## Abstract

Self-compression and spectral supercontinuum (SC) generated by filamentation of femtosecond laser pulses with duration from 45 fs down to 6 fs in argon gas have been numerically investigated. A 45-fs pulse can be self-compressed into a few-cycle pulse with duration of 12 fs at the post-filamentation region. By properly employing a high-pass filter to select the broadening high-frequency spectra which are almost in phase, the pulse can be further shortened to about 7 fs. By contrast, a 6-fs pulse cannot be further self-compressed into a shorter pulse by filamentation although it can generate much broader SC extending from 200 nm to 1300 nm. It is also found that a separate and strong SC in the ultraviolet (UV) region extending from 220 nm to 300 nm and peaked at about 255nm can be generated at proper propagation distances, which corresponds to a pulse with duration of about 5 fs.

© 2009 OSA

## 1. Introduction

The effect of self-compression of femtosecond laser pulse during filamentation has attracted extensive attention owing to the far-ranging applications of few-cycle pulses in attosecond physics [1,2] and relativistic nonlinear optics [3]. Extensive experimental [4–17] and theoretical [5–7,12–14,16–23] investigations of pulse self-shortening have been conducted by exploitation of spatio-temporal shaping during filament formation both in gaseous media as well as in solid media such as fused silica. For the moment, most experimental attention concerning pulse self-compression was paid on cascade filamentation [8,10,12–14]. By using two-cell filament equipment, a pulse of duration as short as ~5 fs has been experimentally demonstrated with excellent beam quality [8,10,14]. Several authors have exploited the four-wave mixing (4WM) to obtain ultraviolet (UV) few-cycle pulses [24–27]. Recently, we have also proposed a spectral filtering method to obtain sub-5 fs pulses based on the self-guided filaments of femtosecond laser beams in fused silica [29].

It has been reported that SC generation from filamentation is highly dependent on the pulse duration [30–33]. By focusing 10-fs laser pulses at 805 nm into argon, Trushin et al. have demonstrated that SC extends from >1000 to 250 nm [30]. The UV cutoff wavelength can extend to 210 nm if a few-cycle pulse of 6 fs is employed [32]. Theoretical simulations indicate that third-harmonic spectra are not required to explain the observed UV spectra [31].

In this paper, by employing a model developed in the frequency domain, we simulated the spatial-temporal dynamics of filamentation in argon gas by 800-nm femtosecond laser pulses with different durations from 45 fs down to 6 fs. For a 45-fs input pulse with energy *E*
_{in}=1.3 mJ, the pulse will undergo self-compression and form a temporally compressed “light bullet” with a duration of ~12 fs at the post-filamentation domain as it is well-known [5,6,28]. By properly employing high-pass filters to cut out the out-phase low-frequency component, we found that the self-compressed pulse can be further shortened to about 7 fs, which is close to the Fourier transform limit of corresponding SC spectra. By contrast, the propagation dynamics of a 6-fs pulse is very different from that of a 45-fs pulse, which cannot be further self-compressed into a shorter pulse by filamentation although it can generate much broader SC extending from 200 nm to 1300 nm. It is interesting that a separate and strong SC in the ultraviolet region extending from 220 nm to 300 nm and peaked at about 255nm can be generated. If the SC generated at proper propagation distance is cut out by a high-pass filter, a pulse with duration of about 5 fs in deep UV can be numerically obtained.

## 2. Simulations

Our model for nonlinear propagation of an ultrashort laser pulse in optical medium is based on the model developed by Brabec *et* al. [34]. Assuming the pulse has radial symmetry and propagates along z axis with a wave vector *k*(*ω*)=n(*ω*)*ω*/*c*, where *ω* is the optical frequency, and *n*(*ω*) is the linear refractive index of the material, the equation of propagation of electric field has the following form [35–38]

Where *E*̂ is the Fourier transform of the forward electric field component, *P*̂* _{NL}* is the nonlinear polarization in frequency domain of

*P*(

_{NL}*r*,

*z*,

*t*) given by

*P*(

_{NL}*r*,

*z*,

*t*)=2

*ε*

_{0}

*n*

_{b}

*n*

_{2}

*I*(

*r*,

*z*,

*t*)

*E*(

*r*,

*z*,

*t*), where

*n*

_{b}and

*n*

_{2}are linear refractive index and nonlinear coefficient at center wavelength

*λ*

_{0}, respectively. The current density

*J*caused by free electrons can be expressed in frequency domain as [36–38]

_{f}where *k*
_{0}=0.5*n _{b}cε*

_{0}. The evolution of free electron density can be described as ∂

_{t}

*ρ*=

*β*

^{(K)}

*I*

*K*+

*η*-

_{cas}ρ*η*

_{rec}ρ^{2}. The initial value of electron density is assumed as

*ρ*

_{0}≡10

^{9}cm

^{-3}[39]. The input pulses are assumed to be Gaussian both in space and time domain, i.e.

Where *P _{in}*,

*w*

_{0},

*T*

_{0}, and

*f*denote input peak power, beam waist, pulse duration, and effective focal length, respectively.

*k*

_{(0)}=

*n*

_{b}

*ω*

_{0}/c is the wave number at the laser frequency. Throughout the paper, we will use full-width at half maximum (FWHM) values with respect to the intensity and beam energy to characterize the pulses, namely, ${t}_{\mathrm{FWHM}}=\sqrt{2\mathrm{ln}\phantom{\rule[-0ex]{.2em}{0ex}}2}{T}_{0},{w}_{\mathrm{FWHM}}=\sqrt{2\mathrm{ln}\phantom{\rule[-0ex]{.2em}{0ex}}2}{w}_{0},\mathrm{and}\phantom{\rule[-0ex]{.2em}{0ex}}{E}_{\mathrm{in}}=\sqrt{\pi \u20442}{T}_{0}{P}_{\mathrm{in}}.$.

The parameters for the propagation of 800-nm laser in argon at atmospheric pressure can be described as follows [14]. The ionization potential is *I _{p}*=15.76 eV and the number of photons required for ionization is

*K*=11. The multiphoton ionization (MPI) coefficient can be calculated as

*β*

^{(11)}=1.34×10

^{-114}

*S*

^{-1}·

*m*

^{19}·

*W*-

^{11}. The collision time can be taken as

*τ*=190 fs. The linear refractive index

_{c}*n*(

*ω*) of argon is expressed as a formula given in Ref [40]. The critical power takes the value

*Pσ*=

*λ*

^{2}

_{0}/2

*π*

*n*b

*n*2 for a nonlinear coefficient

*n*

_{2}=3×10

^{-19}

*cm*2/

*W*.

*η*=3.1×10

_{cas}^{-6}

*s*

^{-1}·

*m*

^{2}·

*W*

^{-1}and

*η*=7×10

_{rec}^{-13}

*m*

^{3}·

*s*

^{-1}represent cascade ionization and electron-ion recombination rate, respectively. The Eq. (1) was solved by a fourth-order Runge-Kutta method with adaptive stepsize control. The transversal Laplace operator ∇

^{2}

_{⊥}is calculated here by a finite difference method [41] instead of Fourier transformation method used in Ref [38].

## 3. Results and Discussions

In this section, we have investigated the self-guiding propagation of 800-nm pulses with different durations initially focused in argon at atmospheric pressure by a lens with a focal length *f*=60 cm. The beam waist of the pulse is assumed as *w*
_{0}=2.5 mm (which corresponds to *w*
* _{FWHM}*=2.94 mm). When the input peak power is higher than the critical power for self-focusing, the beam collapse can be expected to occur near the so-called nonlinear focus z

_{c}, which can be expressed by the well-known Marburger formula

Where *z*
_{0}=*πn*
_{0}
*w*
^{2}
_{0}/*λ*
_{0} is the Rayleigh length of the beam. In the case of a convergent beam, the position of the collapse *z*
*c*,*f* moves to *z*
* _{c}*,

*=*

_{f}*z*

_{c}*/(*

_{f}*z*

*+*

_{c}*f*).

Figure 1(a–c) show the peak intensities and electron densities, respectively, along the propagation axis z for 45-fs input pulses at different input pulse powers. The peak intensity and the plasma density of the laser pulse in the filament are sustained at ~57.5 TW/cm^{2} and about 10^{17} cm^{-3}, respectively, which almost do not change with the input power. This phenomenon has been called intensity clamping. When the input pulse power increases from 5 P_{cr} to 12 P_{cr}, the filament and the plasma channel are elongated from 9 cm to 18 cm. As a comparison and shown in Fig. 1(d), we simulated the propagation of 6-fs pulse at 5 P_{cr} (E_{in}=0.11 mJ). The length of the filament is about 19 cm, which is much longer than the corresponding 45-fs pulse with the same input power. The peak intensity in the filament will alter from about 62 to 70 TW/cm^{2} along the propagation axis z, which is a little higher than that for the corresponding 45-fs pulse. As analyzed in ref [38], the clamped intensity can be empirically expressed as ${I}_{\mathrm{clamping}}\sim {({\omega}_{0}^{2}{n}_{b}{n}_{2}\u2044\sqrt{6{k}^{\left(0\right)}\pi}\pi {T}_{0}{\beta}^{\left(K\right)})}^{1\u2044\left(K-1\right)}.$. For a few-cycle pulse with a shorter duration *T _{0}*, the clamped intensity will be higher than that for a long pulse.

In Figs. 2(a,b,c) we show the on-axis temporal dynamics of the 45-fs pulse in the (t, z) plane at input power 5 P_{cr} (E_{in}=0.8 mJ), 8.2 P_{cr} (E_{in}=1.3 mJ), 12 P_{cr} (E_{in}=2.0 mJ), respectively. As described extensively [5,6,13], the front of the pulse firstly self-focuses near the nonlinear focus and produces plasma to defocus the trailing part of the pulse. On further propagation, the rear part of the pulse will refocus, which result in the pulse splitting on axis. This process can be repeated for each split pulse until the pulse front is completely exhausted. At the post-filamentation region, there is some difference for different input powers. When the input power is as low as P_{in}=5 P_{cr}, this single-sided depletion will eventually give rise to a few-cycle pulse with a duration of ~12 fs after z~63 cm, but the obtained few-cycle pulse is very weak (Fig. 2(a)). When the input power increases to P_{in}=8.2 P_{cr}, an intense temporal-compressed “light bullet” with duration about 12 fs can be obtained from z~63 cm and maintain over 30 cm as shown in Fig. 2(b). The pulse self-compression seems to be less effective with increasing input pulse power. When the input peak power is further increased to P_{in}=12 P_{cr}, a pulse with a duration of ~12 fs can still be obtained at z~65 cm. However, it is very interesting that the resulted few-cycle pulse can filament immediately because its peak power exceeds corresponding critical threshold for self-focusing. The self-compressed pulse will experience another splitting, which also can be seen from Fig. 1(c) and Fig. 2(c).

In order to obtain a shorter few-cycle pulse, we choose input peak power P_{in}=8.2 P_{cr} and analyze the propagation dynamics in detail. Figures 3 shows the typical intensity distribution in time and space domain at z=53.6, 62.6, 67.0, 86.5 cm, respectively. Initially, the pulse is focused by a lens and self-focused due to Kerr effect. The peak intensity will gradually increase along with the way of propagation. The trailing edge of the pulse suffers trivial defocusing due to very weak ionization, which induces the slight asymmetry of the spatial-temporal profile. When the pulse reaches to the nonlinear focus (*z _{c}*,

_{f}~53 cm), the pulse peak intensity quickly increases and reaches to about 57 TW/cm

^{2}on axis, which causes intense ionization and plasma is generated at the trailing edge of the pulse which defocuses the light on-axis. The pulse forms a ‘light cone’ as shown in Fig. 3(a). Figure 3(b,c) show that the rear pulse may refocus, which will generate a typical double-peaked temporal distribution. Eventually, the energy of the leading edge of the pulse will deplete and cause strong pulse shortening as shown in Figs. 3(d), where a pulse with duration of ~12 fs is generated.

The on-axis spectral intensity generated at z=79.9 cm is shown in Fig. 4(a). Spectral SC from 2 to 3 PHz which corresponds to a Fourier transform limited duration of ~5.7 fs (Fig. 4(b)) is mainly induced by SPM and plasma behavior. The corresponding spectral phases are calculated relative to central wavelength 800 nm and shown in Fig. 4 (c) (black line). For clarity, we adjust the spectral phase by subtracting a linear phase shift. It merely corresponds to the shift of the envelope along the time direction and does not change the shape of the pulse envelope. Figure 4(c) (red line) shows the adjusted spectral phase which is detailed from 2.4 to 3.5 PHz by the inset in Fig. 4(c). It can be found that the spectral phase aberration in the region of high frequencies (*ω*>2.4 PHz) is very small while it is large in low frequency region. This kind of asymmetric spectral phase aberration is hard to compensate. However, when we properly cut out low-frequency spectra by using a high-pass filter, we can obtain a broad high-frequency spectrum whose phase aberration is small and can be compensated more easily because its spectral phase is almost quadratic. By using different spectral filters with different cut-off frequency to reshape the temporal profiles of the self-compressed pulses, we calculate the profiles of obtained pulses. As shown in Fig. 4(e), for the self-compressed pulse with a duration of ~12 fs at z=79.9 cm, shorter pulses with durations of about 9 fs, 7 fs, and 7 fs can be obtained by using high-pass filters with cut-off frequency of 2.4, 2.5, 2.6 PHz, respectively. However, when the cut-off frequency increases, the peak intensity of the obtained pulse will become lower.

As a comparison, in the following paragraphs, we have also numerically simulated and analyzed the propagation of 6-fs pulse in argon at 5 P_{cr}. For lower input power such as 2 P_{cr}, the propagation dynamics is similar (not shown here). The on-axis temporal dynamics of 6-fs pulse was shown in Fig. 2(d). Figure 5 shows spatial-temporal intensity profile of 6-fs pulse at 800 nm propagating in argon at z=(a) 40.2, (b) 52.88, (c) 55.5, (d) 61.6 cm, respectively. The propagation dynamics is very different from that of 45-fs pulse as shown in Fig. 3. Initially, the input pulse is assumed as Gaussian both in space domain and time domain. The corresponding profile of spatial-temporal intensity looks like an ‘ellipse’. When the pulse propagates, the profile of the ‘ellipse’ becomes asymmetric (shown in Fig. 5(a)) along time axis mainly due to the external focusing that is numerically applied. When arriving at nonlinear focus (z_{c},_{f}~52.9 cm) the pulse begins to filament (Fig. 5(b)). The trailing edge of the pulse suffers strong self-steepening which can bring about some trivial self-compression effect. When further propagating, the trailing part of the pulse will split and form many short temporal peaks, which will increase in number and peak intensity along with the increase of propagation distance (Fig. 5(c,d)). The short temporal splitting can be explained by the shock profile induced by self-steepening.

Figure 6 (a) shows the spectral SC generated by 6-fs filaments at z=40.2, 52.9, 55.5 and 72.0 cm respectively integrated over the beam cross-section. It is clear that the spectrum SC can extend down to ~200 nm when filamentation begins at z~52.9 cm. On further increasing the path length z, the spectral intensity of the SC near the 255nm will gradually increase and shape into a separate spectral peak extending from 220 nm to 280 nm. This process can be more clearly seen from Fig. 7 where the on-axis spectral intensities generated at z=55.5, 61.6, and 64.0 cm, respectively, are shown.

This separate SC is very interesting because it has small phase aberrations and can correspond to a few-cycle pulse. Figure 7(c) inset details the spectral intensity and phase from 220 nm to 290 nm generated at z=63.96 cm. The spectral phase aberration is small and can be further compensated easily. Figure 8 shows the corresponding temporal profiles generated by the UV SC at frequency window from 210 nm to 310 nm at several different propagation distances. A 5-fs pulse with peak intensity of about 40TW/cm^{2} can be obtained at propagation distances from 63.96 to 65.48 cm just by spectral filtering method.

The reason for generating the UV SC may come from the 4WM of 800-nm (ω_{0}=2.36 PHz) pump pulse with a 400-nm (ω_{s}=4.72 PHz) seed generated by intense self-phase modulation and self-steepening during filamentation following the scheme 2ωs-ω_{0}=ω_{4WM}. This is reasonable because the phase match can be satisfied as discussed by Bergé *et* al. [27]. Interestingly, a completely analogous isolated blueshifted peak has been observed when the laser pulse filamentation in fused silica in the normal group velocity dispersion (GVD) region [17], which has been explained as the blueshifted tail of X wave that is formed in the filaments.

Recently, Aközbek *et al*. have described the experimental observation of the spectral SC generated by different pulse duration in Ref [31], where the pulse was focused by a lens of focal length 1 m and accumulated many pulses by a broad-band spectrometer. Compared with their experimental results (Fig. 1(a) in Ref [31].), the experimentally observed SC generated by a 45-fs pulse is well consistent with our result by simulation as shown in Fig. 6(b). The observed SC generated by a 6-fs pulse can also extend down to 200 nm which is qualitatively consistent with our results shown in Fig. 6(a). However, for the 6-fs pulse, the observed SC intensity did not show an obvious separate peak near 255 nm in the UV but abruptly weakened from 300 nm down to 200 nm. This seems not to be a problem since self-phase modulation is very sensitive to the temporal shape, the spectrum and the spectrum chirp. The experimental pulses may be different from the pulses numerically used.

We have realized that the quadratic phase -*ik*
^{(0)}
*r*
^{2}/2*f* in the wavefront of the initial laser pulse has significant effect on the filamentation dynamics of femtosecond pulse, especially for few-cycle pulse. We also investigated the propagation dynamics of 45-fs and 6-fs pulses with different quadratic phase such as -*ik*(*ω*)*r*
^{2}/2*f* � which in fact can result in much different femtosecond dynamics. Therefore, by adjusting the initial laser parameters, the filamentation process can be controlled. The control of filamentation process by changing the initial laser parameters will be discussed in detail in another paper.

## 4. Conclusion

In conclusion, we have investigated the filamentation and self-compression of femtosecond pulses focused in argon by a lens of focus length 60 cm at different input powers and pulse durations from 45 fs down to 6 fs. The pulse can be self-compressed at the region of post-filamentation at P_{in}=8.2 P_{cr} and last over 30 cm. By properly employing high-pass filter, the pulse can be further shortened to about 7 fs which approaches to the Fourier transform limit of the corresponding spectral SC. As a comparison, the propagation dynamics of a 6-fs pulse is very different from that of a 45-fs pulse, which cannot be further self-compressed into a shorter pulse by filamentation although it can generate much broader SC extending from 200 nm to 1300 nm. It is also found that a separate and strong SC in the ultraviolet region extending from 220 nm to 300 nm and peaked at about 255nm can be generated, which can be explained as 4WM of a 800-nm pump pulse (i.e. input pulse) and a 400-nm seed pulse mainly generated by intense SPM and self-steepening during filamentation. The UV SC generated at proper propagation distance corresponds to a few-cycle pulse with duration of about 5 fs in time domain without spectral phase compensation.

## Acknowledgements

This work was supported by the Chinese National Natural Science Foundations (Contract No 10674145) and the National Basic Research Program of China (Contract No 2006CB806000).

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