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Abstract
A method of temporally compressing picosecond pulses is proposed. To increase the spectral broadening, two picosecond pulses at different central wavelengths are overlapped spatiotemporally to induce a modulation on the intensity envelope, thus leading to a high temporal intensity gradient. The combined pulse is then coupled into a gas-filled hollow-core fiber (HCF) to broaden the spectrum through nonlinear propagation. After that the pulse can be compressed by chirp compensation. This method is demonstrated numerically with two 1-ps/5-mJ pulses centered at 1053- and 1064-nm, respectively, which are coupled into a 250-μm-inner-diameter, 1-m-long HCF filled with 5-bar neon. After nonlinear propagation, the spectrum of the combined pulse is broadened significantly compared with the sum of the broadened spectra of a single 1-ps/10-mJ pulse centered at 1053- and 1064-nm. Under proper initial conditions, the pulse can be compressed down to ~16-fs. The influences of the energy ratio, time delay and wavelength gap between two input pulses, as well as the energy scaling are also discussed. These results show an alternative way to obtain ultrashort laser pulses from the picosecond laser technology, which can deliver both high peak power and high average power, and thus will benefit relevant researches in high-field laser physics.
© 2017 Optical Society of America
1. Introduction
Ultrashort intense laser pulses based on Ti:sapphire (Ti:Sa) have experienced rapid technological advancements and become powerful tools in the investigation of ultrafast science such as time-resolved measurements in atomic and molecular system [1], high-order harmonic generation [2, 3], laser particle acceleration [4, 5] and so on. The advantage of Ti:Sa is its broad gain width, which can support pulses shorter than 30-fs. Although the peak power is high enough to explore many non-perturbative physical processes, the repetition rate (usually 1 kHz or less) and the average power are low due to the relatively higher quantum defect of Ti:Sa [6]. However, many researches and applications require both high peak power and high repetition rate of the driving laser to achieve a better signal-noise-ratio. To obtain the high average power, new laser media with a lower quantum defect are needed, such as Ytterbium [7]. The past decade has seen great development in the high-average-power lasers. For instance, the ultrafast fiber chirped pulse amplification system could generate the energetic laser with 1-kW average power [8]. The single beam with an average output power of 1.1-kW was obtained by the compact Yb:YAG Innoslab laser [9]. The Ytterbium-doped thin-disk laser amplifier was used to deliver 1.1-kW high-average-power, sub-1-ps, 1031-nm pulses at 10-kHz repetition rate [10], and 1.4-kW high-average-power, sub-8-ps, 1030-nm pulses [11].
Although the average power is very high for these lasers, the sacrifice would be the narrower gain width compared with Ti:Sa laser, and the output temporal width of such pulses is usually limited to picosecond level. For most cutting-edge researches, pulses shorter than 100-fs are required. Therefore, nonlinear compression is needed to achieve this goal.
The principle of nonlinear compression for picosecond or femtosecond pulses is that through nonlinear propagation in a transparent medium the pulse can acquire an intensity-gradient-induced spectral broadening which is termed as the self-phase modulation (SPM) effect. The broadened spectrum has a linear chirp for a Gaussian pulse. After chirp compensation, the pulse width can be compressed. There are many implementations of nonlinear compression, such as using one or several pieces of solid transparent medium [12, 13], a gas cell [14–16] or a waveguide [17]. Recently, compression of high-average-power laser has been carried out via a fiber structure with a hollow core due to the advantages of large spectral broadening and good spatial mode control. Emaury et al. [18] experimentally demonstrated that pulses with 10-W average power and 88-fs duration could be obtained from the laser source with near 1-ps pulse duration after transmission through a Kagome fiber. Later, the corresponding output was increased to 46-W and 108-fs [19]. A higher average power in a Kagome fiber for nonlinear compression was shown in [20], where 76-W and 31-fs pulses could be generated. However, the energy of input pulses is limited to tens of microjoules due to the small hollow core of the Kagome fiber. For intense pulses with millijoule-level energy, the spectral broadening can be performed in a gas-filled hollow-core fiber (HCF) whose inner core is hundreds of micrometers. The gas-filled HCF has become an important compressor for femtosecond pulses in the visible and near-infrared spectral range [21–23], which can support pulses with not only high average power but also very high peak power [24]. For example, Rothhardt et al. [25] studied the use of two-stage gas-filled HCFs with an inner diameter of 250 μm to compress the 210-fs pulses down to 7.8-fs with 53-W high average power and 25-GW high peak power. Hädrich et al. [26] obtained the sub-2-cycle pulses with 216-W high average power and 17-GW high peak power, from a sub-300-fs laser source with two-stage 250-μm-inner-diameter, 1-m-long HCFs. However, for picosecond pulses, the spectral broadening is small due to the lower intensity gradient of the pulse and smaller nonlinearity provided by HCFs with a large core. Simply increasing the input pulse energy or gas pressure would result in complex spatiotemporal distributions of pulse energy and poor mode profiles.
In this work, we propose an alternative method to compress energetic picosecond pulses. To increase the spectral broadening, two picosecond pulses at different central wavelengths are overlapped spatiotemporally to induce a modulation on the intensity envelope, thus leading to a high temporal intensity gradient. The combined pulse is then coupled into a gas-filled HCF to undergo a nonlinear propagation, which results in a broadened spectrum. After that the pulse can be compressed by chirp compensation. It should be noted that using two-color pulses to achieve a broader spectrum through HCF compression is not a new idea [27–29], in which two femtosecond pulses are coupled into a gas-filled HCF, and both SPM and induced-phase modulation (IPM) effects contribute to spectral broadening. On the other hand, recently Plötner et al. [30] reported the high-power sub-ps pulse generation through compressing a frequency comb, which is obtained by coupling two narrow-linewidth continuous-wave lasers at central wavelengths near 1 μm into a fiber. The generation of the frequency comb is due to cascaded four-wave mixing (FWM). The work in the current manuscript somehow fills the gaps between the above two situations, i.e., the input pulse is of picosecond level compared with femtosecond and continuous-wave; all the SPM, IPM and FWM effects contribute to the spectral broadening, resulting in a spectrum consisted of discrete bands.
The reminder of this paper is organized as follows. In Section 2 the theoretical model is introduced. In Section 3 the simulation results are presented, in which two 1-ps/5-mJ pulses centered at 1053- and 1064-nm respectively are coupled into a gas-filled HCF for compression, showing the feasibility of this method. Then, the influences of the energy ratio, time delay and wavelength gap between two input pulses, as well as the energy scaling on the compression results, are discussed. The paper ends with a conclusion in Section 4.
2. Theoretical model
To simulate the propagation dynamics and nonlinear compression of optical pulses in HCFs, we use the waveguide version of the unidirectional pulse propagation equation (UPPE) [31], which is described as follows:
where Um is the fiber-mode field with the subscript m indicating the fiber mode order; αm and βm are the linear attenuation and dispersion of the fiber mode, respectively [32]; vg is the group velocity of the fundamental mode at the central wavelength of the pulse; J0(umr/a) is a zero-order Bessel function of the first kind, describing the mode field distribution; um is the mth zero point of J0(x); ω, a, c, ε0 are the angular frequency, fiber inner radius, light speed and permittivity in vacuum, respectively; P(r,ω) and j(r,ω) are the nonlinear polarization and plasma effects, respectively. The electric field of pulse in the time domain can be reconstructed as follows:where Re{····} indicates the real part of a quantity; FFT−1 is the inverse Fourier transform. |E|2 is normalized to the values of optical intensity. For the cubic Kerr effect, the nonlinear polarization is , where is the pulse-intensity envelope. The temporal gas-ionization effect is modeled by means of, where W(I) is the ionization rate calculated according to the Ammosov–Delone–Krainov (ADK) model [33]; Ui is the ionization potential; ρ is the electron density; ρnt is the neutral density of the gas. The plasma effect in the frequency domain is modeled by, where e, me, and τc are the electron charge, mass, and collision time, respectively. Assuming that the electrons are created at rest, the electron density ρ evolves aswhere σ is the impact ionization cross section.The initial input pulses are Gaussian and take the form as
where the subscripts 1, 2 indicate the two input pulses centered at different wavelengths; I, T, w, ω are peak intensity, temporal width, transverse waist and central frequency, respectively. It is also assumed that the combined pulse is coupled optimally into the fiber, i.e., with 98% pulse energy coupled to the fundamental mode for efficient transmission. The advantage of treating the two input pulses as a combined one is that it incorporates SPM, IPM and FWM into a simple expression which is the same as that of SPM effect. The beating of the two pulses centered at different frequencies leads to a modulation of the intensity envelope as shown in Fig. 1, which results in an increased intensity gradient. Because the instantaneous frequency of the pulse shifts by during nonlinear propagation [34], the modulation of the intensity envelope promotes the spectral broadening. Below a numerical example will be presented to demonstrate the feasibility of this method.
Fig. 1 Temporal intensity profiles of the combination of two pulses centered at 1053-nm and a varying wavelength.
3. Simulation results and discussions
The central wavelengths of commonly available picosecond lasers near 1-μm wavelength are 1053- and 1064-nm. The single pulse energy can be very large. Therefore in the simulations the inputs are two pulses centered at 1053- and 1064-nm, respectively, both with 1-ps full width at half maximum (FWHM) and 5-mJ energy. The combined pulse is coupled into a 250-μm-inner-diameter, 1-m-long HCF filled with 5-bar neon. For pulse compression, an HCF should be filled with noble gases rather than molecular gases to broaden the laser spectrum, because the noble gases can induce spectral broadening by pure SPM effect which shows linear chirp and can be compressed easily. On the other hand, five kinds of noble gases are available, among which the neon gas shows a good tradeoff between the ionization potential and the nonlinear refractive index, which is suitable to compress high-power picosecond intense pulses in our simulations. We also studied the use of argon. However, the gas pressure should be less than 100 mbar because of the detrimental ionization effect. This low pressure of argon cannot provide enough spectral broadening. If helium gas is employed, much higher pressure is needed for spectral broadening which is not practical. Thus here the neon is used. For comparison reasons, the propagation of a single 1-ps/10-mJ pulse centered at 1053- or 1064-nm under the otherwise same conditions is first simulated. Figures 2(a) and 2(b) show the corresponding spectral evolutions along propagation in the HCF. After 1-m propagation, the broadened spectra only support a Fourier-transform-limited (FTL) width of 164.3- and 163.6-fs. In Fig. 2(c), the spectrum of the combined pulse shows an obvious broadening. It is also interesting to note that the broadened spectrum consists of several discrete bands. This should be due to the (cascaded) FWM effect. The SPM and IPM effects also play an indispensable role in spectral broadening. Figure 2(d) compares the final output spectra of the three situations. The spectrum of the combined pulse is much broader than the sum of spectra of the two single pulses, and the FTL width of the combined pulse is 2.1-fs FWHM.
Fig. 2 Spectral intensity distributions during propagation for (a) a single pulse centered at 1053-nm, (b) a single pulse centered at 1064-nm, and (c) two pulses centered at 1053- and 1064-nm. (d) Output-pulse spectral intensity profiles for these three kinds of initial pulses.
Figure 3(a) shows the energy evolution of the combined pulse along propagation in the HCF. The red-dotted line indicates the pulse energy, and the other solid lines indicate the energies of different modes. The overall transmission efficiency is about 50%; most of the energy is in the fundamental mode. With the nonlinear refractive index [35] and the refractive index n0 [36], the corresponding critical power level of self-focusing is Pcr = λ2/(2πn0·n2) ≈149 GW at 5-bar. Here the peak input power is 10 GW, far less than this Pcr, Therefore, the energy transferred from the fundamental mode to high-order modes is limited, and the waveguide filters most of the high-order modes, allowing the transverse distribution to become smooth over the last 40-cm, which is shown in Fig. 3(b). In Fig. 3(c), it is interesting to note that the rear part of the temporal intensity is attenuated more than the leading part. The reason for this asymmetric evolution of temporal intensity is due to the plasma defocusing effect, which mainly affects the rear part of the pulse with long duration through transferring the pulse energy from the fundamental mode to high-order lossy modes. The spatiospectral distribution of the pulse at the outlet of HCFs is shown in Fig. 3(d). It can be seen that the overall spatial distribution is uniform thanks to the filtering effect of waveguide structure. The output pulse can be temporally compressed with proper chirp compensation. Figure 4(a) shows temporal profiles of the output pulse with different group delay dispersion (GDD) compensation. The intensity envelope consists of a main peak and two sub-peaks. With chirp compensation of −569 fs2, the sub-peaks can be greatly suppressed and the main peak can be compressed to ~16-fs. Figure 4(b) shows the spatiotemporal distribution of the compressed pulse. The spatial uniformity can be seen. The profile of the main peak and the phase are shown in Fig. 4(c). The phase is almost flat for the main peak, indicating that the chirp is well compensated.
Fig. 3 Propagation results of the combined pulse. (a) Evolutions of normalized total energies of the combined pulse (red-dotted line) and different fiber modes (solid lines with various colors represent different fiber mode orders m of Eq. (1)) during propagation; (b) transverse energy distributions; (c) evolutions of normalized total temporal intensity during propagation; (d) spatiospectral distribution of the output combined spectrum.
Fig. 4 (a) Total output-pulse temporal intensity profiles versus compensated GDDs; (b) spatiotemporal distribution of the combined pulse after compression; (c) temporal intensity and phase profiles of the compressed pulse.
Because the gas pressure determines the nonlinear refractive index, the influence of gas pressure on the spectral broadening and temporal nonlinear compression of the combined pulse is also considered. Figure 5 shows profiles of the total output-pulse intensity in time and frequency domains with optimal chirp compensation for the same input pulse in HCFs filled with 2- and 6-bar neon, obtained by integrating along the fiber-radial direction. It can be seen that with the increase of gas pressure, the spectrum of the combined pulse at the outlet of HCFs can be broadened more. At 2-bar, the spectrum is narrower as expected. The compressed pulse is 30 fs with chirp compensation of −1686 fs2. However, the sub-peaks are larger. While for 6-bar, the pulse can be compressed to 14 fs with smaller sub-peaks after compensated of −355 fs2. Higher pressures may result in the splitting of the main peak. This indicates that the proper gas pressure range would be 5- to 6-bar, at which a single short intense peak can be obtained and all the other sub-peaks are suppressed to a low level.
Fig. 5 Total output-pulse intensity profiles in (a, c) time and (b, d) frequency domains for the combined pulse with optimal chirp compensation through neon gas-filled HCFs at different gas pressure: (a, b) 2-bar and (c, d) 6-bar, and the red-dash curve represents initial-pulse spectral profiles.
According to [29], the intensity envelope can be modified by adjusting the energy ratio or delay of two input pulses, which will have an influence on the propagation characteristics of the combined pulse.
Figures 6(a)-6(c) show the total pulse temporal intensity profiles with 5-mJ/5-mJ, 7-mJ/3-mJ, and 9-mJ/1-mJ initial energy ratios at 1053- and 1064-nm central wavelengths. With the increase of the asymmetry of energy ratios, the modulation depth of the intensity envelope of the combined pulse is reduced, resulting in less spectral broadening as shown in Figs. 6(d)-6(f). This result is similar to that of a frequency comb generated by two continuous-wave lasers [37].
Fig. 6 (a)-(c) Total pulse temporal intensity profiles and (d)-(f) spectral intensity distributions along the fiber-radial direction for two pulses with (a, d) 5/5, (b, e) 7/3, and (c, f) 9/1 initial energy ratios at 1053- and 1064-nm central wavelengths.
As for the influence of the relative time delay of the two pulses, the results show that the change of time delay between −200 fs to 200 fs has little influence on the dynamics of the combined pulse due to the relatively long duration of picosecond pulses. Therefore, this method is robust against time jittering.
Another point needs to be considered is the influence of the wavelength gap between the two pulses. As shown in Fig. 1, the modulation of the intensity envelope depends on the wavelength gap. Small wavelength gap leads to low intensity gradient with few sub-peaks, while larger gap results in higher intensity gradient and more sub-peaks, for which it is more difficult to single out a compressed intensity peak. For comparison, two pulses centered at 1053- and 1060-nm, as well as two centered at 1053- and 1030-nm, are chosen to do the same simulations; the results are shown in Fig. 7. It can be seen that for the small wavelength gap, the spectral broadening of the combined pulse becomes weaker in Fig. 7(b) and the compressed pulse is 33 fs in Fig. 7(a). On the other hand, for the large wavelength gap, due to the increasing number of modulated peaks, it is difficult to obtain a single main peak after compression as shown in Fig. 7(c). Hence the central wavelength gap should be carefully chosen.
Fig. 7 (a) Combined temporal intensity profiles of two pulses at 1053-nm and a varying central wavelength. (b)-(e) Total pulse intensity profiles in time and frequency domains for pulses at different types of central wavelengths: (b, c) 1053- and 1060-nm, (d, e) 1053- and 1030-nm.
Last, the simulations with different total initial energies of the two pulses in HCFs filled with neon at proper gas pressures are studied. Figure 8(a) shows the normalized total energy evolutions for two 1-ps input pulses at 1053- and 1064-nm with different energies along propagation in HCFs filled with neon at proper gas pressure. With increasing total pulse energies, the gas pressure needs to be reduced to obtain a decent temporal profile at the output. Notice that the pulse transmission efficiency gradually decreases with higher pulse energies. It should be noted that for two 4-mJ pulses in HCFs filled with neon at 7 bar, the temporal intensity shows less asymmetry compared to that for two 5-mJ pulses at 5 bar and for two 6-mJ pulses at 4 bar, as shown in Figs. 8(d), 3(c) and 8(f), indicating that the plasma defocusing effect weakens with lower energy. The corresponding spectrum is broadened a little more than that for two 6-mJ pulses at 4 bar, due to the higher allowable gas pressure and smaller plasma defocusing effect. In Figs. 8(e) and 8(g), the spatiospectral distributions of the pulse at the outlet of HCF are shown, in which the overall spatial distributions are uniform. The corresponding FWHM durations are 15.5- and 14.9-fs with optimal chirp compensation in Figs. 8(b) and 8(c), respectively, similar with that of two 5-mJ pulses at 5 bar. This indicates that the compression method of using two input pulses with about 1-ps duration in a neon gas-filled HCF allows for a relatively flexible range of initial energies at proper gas pressures.
Fig. 8 (a) Evolutions of normalized total energies during propagation for two 1-ps/4-mJ pulses in HCFs filled with neon at 7 bar, two 5-mJ pulses at 5 bar, and two 6-mJ pulses at 4 bar, respectively. (b, c) Total output-pulse temporal intensity profiles for two 4-mJ pulses at 7 bar, and two 6-mJ pulses at 4 bar with optimal chirp compensation. (d)-(g) Temporal intensity distributions during propagation, and spectral intensity distributions along the fiber-radial direction: (d, e) two 4-mJ pulses at 7 bar, and (d, e) two 6-mJ pulses at 4 bar.
The above discussions are based on simulations with 1-ps pulses. For longer pulses, the choices of the parameters follow the same reasons. The success of this pulse compression method mainly relies on two points: first, the leading part of the combined pulse undergoes spectral broadening and accumulates enough plasma density; second, the plasma defocusing effect on the rear part of the pulse greatly shortens the temporal duration. The remaining leading part of the pulse can be further compressed through chirp compensation. However, the large compression ratio comes with a price, i.e., the transmission efficiency will be no more than 50%. The longer the picosecond pulse is, the lower the efficiency is. Therefore, it is better to be used for pulses with about 1-ps.
4. Conclusions
A method of temporally compressing picosecond pulses is proposed. To increase the spectral broadening, two picosecond pulses at different central wavelengths are overlapped spatiotemporally to induce a modulation on the intensity envelope, thus leading to a high temporal intensity gradient. The combined pulse is then coupled into a gas-filled HCF to undergo a nonlinear propagation, which results in a broadened spectrum. After that the pulse can be compressed by chirp compensation. This method is further studied extensively through numerical simulations. The results show that pulses shorter than 20 fs with several millijoules can be obtained. This shows an alternative way to obtain ultrashort laser pulses from the picosecond laser technology, which can deliver both high peak power and high average power, and thus will benefit relevant researches in high field laser physics.
Funding
National Natural Science Foundation of China (Grant Nos. 61521093); the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB1603); and the International S&T Cooperation Program of China (Grant No. 2016YFE0119300).
References and links
1. Y. Liu, X. Liu, Y. Deng, C. Wu, H. Jiang, and Q. Gong, “Selective steering of molecular multiple dissociative channels with strong few-cycle laser pulses,” Phys. Rev. Lett. 106(7), 073004 (2011). [PubMed]
2. A. Scrinzi, M. Y. Ivanov, R. Kienberger, and D. M. Villeneuve, “Attosecond physics,” J. Phys. B 39(1), R1–R37 (2005).
3. F. Krausz and M. Ivanov, “Attosecond physic,” Rev. Mod. Phys. 81(1), 163–234 (2009).
4. C. G. Geddes, C. S. Toth, J. Van Tilborg, E. Esarey, C. B. Schroeder, D. Bruhwiler, C. Nieter, J. Cary, and W. P. Leemans, “High-quality electron beams from a laser wakefield accelerator using plasma-channel guiding,” Nature 431(7008), 538–541 (2004). [PubMed]
5. A. Gopal, S. Herzer, A. Schmidt, P. Singh, A. Reinhard, W. Ziegler, D. Brömmel, A. Karmakar, P. Gibbon, U. Dillner, T. May, H.-G. Meyer, and G. G. Paulus, “Observation of Gigawatt-Class THz Pulses from a Compact Laser-Driven Particle Accelerator,” Phys. Rev. Lett. 111(7), 074802 (2013). [PubMed]
6. A. L. Cavalieri, E. Goulielmakis, B. Horvath, W. Helml, M. Schultze, M. Fieß, V. Pervak, L. Veisz, V. S. Yakovlev, M. Uiberacker, A. Apolonski, F. Krausz, and R. Kienberger, “Intense 1.5-cycle near infrared laser waveforms and their use for the generation of ultra-broadband soft-x-ray harmonic continua,” New J. Phys. 9(7), 242 (2007).
7. J. Limpert, A. Liem, T. Gabler, H. Zellmer, A. Tünnermann, S. Unger, S. Jetschke, and H.-R. Müller, “High-average-power picosecond Yb-doped fiber amplifier,” Opt. Lett. 26(23), 1849–1851 (2001). [PubMed]
8. M. Müller, M. Kienel, A. Klenke, T. Gottschall, E. Shestaev, M. Plötner, J. Limpert, and A. Tünnermann, “1 kW 1 mJ eight-channel ultrafast fiber laser,” Opt. Lett. 41(15), 3439–3442 (2016). [PubMed]
9. P. Russbueldt, T. Mans, J. Weitenberg, H. D. Hoffmann, and R. Poprawe, “Compact diode-pumped 1.1 kW Yb:YAG Innoslab femtosecond amplifier,” Opt. Lett. 35(24), 4169–4171 (2010). [PubMed]
10. C. Wandt, S. Klingebiel, M. Schultze, S. Prinz, C. Y. Teisset, S. Stark, C. Grebing, M. Häfner, R. Bessing, T. Herzig, A. Budnicki, D. Sutter, K. Michel, T. Nubbemeyer, F. Krausz, and T. Metzger, “1 kW Ultrafast Thin-Disk Amplifier System,” CLEO: Science and Innovations. Optical Society of America (2017).
11. J. P. Negel, A. Loescher, A. Voss, D. Bauer, D. Sutter, A. Killi, M. A. Ahmed, and T. Graf, “Ultrafast thin-disk multipass laser amplifier delivering 1.4 kW (4.7 mJ, 1030 nm) average power converted to 820 W at 515 nm and 234 W at 343 nm,” Opt. Express 23(16), 21064–21077 (2015). [PubMed]
12. C. H. Lu, Y. J. Tsou, H. Y. Chen, B. H. Chen, Y. C. Cheng, S. D. Yang, M. C. Chen, C. C. Hsu, and A. H. Kung, “Generation of intense supercontinuum in condensed media,” Optica 1, 400–406 (2014).
13. M. Seidel, G. Arisholm, J. Brons, V. Pervak, and O. Pronin, “All solid-state spectral broadening: an average and peak power scalable method for compression of ultrashort pulses,” Opt. Express 24(9), 9412–9428 (2016). [PubMed]
14. 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(6), 673–677 (2004).
15. C. 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). [PubMed]
16. J. Liu, X. W. Chen, R. W. Li, and T. Kobayashi, “Polarization-dependent pulse compression in an argon-filled cell through filamentation,” Laser Phys. Lett. 5(1), 45–47 (2008).
17. W. J. Tomlinson, R. H. Stolen, and C. V. Shank, “Compression of optical pulses chirped by self-phase modulation in fibers,” J. Opt. Soc. Am. B 1(2), 139 (1984).
18. F. Emaury, C. J. Saraceno, B. Debord, D. Ghosh, A. Diebold, F. Gèrôme, T. Südmeyer, F. Benabid, and U. Keller, “Efficient spectral broadening in the 100-W average power regime using gas-filled kagome HC-PCF and pulse compression,” Opt. Lett. 39(24), 6843–6846 (2014). [PubMed]
19. F. Emaury, A. Diebold, C. J. Saraceno, and U. Keller, “Compact extreme ultraviolet source at megahertz pulse repetition rate with a low-noise ultrafast thin-disk laser oscillator,” Optica 2(11), 980 (2015).
20. S. Hädrich, M. Krebs, A. Hoffmann, A. Klenke, J. Rothhardt, J. Limpert, and A. Tünnermann, “Exploring new avenues in high repetition rate table-top coherent extreme ultraviolet sources,” Light Sci. Appl. 4(8), e320 (2015).
21. B. F. Mansour, H. Anis, D. Zeidler, P. B. Corkum, and D. M. Villeneuve, “Generation of 11 fs pulses by using hollow-core gas-filled fibers at a 100 kHz repetition rate,” Opt. Lett. 31(21), 3185–3187 (2006). [PubMed]
22. M. Nisoli, S. D. Silvestri, and O. Svelto, “Generation of high energy 10 fs pulses by a new pulse compression technique,” Appl. Phys. Lett. 68(20), 2793–2795 (1996).
23. M. Nisoli, S. De Silvestri, O. Svelto, R. Szipöcs, K. Ferencz, C. Spielmann, S. Sartania, and F. Krausz, “Compression of high-energy laser pulses below 5 fs,” Opt. Lett. 22(8), 522–524 (1997). [PubMed]
24. J. Schulte, T. Sartorius, J. Weitenberg, A. Vernaleken, and P. Russbueldt, “Nonlinear pulse compression in a multi-pass cell,” Opt. Lett. 41(19), 4511–4514 (2016). [PubMed]
25. J. Rothhardt, S. Hädrich, A. Klenke, S. Demmler, A. Hoffmann, T. Gotschall, T. Eidam, M. Krebs, J. Limpert, and A. Tünnermann, “53 W average power few-cycle fiber laser system generating soft x rays up to the water window,” Opt. Lett. 39(17), 5224–5227 (2014). [PubMed]
26. S. Hädrich, M. Kienel, M. Müller, A. Klenke, J. Rothhardt, R. Klas, T. Gottschall, T. Eidam, A. Drozdy, P. Jójárt, Z. Várallyay, E. Cormier, K. Osvay, A. Tünnermann, and J. Limpert, “Energetic sub-2-cycle laser with 216 W average power,” Opt. Lett. 41(18), 4332–4335 (2016). [PubMed]
27. R. R. Alfano and P. P. Ho, “Self-, cross-, and induced-phase modulations of ultrashort laser pulse propagation,” IEEE J. Quantum Electron. 24, 351–364 (1988).
28. M. Yamashita, H. Sone, R. Morita, and H. Shigekawa, “Generation of monocycle-like optical pulses using induced-phase modulation between two-color femtosecond pulses with carrier phase locking,” IEEE J. Quantum Electron. 34, 2145–2149 (1998).
29. C. Manzoni, O. D. Mücke, G. Cirmi, S. Fang, J. Moses, S. W. Huang, K. H. Hong, G. Cerullo, and F. X. Kärtner, “Coherent pulse synthesis: towards sub-cycle optical waveforms,” Laser Photonics Rev. 9(2), 129–171 (2015).
30. M. Plötner, V. Bock, T. Schultze, F. Beier, T. Schreiber, R. Eberhardt, and A. Tünnermann, “High power sub-ps pulse generation by compression of a frequency comb obtained by a nonlinear broadened two colored seed,” Opt. Express 25(14), 16476–16483 (2017). [PubMed]
31. M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: from Maxwell’s to unidirectional equations,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(3 Pt 2), 036604 (2004). [PubMed]
32. E. A. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43(4), 1783–1809 (1964).
33. M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191 (1986).
34. G. P. Agrawal, “Ultrashort Pulse Propagation in Nonlinear Dispersive Fibers. In The Supercontinuum Laser Source (ed. R. R. Alfano,) 2nd ed,” (Springer).
35. M. Nurhuda, A. Suda, M. Kaku, and K. Midorikawa, “Optimization of hollow fiber pulse compression using pressure gradient,” Appl. Phys. B 89(2), 209–215 (2007).
36. https://refractiveindex.info/
37. A. Antikainen and G. P. Agrawal, “Dual-pump frequency comb generation in normally dispersive optical fibers,” J. Opt. Soc. Am. B 32(8), 1705–1711 (2015).
