1. Introduction

The generation of few-cycle pulses with high energies is the basis for a growing number of experiments in strong-field and attosecond physics [1–4]. Filamentation of ultra-short laser pulses in gaseous media involves a variety of nonlinear processes [5–8] including the generation of new optical frequencies in the range from the THz regime [9, 10] to the ultraviolet [11–14] and even the extreme ultraviolet (XUV) spectral domain [15].

During filamentation, the balance of nonlinear effects such as Kerr self-focusing and plasma defocussing forms a dynamic self-guiding mechanism. This leads to a nonlinear interaction length in a narrow light channel, which is much larger than the typical diffraction length in a linear focal geometry [5, 16]. Additionally, the nonlinear effect of self-phase modulation results in a significant spectral broadening. Effects such as self-steepening have additional influence on the shape of the resulting spectrum [5, 7]. Previous works demonstrate the generation of near-infrared few-cycle pulses, using a double-staged filamentation setup. The chirp can be compensated by chirped mirrors in between the first and the second filamentation stage as well as after the filamentary setup [17]. The accumulated chirp can also be compensated by means of self-compression inside the filament itself [18–21]. The pulses inside the filament undergo a complex spatio-temporal evolution as they evolve through several defocusing and refocusing cycles, leading to a splitting of the fundamental pulse into multiple sub-pulse structures with individual pulse durations and chirp characteristics [22–24].

The detection and characterization of sub-pulse structures and pulses entering the single-cycle regime is a challenging procedure, whereby the complete reconstruction of the phase of a broadband spectrum is one of the major barriers. In contrast to spectral methods, measurements of stereographic above-threshold-ionization (ATI) electron spectra enable the characterization of pulses in the sinlgle-cycle regime directly by means of the electric field [25–27]. With regard to sub-pulse structures evolving from the filament with different dispersion characteristics, the stereo-ATI phasemeter turns out to be an ideal tool. Its setup enables a fine tuning of the chirp and enables the individual compression of a sub-pulse according to its accumulated chirp. Other co-propagating pulse structures will be stretched because of their different dispersion characteristics and not be detected due to the sensitivity of the phasemeter [26]. In this paper, we report on the conversion of 35 fs pulses into pulses with a duration below 4 fs in a single-stage femtosecond filament. This short pulse duration corresponds to sub-1.5-cycles of the electric field at a carrier wavelength of 780 nm. We investigate the duration of pulses emerging from a fully propagated filament in dependence of the gas pressure and the input chirp parameters. Moreover, we examine the evolution of sub-pulses inside the filament by truncating the filament at different propagation lengths.

2. Experimental setup

Figure 1 displays the experimental setup. A Ti:sapphire chirped-pulse-amplification system (Dragon, KM-Labs Inc.) delivers 35-fs (FWHM) pulses centered at 780 nm with an energy of 1.3 mJ at a repetition rate of 3 kHz. An aperture B1 with variable diameter is placed after the amplifier exit and transmits about 63 % of the power at a diameter of 6.0 mm. The aperture is used to adjust the input power of the fundamental beam and to spatially clean its beam profile. These pulses, having a peak power of 35 GW, are focused via a curved silver mirror CM1 with a focal length of 2 m through an 1-mm-thick CaF₂ window into a 1-m-long semi-infinite gas cell (SIGC) [28]. The SIGC is filled with argon at different pressures between 700 and 1000 mbar in order to exceed the critical power for self-focusing $P_{\text{cr}}\propto n_2^{-1}\propto p^{-1}$ [29]. A laser-drilled pinhole P1 (diameter about 0.5 mm) terminates the SIGC by a steep transition from the high pressure cell to a rough vacuum of a few millibars. The transmitted pulses propagate about 1 m in vacuum before they leave the cell through a tilted 2-mm-thick CaF₂ window.

 

Fig. 1 Experimental setup. Pulses from a chirped pulse amplifier are focused into a semi-infinite gas cell filled with argon. The filament, generated in argon, is either truncated by a laser-drilled pinhole with further propagation in vacuum or fully propagated. The generated white-light is coupled out and analyzed in a stereo-ATI-phasemeter. (see text for further details)

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An aperture B2 clips the outer part of the beam and transmits only its white-light-core after the filamentation stage. For further propagation, a curved silver mirror CM2 with an focal length of 1.5 m is used to collimate the beam. The dispersion of the pulses is compensated by a pair of double chirped mirrors, DCM1 and DCM2, (Naneo DCM9 series, VENTEON Technologies GmbH) with a bandwidth from 400 to 950 nm. The chirped mirrors induce a dispersion of −80 fs² per bounce pair. Fused silica wedges are applied for fine adjustment of the dispersion, before the pulses are focused by a curved silver mirror CM3 into the phasemeter setup for characterization by stereo-ATI measurements. An aperture B3 is used to adjust the peak intensity in the phasemeter. The phasemeter enables the measurement of the pulse duration as well as the carrier-envelope-phase (CEP) by utilizing the fact, that ATI-spectra in Xe are highly sensitive to the CEP of ultrashort pulses. In brief, the phasemeter works as follows. Photoelectrons emitted parallel to the laser polarization axis in opposite directions are detected by two microchannel plates. The energy of the photoelectrons can be determined by their time-of-flight. For measuring the pulse duration and the CEP, the rescattered (plateau) ATI-electrons are used [30]. Their asymmetry, defined as the difference of electron yield, detected in the left and right detector normalized to the sum, is computed for two energy intervals in the ATI plateau region. For both energy intervalls, the asymmetry varies sine-like, if plotted versus the CEP, however with a different phase [31]. A parametric plot of the two asymmetries (PAP) results in an approximate circle for a suitable choice of the energy intervals. The radius and azimuth then can be used as a measure of pulse duration and CEP, respectively [25–27]. The shorter the pulse, the larger is the induced asymmetry and the corresponding radius r of the PAP. This method is limited to pulse durations below ∼8 fs, such that the radius of the PAP can be measured with sufficient precision. Via the empirical formula [26]

In order to investigate the filament along the propagation direction, a translation stage is placed between the curved mirror CM1 and the pinhole P1. This allows the translation of the whole filament inside the SIGC with respect to the cell-terminating pinhole P1, where the filament ends with a transition to vacuum. This method was previously applied to investigate the filament via spectral analysis of the fundamental beam, observation of the third-order as well as high-order harmonics [12, 13, 15, 24].

3. Pulse investigation along the propagation axis

By using the pinhole P1 to stop the filamentation process through the transition to vacuum, we are able to investigate the pulses coupled out from different positions inside the filament [15, 24]. Figure 2 shows the measured pulse duration as a function of the propagation length in the filament for an argon pressure of 780 mbar in the SIGC. The record time of the asymmetry plots in the stereo-ATI-phasemeter is set to 1000 single shots for the given measurements, corresponding to 0.33 s if not denoted otherwise. The uncertainty of the resulting pulse duration is shown exemplarily by the errorbars of the first measurements. Before the data acquisition, the chirp-compensation is optimized in order to achieve the shortest possible pulse duration. This means that the measured pulses are compressed after they emerge from the filament and the given temporal duration of the pulses does not give insight into the pulse duration directly after the filamentation stage. In the following, the propagation length is denoted as the distance between the focusing mirror CM1 and the truncating pinhole P1. While the Fourier-limit of the generated white-light decreases and saturates around 4.4 ± (0.2) fs after 230 cm of propagation, the measured pulse duration shows a minimum at a distance of about 237 cm behind the focusing mirror. The shortest pulse duration is measured to 5.7 fs. Sub-5-fs pulses cannot be obtained at this stage of propagation. Note that the settings of the dispersion compensation, used to obtain the PAP after the cut filament, are first optimized but then remain constant over the measurement for the first sub-pulse. After a propagation distance of 239 cm, the signature of another pulse is measured by the phasemeter at a different dispersion setting of the fused silica wedges. Here the compensation differs by a value of 66 fs² compared to the first measured pulse. Consequently, we measure a pulse with different chirp characteristics compared to the first sub-pulse. Our observations show evidence of sub-pulses occurring in a femtosecond filament [24] and imply that the dispersion properties of the first sub-pulse do not change significantly during propagation in the filament. The increasing pulse duration may result from spectral losses due to interaction with the arising sub-pulses with different chirp-characteristics.

 

Fig. 2 Pulses from a truncated filament of different propagation lengths in 780 mbar argon. The propagation length denotes the distance between the curved mirror CM1 and the pinhole P1. The black boxes illustrate the Fourier-limited pulse duration evaluated from spectral measurements, the red circles depict the measured pulse duration. The dashed lines denote a moving average of the measured values over an intervall of 4 cm. The blue triangles are measurements of a second sub-pulse appearing at another post-compression setting i.e another fused silica wedge setting.

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The above observations of multiple pulses are confirmed by stereo-ATI measurements of the pulse duration versus dispersion imposed by the fused-silica wedges at a fixed position of the truncating pinhole at a position of 254.5 cm. The red points in Fig. 3 depict the duration of pulses originating from the cut filament in dependence of the chirp-compensation with the fused silica wedges. We clearly observe the occurrence of multiple pulses, which require a different compensation of the chirp, meaning that we observe pulses with different chirp-characteristics at this late stage of filamentary propagation. In order to exclude the measurement of one single pulse, which is stretched due to the additional dispersion, we calculate the effects of an additional propagation through fused silica with an effective thickness of ±0.2mm for the shortest pulse and a thickness of ±0.3mm for the other pulses. Additional propagation through ±0.3mm corresponds to a variation of the dispersion of ±11.3fs² upon the shortest observed pulse in Fig. 3. Taking the complete measured spectrum behind the phasemeter and assuming a flat spectral phase, we obtain a Fourier-limited pulse of 4.4 fs duration. In the calculation, the effects of the dispersive media and our dispersion-compensating mirrors on pulses with different durations are taken into account. We include the propagation through 2 mm CaF₂ tilted at an angle of 30 degrees, propagation through 5194 mm of air behind the output window, four bounces on the dielectric mirrors as well as the effects of the fused silica wedges and the entrance window of the phasemeter. The shortest obtainable pulse after this setup is calculated to have a temporal duration of 5.3 fs, when a propagation through 4.19 mm of fused silica is assumed. This agrees quite well with our shortest measured pulse duration of about 5.5 fs in Fig. 3. For the investigation of the other sub-pulses, we narrow the acquired fundamental spectrum by cutting its blue part in order to obtain pulse durations which equal the measured duration of the sub-pulses. Afterwards, we change the simulated thickness of the fused silica wedges in order to calculate the effects of additional dispersion upon the pulses. The black lines in Fig. 3 show the simulated pulse duration on changing the propagation distance through fused silica. From the calculations it turns out, that an additional dispersion of 10 fs² on a 5.5 fs Fourier-limited pulse results in a pulse as long as 10.5 fs. All observed pulses turn out to rapidly depend on the induced dispersion, as it can be seen in the black lines in Fig. 3. Note, that this behavior is observed also experimentally by slight changes of the wedge overlap, inducing a slight chirp on the pulse. This variation of the dispersion settings results in a rapid change of the observed PAP radius. From the calculations, we can conclude that for example the pulse, which is measured at a dispersion setting of 195 fs², does not result from the stretched neighboring 5.5 fs sub-pulse, but is indeed a distinct temporal structure representing a sub-pulse with different dispersive properties. Only the pulse measured at a dispersion setting of 94 fs² with a pulse duration of 7.8 fs might result from a stretched neighboring pulse.

 

Fig. 3 The red dots show distinct short sub-pulses contained after filamentation in 780 mbar of argon at a position of the filament of 254.5 cm. The different pulses are distinguished by their different chirp-characteristics. The black lines display the calculated effects of additional dispersion upon the various sub-pulses. This excludes the possible assumption of a measurement of the same stretched pulse at different dispersion settings.

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The delayed appearance of the second sub-pulse at a propagation distance of 239 cm, as shown in Fig. 2, can be explained by a simple picture of pulse propagation along femtosecond filaments [3, 22–24]. Let us consider the initial pulse undergoing self-focusing in the gaseous medium. Right after the nonlinear focus, the leading part of the pulse generates a plasma with a free electron distribution which affects the trailing part of the initial pulse. As a consequence, the trailing part will be defocused at an off-axis position before being refocused onto the axis due to nonlinear interaction with the medium. This cyclic event, referred to as focusing cycle in the further discussion, leads to a formation of a second sub-pulse after propagation along the filament and has been predicted in recent studies [18]. During this event, the leading part of the pulse loses spectral components accumulating in the formation of the second sub-pulse [24]. The leading and the trailing sub-pulses are affected differently during their propagation along the filament. The leading part of the pulse experiences more losses and will vanish with ongoing filamentary propagation [22]. During the propagation inside the filament multiple focusing cycle may occur. As a consequence, several distinct sub-pulses can be measured after a long propagation distance inside the filament, which is observed in the experimental results presented in Fig. 3. Since the leading pulses experience more losses, we expect that in a fully propagated filament, the most trailing pulse will be measured [18].

The explanation of the distinct sub-pulses by focusing-cycles corresponds well with the picture in Fig. 2. Before a propagation of approx. 237 cm in the filament, the leading pulse experiences spectral broadening and thus, a decrease of its pulse duration, while afterwards the spectral losses become such large that the pulse duration increases again. At a distance of about 242 cm, a focusing cycle occurs with the emergence of a second sub-pulse. The multiple sub-pulses at a position of 254.5 cm (Fig. 3) indicate that the increasing spectral loss of the first sub-pulse can be explained by further focusing-cycles. Note that we were not able to track the complete picture of the sub-pulse evolution, since our measurements of the pulse duration are limited to durations below 8 fs.

4. Pulses from the fully propagated filament

The fully propagated filament is investigated by removing the pinhole P1 and flooding the SIGC as well as the subsequent chamber with argon at different pressures. The propagation distance in the argon gas covers the whole filament with an approximated length of about 30 cm, before the generated white-light is coupled out for characterization.

Figure 4(a) shows the spectrum obtained from an undisturbed filament initiated in 810 mbar argon with 0.81-mJ input pulses at a diameter of the aperture B1 of 6 mm. For the measured spectrum in Fig. 4(a), the according PAP with a mean radius r = 0.865 ± 0.132 is depicted in Fig. 4(c). The record time of the asymmetry plot for this measurement is 5 s corresponding to 15000 shots. We measure apparently slightly longer pulses for longer exposure times. This is due to intensity fluctuations inside the filament and pointing instabilities of our laser system, which affect the measurement of the pulse duration. This is confirmed by an increasing standard deviation of the measured pulse duration. The Fourier-limited pulse duration is evaluated from the given sensitivity-corrected spectrum. The temporal pulse envelope is shown in Fig. 4(b) and the Fourier-limited pulse duration is 4.18 fs. The Fourier-limit is slightly above the measured pulse duration obtained with the stereo-ATI-phasemeter and lies within the measurement error.

 

Fig. 4 Fully propagated filament in 810 mbar argon. (a) Spectrum after the fully propagated filament. (b) Temporal pulse envelope, calculated from the shown spectrum. The calculated Fourier-limited pulse duration at FWHM is 4.18 fs. (c) Parametric asymmetry plot (PAP) of sub-1.5-cycle pulses from the fully propagated filament. The radius is 0.865 and corresponds to a pulse duration of 3.8 fs.

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The data from Fig. 4(c) is evaluated via equation 1 and gives the average pulse duration of the pulses. The pulses have a temporal duration of 3.8 fs, corresponding to sub-1.5 cycles of the electric field at the central wavelength of 780 nm. The standard deviation of the measured radius in the PAP corresponds to an uncertainty of the obtained pulses duration, namely 3.8 ± 0.5 fs. Note that the CEP-dependent measurement is directly sensitive to the actual time structure of the pulse, in contrast to common spectral measurement techniques. We emphasize again, that the pulses consist of a complex temporal structure with sub-pulses as stated above. This yields a spectrum with a Fourier-limit larger than the shortest sub-pulse measured by the stereo-ATI measurement.

4.1. Chirp dependence

The re-compression settings of the fused-silica wedges do not change, even when the dispersion of the entrance pulses is varied. Figure 5 shows the retrieved pulse duration from stereo-ATI-measurements after the fully propagated filament versus a variation of the dispersion of the entrance pulses by adjusting the compressor chirp in the amplifier. Sub-5-fs pulses are obtained over a wide range of applied initial dispersions for an unchanged post-compression setup. As our measurements indicate, a change of the initial chirp-parameters does not influence the dispersive properties of the resulting pulses by a large degree. Thereby, the entrance pulses vary in their pulse duration from 35 fs to about 45 fs, which correspond to a maximum pulse duration about 28.6% larger than the minimum pulse duration. In contrast, the pulse duration after the full filament ranges between 4.2 fs and 4.9 fs, corresponding to a maximum pulse duration which is only 16.6% higher than the minimum. This leads to the assumption, that the process of filamentary propagation heals itself for the generation of ultrashort pulses, since the variation in pulse duration of the fundamental beam does not directly translate to a similar variation of the output pulse [32]. The complex spatio-temporal dynamics seem to be able to compensate for an increase of the fundamental pulse duration. We assume this to be related to the number of focusing-cycles in a fully propagated filament, meaning an increase of the input pulse duration results in an addition of focusing-cycles to the dynamics.

 

Fig. 5 Pulse duration after the undisturbed filament in dependence of the relative dispersion of the entrance pulses. The dashed curve shows a quadratic fit to the measurement.

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4.2. Pressure dependence

We varied the pressure in the SIGC in the range from 500 to 950 mbar, in order to investigate the pulse duration versus pressure. Figure 6 illustrates the pressure dependence of the spectra and the obtained pulse durations. The broadest spectra with the shortest Fourier-limits are obtained around 800 mbar. Fourier-limits below 5 fs are obtained within a large pressure range from 600 to 950 mbar. However, as it can be seen in Fig.6(b), only in a small region from 750 to 850 mbar, sub-5-fs pulses could be measured with our phasemeter setup. We note that the dispersion compensation, tuned by the fused silica wedges, remains quasi constant over this pressure range with (1.32±0.06) mm fused silica. At pressures smaller than 750 mbar or larger than 850 mbar, the pulses are significantly longer even for a different dispersion compensation, indicating strong effects of the pressure on the focusing-cycles and our obtained pulses. This means that the complex dynamics along filamentary propagation, which lead to the generation of few-cycle pulses, are very sensitive to the applied gas pressure of the femtosecond filament. This limits the optimal pressure range significantly as compared to previous experiments in a double-staged filamentation setup [34].

 

Fig. 6 Investigation of the undisturbed filament versus argon pressure. (a) Normalized spectra after filamentation versus argon pressure in the SIGC. (b) Fourier-limited pulse duration (boxes) from the spectra and measured pulse duration (circles) determined by stereo-ATI measurements versus pressure.

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5. Conclusion

In conclusion, we demonstrated the emergence of 3.8 fs pulses from a single-stage femtosecond filament. The pulse energy of the observed pulses exceeds 30 μJ, since this is the lower limit for measuring the pulse duration with the stereo ATI-phasemeter. The occurrence of sub-5-fs pulses is robust to changes of the chirp-parameters of the initial pulse, whereas the tunability of the pressure is limited to a range of 750 to 850 mbar. In addition, we were able to investigate the complex dynamics of sub-pulses along filamentary propagation and denote effects of focusing-cycles to the pulse duration of the propagating pulses. We could observe multiple sub-pulses originating from the filament and attributed their appearance to focusing-cycles, which come along with spectral losses of the initial pulse. Our findings agree well with recent theoretical models covering the propagation of femtosecond pulses in filaments [23, 33, 35]. Moreover, we observed self-healing effects during filamentary propagation, whereby the pulse duration of the resulting pulses is robust against significant changes of the pressure or the input pulse duration. Consequently, filaments deliver the foundation of attosecond pulse metrology by the generation of few-cycle pulses, as it is shown in this paper, and can also be a source of XUV-light and attosecond pulses itself. The prospect of a direct measurement of the CEP of ultrashort pulses originating from a femtosecond filament is the focus of future works, whereby the formation of intensity spikes inside the filament is predicted to be independent from the input CEP [15, 35].