Open Access
Add to BibSonomyAbstract
The spectral evolution of a high-intensity light channel formed by filamentation is investigated in a detailed experimental study. We also track the spatio-temporal dynamics by high-order harmonic generation along the filament. Both the spectral and temporal diagnostics are performed as a function of propagation distance, by extracting the light pulses directly from the hot filament core into vacuum via pinholes that terminate the nonlinear propagation. We compare the measured spectral shapes to simulations and analyze numerically the temporal dynamics inside the filament.
©2011 Optical Society of America
1. Introduction
Filamentation of ultrashort laser pulses has become a versatile tool for pulse compression in the few-cycle regime [1, 2], for strong-field physics [3, 4], atmospheric applications [5–8], high aspect ratio micromachining [9–11], investigation of Hawking radiation [12, 13], and generation of THz radiation [14–16]. It denotes a nonlinear pulse propagation regime with very long interaction between a medium and an intense laser pulse forming a dynamic structure with an intense narrow core sustained without the help of any external guiding mechanism [17–19]. An interplay between nonlinear effects including Kerr self-focusing, nonlinear absorption of light, and de-focusing due to free electrons, preserves this regime over distances much longer than the typical diffraction length [20]. Filamentation is notably featured by a dynamical reshaping of the laser pulse in space and time, which leads to the generation of new frequencies by self-phase modulation (SPM) ranging from the ultraviolet into the infrared (IR) spectral region. This spectral-broadening effect is used for the temporal compression of multi-cycle driver pulses to the few-cycle regime [1, 18, 21].
Experimental investigations of the filamentation process are in most cases based on the characterization of the output pulses in the far field after the nonlinear interaction with the medium. Various experimental studies have been performed which investigate for example the influence of the pressure of the nonlinear medium and the chirp of the fundamental field on the filamentation process with regard to the spectral broadening and the pulse duration of the output pulses [3, 22–24]. Inside the filament, however, the spatio-temporal dynamics of the pulses are complicated and are known so far only from complex numerical calculations [19,25]. To verify such theoretical predictions, experimental studies have to be performed which are able to track the evolution of the filamentation process in space and time. New diagnostic techniques have been developed to monitor the amplitude and phase of complex pulses [26]. Because of the high intensity in the hot narrow core of the filament, it is impossible to put the diagnostics directly in the filamentation region. A few approaches exist to extract the pulses and spectral shapes from certain positions within a filament. In Refs. [27–29], a time resolved shadowgraphic technique led to the retrieval of the pulse dynamics in a filament from space and time frames for the refractive index and absorption. In Ref. [30], the filamentation process is stopped by a movable aperture in the filamentation tube guiding the beam further in argon. In Ref. [31], a filament in air is tracked by using impulsive vibrational Raman scattering and by measurement of power spectra along the propagation axis. These methods are either indirect methods, or do not guarantee undisturbed pulses due to further propagation in the gas medium after the extraction. To the best of our knowledge, an experimental investigation of spectral and temporal dynamics during filamentation versus propagation has never been demonstrated.
In this paper, we present a novel concept for analyzing pulses directly from different positions within a filament. We have applied two different types of diagnostics. First, the IR spectra from the filament are analyzed versus the position of extraction and serve as a linear diagnostics. Second, high-order harmonics which are generated by intensity spikes inside the filament [32, 33] serve as a highly nonlinear diagnostics. Both techniques rely on the semi-infinite gas cell (SIGC) geometry [34, 35] using a double differential pumping scheme for realizing high pressure gradients. Atmospheric pressure in the SIGC enables filamentation, while a steep transition to vacuum ensures the abrupt ending of all nonlinear effects to the pulse. By scanning the vacuum transition through the filament, both IR pulses and high-order harmonics from different positions in the nonlinear interaction region are extracted and analyzed. Using the linear diagnostics, we investigate the spectral broadening of the IR pulses versus propagation that gives information about the spectral reshaping of the time- and space-integrated pulse during the filamentation process. The nonlinear diagnostics using high-order harmonics serves to track the local intensity inside the filament and to estimate the pulse duration of the pulses which give rise to high-order harmonic generation (HHG). By implementing these methods for the spatially resolved analysis of the filament along its propagation axis, detailed comparisons of experiments and simulations of the direct measurements of spectral and temporal dynamics within a filament allow us to confirm theoretical predictions.
2. Experimental setup
A chirped-pulse-amplification system (Dragon, KM-Labs Inc.) delivers 35-fs-pulses centered at 780 nm with energies of 1.3 mJ at 3 kHz repetition rate (see Fig. 1). The beam quality is measured to have an M2 of 1.4. An aperture with diameter of 7.5 mm is placed behind the amplifier and transmits about 80% of the power. This corresponds to ∼ 27 GW of peak power. The 1.0 mJ pulses are then focused with a concave silver mirror of 2 m focal length through a 1-mm-thick entrance window (CaF2) into a 1-m-long SIGC filled with argon at different pressures. The entrance window is placed far enough from the focus region to avoid nonlinear effect in the material and in the air before the cell. Because of the slight divergence of the beam the linear focus position is shifted to about 2.4 m. In order to initiate a filament in the SIGC the critical power must be exceeded [36], which is reached for our laser parameters at a pressure p above 250 mbar, estimated with the nonlinear index coefficient n 2 = 1.74 × 10−19cm2/W in 1 atm argon [37, 38]. At the end of the SIGC, an abrupt transition to vacuum is realized by a laser-drilled pinhole (diameter ∼ 700μm) in a metal plate. A second laser-drilled pinhole (diameter ∼ 500μm) is installed at 1 cm distance from the first one. The combination of both pinholes establish a double differential pumping stage with a pressure gradient from atmospheric pressure in the SIGC to < 5 × 10−4 mbar in the vacuum chamber. In the intermediate chamber, the gas is diluted below 10−1 mbar, measured in a distance of about 10 cm from the interaction region. Thus, nonlinear effects on the pulses after the extraction can be neglected. A delay line between the entrance window and the focusing mirror allows for displacement of the whole filament over a range of 30 cm. While moving the filament with respect to the exit pinhole of the SIGC, it is truncated at different lengths and pulses from different evolution stages of the filament are extracted. The pulses propagate 1 m in vacuum before they are transmitted through an exit window (2 mm CaF2) to avoid nonlinear effects in the material. The scattered light is recorded through a multi-mode fiber by a spectrometer (Avantes, Ava-Spec-2048-SPU, 273-1100 nm). For the analysis of high-order harmonics, the exit window is replaced by further propagation in vacuum to an extreme ultraviolet (XUV) spectrometer (McPherson, with 300 lines/mm and CCD DH420A-F0, ANDOR Technology) passing a 200-nm-thick aluminum foil [33].
Fig. 1 (Color online) Experimental setup. Pulses from a chirped-pulse amplifier are focused by a curved mirror (CM) into a semi-infinite gas cell (SIGC) filled with argon. The filament formed in the argon gas is truncated by a laser-drilled pinhole with subsequent propagation in vacuum. The generated white-light is coupled out with an exit window for spectral analysis.
3. Numerical methods
We performed simulations based on an unidirectional propagation equation along the z-direction for the envelope ℰ(r,z,t) of the laser field E(r,z,t) = [ℰ(r,z,t)exp(−iω 0 τ + ik 0 z) + c.c.]/2, with frequency components (r,z,ω). The laser pulse propagates in argon, featured by its dispersion relation k(ω) [39]. Therefore, we write the propagation equations in the frame of the local time t ≡ τ – z/vg with τ the time in the frame of the laboratory and the group velocity of the pulse. The propagation of the envelope is governed by:
where the frequency dependent operator 𝒦(ω) = k 0 + k′ 0(ω – ω 0), and Kerr(r,z,ω), (r,z,ω) denote the Fourier transformed nonlinear polarization and current envelopes, respectively. The nonlinear polarization describes the optical Kerr response where n 2|ℰ|2 is the nonlinear (dimensionless) Kerr index, with n 2 = 1.74 × 10−19 cm2/W for argon at 1 atm [37, 38]. The current envelope comprises two terms: 𝒥(r,z,t) = 𝒥Plasma(r,z,t) + 𝒥NLL(r,z,t), where 𝒥Plasma accounts for plasma induced defocusing and absorption, and 𝒥NLL for nonlinear losses associated with optical field ionization. The coupling with the electron plasma of density Ne(r,z,t), generated by optical field ionization, is described by evolution equations for 𝒥Plasma and for the density of argon atoms N Ar(r,z,t), from which the electron density Ne(r,z,t) is obtained by charge conservation Ne(r,z,t) = N 0 – N Ar(r,z,t): where e and me denote the electron charge and mass, τc = 190/p fs is the collision time in argon, and The current for nonlinear losses reads as where Ui denotes the ionization potential of argon and W(|ℰ|) the intensity dependent ionization rate. We finally model the field dependent ionization rate W(|ℰ|) according to the Keldysh formulation [40] later revisited by Perelomov et al. [41], Ammosov et al. [42] and Ilkov et al. [43] with detailed formulas given in Ref. [18].The propagation equation is solved by calculating the nonlinear polarization and currents in the temporal domain at each step along the propagation axis. These sources are then Fourier-transformed to the frequency domain and used to propagate the frequency components of the laser pulse to the next step, before the laser field is Fourier-transformed back into the temporal domain.
Our calculations start with a 30 fs (FWHM), 780 nm Gaussian laser pulse. The input pulse envelope thus reads
where the peak electric field ℰ0 corresponds to an initial energy of 0.78 mJ and . The focusing conditions reproduce the experimental values with zin = 154 cm, f 0 = 150 cm and w 0 = 1.21 mm. The origin of the propagation axis z = 0 coincides with the position of the curved mirror CM as in the experiments.4. Results
4.1. Results: spectral analysis
In our experiment, a filament is initiated in the SIGC and is scanned with respect to the exit pinhole. Pulses are extracted at different positions along the propagation axis and are spectrally analyzed. Figure 2 shows recorded pulse spectra integrated over 50 ms at three positions within the filament in 1000 mbar argon. The distances are measured with respect to the focusing mirror. All recorded spectra are corrected by the spectrometer sensitivity which is high in the blue spectral region and decreases in the IR. We observe that the spectral bandwidth of the pulses grows with increasing propagation length. The spectrum at extraction position z = 227 cm has a full-width Δλ of Δλ = 120 nm at − 20 dB, i.e. at 1 % of the maximum. After z = 253 cm of propagation the spectrum spans over more than one octave (Δλ = 434 nm). The shape of the spectrum extracted at the shortest propagation distance (z = 227 cm) exhibits the typical symmetrical structure from SPM. With increasing propagation length, the spectra become more and more asymmetric and shift into the blue spectral region due to the formation of free electrons [44]. The full spectral evolution is shown in Figs. 3(a)–3(e). For five different pressures, a spectral map is recorded by extracting pulse spectra in steps of 4 mm along the propagation axis with an integration time of 50 ms. In all cases, the spectral bandwidth increases during the propagation in the filament as discussed above. The spectral broadening, which is initially caused by SPM, becomes asymmetric with increasing propagation distance. Higher pressures result in stronger nonlinearities, and the critical power for self-focusing decreases. Thus, the filamentation process and spectral broadening begin at shorter distances from the focusing optic which can be observed by comparing the maps at different pressures.
Fig. 2 (Color online) Measured pulse spectra at three different positions behind the focusing optic extracted from the filament at 1000 mbar.
Fig. 3 (Color online) Spectral evolution of propagating pulses inside a filament on a logarithmic color (gray) scale. (a)–(e) Normalized integrated pulse spectra versus the position of extraction at five different argon pressures. (f) Numerically calculated map of single shot pulse spectra at 1000 mbar argon.
From the measured spectra, we extract a Fourier-limited pulse duration (FWHM) as a function of propagation distance, as shown in Fig. 4(a). The Fourier limit decreases as the propagation distance increases and reaches below 3 fs in the 1000 mbar case. The different offsets of the Fourier-limited pulse durations at position z = 226 cm confirm the earlier onset of the filamentation process at different pressures. In order to investigate the Fourier-limited pulse duration after the full filamentation stage without passing the pinholes, a 1.5-m-long cell without vacuum transition was installed (not shown in Fig. 1). The Fourier-limited pulse duration of the generated white-light after 4 m of propagation is shown by the filled circles in Fig. 4(a). Figure 4(b) shows the center wavelength versus propagation length. The asymmetry of the spectra changes during the propagation so that the central wavelength shifts into the IR and then into the blue spectral region. This behavior can be explained by simulations of the time structure along the filament, presented in section B. After 4 m of propagation the central wavelength reaches the same value for all pressures.
Fig. 4 (Color online) The evolution of (a) the Fourier-limited pulse duration (FWHM), and (b) the central wavelength versus propagation at three different pressures. The dashed lines show the simulation results for an argon pressure of 1000 mbar. The filled circles in (a) and (b) indicate the converged Fourier limit and central wavelength for the fully propagated filament.
The beam profile at two different extraction positions is measured for a filament generated in 800 mbar argon, see Fig. 5. For this end, a charge-coupled device camera (WinCamD, DataRay Inc.) is placed about 1.5 m behind the truncating pinhole. The profiles demonstrate a Gaussian-shaped intensity distribution which corresponds to the white-light core containing most of the energy. A small ellipticity at position 226 cm results from diffraction at the pinhole.
Fig. 5 (Color online) Measured beam profiles in the far field from the truncated filament 226 cm (a) and 256 cm (b) after the focusing optic.
4.2. Comparison to simulation
The experimental results are compared to numerical simulations at conditions comparable to the experiment. Figure 3(f) shows simulated pulse spectra at 1000 mbar argon over the propagation length inside the filament. An increasing spectral broadening is observed for pulses propagating longer in the filament matching well with the corresponding measurements. In the simulations, a considerable spectral amount is generated in the IR spectral region which could not be measured in the experiment due to a low sensitivity of the spectrometer. Thus, only the wavelengths up to 950 nm are considered for the comparison. The Fourier-limited pulse duration of the simulated pulse spectra is shown as dashed line in Fig. 4(a). The spectral broadening results in a decrease of the Fourier limit, reaching similar values as in the experiment. We interpret the difference in the Fourier limited pulse duration in the early stage of the filament as resulting from a comparison of multiple shot spectra with a single shot simulation. As observed in the experiment, the central wavelength shifts into the IR, and subsequently into the blue spectral region, see Fig. 4(b).
To explain this shift and to understand the connection with the filamentation dynamics, we have followed simulated FROG (Frequency-resolved optical gating) traces of the axial pulse envelope along the propagation distance [45]. FROG traces are defined as the spectrograms of the envelope ℰ(r = 0,t,z)
where g(t′ – t) is the gate function at time delay t and ω is the angular frequency. We chose a Gaussian gate function g(t′ – t) with 5 fs FWHM. FROG traces give an indication of the frequency content and of the local chirp carried by a pulse. A positive (resp. negative) slope of the FROG trace in (t,ω) variables corresponds to a positive (resp. negative) pulse chirp that will lead to temporal broadening (resp. compression) by further propagation in a normally dispersive medium.Figure 6 shows the FROG traces of the pulse as it propagates in the gas cell and undergoes filamentation. For an understanding of the newly generated frequencies we simultaneously monitored the evolution of the pulse intensity and electron density distributions. During the initial focusing/self-focusing stage, SPM broadens the spectrum symmetrically and the pulse undergoes a slow temporal compression. Around and immediately after the nonlinear focus (numerically obtained at z = 239 cm), the pulse dynamics exhibits asymmetry along the time axis, mainly resulting from plasma defocusing of its trailing part (positive times, Fig. 6, z = 242 cm). The most intense part of the pulse moves toward negative times where the low frequencies of the spectrum are the most amplified (z = 246 cm). At positive times, the trailing part of the pulse undergoes a refocusing stage which results in a trailing intensity peak that enhances the plasma density. As shown in the FROG trace at z = 252 cm, this results in a dominant redshifted leading peak and a subdominant but growing blueshifted trailing peak, both carrying positive chirps. It is a standard in the filamentation dynamics that the leading peak experiences nonlinear losses and thus disappears earlier than the rest of the pulse [46]. Correspondingly, the weight of the redshifted leading part in the center of mass of the spectrum is progressively replaced by the weight of this growing trailing component. Overall, this leads to a blueshift of the spectrum central wavelength, as observed in Fig. 4(b). This dynamics is recurrent along the propagation distance, leading to new refocused pulses that are shifted temporally toward positive times (delayed) and spectrally towards shorter wavelengths. The FROG trace at z = 284 cm is taken at the position of the second refocusing cycle and shows that the chirp carried by this blueshifted pulse is still positive and becomes stronger. In spite of this positive chirp, few-cycle pulses of duration supported by the spectral broadening are still generated by further propagation as these mainly result from the space-time refocusing dynamics which is much stronger than the temporal broadening that would be induced by group velocity dispersion [45]. We finally note the importance of the weight of the refocusing pulse in the determination of the central wavelength of the spectrum: When a pulse refocusing occurs in the central part (t ∼ 0), a leading peak and a previously focused trailing peak simultaneously undergo the effect of nonlinear absorption, hence the weight of the long- and short-wavelength components in the spectrum, featuring respectively the leading and trailing peaks, decreases to the benefit of the central wavelength components of the refocused pulse. These may be in turn blueshifted if the refocused peak propagates over a sufficiently long distance. This explains the final increase of the central wavelength after the filament shown in Fig. 4(b).
Fig. 6 (Color online) Simulated FROG trace ℱ(t,ω) (first line), intensity I(t,r) (second line) and electron density ρ(t,r) (third line) distributions at four propagation distances within the gas cell.
The different spectral weights associated with each intensity peak also affect the calculated pulse duration. Figure 7 shows FWHM pulse durations for the radially averaged intensity profiles as functions of the propagation distance. At each distance, only the most intense peak of the averaged profile is monitored. Minima of the pulse duration corresponding to few-cycle pulses between 4 and 5 fs are obtained after each refocusing stage averaging over a transmitted beam radius of 50 μm. Closer around the axis (see averaging radius of 20 μm), pulse durations between 1 and 3 fs are reached. The shortest intensity peaks in the temporal profile, however, are mainly obtained with a blueshifted spectral content and a positive chirp.
Fig. 7 (Color online) Simulated pulse duration (FWHM) of the radially averaged pulse intensity over a radius of 20 or 50 μm, as functions of the propagation distance.
4.3. High-order harmonic generation
Due to the complex temporal structure of the pulses formed during the propagation in the filament the temporal characterization with for example SPIDER is difficult. To give an estimation of the pulse duration we analyzed the high-order harmonic radiation generated inside the filament. The measurements are consistent with the theoretical prediction that HHG in a filament occurs at the refocusing stage in the filament in which an ultrashort sub-pulse is formed on-axis with an intensity which can be higher than the clamping intensity. We attribute these sub-pulses to intensity spikes referred to Ref. [32, 33]. For the filament in 1000 mbar argon, we record the harmonic intensity generated along the filament, see Fig. 8, and identify two confined regions where harmonics are generated while the whole filament extends over 30 cm. One region A at z ≈ 236 cm with a discrete harmonic spectrum and another region B at z ≈ 242 cm shows a continuous spectral shape. From the discrete harmonic structure in region A, we conclude that the generating pulse consists of multiple cycles. The continuous XUV spectrum in region B indicates the occurrence of an intense spike with a near single-cycle time structure [33]. The position of both regions A and B can hardly be seen in the linear diagnostics of radially averaged IR spectra (cf. 3(e)), because the intensity spikes appear only on-axis.
Fig. 8 (Color online) High-order harmonics generated directly in the filament extracted at different positions after the focusing optic.
The harmonic radiation constitutes an ideal tool for tracking the filament. We analyzed the harmonic spectral maps (e.g. Fig. 8) at different pressures and diameters of the first aperture regarding the center location z A, z B of the individual regions (A, B). Figure 9(a) shows the position of the two regions versus pressure at two diameters of the aperture (D = 5.5 mm and 7.5 mm). Below a pressure of 700 mbar, region B disappears which can be account to a smaller filament length at a low pressure, associated with an earlier termination [47]. We can explain the shift of region A and B by the pressure dependence of the nonlinear refractive index n 2(p) ∝ p [48]. Increasing the pressure leads to an stronger self-focusing due to higher nonlinearities. The position of the beam collapse can be calculated for a collimated input beam by [36]
with the Rayleigh range z R and the peak power P Peak. Using a focusing optic, Eq. (8) is amended by the focal length f and the beam collapses at a smaller distance given by [18, 49] To consider the different diameters of the aperture, we use for D = 5.5 mm a beam waist of 2.8 mm at the focusing optic and a transmitted pulse energy of 0.75 mJ, corresponding to a peak power of ≈ 20 GW. For D = 7.5 mm, we use a beam waist of 3.9 mm and a transmitted pulse energy of 1 mJ, corresponding to a peak power of ≈ 27 GW. Considering L c,f as the beginning of the filament and defining the location of the spikes by the relative offset L A and L B in the filament, respectively, their absolute positions are given by As shown by the curves in Fig. 9, this model matches the experimental data for values of L A and L B listed in Table 1(a). We observe a good agreement between the spike positions and the model by considering a single offset-pair L A, L B for each input beam width, i.e aperture diameter D. Spike A appears a few cm beyond the nonlinear focus whereas spike B appears at a the position where the trailing peak is refocused. The simulation results of Fig. 6 show that a refocusing cycles requires a propagation distance of 10 – 12 cm in good agreement with the offset-difference L A – L B. This confirms that spike B is associated with a refocusing event and with the shortest pulse durations [32, 33]. We investigate also the dependence of beam collapse on the entrance pulse duration for the two different aperture sizes at a fixed pressure of 1000 mbar, see Fig. 9(b). The chirp is controlled by the separation of the compressor gratings in the amplifier system. We can stretch the pulses by a positive or negative dispersion of ± 1000 fs2 to a duration of approximately 87 fs before the second spike vanishes. The first spike can be observed at an even wider range of chirp values. We use Eqs. (8) and (9) and assume a change in the peak power of the pulse according to for an applied group delay dispersion β and a Fourier-limited pulse duration τp = 35 fs. We find a good agreement of the model for both spikes, again for a single pair of relative offsets listed in Table 1(b). This shows that the nonlinear diagnostic using high-order harmonics directly from the filament allows to track the refocusing stages inside the femtosecond light channel. In particular, we are able to verify a basic filamentation model predicting the position of the nonlinear focus.Fig. 9 (Color online) (a) Position of the intensity spikes A and B versus pressure of the nonlinear medium for the experiment (box, cross) and theory (lines) for two different apertures (D = 5.5 mm and 7.5 mm). (b) Position of intensity spikes versus chirp for 1000 mbar argon. The symbols denote the same as in (a).
Table 1. Offset Values Determined by a Fit to the Data Shown (a) in Fig. 9(a), and (b) in Fig. 9(b), Respectively (see text for details)
5. Conclusion
In summary, we presented a study on the nonlinear light evolution inside a filament using two different diagnostics. The spectral reshaping of pulses within a filament is experimentally observed in a detailed investigation by extracting pulses at different positions directly from the high intensity light channel. Numerical simulations are in excellent agreement with the experimental data supporting the interpretation of the physics of these highly nonlinear and dynamical effects. The peak intensity of the pulses generated within the filament is high enough to produce high-order harmonics directly in the filament [33]. The spectral structure of these harmonics carries information about the duration of the generating pulse and can be tracked along the propagation axis. The beginning of the filament and the occurring refocusing cycles can be monitored by the position of the intensity spikes self-focusing dependence. Our experimental method opens the way for better understanding of the dynamics during the filamentation process.
Acknowledgments
This work was funded by Deutsche Forschungsgemeinschaft within the Cluster of Excellence QUEST, Centre for Quantum Engineering and Space-Time Research. M. B. Gaarde was supported by the National Science Foundation under Grant No. PHY-1019071.
References and links
1. C. P. Hauri, W. Kornelis, F. W. Helbing, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B 79, 673–677 (2004). [CrossRef]
2. O. Kosareva, N. Panov, D. Uryupina, M. Kurilova, A. Mazhorova, A. Savelev, R. Volkov, V. Kandidov, and S. Chin, “Optimization of a femtosecond pulse self-compression region along a filament in air,” Appl. Phys. B: Lasers Opt. 91, 35–43 (2008). 10.1007/s00340-008-2959-9, URL http://dx.doi.org/10.1007/s00340-008-2959-9. [CrossRef]
3. A. Zaïr, A. Guandalini, F. Schapper, M. Holler, J. Biegert, L. Gallmann, A. Couairon, M. Franco, A. Mysyrowicz, and U. Keller, “Spatio-temporal characterization of few-cycle pulses obtained by filamentation,” Opt. Express 15(9), 5394–5405 (2007). [CrossRef] [PubMed]
4. D. S. Steingrube, E. Schulz, T. Binhammer, T. Vockerodt, U. Morgner, and M. Kovačev, “Generation of high-order harmonics with ultra-short pulses from filamentation,” Opt. Express 17(18), 16177–16182 (2009). [CrossRef] [PubMed]
5. J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. André, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003). [CrossRef] [PubMed]
6. T. Fujii, M. Miki, N. Goto, A. Zhidkov, T. Fukuchi, Y. Oishi, and K. Nemoto, “Leader effects on femtosecond-laser-filament-triggered discharges,” Phys. Plasmas 15(1), 013107 (2008). URL http://dx.doi.org/doi/10.1063/1.2830647. [CrossRef]
7. S. Chin, H. Xu, Q. Luo, F. Thberge, W. Liu, J. Daigle, Y. Kamali, P. Simard, J. Bernhardt, S. Hosseini, M. Sharifi, G. Mjean, A. Azarm, C. Marceau, O. Kosareva, V. Kandidov, N. Akzbek, A. Becker, G. Roy, P. Mathieu, J. Simard, M. Chteauneuf, and J. Dubois, “Filamentation remote sensing of chemical and biological agents/pollutants using only one femtosecond laser source,” Appl. Phys. B: Lasers Opt. 95, 1–12 (2009). 10.1007/s00340-009-3381-7, URL http://dx.doi.org/10.1007/s00340-009-3381-7. [CrossRef]
8. P. Rohwetter, J. Kasparian, K. Stelmaszczyk, Z. Hao, S. Henin, N. Lascoux, W. M. Nakaema, Y. Petit, M. Queiszer, R. Salame, E. Salmon, L. Woste, and J.-P. Wolf, “Laser-induced water condensation in air,” Nat. Photonics 4(7), 451–456 (2010). [CrossRef]
9. M. K. Bhuyan, F. Courvoisier, P. A. Lacourt, M. Jacquot, R. Salut, L. Furfaro, and J. M. Dudley, “High aspect ratio nanochannel machining using single shot femtosecond Bessel beams,” Appl. Phys. Lett. 97, 081102 (2010). [CrossRef]
10. F. Courvoisier, P.-A. Lacourt, M. Jacquot, M. K. Bhuyan, L. Furfaro, and J. M. Dudley, “Surface nanoprocessing with nondiffracting femtosecond Bessel beams,” Opt. Lett. 34(20), 3163–3165 (2009). URL http://ol.osa.org/abstract.cfm?URI=ol-34-20-3163. [CrossRef] [PubMed]
11. M. K. Bhuyan, F. Courvoisier, P.-A. Lacourt, M. Jacquot, L. Furfaro, M. J. Withford, and J. M. Dudley, “High aspect ratio taper-free microchannel fabrication using femtosecond Bessel beams,” Opt. Express 18(2), 566–574 (2010). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-18-2-566. [CrossRef] [PubMed]
12. F. Belgiorno, S. L. Cacciatori, G. Ortenzi, V. G. Sala, and D. Faccio, “Quantum radiation from superluminal refractive-index perturbations,” Phys. Rev. Lett. 104(14), 140403 (2010). [CrossRef] [PubMed]
13. F. Belgiorno, S. L. Cacciatori, M. Clerici, V. Gorini, G. Ortenzi, L. Rizzi, E. Rubino, V. G. Sala, and D. Faccio, “Hawking radiation from ultrashort laser pulse filaments,” Phys. Rev. Lett. 105(20), 203901 (2010). [CrossRef]
14. S. Tzortzakis, G. Méchain, G. Patalano, Y.-B. André, B. Prade, M. Franco, A. Mysyrowicz, J.-M. Munier, M. Gheudin, G. Beaudin, and P. Encrenaz, “Coherent subterahertz radiation from femtosecond infrared filaments in air,” Opt. Lett. 27(21), 1944–1946 (2002). [CrossRef]
15. C. D’Amico, A. Houard, S. Akturk, Y. Liu, J. L. Bloas, M. Franco, B. Prade, A. Couairon, V. T. Tikhonchuk, and A. Mysyrowicz, “Forward THz radiation emission by femtosecond filamentation in gases: theory and experiment,” N. J. Phys. 10, 013015 (2008). [CrossRef]
16. A. Houard, Y. Liu, A. Mysyrowicz, and B. Leriche, “Calorimetric detection of the conical terahertz radiation from femtosecond laser filaments in air,” Appl. Phys. Lett. 91, 241105 (2007). [CrossRef]
17. M. Mlejnek, M. Kolesik, J. V. Moloney, and E. M. Wright, “Optically turbulent femtosecond light guide in air,” Phys. Rev. Lett. 83(15), 2938–2941 (1999). [CrossRef]
18. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–189 (2007). [CrossRef]
19. E. Lorin, S. Chelkowski, and A. D. Bandrauk, “The WASP model: a micro-macro system of wave-Schrodinger-plasma equations for filamentation,” Commun. Comput. Phys. 9, 406–440 (2011).
20. V. P. Kandidov, Svyatoslav, A. Shlenov, and O. G. Kosareva, “Filamentation of high-power femtosecond laser radiation,” Quantum Electron. 39, 205–228 (2009). [CrossRef]
21. E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320, 1614–1617 (2008). [CrossRef] [PubMed]
22. C. P. Hauri, A. Guandalini, P. Eckle, W. Kornelis, J. Biegert, and U. Keller, “Generation of intense few-cycle laser pulses through filamentation–parameter dependence,” Opt. Express 13(19), 7541–7547 (2005). [CrossRef] [PubMed]
23. J. Y. Park, J. H. Lee, and C. H. Nam, “Laser chirp effect on femtosecond laser filamentation generated for pulse compression,” Opt. Express 16, 4465–4470 (2008). [CrossRef] [PubMed]
24. O. Varela, B. Alonso, I. J. Sola, J. S. Román, A. Zaïr, C. Méndez, and L. Roso, “Self-compression controlled by the chirp of the input pulse,” Opt. Lett. 35(21), 3649–3651 (2010). URL http://ol.osa.org/abstract.cfm?URI=ol-35-21-3649. [CrossRef]
25. M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations,” Phys. Rev. E 70(3), 036604 (2004). [CrossRef]
26. B. Alonso, I. J. Sola, O. Varela, J. Hernández-Toro, C. Méndez, J. S. Román, A. Zaïr, and L. Roso, “Spatiotemporal amplitude-and-phase reconstruction by Fourier-transform of interference spectra of high-complex-beams,” J. Opt. Soc. Am. B 27(5), 933–940 (2010). URL http://josab.osa.org/abstract.cfm?URI=josab-27-5-933. [CrossRef]
27. S. Minardi, A. Gopal, M. Tatarakis, A. Couairon, G. Tamošauskas, R. Piskarskas, A. Dubietis, and P. D. Trapani, “Time-resolved refractive index and absorption mapping of light-plasma filaments in water,” Opt. Lett. 33(1), 86–88 (2008). URL http://ol.osa.org/abstract.cfm?URI=ol-33-1-86. [CrossRef]
28. S. Minardi, A. Gopal, A. Couairon, G. Tamoašuskas, R. Piskarskas, A. Dubietis, and P. D. Trapani, “Accurate retrieval of pulse-splitting dynamics of a femtosecond filament in water by time-resolved shadowgraphy,” Opt. Lett. 34(19), 3020–3022 (2009). URL http://ol.osa.org/abstract.cfm?URI=ol-34-19-3020. [CrossRef] [PubMed]
29. D. G. Papazoglou and S. Tzortzakis, “In-line holography for the characterization of ultrafast laser filamentation in transparent media,” Appl. Phys. Lett. 93, 041120 (2008). [CrossRef]
30. D. Uryupina, M. Kurilova, A. Mazhorova, N. Panov, R. Volkov, S. Gorgutsa, O. Kosareva, A. Savel’ev, and S. L. Chin, “Few-cycle optical pulse production from collimated femtosecond laser beam filamentation,” J. Opt. Soc. Am. B 27(4), 667–674 (2010). [CrossRef]
31. J. Odhner, D. Romanov, and R. J. Levis, “Self-shortening dynamics measured along a femtosecond laser filament in air,” Phys. Rev. Lett. 105, 125001 (2010). [CrossRef] [PubMed]
32. M. B. Gaarde and A. Couairon, “Intensity spikes in laser filamentation: diagnostics and application,” Phys. Rev. Lett. 103, 043901 (2009). [CrossRef] [PubMed]
33. D. S. Steingrube, E. Schulz, T. Binhammer, M. B. Gaarde, A. Couairon, U. Morgner, and M. Kovačev, “High-order harmonic generation directly from a filament,” N. J. Phys. 13, 043022 (2011). [CrossRef]
34. N. Papadogiannis, C. Kalpouzos, E. Goulielmakis, G. Nersisyan, D. Charalambidis, F. Augé, F. Weihe, and P. Balcou, “Kilohertz extreme-ultraviolet light source based on femtosecond high-order harmonic generation from noble gases,” Appl. Phys. B 73, 687–692 (2001). [CrossRef]
35. D. S. Steingrube, T. Vockerodt, E. Schulz, U. Morgner, and M. Kovačev, “Phase-matching of high-order harmonics in a semi-infinite gas cell,” Phys. Rev. A 80, 043819 (2009). [CrossRef]
36. J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975). [CrossRef]
37. D. P. Shelton, “Nonlinear-optical susceptibilities of gases measured at 1064 and 1319 nm,” Phys. Rev. A 42(5), 2578–2592 (1990). [CrossRef] [PubMed]
38. A. Couairon, H. S. Chakraborty, and M. B. Gaarde, “From single-cycle self-compressed filaments to isolated attosecond pulses in noble gases,” Phys. Rev. A 77(5), 053814 (2008). [CrossRef]
39. A. Dalgarno and A. E. Kingston, “The refractive indices and Verdet constants of the inert gases,” Proc. R. Soc. London Ser. A 259, 424–429 (1966).
40. L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Sov. Phys. JETP 20(5), 1307–1314 (1965).
41. A. M. Perelomov, V. S. Popov, and M. V. Terent’ev, “Ionization of atoms in an alternating electric field,” Sov. Phys. JETP 23(5), 924–934 (1966).
42. M. Ammosov, N. Delone, and V. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64(6), 1191–1194 (1986).
43. F. A. Ilkov, J. E. Decker, and S. L. Chin, “Ionization of atoms in the tunnelling regime with experimental evidence using Hg atoms,” J. Phys. B: At. Mol. Opt. Phys. 25(19), 4005–4020 (1992). [CrossRef]
44. W. Wood, C. Siders, and M. Downer, “Measurement of femtosecond ionisation dynamics of atmospheric density gases by spectral blueshifting,” Phys. Rev. Lett. 67, 3523–3526 (1991). [CrossRef] [PubMed]
45. S. Akturk, A. Couairon, M. Franco, and A. Mysyrowicz, “Spectrogram representation of pulse self-compression by filamentation,” Opt. Express 16, 17626–17636 (2008). [CrossRef] [PubMed]
46. M. Mlejnek, E. M. Wright, and J. V. Moloney, “Dynamic spatial replenishment of femtosecond pulses propagating in air,” Opt. Lett. 23(5), 382–384 (1998). URL http://ol.osa.org/abstract.cfm?URI=ol-23-5-382. [CrossRef]
47. G. Méchain, T. Olivier, M. Franco, A. Couairon, B. Prade, and A. Mysyrowicz, “Femtosecond filamentation in air at low pressures. Part II: Laboratory experiments,” Opt. Commun. 261(2), 322–326 (2006). URL http://www.sciencedirect.com/science/article/B6TVF-4HRN3HK-4/2/c167a65f6ec2f419c301db92f492e60c. [CrossRef]
48. A. Börzsönyi, Z. Heiner, A. Kovács, M. P. Kalashnikov, and K. Osvay, “Measurement of pressure dependent nonlinear refractive index of inert gases,” Opt. Express 18(25), 25847–25854 (2010). URL url = http://www.opticsexpress.org/abstract.cfm?URI=oe-18-25-25847. [CrossRef] [PubMed]
49. G. Fibich and G. Papanicolaou, “Self-focusing in the perturbed and unperturbed nonlinear Schrdinger equation in critical dimension,” SIAM J. Appl. Math. 60, 183–240 (1999). [CrossRef]
