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We report on a new technique to produce high power sub-picosecond pulses from a fiber amplifier. We use parabolic pulse shaping of 27ps transform-limited pulses in order to control nonlinear effects in the fiber amplifier. 63MW, 780fs pulses with 25W average power were obtained, and ways to scale the technique to higher peak powers were identified.
© 2013 Optical Society of America
1. Introduction
Lasers based on Yb doped active fibers are particularly suitable for high average power generation, thanks to fibers geometry that ensures excellent heat dissipation. Ultrafast fiber lasers are thus of interest in many fields of physics and industry, for high speed process in micromachining, scribing, cutting of various materials. Transition between thermal and non-thermal ablation is known to be in the 1–10 ps range, which makes it particularly interesting to work with pulses shorter than 1 ps.
Many years of research lead to impressive performance demonstrations of ultrafast fiber lasers, and fibers ability to amplify short pulses to very high average powers is now well-established [1]. Besides, recent development of new fiber design enabled to manufacture fibers with exceptionally large mode area, corresponding to a core diameter size in excess of one hundred times the guided light wavelength [2]. These so-called large pitch fibers (LPF), once doped, enable to amplify pulses to energies in the millijoule range [3] and to high average powers in excess of 200W [4].
Providing reliable high power picosecond and femtosecond sources both for academic laser-matter interaction investigation, and for industrial applications constitutes the next step of these lasers development. The simplest laser architecture including the lowest number of components is preferred to build compact and robust laser sources. Investigating set-up simplifications requires a perfect control and understanding of phenomena at stake in short pulses amplification, which may open the way to innovative amplifying techniques development.
Up to now, the chirped pulse amplification (CPA) technique introduced in 1985 [5] has been mainly used to generate high energy and/or high power sub-picosecond pulses. In this technique temporally stretched pulses are used to stay below the damage fluence threshold of the active medium and to avoid nonlinear effects, responsible for dramatic spectral and temporal phase distortions, and spatial self-focusing. However, because of light tight confinement over long distances in fibers, fiber based systems may suffer from nonlinear spectral phase distortions even if the CPA scheme is implemented. It has been shown that accumulated B integral as low as 1rad could be enough to significatively degrades compressed pulse temporal quality [6].
To generate high contrast short pulses using a fiber based active medium, it is therefore desirable to control nonlinear effects, and find methods to handle them rather than avoid them. Stretching ratios in most fiber chirped pulse amplification (FCPA) systems are indeed often closed to the pratical limit, sometimes as high as 1:5000 [7], and yet non negligible nonlinear phase is still accumulated during amplification, leading to a compressed pulse contrast degradation [7, 8, 1]. Considerable attention has been paid over the past years to develop self-phase modulation (SPM) control solutions in FCPA systems, since the sole use of a stretcher appears to be ineffective to eliminate nonlinear phase distorsions. Most of them are only relevant for initial pulse duration range below 500fs. For instance it has been shown that a large amount of third-order dispersion (TOD) can be used to flatten the spectral phase in presence of high nonlinearities [9, 10]. This technique enabled to obtain high contrast 6μJ 240fs pulses in presence of 17π B integral [11], and 100μJ 270fs pulses despite 5.4π B integral [12]. More classical methods consist in the use of active spectral phase pre-control to compensate for self-phase modulation, either with an adaptive control loop [13, 14], or without one [15]. These methods rely on broad-band oscillators: the latter are needed to accumulate sufficient amount of TOD on the one hand, or to modulate spectral phase or intensity on the other hand. However, broad linewidths introduce complexity in a laser system, as they involve precise dispersion management, and in some cases spectral deformation due to the limited active medium gain bandwidth, that also induces temporal deformation on chirped pulses, detrimental for instance in parabolic amplification [16]. One can get rid of these limitations by using narrow linewidths. Stretching pulses with conventional means is uneasy while using narrow linewidth long pulses, that paradoxically makes CPA scheme difficult to implement, and opens the way to new amplification schemes investigation.
In this paper, we demonstrate a method based on initially narrow linewidth pulses, i.e. transform-limited (TL) pulses in the tens of picoseconds pulse duration range, that enables to control nonlinear spectral phase accumulated during amplification in Yb-doped rod-type LPF, and then to compress pulses to sub-picosecond duration. Thanks to the relatively long initial pulse duration, very small stretching ratio can be used, which differentiates the method from a classical CPA scheme.
2. Temporal pulse shaping
TL pulses longer than 15ps have a spectral full width at half maximum (FWHM) below 0.1nm. These pulses are short enough to be amplified to MW peak powers in rod-type fibers [17], where they accumulate significant amount of nonlinear spectral phase, but too long to be easily stretched before amplification. As a result nonlinear spectral phase cannot be avoided during amplification, and the amplifier configuration based on initial picosecond pulses can be used to investigate the nonlinear effects control issue.
One method in order to control nonlinear effects is to shape pulses temporal profile. The SPM induced temporal phase ϕNL and chirp ωNL are indeed proportional respectively to the intensity temporal profile I and to its first derivative:
As a consequence, temporal pulse shaping allows one to directly act on the pulse temporal chirp ωNL. It is well-known that parabolic temporal profile leads to a SPM induced positive linear chirp. Amplification of parabolic pulses in fibers has already been studied starting from initial pulses shorter than 500fs, both theoretically [18], and experimentally [19]. In these two examples the method has been found to systematically hit the gain bandwidth limitation, that induces spectral and temporal intensity profile distorsions, detrimental in order to keep the SPM chirp linear during amplification. This limitation may be overcome using pre-compensating spectral phase active control or TOD compensation. However, one can also avoid it by using narrow linewidth initial pulses. We show here that using tens of picoseconds parabolic pulses with moderate bandwidth as the input of an amplifier makes it possible to accumulate a well-controlled linear SPM induced chirp during amplification, as the parabolic shape is maintained in the absence of gain saturation.To produce parabolic pulses starting from narrow linewidth pulses, mainly two methods can be used: parabolic amplification where an asymptotic self-similar pulse is generated [20, 21], or using passive fibers [22]. The former imposes use of an amplifying medium, while the latter does not, making it more attractive for our use.
Pulse evolution in a passive fiber can be described by the nonlinear Schrödinger equation (NLSE) [23]:
where A is the slowly varying pulse envelope in a co-moving frame, β2 and β3 respectively the group velocity dispersion (GVD) and third-order dispersion parameters, γ the nonlinearity parameter, and α the loss parameter. Usually the following parameters are used to describe the pulse propagation regime: N is the soliton number, LD the dispersion length, LNL the nonlinearity length, T0 the 1/e pulse width, and Pc the peak power. The soliton number is helpful to estimate whether the propagation regime is dominated by dispersion (N<1) or by nonlinearities (N≫1).The joint action of dispersion and nonlinearities acts to shape temporal and spectral pulse profiles [24], with a linearized chirp compared to a pure SPM induced chirp.
By solving the NLSE equation using the Split-Step Fourier method, one can describe pulse evolution in a passive fiber. We consider a passive fiber characterized by the following parameters: β2= 25.10−3ps2.m−1, β3=6.10−5ps3.m−1 and γ= 8.10−3 W−1.m−1. The loss parameter α is computed considering an attenuation of 2dB.km−1 at 1030nm in the passive fiber. All the theoretical and experimental results presented in this paper have been obtained using this particular passive fiber. Starting from 1.3nJ, 27ps gaussian pulses, pulse evolution in a segment of passive fiber is represented in Fig. 1. The input pulse soliton number for these parameters is N=64.
Fig. 1 Simulated temporal pulse profile (a) and spectrum (b) evolution for a N=64 27ps gaussian shaped pulse propagating in a segment of passive fiber.
In Fig. 1 the final pulse width is ∼57ps and the total accumulated B integral ∼18π. One can see that the temporal pulse profile evolves during propagation in the passive fiber from a gaussian shape to an almost parabolic shape.
It has been shown that a quasi perfect parabolic temporal shape can be obtained using passive fibers, providing initially gaussian shaped pulses with a soliton number N=2.6 are propagated through 0.42LD length of fiber [22].
In the previous example corresponding to Fig. 1, the input pulse soliton number is equal to 64, leading to a non perfect parabolic pulse shape at the output of the passive fiber. A lower soliton number could be obtained by reducing input energy. However, as our goal is to amplify temporally shaped pulses in rod-type LPF, it is desirable to use the highest energy possible at the input of the amplifier, both to reduce gain and to reach higher final output energy. Working with soliton numbers higher than 2.6 is beneficial in order to obtain sufficient energy at the LPF amplifier input on the one hand, and a pulse shape thanks to which nonlinear effects can be controlled in the amplifier on the other hand.
Let us note that in Fig. 1 pulse propagation has been stopped before the “wave-breaking” (WB) phenomenon occurs [25, 26], that would generate side lobes in the spectrum and oscillatory features in the temporal shape. If the pulse at the 200m passive fiber output presented in Fig. 1(a) is further propagated in the same normally dispersive fiber, it will eventually be affected by WB. Nevertheless it has been shown that such a pulse shape can be preserved during propagation, provided it is propagated in a high intensity regime [27, 22]. Our amplifers work in this high intensity regime, and no WB is expected along the whole system.
3. Temporally shaped pulses compression
An initially TL gaussian pulse acquires an almost linear chirp while propagating in a passive fiber. High temporal quality pulses can therefore be obtained by compressing pulses chirped by self-phase modulation in fibers [28]. However, depending on initial pulse duration, TOD may have detrimental impact on the final pulse contrast ratio. To evaluate the TOD impact we performed simulations which are presented in this section.
We consider a particular case in which N=64 gaussian pulses are propagated in a segment of passive fiber, until they reach 18π B integral. With constant soliton number and accumulated B integral, one can show that pulses longer than 5ps have similar pulse and spectrum shape evolution in the passive fiber. For shorter pulses, TOD in the fiber induces spectral and temporal assymetry. In Fig. 2, we present pulse shape and spectrum for these N=64 gaussian pulses with 18π of B integral at the output of the passive fiber, and initial pulse duration longer than 5ps. Time axis is normalized with respect to pulse FWHM τp in Fig. 2(a), and wavelength axis with respect to spectral width Sλ in Fig. 2(b).
Fig. 2 Simulated temporal intensity profile (a) and spectrum (b) of N=64 pulses with 18π B integral, and initial pulse duration >5ps after propagation in a segment of passive fiber.
Non negligible amount of TOD can be added in the compressor, based in our case on a pair of 1800 l.mm−1 gratings. By taking TOD in the compressor into account in our simulation, we numerically compress N=64 pulses that have accumulated 18π of B integral in the passive fiber. The compressor is used at Littrow angle. In Fig. 3 we present compression results for two initial pulse durations, τp=5ps and τp=27ps.
Fig. 3 Simulated compressed pulse intensity profile obtained using 1800 l.mm−1 gratings at the output of a segment of passive fiber, for N=64 pulses with 18π B integral, and initial pulse duration τp=5ps (a) τp=27ps (b).
We estimate that the compressed pulse relative peak power is about 70% for the 5ps initial FWHM pulse, whereas it is 82% for the 27ps initial FWHM one, compared to perfect gaussian pulses. Furthermore, one can see in Fig. 3(b) that the final pulse profile is almost symmetrical, which is not the case of the Fig. 3(a) pulse profile. This means we expect TOD coming either from the fiber or the compressor to have only a minor impact when N=64, 27ps pulses accumulate 18π B integral in a normally dispersive passive fiber, even if we use highly dispersive 1800 l.mm−1 gratings for pulse compression.
4. Experimental results
In our experiment described in Fig. 4, we use 27ps TL gaussian pulses at 1030nm and propagate them through 200m of passive fiber. The 27ps pulses are produced by a mode-locked Yb doped fiber laser.
The passive fiber input energy is fixed to 1.3nJ, and the resulting soliton number N=64 is consistent with the simulation parameters used in sections 2 and 3. The low input energy is below Raman effect threshold, no Stokes wave is identified in the spectrum measured at the output of the pulse shaping stage.
In a preliminary experiment, the pulses are compressed directly at the output of the passive fiber in order to measure the stretched quasi-parabolic pulse temporal shape using a cross-correlation measurement between the shaped pulses and the compressed pulses. Temporal profile resulting from cross-correlation and spectrum measurements are given in Fig. 5.
Fig. 5 Cross-correlation (a) and spectrum (b) measurements of N=64, 27ps gaussian shaped pulses after propagation through 200m of passive fiber.
As seen in Fig. 5(a) the temporal shape is almost parabolic. The blue curves correspond to the calculated temporal and spectral pulse profiles as presented in section 2. Thanks to the model in good accordance with the measurements, we are able to estimate the total accumulated B integral in the passive fiber to 18π. Pulses are then amplified through two stages of LPF amplifiers used in single pass configuration, and compressed as shown in Fig. 4. The first LPF corresponds to the so-called LPF30 [29] and has a core diameter of 53μm. The second one has a core diameter of 66μm. Their lengths are both equal to 75cm, and they are pumped by independent 100W diodes. In the first amplifier stage the pump power is fixed to obtain an output average power of 200mW at 500kHz. Output energy and average power at 500kHz as a function of the pump power in the second amplifier stage are given in Fig. 6.
Fig. 6 Output energy and average power at 500kHz as a function of pump power in the second amplifier stage.
At the maximum pump power we obtain 30W and 60μJ with a polarization extinction ratio of 21dB. High power compression has been performed using transmission gratings with 1800 l.mm−1. A constant compression efficiency of 82.5% has been measured for incident average powers from 3 to 30W. Cross-correlation and spectrum measurements after amplification are performed for each output power level, and are shown in Fig. 7.
As expected we observe that the pulse shape is maintained during amplification. On the other hand the spectrum broadens to reach the final spectral width of ∼3.7nm. The B integral accumulated in the amplifier is evaluated to ∼3.8π. As 18π B integral has already been accumulated in the passive fiber, the total B integral at the maximum output energy of 60μJ is estimated to ∼22π.
In the Fig. 8 we give the autocorrelation traces of compressed pulses for the lowest and highest output energy levels. FWHM given in the figure are autocorrelation trace widths and not pulse durations.
In Fig. 8 (b), the autocorrelation FWHM is slightly shorter than the one in Fig. 8(a) owing to SPM spectral broadening during amplification (see Fig. 7(b)). The autocorrelation trace fit presented in Fig. 8(b) is obtained by solving the NLSE using the Split-Step Fourier method to fit pulse and spectrum of Fig. 5, by adding 3.8π nonlinear phase on these pulses that corresponds to the nonlinear phase accumulated during pulses amplification, and then by numerically compressing them. It enables to estimate the final pulse FWHM to 780 ± 40 fs. At the maximum output energy of 60μJ we obtained 49μJ after compression, and the compressed pulse duration of 780fs leads to 63MW peak power.
At the maximum output energy level, the time-bandwidth product is evaluated to 0.84, and we estimate that about 10% of the energy is contained in the pulse wings. The pulse contrast ratio is slightly improved compared to Fig. 3(b), we believe this is due to the small spectral reshaping that occurs in the amplifier, as seen in Fig. 7(b).
Let us note that even a perfect parabolic pulse cannot lead to a sidelobe-free pulse, and in particular when the pulse is propagated in a pure self-phase modulation regime [30]. However in our experiment we work with a non perfect parabolic shape as a soliton number of 64 is used, meaning the final pulse contrast ratio can still be improved by working with a soliton number in the passive fiber closer to 2.6.
To conclude we manage to obtain 780fs, 25W and 63MW peak power pulses starting from 27ps TL pulses, and with a very simple laser architecture. In the next section we will show that the final compressed pulse contrast obtained is improved compared to 27ps TL gaussian pulses amplified and compressed without initial temporal pulse shaping.
5. Discussion
To demonstrate the previously presented technique potential, we compare compressed pulse contrasts after amplification with and without the pulse shaping stage. For the comparison 27ps TL pulses are amplified through two LPF amplifier stages without initial pulse shaping. We used the same LPF fibers as presented in Fig. 4, the first one in double pass configuration and the second one in single pass configuration. The goal is to accumulate similar amount of nonlinear phase compared to the case where the pulse shaping stage is used, i.e. a B integral in the order of 20π and a final spectral width around 3nm, and then compress pulses to compare pulse contrasts. At a repetition rate of 300kHz, 27ps TL pulses are then amplified to 40μJ, corresponding to a final B integral of 17π and a spectral width of 3.1 ± 0.1nm. The energy is limited to 40μJ to stay below the amplifier damage threshold. In Fig. 9 the autocorrelation trace of the gaussian 27ps amplified pulses and spectrum at 40μJ are presented.
Fig. 9 Autocorrelation (a) and spectrum (b) measurements of 40μJ and 27ps gaussian shaped pulses directly amplified in two LPF amplifier stages.
The 38ps autocorrelation FWHM corresponds to 27ps FWHM pulses.
Figure 10 gives simulated Polarization-gate (PG) FROG traces [31, 32] for pulses represented respectively in Fig. 7 for 60μJ output energy and Fig. 9.
Fig. 10 Simulated PG FROG traces (a) 1.3nJ, 27ps TL pulses after propagation through 200m of passive fiber, and amplification to 60μJ (22π B integral). (b) 27ps gaussian pulses directly amplified to 40μJ (17π B integral).
One can see the linearized chirp in Fig. 10(a) compared to Fig. 10(b).
Pulses corresponding to Fig. 9 are compressed using a pair of 1800 l.mm−1 gratings, and autocorrelation traces of compressed pulses amplified with and without the initial temporal shaping stage are compared in Fig. 11.
Fig. 11 Compressed pulse autocorrelations comparison for pulses amplified in rod-type fibers with (black line) and without (red line) initial temporal pulse shaping stage. (a) Linear scale (b) Logarithmic scale.
Time axis is normalized with respect to pulses FWHM τp.
The experimental autocorrelation trace FWHM corresponding to the compressed gaussian pulse (red curve) is estimated to 1.13 ± 0.06 ps versus 1.03 ± 0.05 ps for the compressed shaped pulse (black curve).
We can see in Fig. 11 that even if the accumulated B integral is higher when a temporal pulse shaping stage is used before amplification, the final compressed pulse contrast is improved compared to gaussian shaped pulses directly amplified before compression. In the present case we compress pulses to a 35 times shorter pulse duration than the initial 27ps pulse duration. The N=64 soliton number is responsible for a part of the energy contained in the pulse wings, that could be reduced by working on the initial pulse shaping stage.
The technique presented here offers promising results for high peak power subpicosecond pulses generation in rod type fibers, as it constitutes a very simple way to build a femtosecond laser starting from a picosecond laser.
6. Conclusion
Ultra-short pulses production is usually performed using seed lasers that produce pulses in the same pulse duration range than the desired final one, and using a CPA architecture.
In this paper we demonstrate a technique in which initially narrow linedwidth pulses, in the tens of picoseconds pulse duration range, can be amplified, accumulate high nonlinearities and then be compressed 20 to 50 times shorter than the initial pulse duration, with an improved contrast compared to compressed gaussian pulses that have accumulated a similar amount of B integral and have a similar final pulse duration.
The use of passive fiber to temporally shape pulses is straightforward and constitutes the key element that ensures nonlinear effects control in the amplifier. We have experimentally demonstrated that the pulse shape is maintained during amplification, since the amplifier is not saturated, the narrow linedwidths are below gain bandwidth, GVD is negligible in our short rod-type fiber amplifier and pulses are propagated in a high intensity regime. Hence, nonlinear spectral phase added during amplification only depends on the initial temporal pulse shape, which gives a powerful way to control nonlinear effects in any amplifier stage.
The use of relatively high initial soliton number, N=64 for 27ps TL pulses, leads to a compressed pulse contrast ratio that is not notably degraded after amplification. This contrast ratio could be improved by working with a more parabolic pulse shape at the output of the passive fiber, using an initial soliton number closer to 2.6 in the passive fiber. The presented technique can therefore be scaled to higher peak power.
References and links
1. T. Eidam, S. Hanf, E. Seise, T. V. Andersen, T. Gabler, C. Wirth, T. Schreiber, J. Limpert, and A. Tünnermann, “Femtosecond fiber CPA system emitting 830W average output power,” Opt. Lett. 35, 94–96 (2010). [CrossRef] [PubMed]
2. F. Stutzki, F. Jansen, A. Liem, C. Jauregui, J. Limpert, and A. Tünnermann, “26mJ, 130W Q-switched fiber-laser system with near-diffraction-limited beam quality,” Opt. Lett. 37, 1073–1075 (2012). [CrossRef] [PubMed]
3. T. Eidam, J. Rothhardt, F. Stutzki, F. Jansen, S. H¨adrich, H. Carstens, C. Jauregui, J. Limpert, and A. Tünnermann, “Fiber chirped-pulse amplification system emitting 3.8GW peak power,” Opt. Express 19, 255–260 (2011). [CrossRef] [PubMed]
4. F. Stutzki, F. Jansen, T. Eidam, A. Steinmetz, C. Jauregui, J. Limpert, and A. Tünnermann, “High average power large-pitch fiber amplifier with robust single-mode operation,” Opt. Lett. 36, 689–691 (2011). [CrossRef] [PubMed]
5. D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56, 219–221 (1985). [CrossRef]
6. M. D. Perry, T. Ditmire, and B. C. Stuart, “Self-phase modulation in chirped-pulse amplification,” Opt. Lett. 19, 2149–2151 (1994). [CrossRef] [PubMed]
7. F. Röser, T. Eidam, J. Rothhardt, O. Schmidt, D. N. Schimpf, J. Limpert, and A. Tünnermann, “Millijoule pulse energy high repetition rate femtosecond fiber chirped-pulse amplification system,” Opt. Lett. 32, 3495–3497 (2007). [CrossRef] [PubMed]
8. F. Röser, D. Schimpf, O. Schmidt, B. Orta, K. Rademaker, J. Limpert, and A. Tünnermann, “90W average power 100μ J energy femtosecond fiber chirped-pulse amplification system,” Opt. Lett. 32, 2230–2232 (2007). [CrossRef]
9. L. Shah, Z. Liu, I. Hartl, G. Imeshev, G. C. Cho, and M. E. Fermann, “High energy femtosecond Yb cubicon fiber amplifier,” Opt. Express 13, 4717–4722 (2005). [CrossRef] [PubMed]
10. S. Zhou, L. Kuznetsova, A. Chong, and F. W. Wise, “Compensation of nonlinear phase shifts with third-order dispersion in short-pulse fiber amplifiers,” Opt. Express 13, 2149–2151 (2005). [CrossRef]
11. L. Kuznetsova and F. W. Wise, “Scaling of femtosecond Yb-doped fiber amplifiers to tens of microjoule pulse energy via nonlinear chirped pulse amplification,” Opt. Lett. 32, 2671–2673 (2007). [CrossRef] [PubMed]
12. Y. Zaouter, J. Boullet, E. Mottay, and E. Cormier, “Transform-limited 100 μ J, 340MW pulses from a nonlinear-fiber chirped-pulse amplifier using a mismatched grating stretcher-compressor,” Opt. Lett. 33, 2149–2151 (2008). [CrossRef]
13. F. He, H. S. S. Hung, J. H. V. Price, N. K. Daga, N. Naz, J. Prawiharjo, D. C. Hanna, D. P. Shepherd, D. J. Richardson, J. W. Dawson, C. W. Siders, and C. P. J. Barty, “High energy femtosecond fiber chirped pulse amplification system with adaptive phase control,” Opt. Express 16, 5813–5821 (2008). [CrossRef] [PubMed]
14. D. Nguyen, M. U. Piracha, and P. J. Delfyett, “Transform-limited pulses for chirped-pulse amplification systems utilizing an active feedback pulse shaping technique enabling five time increase in peak power,” Opt. Lett. 23, 4913–4915 (2012). [CrossRef]
15. D. N. Schimpf, E. Seise, T. Eidam, J. Limpert, and A. Tünnermann, “Control of the optical Kerr effect in chirped-pulse-amplification systems using model-based phase shaping,” Opt. Lett. 34, 3788–3790 (2009). [CrossRef] [PubMed]
16. D. B. S. Soh, J. Nilsson, and A. B. Grudinin, “Efficient femtosecond pulse generation using a parabolic amplifier combined with a pulse compressor. II. Finite gain-bandwidth effect,” J. Opt. Soc. Am. B 23, 10–19 (2006). [CrossRef]
17. J. Saby, D. Sangla, S. Pierrot, P. Deslandes, and F. Salin, “High power industrial picosecond laser from IR to UV,” Proc. SPIE 8601 (2013).
18. T. Schreiber, D. Schimpf, D. Müller, F. Röser, J. Limpert, and A. Tünnermann, “Influence of pulse shape in self-phase-modulation-limited chirped pulse fiber amplifier systems,” J. Opt. Soc. Am. B 24, 1809–1814 (2007). [CrossRef]
19. D. N. Papadopoulos, Y. Zaouter, M. Hanna, F. Druon, E. Mottay, E. Cormier, and P. Georges, “Generation of 63fs 4.1MW peak power pulses from a parabolic fiber amplifier operated beyond the gain bandwidth limit,” Opt. Lett. 32, 2520–2522 (2007). [CrossRef] [PubMed]
20. M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84, 6010–6013 (2000). [CrossRef] [PubMed]
21. S. Wang, B. Liu, C. Gu, Y. Song, C. Qian, M. Hu, L. Chai, and C. Wang, “Self-similar evolution in a short fiber amplifier through nonlinear pulse preshaping,” Opt. Lett. 38, 296–298 (2013). [CrossRef] [PubMed]
22. C. Finot, L. Provost, P. Petropoulos, and D. J. Richardson, “Parabolic pulse generation through passive nonlinear pulse reshaping in a normally dispersive two segment fiber device,” Opt. Express 15, 852–864 (2007). [CrossRef] [PubMed]
23. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, 2007).
24. H. Nakatsuka, D. Grischkowsky, and A. C. Balant, “Nonlinear Picosecond-Pulse Propagation through Optical Fibers with Positive Group Velocity Dispersion,” Phys. Rev. Lett. 47, 910–913 (1981). [CrossRef]
25. W. J. Tomlinson, R. H. Stolen, and A. M. Johnson, “Optical wave breaking of pulses in nonlinear optical fibers,” Opt. Lett. 10, 457–459 (1985). [CrossRef] [PubMed]
26. C. Finot, B. Kibler, L. Provost, and S. Wabnitz, “Beneficial impact of wave-breaking for coherent continuum formation in normally dispersive nonlinear fibers,” J. Opt. Soc. Am. B 25, 1938–1948 (2008). [CrossRef]
27. D. Anderson, M. Desaix, M. Karlsson, M. Lisak, and M. L. Quiroga-Teixeiro, “Wave-breaking-free pulses in nonlinear-optical fibers,” J. Opt. Soc. Am. B 10, 1185–1190 (1993). [CrossRef]
28. 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, 139–149 (1984). [CrossRef]
29. F. Jansen, F. Stutzki, H.-J. Otto, T. Eidam, A. Liem, C. Jauregui, J. Limpert, and A. Tünnermann, “Thermally induced waveguide changes in active fibers,” Opt. Express 20, 3997–4008 (2012). [CrossRef] [PubMed]
30. C. Finot, F. Parmigiani, P. Petropoulos, and D. J. Richardson, “Parabolic pulse evolution in normally dispersive fiber amplifiers preceding the similariton formation regime,” Opt. Express 14, 3161–3170 (2006). [CrossRef] [PubMed]
31. 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]
