defocused all the while along the propagation axis from z = −5 to 0 cm where the pulse intensity is well clamped. In this region, the spatio-temporal profile takes on a $Λ$ shape as shown in Fig. 2(a) , which in fact has been shown in many published works [17–23]. When the peak intensity of the pulse begins to decline near the position of z = 0, the rear of the pulse can refocus and therefore the pulse splitting appears. Figure 2(b) shows the typical dynamics of on-axis pulse-splitting. By contrast, we show the on-axis temporal profile as a function of propagation distance in the non-uniform case in Fig. 1(f), the typical spatio-temporal intensity distributions are shown in Fig. 2(c), 2(d). The refocusing effect appears very early due to its sensitivity to the plasma density, which result in the co-propagation of intense on-axis double pulses. In fact, the refocusing begin to appear when the plasma density decline to just half of its peak value. The spatio-temporal profile takes on a double $Λ$ shape in the typical co-propagation region as shown in Fig. 2(c), which is different with the traditional refocusing dynamics as shown in Fig. 2(b) and some previously published works [17–23]. In the non-uniform density case, it is the rear of the pulse that is firstly exhausted rather than the pulse front. The pulse self-compression is achieved by isolating the pulse front. Although the pulse rear can refocus when the pulse front is exhausted, there is almost no self-compression in this region. Figure 2(d) show the typical dynamics in the region from z = −1 to 0 cm where the rear part of the pulse has been exhausted and the self-compression effect is achieved.

Fig. 2 Intensity distribution in time and space domain at two typical distances. (a,b) in the case of uniform density, (c,d) in the case of non-uniform density.

Download Full Size | PDF

To obtain a deeper insight into the filamentation dynamics in the non-uniform case, we show the on-axis temporal profile in Fig. 3(a) , 3(c) and the on-axis spectral intensity in Fig. 3(b), 3(d) corresponding to two typical distances of z = −1.6 and −0.5 cm, respectively. Figure 3(a) shows the process of pulse-splitting into two intense pulses, which can sustain about 1 cm along the propagation direction. Their corresponding spectra extend from 700 to 1000 nm and show a complete splitting. The front pulse corresponds to the low-frequency components and the back pulse the high-frequency components, which can be furtherly confirmed from the calculated XFROG spectrograms (see Ref [34] for the calculation method) shown in Fig. 3(c), where the gate function is a Gaussian with 30 fs FWHM. When the rear of the pulse is exhausted, an intense isolated self-compressed pulse appears, which corresponds to the low-frequency components of the generated supercontinuum as confirmed from Fig. 3(f).

Fig. 3 (a,d) Temporal profiles, (b,e) spectral intensity, and (c,f) spectrograms of on-axis pulse in the non-uniform argon at two typical distances consistent with Fig. 2(c,d), respectively. The values are normalized relative to its maximum in (c,f).

Download Full Size | PDF

4. Summary

In summary, we numerically investigated the dynamics of filamentation in an cell filled with argon of uniform and non-uniform density distributions, respectively. The non-uniform density distribution is assumed as Gaussian profile along the propagation axis in the argon cell. We found the pulse-splitting appeared earlier due to the sensitivity of rear-part refocusing to the plasma density and the pulse self-compression is achieved mainly by the exhausting of pulse rear in the case of non-uniform density distribution. In the double-peak pulse splitting region, a double $Λ$ shape can appear in the spatio-temporal intensity profile in the non-uniform density case, which is a kind of novel spatio-temporal dynamics.

Acknowledgments

Z. Wang gratefully thanks Prof. Ya Cheng for his fruitful discussions. This work was supported by the National Basic Research program (Contract No. 2010CB923203), the National Natural Science Foundation (Contracts No. 11104236, 61008061, 10974214), Jiangsu Province Natural Science Foundation (Contract No. BK2010173), and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD) of China.

References and links

1. A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, “Self-channeling of high-peak-power femtosecond laser pulses in air,” Opt. Lett. 20(1), 73–75 (1995). [CrossRef] [PubMed]

2. E. T. J. Nibbering, P. F. Curley, G. Grillon, B. S. Prade, M. A. Franco, F. Salin, and A. Mysyrowicz, “Conical emission from self-guided femtosecond pulses in air,” Opt. Lett. 21(1), 62–65 (1996). [CrossRef] [PubMed]

3. J. Kasparian and J.-P. Wolf, “Physics and applications of atmospheric nonlinear optics and filamentation,” Opt. Express 16(1), 466–493 (2008). [CrossRef] [PubMed]

4. J. Schwarz and J.-C. Diels, “UV filaments and their application for laser-induced lightning and high-aspect-ratio hole drilling,” Appl. Phys., A Mater. Sci. Process. 77(2), 185–191 (2003).

5. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2–4), 47–189 (2007). [CrossRef]

6. L. Berge, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007). [CrossRef]

7. P. Béjot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J.-P. Wolf, “Higher-order Kerr terms allow ionization-free filamentation in gases,” Phys. Rev. Lett. 104(10), 103903 (2010). [CrossRef] [PubMed]

8. M. Kolesik, E. M. Wright, and J. V. Moloney, “Femtosecond filamentation in air and higher-order nonlinearities,” Opt. Lett. 35(15), 2550–2552 (2010). [CrossRef] [PubMed]

9. P. Polynkin, M. Kolesik, E. M. Wright, and J. V. Moloney, “Experimental tests of the new paradigm for laser filamentation in gases,” Phys. Rev. Lett. 106(15), 153902 (2011). [CrossRef] [PubMed]

10. Z. Wang, C. Zhang, J. Liu, R. Li, and Z. Xu, “Femtosecond filamentation in argon and higher-order nonlinearities,” Opt. Lett. 36(12), 2336–2338 (2011). [CrossRef] [PubMed]

11. O. Kosareva, J.-F. Daigle, N. Panov, T. Wang, S. Hosseini, S. Yuan, G. Roy, V. Makarov, and S. L. Chin, “Arrest of self-focusing collapse in femtosecond air filaments: higher order Kerr or plasma defocusing?” Opt. Lett. 36(7), 1035–1037 (2011). [CrossRef] [PubMed]

12. Y.-H. Chen, S. Varma, T. M. Antonsen, and H. M. Milchberg, “Direct measurement of the electron density of extended femtosecond laser pulse-induced filaments,” Phys. Rev. Lett. 105(21), 215005 (2010). [CrossRef] [PubMed]

13. P. Béjot, E. Hertz, J. Kasparian, B. Lavorel, J.-P. Wolf, and O. Faucher, “Transition from plasma-driven to Kerr-driven laser filamentation,” Phys. Rev. Lett. 106(24), 243902 (2011). [CrossRef] [PubMed]

14. A. L. Gaeta, “Catastrophic collapse of ultrashort pulses,” Phys. Rev. Lett. 84(16), 3582–3585 (2000). [CrossRef] [PubMed]

15. L. Bergé and A. Couairon, “Gas-induced solitons,” Phys. Rev. Lett. 86(6), 1003–1006 (2001). [CrossRef] [PubMed]

16. C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B 79(6), 673–677 (2004). [CrossRef]

17. S. Skupin, G. Stibenz, L. Bergé, F. Lederer, T. Sokollik, M. Schnürer, N. Zhavoronkov, and G. Steinmeyer, “Self-compression by femtosecond pulse filamentation: experiments versus numerical simulations,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(5), 056604 (2006). [CrossRef] [PubMed]

18. A. Mysyrowicz, A. Couairon, and U. Keller, “Self-compression of optical laser pulses by filamentation,” New J. Phys. 10(2), 025023 (2008). [CrossRef]

19. L. Bergé, S. Skupin, and G. Steinmeyer, “Temporal self-restoration of compressed optical filaments,” Phys. Rev. Lett. 101(21), 213901 (2008). [CrossRef] [PubMed]

20. Z. Wang, J. Liu, R. Li, and Z. Xu, “Supercontinuum generation and pulse compression from gas filamentation of femtosecond laser pulses with different durations,” Opt. Express 17(16), 13841–13850 (2009). [CrossRef] [PubMed]

21. C. Brée, J. Bethge, S. Skupin, L. Bergé, A. Demircan, and G. Steinmeyer, “Cascaded self-compression of femtosecond pulses in filaments,” New J. Phys. 12(9), 093046 (2010). [CrossRef]

22. G. Stibenz, N. Zhavoronkov, and G. Steinmeyer, “Self-compression of millijoule pulses to 7.8 fs duration in a white-light filament,” Opt. Lett. 31(2), 274–276 (2006). [CrossRef] [PubMed]

23. A. Couairon, M. Franco, A. Mysyrowicz, J. Biegert, and U. Keller, “Pulse self-compression to the single-cycle limit by filamentation in a gas with a pressure gradient,” Opt. Lett. 30(19), 2657–2659 (2005). [CrossRef] [PubMed]

24. A. Couairon, J. Biegert, C. P. Hauri, W. Kornelis, F. W. Helbing, U. Keller, and A. Mysyrowicz, “Self-compression of ultra-short laser pulses down to one optical cycle by filamentation,” J. Mod. Opt. 53(1–2), 75–85 (2006). [CrossRef]

25. A. Zaïr, A. Guandalini, F. Schapper, M. Holler, J. Biegert, L. Gallmann, A. Couairon, M. Franco, A. Mysyrowicz, and U. Keller, “Spatio-temporal characterization of few-cycle pulses obtained by filamentation,” Opt. Express 15(9), 5394–5404 (2007). [CrossRef] [PubMed]

26. O. G. Kosareva, N. A. Panov, D. S. Uryupina, M. V. Kurilova, A. V. Mazhorova, A. B. Savel’ev, R. V. Volkov, V. P. Kandidov, and S. L. Chin, “Optimization of a femtosecond pulse self-compression region along a filament in air,” Appl. Phys. B 91(1), 35–43 (2008). [CrossRef]

27. H. S. Chakraborty, M. B. Gaarde, and A. Couairon, “Single attosecond pulses from high harmonics driven by self-compressed filaments,” Opt. Lett. 31(24), 3662–3664 (2006). [CrossRef] [PubMed]

28. M. B. Gaarde and A. Couairon, “Intensity spikes in laser filamentation: diagnostics and application,” Phys. Rev. Lett. 103(4), 043901 (2009). [CrossRef] [PubMed]

29. J.-C. Diels, J. Yeak, D. Mirell, R. Fuentes, S. Rostami, D. Faccio, and P. Trapani, “Air filaments and vacuum,” Laser Phys. 20(5), 1101–1106 (2010). [CrossRef]

30. W. Ettoumi, P. Béjot, Y. Petit, V. Loriot, E. Hertz, O. Faucher, B. Lavorel, J. Kasparian, and J.-P. Wolf, “Spectral dependence of purely-Kerr-driven filamentation in air and argon,” Phys. Rev. A 82(3), 033826 (2010). [CrossRef]

31. V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components: erratum,” Opt. Express 18(3), 3011–3012 (2010). [CrossRef]

32. A. Dalgarno and A. E. Kingston, “The refractive indices and Verdet constants of the inert gases,” Proc. R. Soc. Lond. A Math. Phys. Sci. 259(1298), 424–431 (1960). [CrossRef]

33. A. Couairon, E. Brambilla, T. Corti, D. Majus, O. Ramírez-Góngora, and M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top. 199(1), 5–76 (2011). [CrossRef]

34. S. Akturk, A. Couairon, M. Franco, and A. Mysyrowicz, “Spectrogram representation of pulse self compression by filamentation,” Opt. Express 16(22), 17626–17636 (2008). [CrossRef] [PubMed]

Dynamics of femtosecond laser filamentation in argon with non-uniform density distribution

Abstract

1. Introduction

2. Models and parameters

3. Results and discussion

4. Summary

Acknowledgments

References and links

Cited By

Figures (3)

Equations (2)

Optics Express