We report on the direct experimental observation of pulse-splitting dynamics along a femtosecond filament. The fundamental pulse experiences a significant self-shortening during the propagation leading to pulse durations of 5.3 fs, corresponding to sub-3 cycles, which is measured without external pulse compression. A compression factor of eight could be achieved in a single filamentary stage. Theoretical modeling of the fundamental pulse propagation confirms our observed pulse structures and durations and gives further insight into the nonlinear dynamics during filamentation.
© 2014 Optical Society of America
The generation of few-cycle pulses is the foundation for a growing number of experiments in strong-field and attosecond physics [1–3]. Ultashort pulses are commonly generated in hollow-core fibers or filaments, where nonlinear effects, such as self-phase modulation (SPM), broaden the spectrum and the obtained pulse is temporally compressed by dispersion compensating optics .
The self-guiding mechanism in filaments overcomes the need for external guiding setups. Instead, dynamic self-guiding results from a balance of nonlinear effects such as Kerr self-focusing and plasma defocusing. In this case, a filament is created, i.e., a narrow channel of light and free electrons, which is much longer than the typical diffraction length in a linear focal geometry . Filamentation in gaseous media involves a variety of nonlinear processes leading to the generation of new optical frequencies ranging from the THz regime [6, 7] to the ultraviolet [8, 9] and even the extreme ultraviolet (XUV) spectral domain .
Besides SPM, effects such as self-steepening and ionization-induced blueshift influence the evolution of the fundamental spectrum in femtosecond filaments [11–14]. The pulse inside the filament undergoes a complex spatio-temporal evolution as it evolves through several defocusing and refocusing cycles, leading to a splitting of the fundamental pulse into a structure of multiple sub-pulses with individual pulse durations and chirp characteristics [15–17]. These phenomena are a general feature of filamentation, being observed in different media such as gases, liquids and solids [18–20]. It has been demonstrated, that the resulting pulses can be compressed to pulse durations in the few-cycle regime [21,22]. The chirp is usually compensated by chirped-mirrors and fused-silica wedges after coupling the beam out of the filamentation cell. Alternatively, the accumulated chirp can also be compensated by means of self-compression inside the filament itself [23, 24].
Early experimental setups relied on delivering the spectrally broadened and self-compressed pulse through an output window and a subsequent propagation stage in air to the pulse diagnostics setups [25–27]. Consequently, the chirp induced behind the filament had to be either taken into account theoretically or removed experimentally by a suitable dispersion compensating setup. Unfortunately, the components used for chirp management can imprint phase-distortions on the fundamental pulse and influence the measurements . Furthermore, a deduction from the experimentally measured pulses after a fully propagated filament to pulse dynamics inside the filament by a comparison with simulations strongly depends on the quality of the theoretical model. Therefore, a measurement directly inside the filament is an optimal approach to study the complex dynamics occurring along filamentary propagation and can contribute to the understanding of the nonlinear interaction along the filament.
Recent experiments aimed at the direct investigation of the fundamental pulses, using different approaches to be able to measure the fundamental pulse shape in the high-intensity region along the filaments core. Pinhole setups have been used to cut the filament, enabling the observation of pulse-splitting dynamics along the filament [29, 30]. Impulsive vibrational Raman scattering can be used to draw conclusions on the self-compression of the fundamental beam in molecular gases . A TG-XFROG setup and the STARFISH-method are able to probe the pulses directly in the filament and have shown self-compression of the fundamental pulses up to a factor of four [32–34]. Also, a tomographic method has been developed to probe light bullets during their evolution in fused silica . The observation of high-order harmonic radiation, generated directly in the filament, indicates the occurrence of ultrashort pulses close to the single-cycle regime since a continuous spectrum of high-order harmonic radiation has been observed . However, a detailed scan of the pulse evolution along the filament remained an open task for experimentalists.
In this paper, we report on the self-shortening of 45 fs pulses to a duration below 5.5 fs in a single femtosecond filament stage without using pulse compression elements. The pulse-shapes and durations are tracked with centimeter resolution along the propagation direction of the filament, having an excellent agreement with theoretical results. We are able to localize the point at which the fundamental pulse breaks into two temporally separated parts and relate it to its spectral evolution.
2. Experimental setup
The experimental arrangement is shown in Fig. 1. A Ti:Sa amplifier system delivers 35-fs pulses with 2.5 mJ pulse energy at a central wavelength of 780 nm. An iris and attenuator decrease the pulse energy to 0.9 mJ in order to avoid multiple filamentation.
The pulses are focused via a curved mirror (f=2 m) into a semi-infinite gas cell (SIGC), filled with argon at a pressure of 1000 mbar to ignite a filament. The SIGC is closed by differential pumping stage incorporating two sequential metal plates, which truncate the propagation of the filament at a certain position . The first pinhole has an inner diameter of approximately 500 μm, leaving the on-axis region of the filamentary beam undisturbed for further propagation. The second pinhole, located 1 cm after the first one, has a diameter of 150 μm, cutting away all remaining off-axis components, which can influence the pulse during its subsequent propagation [36, 37]. Translating the focus in the gas cell by varying the path length between the focusing mirror and the cell entrance, moves the point at which the filament starts in the gas cell. This allows the extraction of the fundamental pulse at different positions along the filament. The size of the pinholes remains the same for all measuring positions along the propagation direction of the filament. Note, that the labeling of positions in this paper refer to total distances from the curved mirror to the truncating pinhole. The distance to the pinhole has been measured at an intermediate position of the translation stage and is determined to be 205 cm. This is close to the linear focus of 200 cm. The translation stage is able to move the fundamental beam by ±15 cm from its initial position, meaning our total scanning range covers a distance from 190 cm up to 220 cm, ensuring that most of the filaments length is covered within. The input pulses are slightly positively chirped to a duration of approx. 45 fs, which yields a maximal spectral broadening in the filament.
The data of the pulse evolution is recorded in two steps. First, the spectrum is recorded in dependence of the filament’s propagation length. The spectrometer (Avantes) allows a quick scanning of the spectrum along the filament and it has been placed directly after a 17.5° fused-silica wedge W1, whose reflection has been used for coupling out the white-light beam after the filament. The recorded spectra are used for comparison with the measured spectra after the SHG-FROG device.
Second, a SHG-FROG device is placed inside the subsequent vacuum-setup in order to analyze the emerging pulses from different positions along the filament, enabling the observation of the pulse duration without any perturbation by propagation through air or an exit window. Therefore, no chirp management is needed for the sake of dispersion-compensation and the observed pulses are the self-compressed pulses extracted at the position of the laser drilled pinholes. The emerging fundamental beam is guided and attenuated by the wedge W1 and collimated by a spherical silver-mirror (R=−4000 mm) and coupled into the FROG-setup. The FROG device is located 1.75 m after the truncating pinhole, allowing the beam to spatially expand, so that only radiation from the on-axis region of the filament is coupled into the pulse detection setup. We ensured to avoid the detection of off-axis components, since they are known to have differing pulse-characteristics than the on-axis radiation . A mask made of two holes with a diameter of 0.5 mm and a separation of 2 mm cuts out two spatially separated copies of the central part of the beam profile. These beams are reflected by a curved split-mirror, allowing for a precise control of the temporal delay between the two pulse replicas via a piezo-actuated translation stage. The two beams are focused non-collinearly on a 12 μm thick Beta-Bariumborat-crystal to generate the second-harmonic signal. The signal is recorded with a spectrometer (HR4000, OceanOptics) as a function of the temporal delay, generating SHG-FROG traces for pulse reconstruction. Also, the fundamental spectrum after the SHG-FROG device has been taken after each trace for the purpose of marginal-correction of the traces. The traces are recorded in step sizes of 1 cm along the filamentary propagation direction.
3. Experimental and theoretical results
In this paper, our experimental results are directly compared to simulations. For the theoretical modeling of the pulse propagation, numerical simulations of the generalized nonlinear Schrödinger equation are performed [11, 23, 27]. The evolution of the electric field envelope E is governed by the propagation equation
Here, I = |E|2 is the cycle-averaged field intensity. Space-time focusing and self-steepening are introduced by the operator T = 1 + (i/ω0)∂t, while the operator T−1 is evaluated in the Fourier domain. Correspondingly, the operator D modeling dispersion in argon is treated in the Fourier domain according to , where k(ω) = n(ω)ω/c is the wavenumber of the angular frequency ω, and n(ω) is the frequency-dependent refractive index in argon according to Dalgarno and Kingston . The nonlinear refractive index is denoted as n2 = 1 × 10−19 cm2/W and Ui corresponds to the ionization potential of the medium . Moreover, ρc = 1.73 × 1021 cm−3 is the critical plasma density at ω0, and ρnt denotes the neutral density of the medium. σ = 1 × 10−19 cm2 is the cross section for collisional ionization. The ionization rate W(I) is modeled according to Perelomov-Popov-Terentev (PPT) and adequately describes both multiphoton and tunneling ionization processes .
The parameters chosen for simulating the pulse propagation correspond to a focal length of 200 cm, a pulse duration of 35 fs (FL) and a peak-power of 30.1 GW, which is comparable to our experimental parameters. The initial radial pulse profile was clipped by a diaphragm with 7 mm diameter, which was modeled in theory by multiplying the initial spatial profile with a N=32 super-gaussian exp(−(2r/d)32). The evolution of the spectrum has also been modeled for the case of a chirped input pulse, showing the same spectral characteristics as the Fourier-limited input pulse.
Figures 2(a) and 2(b) show the evolution of the fundamental spectrum in dependence of the filament length. The central frequency of the spectra is plotted in red on top of the according spectra. The left hand side of the figure corresponds to the experimental results, whereas the right hand side refers to theoretical simulations. Experimentally, the spectrum has been detected using the surface reflection of wedge W1. The spectrum continuously broadens with ongoing propagation length of the filament. The broadening rapidly sets in around a position of 205 cm leading to a maximum spectral width (at 1/e2) of 267 nm at a position of 212 cm. In contrast to spectra which are purely influenced by self-phase modulation, the observed spectral broadening is strongly asymmetric, favoring the blue spectral components. Consequently the central frequency decreases with ongoing propagation to 694 nm in the experiment.
The simulations in Fig. 2(b) show a well agreeing spectral behavior along filamentary propagation. Here, the spectral broadening increases strongly between positions 195 and 205 cm. The maximal spectral width at 1/e2-level is determined to be 252 nm at a position of 214 nm and the central frequency has a minimum of 735 nm at a position of 211 cm.
Figures 2(c) and 2(d) show the corresponding evolution of the Fourier-limited pulse duration in dependence of the propagation length. Both, experiment and simulation, show a steady decline in the pulse duration and a saturation after an approximated distance of 205 cm, as it can be seen in the inlets. At distances between 205 and 220 cm, the Fourier-limited pulse duration drops below 5 fs and levels at 2 fs around a position of 207 cm.
Using the SHG-FROG device allows us to complement the spectral evolution with the evolution of the pulse shape in the temporal domain. Connecting the results of both measurements gives a detailed insight into the nonlinear dynamics during filamentation. Figure 3(a) shows the spectral evolution measured after the SHG-FROG device. The overall behavior of the spectrum in dependence of the propagation length is not altered. However, the spectral width decreases due to the change of the detection position. It can be derived, that the measured pulse looses spectral components due to the beam steering into the pulse detection setup. The maximum spectral width decreases from initially 268 nm to 165 nm. The red-dotted lines in panel (a) show three exemplary positions from which the recorded FROG-traces and their reconstructed pulses are depicted in panels (b)–(d). For each position, the measured and reconstructed traces are shown together with the reconstructed spectrum and pulse shape. The pulse envelope becomes more complex and structured with ongoing propagation along the filament. A narrow feature in the temporal shape is measured at later positions of filamentary propagation. It can be deducted that the increasing complexity of the pulse shapes with ongoing propagation direction also leads to an increase in spectral features in the SHG-FROG traces, as it can be seen in the traces shown in Figs. 3(b) and 3(d). The SHG-FROG traces are symmetric to the temporal axis leading to a ambiguity in the temporal direction of the pulse. The pulses at later positions along the filament are interpreted to consist of a weak leading pulse and a sharp and intense trailing part. This temporal direction is chosen because the sharp feature of the propagating pulse is confined to the trailing part of the pulse [11, 41]. The trailing part can shorten down to durations around 5.3 fs at a position of 212 cm, which corresponds to sub-3 cycles at the related spectral center-of-gravity (730 nm) at this position in the filament. The compression from initially 45 fs pulses down to 5.3 fs equals a factor of 8.5.
Figure 4 shows a detailed evolution of the experimentally reconstructed and simulated fundamental pulse shapes along the filament and the corresponding pulse duration (FWHM) for each pulse.
In the experiment (c.f. Fig. 4(a)), the pulse is measured in steps of 1 cm from a position of 197 cm to a position of 220 cm along the filament. The temporal direction of the reconstructed pulse shapes are adapted to our interpretation above. A splitting of the pulses into two separate parts occurs at a distance of approximately 202 cm. At this position, a jump of the peak along the temporal pulse profile towards the trailing edge is observed. The maximum shifts from −1 fs at position 200 cm to 15 fs at position 203 cm. From this point on, two temporally separated pulses co-propagate with the leading pulse being the weaker one and the trailing pulse the stronger and sharper pulse feature. The temporal splitting increases with ongoing propagation in the filament, leading to a peak to peak separation of 41 fs at position 220 cm.
The corresponding simulations (c.f. Fig. 4(b)) are in good agreement with the measured evolution of the pulse. Note, that the range over which the propagation of the pulse is modeled exceeds the experimentally covered range. Each calculated pulse-shape along the propagation direction of the filament has been normalized in order to enable a direct comparison with the experimentally retrieved pulses. The splitting of the fundamental pulse into two temporally separated pulses can be clearly observed at a position of 197 cm. At this point, the pulse maximum rapidly jumps from −4.1 fs to 12.7 fs along the temporal axis. The emerging pulses temporally split further with ongoing propagation to a pulse separation of 45 fs at 220 cm.
Figures 4(c) and 4(d) show the pulse duration at FWHM for the experimentally retrieved and simulated pulse-shapes. The pulse duration of the reconstructed pulse-shapes decreases with ongoing propagation distance and decreases rapidly between positions 200 cm and 205 cm. The pulse shortens below 6 fs between positions 209 cm and 215 cm, reaching its minimum of 5.3 fs at 212 cm. The results of the theoretical modeling show a similar behavior (Fig. 4(d)). The increased range of the simulations first show a steady decrease of the pulse duration at propagation distances between 180 and 192 cm to 20 fs, which are not covered by the experimental observations. Afterwards, the length of the pulse increases up to a position of 197 cm. This is followed by a region with a moderate shortening until a propagation distance of 200.6 cm, where the pulse duration drastically drops from 25.9 fs to 8.8 fs within a few millimeters (≈10 mm) of propagation. From there on, the pulse shortens to 4 fs at 214.5 cm.
Comparing key-features of the simulations and the experimental results in Table 1, we find a very good agreement between the evolution of both spectra and the fundamental pulse shapes. Consequently, the simulations give an adequate insight to the nonlinear dynamics along the experimentally investigated filament.
Previous works indicate, that the free-space propagation of the beam behind the filament can considerably change the observed pulse dynamics due to spatial effects . Therefore, we systematically investigated numerically the effect of free propagation on the observed pulse shapes and found that the propagation of the full beam-profile in vacuum can indeed alter the measurement of the beam profile considerably. However, if the beam-profile is directly cut by an aperture, as it is the case for our setup, the propagation effect can be drastically reduced. The size of the second pinhole effectively cuts away the spatio-temporal inhomogeneous off-axis part of the beam profile and only selects a confined area of the on-axis components. This ensures that the on axis temporal profile at the pinhole is transmitted into the far field nearly unchanged. Figure 5 shows the calculated near-field pulse-shape extracted at a propagation distance of 220 cm in comparison to the far-field (z=∞) intensity profile, which has propagated through a setup consisting of a 425 μm and a 150 μm pinhole. Thanks to the small diameter of the second pinhole, the effect of subsequent propagation can be neglected in the further discussion.
At the early stage of propagation, in between positions 190 cm and 195 cm, the spectrum broadens symmetrically and its COG changes by less than 5 nm for the experimental as well as the simulated results (comp. Fig. 2). Therefore, the spectral evolution is dominantly influenced by self-phase modulation. Further propagation leads to an increasingly unbalanced spectrum, indicating that additional nonlinear effects start to influence the pulse dynamics.
The investigated evolution of the pulse-shape serves as an explanation for the progressive blueshift of the spectrum. At a position of 197 cm/200 cm (sim./exp.), a second pulse structure occurs on-axis. This pulse originates from a plasma defocusing-refocusing cycle . The leading part of the initial pulse ionizes the guiding medium and generates free-electrons through multi-photon ionization (MPI). These electrons act as a defocusing lens for the trailing part of the pulse, guiding it off-axis. At a later position along the propagation direction, this pulse can again be refocused and re-appear on-axis. This evolution involves a spatio-temporal refocusing, meaning that the pulse is not only refocused spatially on the optical axis, but also is confined to a short time interval [11, 12].
After the splitting, the pulse shape consists of a weak leading pulse, which is quickly depleted due to MPI, and a second, trailing pulse with a sharp feature in its trailing edge, as it can be seen for propagation distances larger than 205 cm (comp. Fig. 4). Owing to self-steepening , the slope dI/dt of the intensity distribution in the trailing edge further increases along the filament axis, which leads to an increasing spectral blueshift of the generated spectrum, which sets in at z=205 cm, approximately 5 cm after the splitting, and ceases at z=218 cm.
The pulse dynamics not only leaves fingerprints in the spectrum originating from the filament, but also on the pulse duration along the propagation direction. Following the simulated propagation of the pulse shapes and their related FWHM-durations in Fig. 4(b) and 4(d), it can be observed, that the pulse duration initially decreases slightly up to 187 cm and increases afterwards up to a local maximum at the point where the pulse splits into two parts (197 cm). The initial decrease observed in the simulations can be explained by self-steepening of the fundamental pulse. In addition the defocusing of the trailing part into the off-axis region of the filament starts to influence the pulses shape. On-axis of the filament, this leads to an effective pulse shortening. After the minimum at 187 cm up to a position of 197 cm, the pulse duration increases due to off-axis radiation which is refocused towards the optical axis. This leads to a strongly increasing pedestal at the trailing part of the pulse between positions 187 cm and 197 cm.
Simultaneously, a spectral redshift and the corresponding increase of the group velocity push the leading pulse towards negative delays. In fact, the spectral redshift is a consequence of plasma defocusing, which, while acting in the trailing part of the pulse, also results in a steep-ening of the leading pulse front and a corresponding SPM induced redshift . After the local maximum at position 197 cm, the pulses begin to separate from each other temporally and the intensity of the leading part decreases strongly, meaning that the pulse duration at FWHM decreases with ongoing propagation. This leads to the steep drop-off at position 200.6 cm, shown in Fig. 4(d). At a position of 202 cm, the trailing part isolates completely from the leading pulse feature and dominates the ongoing FWHM pulse duration since the leading parts intensity decreases rapidly. Concluding, the dominant effect leading to the direct generation of ultrashort pulses is the spatio-temporal refocusing of the leading part of the pulse during nonlinear propagation. Self-compression due to plasma-effects in the guiding medium appears to be the sub-dominant contributor to the rapid onset of the fundamental pulse-shortening. Interestingly, the shortest pulse-duration is not obtained at the point of pulse separation at 202 cm, but rather 10 cm later at a 212 cm. This is due to the already mentioned fact, that the trailing pulse-feature is strongly influenced by the effect of self-steepening and to some extent by self-compression due to free electrons, leading to a steady decrease of the pulse duration up to a position of 212 cm.
The slight difference between the simulated and experimentally obtained pulse durations can be explained by the experimental setup. The spectral losses, induced by the beam-steering into the SHG-FROG device, can be accounted for the measurement of longer pulse durations in the experiment, compared to the simulations.
The rapid occurrence of the sharp trailing pulse on the optical axis agrees with previous simulations and experiments concerning intensity spikes and high-order harmonic generation along femtosecond filaments [10,17,24,44]. The findings of a self-shortening, dominated by the spatio-temporal dynamics rather than the negative group-velocity dispersion of free electrons also coincide with previous theoretical publications [16, 45, 46].
In conclusion, we are able to directly measure ultrashort self-compressed pulses of 5.3 fs duration, emerging from a truncated filament. The maximal compression factor is estimated to be close to eight. The dispersion-free setup enables a direct observation of the evolution of the spectrum as well as the pulse shapes and durations along the propagation direction of a femtosecond filament. Using a double-staged pinhole-setup plays an important role in the comparability between experiments and direct simulations, because it strongly reduces the effects of post-filament vacuum propagation on the observed pulse shapes.
Comparing the experimental results with simulations support our measured pulse-features as well as durations and help us to connect the onset of rapid spectral broadening with the splitting of the fundamental pulse and subsequent self-steepening. The shortening of the pulse is dominated by the spatio-temporal dynamics along filamentary propagation and to a lesser extend influenced by the negative group-velocity dispersion of the generated plasma.
The authors would like to acknowledge the funding by Deutsche Forschungsgemeinschaft under grant number KO 3798/3-1, funding from the Cluster of Excellence QUEST, Centre for Quantum Engineering and Space-Time Research and support by the Open Access Publishing Fund of Leibniz Universität Hannover.
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