We report on numerical simulations and experiments of pulse self-compression by filamentation. Spectral and temporal evolution during ultrashort-pulse laser filamentation can be intuitively represented using spectrograms, which display spectra at different time delays of a gate pulse. This representation makes evident the features of self-compression by filamentation, namely spectral broadening and pulse shortening. In addition, it allows for an analysis of the spectral phase during the nonlinear propagation. In our simulations we observe occurrence of negative chirp over a few cm before the shortest pulse is obtained during filamentation; this provides an important basis for the understanding of the mechanisms leading to self-compressed filaments. We attribute the negative chirp to spatio-temporal reshaping due to the competition between self-phase modulation and group velocity dispersion. We show that the plasma induced dispersion plays a minor role in establishing the negative chirp.
©2008 Optical Society of America
With the progresses on the pulse compression techniques in the last decades, ultrashort laser pulse durations approach the period of oscillation of the pulse electric field. These few cycle laser pulses provide indispensable tools for generating coherent, attosecond-length ultraviolet (UV) and extreme ultraviolet (XUV) radiation through high harmonic generation [1, 2]. Since the pulse peak intensity is increased by the factor of compression in duration, pulse compression also potentially increases the efficiency and range of nonlinear optical applications of a given laser system [3, 4]. For the pulse energies in the milijoule range and above, Ti:sapphire based chirped pulse amplification (CPA) systems deliver pulses with durations longer than ~20 fs. Further compression of the duration is possible using hollow fibers [5, 6], where the pulse spectrum is broadened through self-phase modulation (SPM), and the resulting phase is compensated using chirped mirrors. Filamentation of femtosecond pulses in noble gases is also shown to be an efficient way of pulse compression [7, 8]. This scheme is particularly advantageous since it requires no critical alignment, and it also offers the possibility of self compression i.e. compensation of the phase during the process itself [9–11].
Filamentation is a dynamical process reflecting a competition between several physical effects which maintain high intensities confined within a narrow diameter over long distances . This regime occurs for laser pulses with peak powers greater than a critical value, such that the effect of self-focusing overcomes that of diffraction [13, 14] and the beam would undergo a catastrophic collapse in the absence of any saturation . In real media, however, this collapse is not observed since the intensity becomes sufficiently large so as to induce significant multi-photon absorption (MPA) and plasma generation which in turn reshape the beam and prevent further self-focusing. Beyond the nonlinear focus, the competition between these high-order saturating nonlinear effects and self-focusing maintains high intensities over a relatively long distance. As a result, SPM and hence the spectral broadening becomes significant.
While the SPM is one of the principle mechanisms for spectral broadening, there are various other factors that affect the spectral, spatial and temporal propagation dynamics during filamentation . Self steepening, for example causes the trailing edge of the pulse to reshape as a conical optical shock , leading to further spectral broadening towards UV. The accompanying plasma plays a significant role both in reshaping the trailing part of the pulse and in generating plasma-enhanced SPM. It was also proposed to compensate the normal dispersion of the medium  and to induce negative chirp responsible for the pulse shortening [18, 19]. However, filaments are associated with pulse splitting, a phenomenon which generates in itself shorter sub-pulses and which can arise either because of a competition between group velocity dispersion and the Kerr effect  or because of MPA of the most intense part of laser pulses with large powers . Plasma, which is absent from the modeling of these pulse splitting phenomena does not play any role in the process and another mechanism must be identified to explain a possible negative chirp in self-compressed filaments. In order to better understand the pulse compression process, have an intuitive picture of the pulse evolution, and identify the physical effects generating the chirp which may promote or prevent self-compression, it is necessary to follow the pulse spectrum, spectral phase and temporal intensity during the filamentary propagation.
Numerical simulations of filamentation (by solving the nonlinear envelope equation -NEE-) yield reliable results for the pulse evolution . The results of the simulations are usually presented in time and frequency domains, separately [22, 23]. Due to the complexity of the evolution and importance of the pulse spectral phase, a time-frequency display of the process would yield a more intuitive and easier to interpret representation. This intermediate domain is commonly represented using spectrograms, which display the signal spectrum at different time gates. As a result, spectrograms can provide intuitive representation of the pulse propagation dynamics during filamentation.
In this work, we show simulation results of pulse propagation during filamentation, expressed using spectrograms. With the generated movies, we show that, apart from the positive chirp caused by GVD and SPM, the pulse may also exhibit negative chirp, as required for self compression. We interpret the origin of this negative chirp primarily as an effect of the spatio-temporal reshaping occurring before maximum compression. In following sections, we first present a brief discussion on spectrograms, and then we present our simulation method, results and their interpretation. We also provide an experimental evidence of self compression by filamentation.
The goal of a spectrogram is to represent the variation of the intensity of a signal in time and frequency, simultaneously [24, 25]. This is done by calculating the signal spectrum at consequent time gates. Mathematically, the spectrogram of a function, S(t) is :
where P(τ,ω) is the spectrogram, h(t-τ) is the gate function at time delay τ and ω is the angular frequency. Figure 1 shows examples of spectrograms of ultrashort pulses with different spectral phases (at 800 nm center wavelengths). The gate function is a Gaussian with 40 fs FWHM. Figure 1a represents a flat-phase pulse with duration of 25 fs; Fig.1b a positively chirped pulse with second order spectral phase coefficient of 390 fs2 and duration of 50 fs and Fig.1c pulse with third order spectral phase coefficient of 3×104 fs3 and duration of 44 fs. In these illustrations, it is interesting to note that the width of the gate pulse does not impose a strong limitation on the temporal resolution, since the spectrograms are calculated in the simultaneous time-frequency domain. A gate pulse shorter than the pulse to be characterized yields good temporal resolution and it blurs the features in the frequency domain; while a narrow spectrum (hence longer) gate pulse yields good spectral resolution and blurs the temporal features. Both cases, however, provide proper measurements, since the information lost in one domain is compensated in the other . In order to make the traces more intuitive, the information can be distributed evenly between the time and frequency domains, as we have done in Fig. 3. Particularly in Fig. 3b, one can observe that the gate pulse partially resolves the satellite pulses (in time) resulting from the third order spectral phase; and it also reveals the overall parabolic dependence of the instantaneous frequency versus time.
Spectrograms also provide important base for characterization of ultrashort laser pulses. Inversion algorithms can retrieve a signal from its spectrogram when the gate function is known [27, 28]. When the gate function is the pulse itself (hence unknown), two-dimensional phase retrieval algorithms can be used to retrieve the pulse intensity and phase . This is the main concept of a well established pulse measurement method: frequency resolved optical gating (FROG) [26, 29, 30], which is shown to be capable of measuring pulses in the single-cycle regime, as well [31, 32].
As a result, the spectrogram representation of pulse compression not only yields and intuitive picture of the propagation dynamics but also allows a direct comparison with the measured traces if FROG method is used. Since the pulse electric field typically goes through complicated spectral and temporal reshaping during filamentation, such representation would be particularly advantageous. Spectrograms were also used to simulate nonlinear propagation in optical fibers , which exhibits complicated reshaping of initial electric field, as well.
3. Simulated spectrograms of ultrashort pulses during filamentation
In this section we present our simulation results on pulse compression by filamentation, represented by spectrograms. We calculated the electric field of the pulse as a function of frequency and position, by numerically solving the NEE :
Where εω=ε(x,y,z,ω) and εt=ε(x,y,z,t) are the electric fields in time and frequency domains, Δ⊥ is the transverse Laplacian and F is the Fourier transform operator. The various operators in Eq. (2) are: κ(ω)=k0+k′0(ω-ω0), D(ω)=k2(ω)-κ2(ω), where ω 0 is the pulse center frequency k(ω) denotes the dispersion relation of the medium and k 0 is the wavevector at the center frequency. The function N includes the Kerr effect, plasma defocusing and energy losses resulting from photoionization, as expressed by:
Here, the operator T accounts for space-time focusing, self-steepening and the deviations from the slowly varying envelope approximation . In the plasma terms, τc is the electron collision time, ρ is the electron density in the plasma, ρc is the critical plasma density, Ui is the ionization potential, ρat is the density of neutral atoms and W describes the photoionization rate which depends on intensity.
We numerically solved Eq. (2) by using an extended Crank-Nicholson scheme that consists of space marching in the frequency domain and at each step in the propagation direction, using the resulting time-dependent electric field to evaluate the electron density and nonlinear source terms, as described in . The input pulses have 0.8 mJ energy and 27 fs duration. The beam of initial diameter 2.5 mm was focused at 1 m. The nonlinear medium is argon at 0.8 atm. Figure 2 shows the simulated on-axis intensity and electron density as a function of propagation distance. The filamentation begins at ~80 cm and continues until ~140 cm, in two successive stages.
In order to fully characterize the pulse compression process during filamentation, we have also calculated the pulse spectrograms using on axis electric field. As a gate function we used the pulse intensity profile (as in polarization gating -PG- FROG ), as well as the complex electric field (as in second harmonic generation -SHG-) FROG ). Interpretation of results from SHG-FROG is less intuitive due to the ambiguity in the direction of time. However, experimentally, SHG is the most sensitive nonlinearity (measures weakest energies), hence we included it in the calculations.
Figure 3 shows the movie of PG and SHG traces of the pulse as it propagates through the filament. The evolution of the pulse duration is shown in Fig. 4. The initial decrease of the pulse duration, up to ~80 cm results from the self steepening. At the beginning of the movie (Fig.3), which starts in the middle of the filament (z=87 cm), the pulse is positively chirped, as a result of GVD and SPM accumulated so far. Accordingly, the pulse spectrum and duration continue to increase until z~95 cm. After this point, the pulse duration starts to fall due to the defocusing of the leading edge (as explained below), even though the pulse keeps positively chirped until ~100 cm. Right after this point, the PG trace starts to exhibit a left tilt; a clear signature of occurrence of negative chirp. Within ~8 cm, the negative chirp is totally cancelled and the trace shows a near-flat-phase pulse with duration of only ~4.9 fs. After further propagation, the positive chirp becomes dominant again and the pulse duration increases continuously.
In order to understand the source of the occurrence of this unusual negative chirp, we analyze contribution of different effects to the pulse’s phase. As recalled in refs [11, 36], the contribution to the chirp induced by SPM for a pulse of intensity Isat and of duration τ, is found from:
and is usually much larger than the chirp induced by GVD. For instance, the intensity in the filament which corresponds to saturation of Kerr self-focusing by multiphoton processes can be estimated as [12, 37],
where for argon, the MPA coefficient is σK=5×10-140 s -1 cm 22 W -11 with K=11 photons, the Kerr coefficient is n2=4×10-19 cm 2 W -1, the critical plasma density is ρc=1.8×1021 cm -3 and the neutral density is ρat=2×1019 cm -3. For τ=25 fs pulse duration, we obtain Isat≡4.3×1013 Wcm -2, in agreement with the simulation results shown above in Fig. 2. This corresponds to an effective chirp of k″eff-SPM=163fs 2 cm -1, whereas the GVD coefficient for argon is k″GVD=0.2fs 2 cm -1. For a comparison, the contribution of the plasma to the chirp via plasma-induced SPM is given by [17, 37]:
where vg is the group velocity and n0 is the linear index of refraction. The electron density is estimated by:
and is also in good agreement with the simulation results shown above (Fig. 2). As a result, Eq. (6) yields k″eff-plasma=-0.18fs 2 cm -1. It is therefore clear that the chirp induced by the plasma alone is barely sufficient to compensate for the dispersion of the medium but in any case, it cannot be expected to compensate the SPM-induced chirp. The spectrograms shown in Fig. 3, however, show that after the distance of z=100 cm, the pulse chirp (the tilt in PG trace) changes sign and we observe a negatively chirped pulse. In view of the above analysis, this is rather unexpected even if the different contributions to the chirp must be weighted due to the fundamental difference that the plasma and Kerr induced SPM only act on the most intense, saturated part of the filament whereas the dispersion of the medium acts on the whole pulse and leaves its signature on the conical emission accompanying filaments [21, 38]. The contribution of the plasma to the chirp via plasma-enhanced SPM is nevertheless ruled out since the plasma density is well below 1016 cm-3 at the position where we observe the most significant negative chirp.
An explanation of the possible occurrence of negatively chirped pulses reshaped by filamentation was brought by Liu et al , who showed that the competition between GVD and Kerr-SPM can lead to a negative effective dispersion, provided that the spectrum is sufficiently broadened. With the parameters of our simulations, this approach reproduces well the occurrence of a region with negative chirp resulting from the action of GVD and Kerr-SPM over a sufficiently long distance. However, quantitative agreement is obtained only by considering the complete spatio-temporal reshaping of the pulse which is the main responsible for the pulse compression obtained between 105 and 108 cm (see Fig. 4).
An intuitive picture of the occurrence of negative chirp is obtained by carefully analyzing the spatio-temporal evolution of the pulse during nonlinear propagation. Note first, that normal GVD will compress the pulse temporally if bluer frequencies are found in the leading part and redder frequencies in the trailing part of the pulse. This is exactly the situation that can arise after an asymmetric pulse splitting during which a moderately intense leading pulse does not survive over a long distance allowing for a refocusing event and a much more intense trailing pulse, a standard situation in ultrashort laser pulse filamentation leading to pulse reshaping/compression . For a bell shaped pulse Kerr SPM generates red frequencies in the leading part and blue frequencies in the trailing part. Plasma induced SPM only generates blue frequencies localized around the steepest part of the electron density profile. Overall, this results in spectral broadening with new red frequencies ahead of the intensity spike and new blue frequencies covering the spike and the trailing part. This frequency distribution is localized at negative times, around the position of the leading split pulse which appears after the beginning of filamentation; the refocusing stage for the trailing split pulse will generate a similar frequency distribution but with mainly red frequencies localized in the trailing part, at positive times, until it becomes sufficiently intense to take over ionization due to the leading split pulse. This situation leads precisely to a negative effective chirp, with red frequencies in the trailing part and blue frequencies in the leading part as produced by the reshaping of the pulse in space and time, which is therefore very favorable for pulse self-compression.
These phenomena become clearer by inspecting the simulated pulse profiles in Fig. 5. This figure shows the on-axis intensity and instantaneous frequency vs. time, as well as the intensity vs. time and radius, at 90 cm, 101.5 cm, 105 cm and 108 cm. At 90 cm, positive chirp is dominant, resulting lower frequencies ahead of higher ones. At this position, the leading part of the intensity starts to decrease due to energy losses and plasma defocusing. The trailing part is then refocused, as can be seen from the profiles at further positions. However, due to the delay caused by defocusing and refocusing, the lower-frequency-shifted refocused portion lags behind the higher-frequency-shifted, originally trailing portion of the pulse. As a result, the pulse intensity accumulates at positives times, with instantaneous frequency decreasing over time; hence the occurrence of the dominant negative chirp.
In agreement with this scenario, the movie (Fig.3) shows a negatively chirped pulse at ~100 cm and self-compressed pulse with duration of 4.9 fs fs appearing at 108 cm, while the negative chirp becomes totally compensated as a result of the standard SPM induced chirp due to the isolated trailing pulse. Beyond this point, the positive chirp prevails again, and the pulse continues to broaden spectrally and temporally, although the temporal broadening is slow and the pulse remains much shorter than its initial duration over extended distances, as shown in the evolution of the pulse duration in Fig. 4. In contrast with refs [18, 19], however, the chirp after the end of the filament is positive and increasing so that the pulse is not expected to be further compressed by propagation in normally dispersive media. We also note that the negative group delay dispersion (GDD) that we obtain with the simulations is of the order -10 fs2, hence the reported value of -500 fs2  does not find an explanation in the scenario proposed in our work.
4. Experimental evidence of self compression
In the previous section, we have shown our simulation results, which indicate occurrence of negative chirp, hence self-compression during filamentation. In this section we provide an experimental evidence of self compression by filamentation. We focused 70 fs (FWHM), 3 mJ pulses into a gas cell filled with argon at 0.9 bar. At the output of the cell, we sent the pulses directly to diagnostics. We measured the input and output pulse intensities and phases using SHG FROG . In order to have optimal compression, we tuned the initial spectral phase and adjusted an iris before the gas cell.
Figure 6 shows the measured temporal intensity and phase profiles before and after the filamentation. Even though we did not use any element with negative dispersion after the cell, it is evident that the output pulse is significantly compressed with respect to the input (~70 at the input versus ~10 fs at the output). The input pulse is positively chirped, with the second order spectral phase coefficient of φ2=700 fs2. The output is also slightly positively chirped, with φ2=100 fs2. As calculated in detail above, during filamentation, the SPM-induced spectral phase would increase the φ2, while the contributions from the plasma and normal dispersion roughly cancel each other. Our measurements indicate a substantial reduction of the φ2, at the output. As detailed in section 3 above, we attribute the compensation of the phase to the spatio-temporal reshaping of the pulse during the nonlinear propagation. While part of the spectral phase is cancelled during filamentation, the output pulse still exhibits positive chirp and is longer than transform-limited, in consistence with Fig.3 and Fig. 4. This measurement provides a clear experimental evidence of self compression, indicating that part of the spectral phase is compensated during filamentation, as observed in the simulation above.
We have shown that spectrograms provide intuitive and useful representation of complicated temporal and spectral reshaping during filamentation. From our simulated spectrograms, we have observed the presence of negative chirp in a localized zone of the filament and subsequent self compression. We attribute the self compression mainly to the spatio-temporal reshaping occurring in several stages which eventually cause the sign of the effective chirp to change: (i) Propagation starts by beam self-focusing which ends up by an asymmetric pulse splitting. (ii) The leading split-pulse generates more blue frequencies than red frequencies while the plasma defocuses it. (iii) The defocused leading split pulse refocuses progressively, which in turn yields delayed red frequencies in the trailing part, i.e, the effective chirp is negative. (iv) GVD and Kerr-SPM compress the refocused trailing pulse. We have also provided an experimental example where we observed self compression by filamentation from 70 to 10 fs.
References and links
1. M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. B. Corkum, U. Heinzmann, D. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414, 509–513 (2001). [CrossRef] [PubMed]
2. I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “High-Harmonic Generation of Attosecond Pulses in the “Single-Cycle” Regime,” Phys. Rev. Lett. 78, 1251–1254 (1997). [CrossRef]
4. T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000). [CrossRef]
5. M. Nisoli, S. De Silvestri, O. Svelto, R. Szipocs, K. Ferencz, C. Spielmann, S. Sartania, and F. Krausz, “Compression of high-energy laser pulses below 5 fs,” Opt. Lett. 22, 522–524 (1997). [CrossRef] [PubMed]
6. B. Schenkel, J. Biegert, U. Keller, C. Vozzi, M. Nisoli, G. Sansone, S. Stagira, S. De Silvestri, and O. Svelto, “Generation of 3.8-fs pulses from adaptive compression of a cascaded hollow fiber supercontinuum,” Opt. Lett. 28, 1987–1989 (2003). [CrossRef] [PubMed]
7. C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, 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]
8. A. Zair, A. Guandalini, F. Schapper, M. Holler, J. Biegert, L. Gallmann, U. Keller, A. Couairon, M. Franco, and A. Mysyrowicz, “Spatio-temporal characterization of few-cycle pulses obtained by filamentation,” Opt. Express 15, 5394–5404 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-9-5394. [CrossRef] [PubMed]
9. A. Couairon, J. Biegert, C. P. Hauri, W. Kornelis, F. W. Helbing, U. Keller, and A. Mysyrowicz, “Selfcompression of ultra-short laser pulses down to one optical cycle by filamentation,” J. Mod. Opt. 53, 75–85 (2006). [CrossRef]
10. A. Couairon, M. Franco, A. Mysyrowicz, J. Biegert, and U. Keller, “Pulse self-compression to the single-cycle limit by filamentation in a gas with a pressure gradient,” Opt. Lett. 30, 2657–2659 (2006). [CrossRef]
11. A. Mysyrowicz, A. Couairon, and U. Keller, “Self-compression of optical laser pulses by filamentation,” New J. Phys. 10, 025023 (2008). [CrossRef]
12. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–189 (2007). [CrossRef]
13. S. Akturk, C. D’Amico, M. Franco, A. Couairon, and A. Mysyrowicz, “A simple method for determination of nonlinear propagation regimes in gases,” Opt. Express 15, 15260–15267 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-23-15260. [CrossRef] [PubMed]
14. J. H. Marburger, “Self-focusing: Theory,” Prog. Quantum. Electron. 4, 35–110 (1975). [CrossRef]
16. F. Bragheri, D. Faccio, A. Couairon, A. Matijosius, G. Tamosauskas, A. Varanavicius, V. Degiorgio, A. Piskarskas, and P. Di Trapani, “Conical-emission and shock-front dynamics in femtosecond laser-pulse filamentation,” Phys. Rev. A 76, 025801 (2007). [CrossRef]
17. I. G. Koprinkov, “Ionization variation of the group velocity dispersion by high-intensity optical pulses,” Appl. Phys. B 76, 359–361 (2004). [CrossRef]
18. C. P. Hauri, A. Trisorio, M. Merano, G. Rey, R. B. Lopez-Martens, and G. Mourou, “Generation of high-fidelity, down-chirped sub-10 fs mJ pulses through filamentation for driving relativistic laser-matter interactions at 1 kHz,” Appl. Phys. Lett. 89 (2006). [CrossRef]
21. D. Faccio, A. Averchi, A. Lotti, P. Di Trapani, A. Couairon, D. Papazoglou, and S. Tzortzakis, “Ultrashort laser pulse filamentation from spontaneous X-Wave formation in air,” Opt. Express 16, 1565–1570 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-3-1565. [CrossRef] [PubMed]
22. A. Couairon, S. Tzortzakis, L. Berge, M. Franco, B. Prade, and A. Mysyrowicz, “Infrared femtosecond light filaments in air: simulations and experiments,” J. Opt. Soc. Am. B 19, 1117–1131 (2002). [CrossRef]
23. L. T. Vuong, R. B. Lopez-Martens, C. P. Hauri, and A. L. Gaeta, “Spectral reshaping and pulse compression via sequential filamentation in gases,” Opt. Express 16, 390–401 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-1-390. [CrossRef] [PubMed]
24. R. A. Altes, “Detection, estimation, and classification with spectrograms,” J. Acoust. Soc. Am. 67, 1232–1248 (1980). [CrossRef]
25. L. Cohen, “Time-Frequency Distributions-A Review,” Proceedings of the IEEE 77, 941–981 (1989). [CrossRef]
26. R. Trebino, Frequency-Resolved Optical Gating (Kluwer Academic Publishers, Boston, 2002). [CrossRef]
27. H. S. Nawab, T. F. Quatieri, and J. S. Lim, “Signal reconstruction from short-time fourier transform magnitude,” IEEE Trans. Acoust. Speech. Signal. Process. ASSP-31 , 986–998 (1983). [CrossRef]
28. S. Linden, J. Kuhl, and H. Giessen, “Amplitude and phase characterization of weak blue ultrashort pulses by downconversion,” Opt. Lett. 24, 569–571 (1999). [CrossRef]
29. R. Trebino and D. J. Kane, “Using phase retrieval to measure the intensity and phase of ultrashort pulses: frequency-resolved optical gating,” J. Opt. Soc. Am. A 10, 1101–1111 (1993). [CrossRef]
30. R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, and B. A. Richman, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997). [CrossRef]
31. S. Akturk, C. D’Amico, and A. Mysyrowicz, “Measuring ultrashort pulses in the single-cycle regime using frequency-resolved optical gating,” J. Opt. Soc. Am. B 25 (2008). [CrossRef]
32. A. Baltuska, M. S. Pshenichnikov, and D. A. Wiersma, “Second-Harmonic Generation Frequency-Resolved Optical Gating in the Single-Cycle Regime,” IEEE J. Quant. Electron. 35, 459–478 (1999). [CrossRef]
33. A. Efimov, A. J. Taylor, F. G. Omenetto, A. V. Yulin, N. Y. Joly, F. Biancalana, D. V. Skryabin, J. C. Knight, and P. S. J. Russel, “Time-spectrally-resolved ultrafast nonlinear dynamics in small-core photonic crystal fibers: Experiment and modelling,” Opt. Express 12, 6498–6507 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-26-6498. [CrossRef] [PubMed]
34. R. W. Boyd, Nonlinear Optics (Academic Press, 2003).
35. T. Brabec and F. Krausz, “Nonlinear Optical Pulse Propagation in the Single-Cycle Regime,” Phys. Rev. Lett. 78, 3282–3285 (1997). [CrossRef]
36. J. Liu, H. Schroeder, S. L. Chin, R. Li, W. Yu, and Z. Xu, “Space-frequency coupling, conical waves and small-scale filamentation in water,” Phys. Rev. A 72, 053817 (2005). [CrossRef]
37. A. Couairon, “Dynamics of femtosecond filamentation from saturation of self-focusing laser pulses,” Phys. Rev. A 68, 015801 (2003). [CrossRef]
38. D. Faccio, M. A. Porras, A. Dubietis, G. Tamosauskas, E. Kucinskas, A. Couairon, and P. Di Trapani, “Angular and chromatic dispersion in Kerr-driven conical emission,” Opt. Commun. 265, 672–677 (2007). [CrossRef]
39. S. Akturk, C. D’Amico, M. Franco, A. Couairon, and A. Mysyrowicz, “Pulse shortening, spatial mode cleaning, and intense terahertz generation by filamentation in xenon,” Phys. Rev. A 76, 063819 (2007). [CrossRef]