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Abstract
We present an experimental method capable of capturing the complete spatio-temporal dynamics of filamenting ultrashort laser pulses. By employing spatially resolved Fourier transform spectrometry in combination with the capability to terminate the filament at any length, we can follow the nonlinear dynamics in four dimensions, i.e. the transverse domain, time and filament length. Our method thus not only enables the full characterization of the filamentation process throughout its evolution, but also allows to identify and select laser pulses with desired parameters.
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1. Introduction
Ever since the very first observation of filaments formed by intense ultrashort light pulses [1], the complex nonlinear dynamics of the filamentation process have intrigued researchers worldwide. In the simplest interpretation, femtosecond filaments are formed by a dynamic balance between self-focusing due to the Kerr effect and plasma-induced defocusing during the propagation of an intense ultrashort laser pulse inside a dielectric medium [2]. High pulse intensity and a small beam diameter can be sustained over distances much longer than the corresponding Rayleigh length, promoting substantial spectral broadening and re-shaping. Filamentation is widely used as a versatile source of radiation, including supercontinuum, few-cycle pulses as well as terahertz pulses [3–8], and for diverse applications like discharge control and environmental monitoring [9,10]. However, fine control of the filamentation process remains limited due to the intricate nonlinear interplay of the various effects involved [11–15]. Furthermore, the high intensity attained in the filament core, in excess of the ionization threshold for most materials, precludes direct observation and characterization. This hinders the possibility of finding optimal experimental conditions and complicates the identification of spatio-temporal couplings, which compromise the output beam quality and may be detrimental for applications [16,17].
Here, we present the complete 4D characterization of ultrashort pulses undergoing filamentation. Using spatially resolved Fourier transform spectrometry [18], we measure the electric field of the pulse with full 3D resolution. To gain access to the nonlinear dynamics taking place along the filament length, we terminate the filament at various points using differential pumping through a pinhole [19]. We thus demonstrate rigorous three-dimensional spatio-temporal electric field characterization along the entire length of the filament. Compared to earlier work by Alonso et al. [20], using the STARFISH technique [21], similarly to spatially resolved Fourier transform spectrometry also capable of measuring amplitude and phase in 3D, the filament termination in our work strictly ensures linear propagation after the pinhole, granting us certainty in spectral phase reconstruction. Other approaches investigated spatio-temporal dynamics on a limited propagation range [22] or focused on tracking the pulse energy along the propagation length [23,24]. Prominent studies have addressed the pulse intensity in the temporal or spectral domains, often with the addition of spatially resolving quantities such as the spectrum or the pulse duration [19,25–33]. Three-dimensional imaging techniques, such as recording the spatially resolved cross-correlation function [34] or femtosecond time-resolved optical polarigraphy (FTOP) [35], have been used to reconstruct the spatio-temporal intensity profile or the wavefront at various positions in a filament [36–39]. Complete spatio-temporal characterization, i.e. applying methods that can obtain the electric field of the pulse in space and frequency (or time) in the near-field [21,40–43], have only rarely been applied to pulses originating from a filament [20]. Moreover, in contrast to characterization techniques that capture the filament pulse in the far field, after the nonlinear propagation has finished, our technique can follow the nonlinear evolution of the pulse inside the filament and thus give important information on the dynamics and a potential direct comparison with simulations. Apart from the exact knowledge of the pulse in the measurement plane, access to the electric field allows to numerically propagate the pulse to any plane of interest.
2. Materials and methods
2.1 Experimental setup and procedure
In this work, single femtosecond filaments are generated using pulses from a Ti:sapphire-based CPA laser, operating at a central wavelength of 800 nm, with a repetition rate of 1 kHz and pulse energy of up to 5 mJ. The experimental setup is illustrated in Fig. 1. An $f={1}\;\textrm{m}$ lens, mounted on a translation stage with a 15 cm travel range, is used to focus the beam into a 1.2 m long gas tube with 0.5 mm thick, anti-reflection coated entrance and exit windows. An adjustable aperture is used to fine-tune the input energy and beam size. The gas tube is divided into two compartments by a 500 µm thick aluminium plate located closely behind the geometrical focus of the beam when the lens is positioned in the middle of the translation stage. A pinhole of around 200 µm in diameter is drilled by the focused laser beam, while the gas tube is evacuated. To generate filaments, the first compartment of the tube is filled with 1.2 bar of argon, whereas the second one is continuously pumped, maintaining a residual pressure of 10−1 mbar. The sharp pressure gradient created by the pinhole abruptly stops the filamentation process. Translating the focusing lens thus allows us to vary the position at which the filament is terminated with sub-millimeter precision. The pinhole selects the filament core, discriminating against the surrounding conical emission [2]. After passing the pressure gradient created by the pinhole, all subsequent pulse propagation is linear, which is crucial for accurate characterization and numerical backpropagation from the plane of detection in the far field to the termination pinhole. After leaving the filamentation stage, the beam is collimated with an $f={1}\;\textrm{m}$, silver-coated mirror and sent to the characterization setup. Optimum conditions for generating stable, single filaments were found empirically for an input iris diameter of 4.2 mm, corresponding to a pulse energy of 0.7 mJ. A programmable acousto-optic filter (Fastlite DAZZLER) was used to obtain transform-limited pulses with a duration of $\approx \! {22}\;\textrm{fs}$ entering the gas tube. The power corresponded to $P\approx 2.7\,P_{\text {cr}}$ for argon, where $P_\text {cr}$ is the critical power for self-focusing [2].
Fig. 1. Filamentation setup. Laser pulses are focused by a lens on a translation stage; in combination with a termination plate, this enables scanning along the length of the filament. The distance between the lens and the plate is varied across a range of $95-110$ cm.
In the present work, spatio-temporal pulse characterization is performed by employing spatially resolved Fourier transform spectrometry [18] in combination with the dispersion scan technique [44], the principle of each is illustrated in Fig. 2. In Fourier transform spectrometry, shown in Fig. 2 (a), the beam is sent into a Mach-Zehnder interferometer, where it is split with a 10:90 beamsplitter (BS) into the signal and reference arms, later recombined with a 50:50 beamsplitter. While the beam in the signal arm is not manipulated, there is an $f={2.5}\;\textrm{cm}$ off-axis parobolic mirror in the probe arm that tightly focuses the beam and generates a rapidly expanding reference wave. After recombination at the second beam splitter, the signal and reference beams are brought to interference on the chip of a CCD camera. The reference is much larger than the signal beam and has a high degree of spatial and spectral homogeneity. The size of the chip (${12}\;\textrm{mm} \times {8}\;\textrm{mm}$) was chosen to well contain the interference pattern, and the pixel size (3.6 µm) is sufficiently small to sample the relevant angular components with good resolution, essential for later numerical backpropagation to the termination plate. Scanning the delay $\tau$ between the two interferometer arms by means of a piezo stage in the probe arm (Piezosystem Jena GmbH), an interferogram $I(x,y,\tau )$ is recorded for every pixel of the CCD chip, encoding the phase difference between the signal and the reference in a spatially resolved manner. Knowledge of the spectral phase of the reference $\varphi _r(\omega )$ allows to unambiguously obtain the electric field of the pulse [18]. The reference characterization (Fig. 2(b)) is performed by means of the dispersion scan (d-scan) method, a well-established technique for ultrashort pulse characterization [44,45]. The beam is sent to the d-scan setup by inserting a flip-mirror behind the gas tube (See Fig. 1). It is essential to ensure that the part of the beam characterized by the d-scan match the one used as the reference in the spatio-temporal characterization, i.e. the spatially enlarged beam center. To facilitate the alignment, a cross-hair element is temporarily inserted into the beam immediately after the exit from the filamentation tube and special care is taken that the center of the created diffraction pattern is precisely selected in both setups. The spectral phase of the reference $\varphi _r(\omega )$ is obtained by retrieving the pulse from the recorded d-scan trace and subtracting the dispersion offset between the d-scan and the spatio-temporal characterization setup, including fused silica (FS) wedges and double chirped mirrors (DCM, Fig. 2(b)). It should be noted that both the d-scan and the spatio-temporal characterisation used in the present work are inherently averaging multiple laser shots and rely on conditions being stable for the duration of the measurement. The integration times were adapted to obtain a reasonable signal-to-noise ratio while minimizing the measurement time. We made sure that both the reference and the sample beam were stable and there was no obvious significant shot-to-shot fluctuation of the spectral phase, which would have also been visible from the recorded d-scan traces. While recording a d-scan trace took a few seconds, the measurement of the spatio-temporal pulse profile at each filament termination point took about 30 minutes. To minimize the impact of slow drift and to assess the data quality on the fly, d-scan traces of the reference pulse before and after each interferometric measurement were compared.
Fig. 2. Pulse diagnostics setup. (a) Spatially resolved Fourier transform spectrometer. The CCD sensor records an interferogram formed by the interference of a spatially expanded reference beam and the signal beam as the delay is scanned. (b) Dispersion scan setup, consisting of a pair of movable glass wedges, double chirped mirrors (DCMs, Ultrafast Innovations PC70), a thin beta-barium borate (BBO) crystal for second harmonic generation and a spectrometer (S).
2.2 Data treatment
The recorded spatio-temporal data for every chosen termination point corresponds to a spatially resolved interferogram pattern $I(x,y,\tau )$. If $\omega _0$ is the carrier frequency of the pulse to be characterized, the Fourier transform of $I(x,y,\tau )$ contains three contributions, i.e. one around $-\omega _0$, one around zero frequency and one around $+\omega _0$. Filtering a relevant spectral range around $+\omega _0$ gives $A(x,y,\omega )$ and the electric field $\tilde {U'}(x,y,\omega )$ is determined by [18]:
2.3 Supporting simulations
Nonlinear pulse propagation simulations were performed to support and reference the experimental results. The simulations are based on the numerical integration of the unidirectional pulse propagation equation (UPPE) [46] for the carrier-resolved electric field in cylindrical symmetry, as detailed in [47]. All relevant processes to describe filamentation dynamics, such as self-focusing, self-phase modulation and plasma-related effects are taken into account. The generation of free electrons is governed by the formulation derived by Perelomov, Popov and Terent’ev [48]. The linear refractive index of argon follows from Dalgarno and Kingston [49]. For the nonlinear refractive index, $n_2=1 \times 10^{-19}\text {cm}^2/\text {W}$ was used in the simulations following Nibbering et al. [50]. A transform-limited pulse, corresponding to the experimentally measured spectrum, together with the experimentally measured energy and beam size was used as input.
3. Results
3.1 Macroscopic temporal and spectral properties
The experimentally generated filaments had a length of about 20 cm, assessed by plasma luminescence. The filaments were terminated at nine positions $d$ within a total span of 15 cm, covering the range within which reliable pulse characterization was possible. The measured spectral evolution along the filament is shown in Fig. 3 (a), supplemented by the simulated spectra for direct comparison of the macroscopic dynamics in (b). The experimental spectrum exhibits clear broadening, reaching a maximum width of 575 - 925 nm for the longest measured filament. The broadening mainly takes place towards higher frequencies, indicating a significant influence of plasma-related blue-shifting [51].
Fig. 3. Spectral evolution of the beam center as a function of relative filament length d: (a) Experimental spectrum from measurements at 9 filament lengths d (The data were interpolated between measurement points for smoother plotting.), (b) Simulated spectrum.
Fig. 4 shows the on-axis temporal pulse evolution along the filament length. As seen in panel (a), for the shortest of the measured filaments, the pulse duration still is close to the one of the input pulse, indicating that the initial stage of the filamentation process was captured, before significant temporal reshaping occurred. The pulse consists of a main peak with a duration of 19 fs as well as a pre- and a post-pulse, 35 fs before and 25 fs after the main pulse, respectively. We observe the satellite pulses consistently in all measurements at all filament termination points. They are likely to originate from an optical component prior to the setup (e.g. a coating). They seem to neither be related to a non-Gaussian spectrum nor to the filamentation process itself and are not observed in the simulations, shown in panel (b). As the nonlinear dynamics evolve along the filament, the main pulse splits until the delay between its two components reaches around 20 fs, in excellent agreement with the simulations. The observed splitting is an indication of dynamic spatial replenishment [52]. This process stems from the different dynamics of the peak of the pulse and its trailing edge. The peak loses energy due to plasma formation. Due to the defocusing effect of the plasma, the self-focusing of the trailing part of the pulse requires a larger distance compared to the peak. The higher intensity of the trailing pulse is characteristic for the unfolding of this process. Governed by the low power compared to the critical power for self-focusing in the studied conditions, in combination with the low dispersion of the gaseous medium, distinct X-wave structures, otherwise characteristic for pulse splitting [53,54], were neither observed in the experiment nor in the simulations.
Fig. 4. Temporal evolution of the beam center as a function of the relative filament length d: (a) experimental data interpolated from measurements at 9 termination points, (b) simulated values. The temporal profiles in (a) have been arbitrarily aligned to the pre-pulse, as the individual temporal characterisations have no absolute reference.
The pre- and post-pulses are likely to be too weak to undergo self-driven nonlinear dynamics. The observation that the pre-pulse seems to gain power along the filament length is a normalization effect, ascribed to the fact that the collection efficiency for the pre-pulse through the pinhole varies with the position of the filament termination point with respect to the geometrical focus. The post-pulse, on the other hand, most likely experiences defocusing by the plasma generated by the main pulses. However, from pure spectral or temporal domain observations, it is not clear whether or not the pre- and post-pulse contribute to the nonlinear dynamics.
To elucidate the role of the pre-pulse, we apply time-frequency analysis by means of short-time Fourier transform. While imposing a compromise between the spectral and temporal resolutions, time-frequency analysis gives access to the spectral content of the different features in the temporal profile and is thus capable of tracing spatio-temporal couplings, and helps understand what parts of the pulse drive the nonlinear dynamics [55]. The time-frequency representation of the longest filament ($d={150}\;\textrm{mm}$) is shown in Fig. 5(b), together with the spectral and temporal projections, panels (a) and (c), respectively. While the temporal resolution is not sufficient to resolve the splitting of the main pulse, it can be clearly observed that the pre-pulse appearing at −35 fs has an unbroadened spectrum. This finding corroborates the hypothesis that the pre-pulse does not participate in the nonlinear dynamics, underlining the value of time-frequency analysis for filament dynamics investigation.
Fig. 5. Time-frequency representation of the measured pulse at the longest termination point (i.e. $d=150$ mm). Short-time Fourier transform of the on-axis field $U(x_\text {c},y_\text {c},t)$ (b). Frequency- (a) and temporal projections (c). A 34 fs long sliding Kaiser window with a shape factor 10 was used to gate the electric field of the pulse. Frequency $\omega _0=2.31$ rad/fs corresponds to the central frequency of the input spectrum.
3.2 Observation of filament dynamics in complementary domains
The complete spatio-temporal characterization of the filament pulses by the method introduced in this work offers the possibility to trace the nonlinear dynamics in different relevant domains. To illustrate the capacity of the method, we follow the evolution of the pulses at four relative filament lengths. In Fig. 6, we present the intensity of the filamenting pulse in one spatial dimension $I(x,y_c,t)$ as a function of time. Different behaviors of the pre-, main- and post-pulses, indicated by colored ticks, can be distinguished. The front of the main pulse (green tick) becomes gradually weaker, while its trailing part (red tick) continuously gains intensity and spatially contracts as the temporal splitting proceeds. The pre-pulse (blue tick) does not exhibit a significant temporal evolution, consistent with the fact that it does not participate in the nonlinear dynamics. Contrary to that, the post pulse (yellow tick) becomes gradually weaker, most likely due to plasma defocusing. It should be noted that subtle dynamics away from the optical axis, such as e.g. plasma defocusing, could not be observed in the spatial domain. We identify the size of the pinhole as the cause of this shortfall. In particular, the outer part of the filament was not intense enough to expand the laser-drilled pinhole, which thus only selected the core. Furthermore, beam-pointing fluctuations, which had in earlier work contributed to pinhole expansion [19], were absent from our experiment. Using larger pinholes would be possible, however this would require more sophisticated two- or multiple-stage differential pumping to prevent turbulent gas flow and to achieve a rapid pressure drop.
Fig. 6. Intensity in the spatio-temporal domain $I(x,y_c,t)$ at four filament termination positions. The four peaks present in the pulse in the early stages of filament dynamics (panel a) are marked by colored ticks for ease of comparison.
Fig. 7 shows the intensity $I(x,y_c,\lambda )$ in the spatio-spectral domain, for the same four termination points as in Fig. 6. The spectral broadening towards shorter wavelengths is evident. The power of the main peak, centered around 800 nm, is redistributed as the spectrum spreads. The long wavelength peak around 850 nm acquires power and grows radially. The chosen visualization can also reveal spatio-temporal couplings. One obvious trend throughout all observation points is a slight spatial shift, corresponding to about 10 % of the beam size, of one spectral component (770 nm). The emergence of the shift is evident from the white dashed curve representing the center of mass of the beam at each wavelength. While the spatio-temporal coupling itself most likely originates from the laser (e.g. imperfection of the compressor gratings) and does not result from the nonlinear interaction, the consistent observation of this small feature at all termination points underlines the sensitivity of the characterization technique.
Fig. 7. Intensity in the spatio-spectral domain $I(x,y_c,\lambda )$ at the same four filament termination positions as in Fig. 6. The white dashed lines indicate the position of the spatial center of mass vs. wavelength.
Lastly, we show the intensity $I(k_x,k_y=0,\lambda )$ in the angular spectral domain in Fig. 8. The observed angular spread exceeds the transverse angular components originating from geometrical focusing (i.e. $(k_{x,\text{max}},k_{y,\text{max}})/k_0 \approx 0.002$), indicating that the pulse indeed has undergone nonlinear dynamics in the spatial domain. The angular spread contracts for the longest plotted filament length. It should be noted that due to the finite size of the pinhole, strongly divergent portions of the beam are not correctly sampled. The spatio-temporal coupling that was already observed in the spatio-spectral domain in Fig. 6 is also evident here as a small angular shift also around 770 nm, indicating a slight deviation in the angle of propagation. It is a characteristic feature of spatio-temporal couplings that angular and spatial chirp occur together [16].
Fig. 8. Intensity $I(k_x,k_y=0,\lambda )$ in the angular-spectral domain at the same four filament termination positions as in Fig. 6. The white dashed lines indicate the position of the angular center of mass vs. wavelength.
3.3 Three-dimensional view
The presented method provides all the information for a complete 3D reconstruction of the pulse for every filament termination point. As an example, we present the pulse from the longest measured filament ($d=150$ mm), resolved in space and time, $I(x,y,t)$, in Fig. 9. The splitting of the main pulse as a result of the nonlinear dynamics is evident. Also, the pre-pulse can be clearly identified. In addition, using the 3D information, the pulse can be numerically propagated to any plane of interest, and macroscopic beam parameters, such as the focusability (3D Strehl ratio), can be obtained [56].
Fig. 9. 3D intensity profile $I(x,y,t)$ for the pulses originating from the last termination point ($d=150$ mm), corresponding to the longest filament. Isosurfaces are rendered at 25, 50, and $75\%$ of the maximum intensity.
While a 3D view may be instructive for observing macroscopic features, local inhomogeneities in the beam profile can easily be overlooked. In Fig. 10, we investigate the pulse structure across the beam profile for the longest measured filament ($d={150}\;\textrm{mm}$). Using time-frequency representation for different $(x,y)$-positions can reveal small differences. An interesting feature to notice is that the pulse is rather homogeneous along the y-direction (panels (b), (d) and (f)), while in the x-direction (panels (c), (d) and (e)) the spectro-temporal structure varies more strongly across the beam profile, most evidently the energy sharing between the pre- and the main pulses. This is consistent with the observation of the angular deviation along the x-axis, shown in Fig. 7 and 8.
Fig. 10. Time-frequency analysis at different spatial positions for the longest measured filament ($d={150}\;\textrm{mm}$). (a) transverse beam profile (intensity integrated over time); (b-f) time-frequency representation of the pulse at different positions within the beam, indicated by the colored points. Each panel is normalized to unity.
4. Conclusion
We have presented a new technique for complete 4D characterization of filamenting ultrashort pulses in gases by combining spatially resolved Fourier transform spectrometry with filament termination. In contrast to approaches based on intensity measurements, we explicitly measure the electric field in space and time. This spatially resolved electric field characterization opens the possibility for thorough investigation of complicated filament dynamics, including spatio-temporal couplings, as well as for numerical propagation of the pulses to any plane of interest, e.g. to the interaction point in an application.
We have shown that the presented characterization method can reveal basic filament properties, such as spectral broadening, blue-shifting as well as pulse-shortening and splitting, all of which can be reproduced by numerical simulations. Furthermore, insights into the complex spatio-temporal filament dynamics can be obtained via an analysis of the filament pulse in 2D or even in complete 3D representations. We have illustrated the potential of combining spatio-temporal characterization and time-frequency analysis to understand the contributions of different constituents of the pulse to the nonlinear dynamics and identify inhomogeneities of the pulse parameters across the transverse beam profile.
Our spatio-temporal pulse characterization is universal and can be adapted to studies of a large range of nonlinear phenomena. In order to follow the build-up of a specific process, it needs to be combined with an adequate termination procedure. For filamentation in non-gaseous media, termination can be achieved straightforwardly by varying the sample length or alternatively by using a variable-length cell for liquid samples [37] or a wedge-shaped solid sample [57,58]. For gas filamentation, as studied in the present work, more sophisticated two- or multi-stage differential pumping schemes could help to avoid adverse effects that arise from a small-sized pinhole. This would allow the detection of off-axis structures further away from the beam center than the edge of the pinhole (for example, ring structures arising upon intensity saturation and aberrational defocusing of the pulse tail induced by the laser plasma, observed in previous studies [59,60]), which might have been present yet remained undetected in the present work. The completeness and adaptability of our method suggest its high potential, not only in future investigations of nonlinear dynamics, but also for experiments demanding precise characterization and extraction of filament pulses with particular qualities.
Funding
European Regional Development Fund (GINOP-2.3.6-15-2015-00001); Deutsche Forschungsgemeinschaft (EXC2122 390833453, EXC-2123-A05 390837967, QUTIF IV 152/7-1); Laserlab-Europe (654148); Vetenskapsrådet (2016-04907).
Acknowledgments
The ELI-ALPS project (GINOP-2.3.6-15-2015-00001) is supported by the European Union and co-financed by the European Regional Development Fund.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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