Abstract

A high-energy and low repetition rate dispersion-mapped amplifier similariton oscillator with a large net intracavity anomalous dispersion and a linear cavity configuration is demonstrated experimentally at 1 μm. The numerical results confirm that self-similar evolution is accomplished in the gain fiber, and both the parabolic- and Gauss-shaped pulses can be emitted at different ports of the cavity, respectively. The maximum output power of 820 mW at a repetition rate of 8.6 MHz under a pump power of 12.76 W, corresponding to a pulse energy as high as of 95 nJ has been obtained.

© 2015 Optical Society of America

1. Introduction

With the advantages of short duration, high peak power, and broad spectrum, femtosecond lasers have been widely used for ultra-precision measurement, information and communication and micromachining. Among them, passively mode-locking fiber lasers have been investigated extensively. Pulse shape is also an important property for practical applications. Due to advantages [1,2 ] such as clean wings and strictly linear chirp, which has a potential to get higher pulse energy and peak power by amplifying and recompressing, the parabolic pulse has attracted wide interests in recent years [3,4 ]. The parabolic pulse was first observed in a fiber amplifier during self-similar evolution [5], wherein the seed pulse was evolved into parabolic pulse inside the gain fiber under the complex interaction of gain, dispersion and nonlinear effects [6–8 ]. It has identified in these reports that a longer gain fiber is needed to accomplish the self-similar evolution during amplification. However, the limitations for the self-similar amplifier with a longer gain fiber are the gain-bandwidth and stimulated Raman scattering [9], so that the output energy is restricted, the linear chirp is disturbed, and the beam quality is degraded at a high enough pulse energy. Therefore, a seed source at a lower repetition rate and a parabolic pulse shape is necessary and more suited for the self-similar amplifier to eliminate the limitations and scaling up the pulse energy [10].

The self-similar amplifier with feedback and spectral filtering has become an oscillator of interest (so called the self-similar laser or the similariton laser), where the parabolic pulses can be generated in the laser cavity under a net normal dispersion and was first reported by Ildayet al [11]. Since then, an avenue was opened to achieve high energy femtosecond pulse in this manner from fiber lasers. Many numerical and experimental progresses have been made in self-similar lasers [12,13 ]. It is well known that the pulses can evolve into parabolic pulses either in active fibers (so called the amplifier similaritons) [14,15 ] or in passive fibers (so called the passive similaritons) [16] in the self-similar lasers.

Recently, a novel mode-locking regime in an Er-doped fiber laser with similariton and soliton propagating respectively in each half of the cavity consisting of a segment gain fiber of normal dispersion and a segment passive fiber of negative dispersion, was reported by Oktem et al [17]. This laser with two pulse evolution mechanism in the same cavity was called the soliton-similariton laser. In practice, the laser can operate in the net normal or negative dispersion regions, depending on the intracavity dispersion map and dissipative mechanism. Subsequently, systematic discussions of the amplifier similaritons with the dispersion map were reported by Rennerger et al. [12,14 ], where the term was renamed as the dispersion-mapped amplifier similariton (DMAS). The pulse energy has been scaled to dozens of nJ in the self-similar lasers to be comparable to the similariton amplifier [18,19 ].

In this paper, a high-energy and low repetition rate DMAS oscillator with a long normal dispersion gain fiber, a large linear anomalous dispersion segment, a filter and a linear cavity configuration was designed and demonstrated at 1 μm. The numerical model for the cavity configuration is based on the nonlinear Schrödinger equation and the whole pulse shaping dynamics within one round-trip in the cavity is simulated. The numerical results confirm that the self-similar evolution and soliton-like pulses can be simultaneously achieved in the laser. Self-similar pulses with an average power of 820 mW at a repetition rate of 8.6 MHz, corresponding to a pulse energy of 95 nJ, are obtained. To the best of our knowledge, this is the first report to achieve amplifier similaritons at a repetition rate below 10 MHz and with a pulse energy as high as 95 nJ in the DMAS laser. The features of the laser make it ideal for micromachining [20] and as a seed source for self-similar amplification.

2. Experimental setup

A long enough gain fiber and the linear cavity configuration are more favorable to get a high energy amplifier similariton. The experimental setup of the DMAS mode-locking fiber laser is schematically shown in Fig. 1 . A segment of 10 m Yb-doped large-mode-area (LMA) polarization-maintaining (PM) photonic crystal fiber (PCF) is used in the linear cavity, which dominates the amplifier similariton evolution by double-pass within one intracavity round-trip. The Yb-doped double-clad LMA-PM-PCF (NKT Company) has a mode area of 660 μm2 and a NA of 0.03. Both of the fiber ends are fused and polished at 8° to suppress parasitic lasing. The laser is pumped through a dichroic mirror (DM) using a 20 W, 200 μm (NA = 0.22) fiber-coupled laser diode (LD, N-light) emitting at 976 nm. The pump laser is collimated and focused by a lens system composed of two aspheric lens (AL) with a focal length of 11 mm before coupled into the PCF. The laser beam, through the gain fiber, is collimated by an AL with a focal length of 18 mm, and separated by two DMs from the residual pump laser. A half wave plate (HWP) in combination with a polarization beam splitter (PBS) is used to improve the intracavity laser polarizability. A commercial semiconductor saturable absorption mirror (SESAM) is employed at one end of the cavity to initiate and maintain mode-locking. The SESAM (BATOP GmbH) has a fast relaxation time of 500 fs and a maximum modulation depth of 40% at 1060 nm. The dispersion map is provided by a pair of gratings (600 line/mm) at the other end of the cavity. The HWP placed before the grating pair is use to optimize the diffraction efficiency and to adjust output power. A slit is inserted between the grating pair and the closed end mirror, which is apt to adjust the operating wavelength and offer the dissipation.

 figure: Fig. 1

Fig. 1 Experimental setup.AL: aspheric lens; DM: dichroic mirror; PBS: polarization beam splitter; HWP: half-wave plate; GR: grating; HR: high-reflectivity mirror; OC: output coupler; SL: slit

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3. Round-trip pulse shaping dynamics

In contrast to previous amplifier similaritons with a single-pass amplifier in the ring lasers [14,17 ], there are forward and backed passes through the gain fiber within one round-trip in the linear cavity. How is the evolution process of DMAS for a double-pass of the gain fiber in the laser? Where is the optimal output port for a self-similar pulse in the cavity? To answer these questions, a numerical model for the laser is built based on the nonlinear Schrödinger equation (NLSE) with gain as follows [21]:

Az+iβ222At2=G2A+iγ|A|2A
where A is the electric field envelope, β2 represents the GVD, γ represents the Kerr nonlinearity, z is the propagation coordinate, and t is the local time. The linear gain coefficient is defined as:
G=G01+EpulseEsat/gain
where G0 corresponds to the small-signal gain determined by the pump power distribution along the gain fiber (see Table 1 ), Esat/gain is the gain saturation energy. Epulse is the pulse energy defined as:
Epulse=TR2TR2|A|2dt
where TR is the cavity round trip time. The SESAM is modeled as a reflectivity function:
R(t)=eδs(t)δns
where δns is the nonsaturable loss and δs(t) is the saturable loss. The δs(t) can be described by the equation [22]:
δs(t)=ΔRf(t)[1τrelf(t)dt+1]
where ΔR and τrel are the absorber’s modulation depth and the relaxation time, respectively. f(t) is expressed as:
f(t)=exp[(1τrel+|A|2Esat/abs)dt]
where Esat/abs is the saturation energy of the absorber.

Tables Icon

Table 1. Simulation parameters

The model is solved by a symmetric split-step Fourier method and the initial field is assumed to be white noise. The parameters related to the stable operation of mode-locking dynamics in the DMAS laser experiment are listed in Table 1.

For the gain fiber, L = 10 m, β2 = 0.024 ps2/m and γ = 9.1 × 10−5 m−1W−1. The slit (filter) has a rectangular spectrum width of 2 nm. The simulated results within one round-trip in the DMASlaser are shown in Fig. 2 . Chirp management in the laser by the dispersion map is more insightful for self-similar lasers [23], and so it is shown in Fig. 2(a). The pulse evolution towards to a parabola can be quantified with the misfit parameter M defined as [24]:

M2=(|A|2+|Apfit|2)2dt|A|4dt
where Ap-fit is the electric field envelope of a parabolic fit. The M evolution within one intracavity round-trip is indicated in Fig. 2(b). From Fig. 2(a), the sign symbol of the round-trip chirp in the cavity can be deduced from the dispersion compensation segment. The pulse chirp is negative before getting into the gain fiber (PCF-1, co-propagating between the pump and the pulse) and becomes normal after the PCF, i.e., there is a transition point for the chirp sign symbol, which can be explained by the power nature of the nonlinear attractor [14,21 ]. In the meantime, from Fig. 2(b), M is decreased quickly from 0.14 to 0.75. The chirp further drops because of spectral filtering by the SESAM, and M increases to 0.09. Then the pulse begins backward propagation into the gain fiber (PCF-2, counter-propagating between the pump and the pulse) again. In the PCF-2 segment, the chirp is always normal and accumulated gradually (Fig. 2(a)), while M is reduced slowly from 0.09 to 0.035 as the pulse is at the output position (OC, the zero-order reflection of the grating shown in Fig. 1). This is a clear indication that the output pulse has evolved towards a parabola very well, with the result shown in Fig. 2(c), which proves that self-similar evolution can be accomplished in this segment. At the 80%OC in the dispersion-compensation segment (DDL, shown in Fig. 2(a) and 2(b)), the positive chirp decreases and M rises quickly, because the chirp is balanced by the negative dispersion. The chirp reaches the lowest point as M rises to the largest value of 0.22 within the DDL segment. From there on, the chirp is reversed from positive to negative, and the negative chirp increases linearly by the over-compensation of the DDL. In contrast to the chirp rising monotonically after the reversal point in the DDL segment, M is first decreased quickly and then tends to be a constant between 0.13 and 0.14. This indicates that the operation state with a large anomalous dispersion after the DDL can be distinguished clearly from the previous pulse-propagation in the gain fiber for the similariton formation. The pulse just passing the spectral filter (SF, shown in Fig. 2) can be verified as having a perfect Gaussian shape by fitting, which should be a soliton-like pulse, as shown in Fig. 2(d). In addition, in Fig. 2(a), the fact that the negative chirp has a rise after the narrow SF is because the positive high-order chirp accumulated at both edges of the bandwidth is cut off. As a result, the DMAS laser is also a soliton-similariton laser, where the similariton evolves in the gain fiber and the soliton is formed in the anomalous dispersion section, respectively.

 figure: Fig. 2

Fig. 2 Simulated results: evolution of (a) pulse chirp and (b) M parameter within one round-trip in the cavity (SA: saturable absorber; OC: output coupler; DDL: dispersion delay line; SF: spectral filter); simulated pulse intensity and instantaneous frequency (c) at the OC and (d) at the position after SF.

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4. Experimental results

The DMAS laser is designed and built with parameters similar to those used in the simulations. The schematic setup is shown in Fig. 1. In the experiment, the zero-order reflected laser from the first grating is taken as the output port in order to obtain the similariton pulse. The slit here has two functions, both as a narrow spectral filter and a controller of the operation wavelength. Without the slit, the laser operates at a central wavelength of 1082 nm, which is far from the working wavelength of the SESAM at 1060 nm, preventing the initiation of the mode-locking operation. Mode-locking operation is self-started as the slit is inserted and the central wavelength of the laser is blue shifted to 1074 nm by moving the slit laterally. The other important parameter is dispersion map [25]. The laser operates only in the double-pulse state if the intracavity negative dispersion is not sufficient enough, while mode-locking operation becomes unstable with too much intracavity negative dispersion. For the laser, a stable single pulse operation can be achieved with the net intracavity negative dispersion around-0.89 ps2.

The mode-locking operation dynamics is illustrated in Figs. 3 and 4 . Figure 3 shows the output laser power recorded from the OC as a function of the pump power. The mode-locking operation threshold is 6.8 W of pump power, corresponding to an output power of 82 mW. With an increase of the pump power, the output power is increased linearly. The highest mode-locked single pulse output power of 820 mW is obtained for a pump power of 12.76W. The double-pulse state would occur when pump power is increased further. In order to gain an insight into the dynamics of the single pulse state in the DMAS laser, the pulse characteristics in the spectral and time domains as a function of the pump power are measured. The pulse durations and the spectral profiles versus the different pump power are demonstrated in Fig. 4. From in Figs. 4(a) and 4(b), both the pulse duration and spectral width broaden proportionally with the pump power, which indicates the feature of evolution in the amplifier fiber with a normal chirp. The maximum output power of 820 mW is generated with a duration of 6.1 ps and spectral width of 3.84 nm. The measured pulse train and its radio frequency (RF) spectrum are shown in Fig. 5 . A signal-to-noise ratio of 62 dB and RF of 8.6 MHz are identified, which corresponds to a highest pulse energy of 95nJ.

 figure: Fig. 3

Fig. 3 Output power as a function of pump power.

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 figure: Fig. 4

Fig. 4 (a) Autocorrelations and (b) spectra at different pump power.

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 figure: Fig. 5

Fig. 5 (a) Measured pulse train and (b) its radio frequency spectrum under pump power of 12.76W.

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To illustrate whether a parabolic pulse is achieved at OC or not, a pulse intensity retrieved by PICASO and a measured spectrum are plotted with black lines in Figs. 6(a) and 6(b) respectively. For comparison, a parabolic fit (red dash line), a hyperbolic secant fit (purple dash dot line) and a Gauss fit (olive dot line) are shown in same figure. The excellent agreement between a parabolic fit and the pulses both in temporal and spectral domains suggests that the output pulse is nearly parabolic. Highlighting the soliton-similariton laser with a linear cavity and large net intracavity anomalous dispersion, a DMAS with a low repetition rate and a pulse energy as high as 95nJ is generated, which is the highest energy from a DMAS laser, to the best of our knowledge. The pulse is dechirped from 6.1ps to a nearly transform-limited duration of 635 fs, as shown in Fig. 7 .

 figure: Fig. 6

Fig. 6 (a) Pulse intensity (black solid line) and instantaneous frequency (blue solid line) retrieved by PICASO and (b) measured spectrum (black solid line) on log scale under the max output pulse energy, with the parabola fit (red dash line), hyperbolic secant fit (purple dash dot line) and Gauss fit (olive dot line).

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 figure: Fig. 7

Fig. 7 Dechirped pulse intensity (black line) and the instantaneous frequency (blue line) retrieved by PICASO.

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5. Conclusion

A high-energy and low repetition rate DMAS oscillator with a large net intracavity anomalous dispersion and a linear cavity configuration is demonstrated experimentally. For the laser with a net intracavity dispersion of −0.89 ps2, stable DMAS operation is achieved. The maximum output power of 820 mW at a repetition rate of 8.6 MHz under a pump power of 12.76 W, corresponding to a pulse energy as high as of 95 nJ has been obtained. Numerical results confirm that the self-similar evolution is accomplished in the gain fiber, and parabolic- and Gauss-shaped pulses can be emitted at the zero-order reflection of the grating and after the slit, respectively. This low repetition rate and high energy DMAS laser is expected to be a perfect seed source of self-similar amplification and find a wide range of important applications such as in micromachining.

Acknowledgments

This work was supported in part by the National Basic Research Program of China (Grants 2010CB327604, 2011CB808101, 2014CB339800), the National Natural Science Foundation of China (Grants 61377041,61377047, 61322502, 61027013), and Program for Changjiang Scholars and Innovative Research Team in University (Grant IRT13033).

References and links

1. K. Tamura and M. Nakazawa, “Pulse compression by nonlinear pulse evolution with reduced optical wave breaking in erbium-doped fiber amplifiers,” Opt. Lett. 21(1), 68–70 (1996). [CrossRef]   [PubMed]  

2. V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, “Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers,” J. Opt. Soc. Am. B 19(3), 461–469 (2002). [CrossRef]  

3. J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3(9), 597–603 (2007). [CrossRef]  

4. C. Finot, J. M. Dudley, B. Kibler, D. J. Richardson, and G. Millot, “Optical parabolic pulse generation and applications,” IEEE J. Quantum Electron. 45(11), 1482–1489 (2009). [CrossRef]  

5. 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(26), 6010–6013 (2000). [CrossRef]   [PubMed]  

6. J. Limpert, T. Schreiber, T. Clausnitzer, K. Zöllner, H. Fuchs, E. Kley, H. Zellmer, and A. Tünnermann, “High-power femtosecond Yb-doped fiber amplifier,” Opt. Express 10(14), 628–638 (2002). [CrossRef]   [PubMed]  

7. I. A. Sukhoivanov, S. O. Iakushev, O. V. Shulika, A. Díez, and M. Andrés, “Femtosecond parabolic pulse shaping in normally dispersive optical fibers,” Opt. Express 21(15), 17769–17785 (2013). [CrossRef]   [PubMed]  

8. D. N. Papadopoulos, Y. Zaouter, M. Hanna, F. Druon, E. Mottay, E. Cormier, and P. Georges, “Generation of 63 fs 4.1 MW peak power pulses from a parabolic fiber amplifier operated beyond the gain bandwidth limit,” Opt. Lett. 32(17), 2520–2522 (2007). [CrossRef]   [PubMed]  

9. G. Chang, A. Galvanauskas, H. G. Winful, and T. B. Norris, “Dependence of parabolic pulse amplification on stimulated Raman scattering and gain bandwidth,” Opt. Lett. 29(22), 2647–2649 (2004). [CrossRef]   [PubMed]  

10. W. H. Renninger, A. Chong, and F. W. Wise, “Giant-chirp oscillators for short-pulse fiber amplifiers,” Opt. Lett. 33(24), 3025–3027 (2008). [CrossRef]   [PubMed]  

11. F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902 (2004). [CrossRef]   [PubMed]  

12. W. H. Renninger, A. Chong, and F. W. Wise, “Pulse shaping and evolution in normal-dispersion mode-locked fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 18(1), 389–398 (2012). [CrossRef]   [PubMed]  

13. V. G. Bucklew, W. H. Renninger, F. W. Wise, and C. R. Pollock, “Average cavity description of self-similar lasers,” J. Opt. Soc. Am. B 31(4), 842–850 (2014). [CrossRef]  

14. W. H. Renningar, A. Chong, and F. W. Wise, “Amplifier similaritons in a dispersion-mapped fiber lasers [Invited],” Opt. Express 19(13), 12074–12080 (2011). [PubMed]  

15. S. Boscolo, S. K. Turitsyn, and C. Finot, “Amplifier similariton fiber laser with nonlinear spectral compression,” Opt. Lett. 37(21), 4531–4533 (2012). [CrossRef]   [PubMed]  

16. C. Finot, B. Barviau, G. Millot, A. Guryanov, A. Sysoliatin, and S. Wabnitz, “Parabolic pulse generation with active or passive dispersion decreasing optical fibers,” Opt. Express 15(24), 15824–15835 (2007). [CrossRef]   [PubMed]  

17. B. Oktem, C. Ülgüdür, and F. Ö. Ilday, “Soliton-similaritonfibre laser,” Nat. Photonics 4(5), 307–311 (2010). [CrossRef]  

18. B. Nie, D. Pestov, F. W. Wise, and M. Dantus, “Generation of 42-fs and 10-nJ pulses from a fiber laser with self-similar evolution in the gain segment,” Opt. Express 19(13), 12074–12080 (2011). [CrossRef]   [PubMed]  

19. S. Lefrancois, C.-H. Liu, M. L. Stock, T. S. Sosnowski, A. Galvanauskas, and F. W. Wise, “High-energy similariton fiber laser using chirally coupled core fiber,” Opt. Lett. 38(1), 43–45 (2013). [CrossRef]   [PubMed]  

20. M. E. Fermann and I. Hartl, “Ultrafast fibre lasers,” Nat. Photonics 7(11), 868–874 (2013). [CrossRef]  

21. W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A 82(2), 021805 (2010). [CrossRef]   [PubMed]  

22. N. N. Akhmediev, A. Ankiewicz, M. J. Lederer, and B. Luther-Davies, “Ultrashort pulses generated by mode-locked lasers with either a slow or a fast saturable-absorber response,” Opt. Lett. 23(4), 280–282 (1998). [CrossRef]   [PubMed]  

23. F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasersbased on pulse propagation at normal dispersion,” Laser Photonics Rev. 2(1–2), 58–73 (2008). [CrossRef]  

24. C. Finot, F. Parmigiani, P. Petropoulos, and D. Richardson, “Parabolic pulse evolution in normally dispersive fiber amplifiers preceding the similariton formation regime,” Opt. Express 14(8), 3161–3170 (2006). [CrossRef]   [PubMed]  

25. M. Baumgartl, B. Ortaç, J. Limpert, and A. Tünnermann, “Impact of dispersion on pulse dynamics in chirped-pulse fiber lasers,” Appl. Phys. B 107(2), 263–274 (2012). [CrossRef]  

References

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  • |

  1. K. Tamura and M. Nakazawa, “Pulse compression by nonlinear pulse evolution with reduced optical wave breaking in erbium-doped fiber amplifiers,” Opt. Lett. 21(1), 68–70 (1996).
    [Crossref] [PubMed]
  2. V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, “Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers,” J. Opt. Soc. Am. B 19(3), 461–469 (2002).
    [Crossref]
  3. J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3(9), 597–603 (2007).
    [Crossref]
  4. C. Finot, J. M. Dudley, B. Kibler, D. J. Richardson, and G. Millot, “Optical parabolic pulse generation and applications,” IEEE J. Quantum Electron. 45(11), 1482–1489 (2009).
    [Crossref]
  5. 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(26), 6010–6013 (2000).
    [Crossref] [PubMed]
  6. J. Limpert, T. Schreiber, T. Clausnitzer, K. Zöllner, H. Fuchs, E. Kley, H. Zellmer, and A. Tünnermann, “High-power femtosecond Yb-doped fiber amplifier,” Opt. Express 10(14), 628–638 (2002).
    [Crossref] [PubMed]
  7. I. A. Sukhoivanov, S. O. Iakushev, O. V. Shulika, A. Díez, and M. Andrés, “Femtosecond parabolic pulse shaping in normally dispersive optical fibers,” Opt. Express 21(15), 17769–17785 (2013).
    [Crossref] [PubMed]
  8. D. N. Papadopoulos, Y. Zaouter, M. Hanna, F. Druon, E. Mottay, E. Cormier, and P. Georges, “Generation of 63 fs 4.1 MW peak power pulses from a parabolic fiber amplifier operated beyond the gain bandwidth limit,” Opt. Lett. 32(17), 2520–2522 (2007).
    [Crossref] [PubMed]
  9. G. Chang, A. Galvanauskas, H. G. Winful, and T. B. Norris, “Dependence of parabolic pulse amplification on stimulated Raman scattering and gain bandwidth,” Opt. Lett. 29(22), 2647–2649 (2004).
    [Crossref] [PubMed]
  10. W. H. Renninger, A. Chong, and F. W. Wise, “Giant-chirp oscillators for short-pulse fiber amplifiers,” Opt. Lett. 33(24), 3025–3027 (2008).
    [Crossref] [PubMed]
  11. F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902 (2004).
    [Crossref] [PubMed]
  12. W. H. Renninger, A. Chong, and F. W. Wise, “Pulse shaping and evolution in normal-dispersion mode-locked fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 18(1), 389–398 (2012).
    [Crossref] [PubMed]
  13. V. G. Bucklew, W. H. Renninger, F. W. Wise, and C. R. Pollock, “Average cavity description of self-similar lasers,” J. Opt. Soc. Am. B 31(4), 842–850 (2014).
    [Crossref]
  14. W. H. Renningar, A. Chong, and F. W. Wise, “Amplifier similaritons in a dispersion-mapped fiber lasers [Invited],” Opt. Express 19(13), 12074–12080 (2011).
    [PubMed]
  15. S. Boscolo, S. K. Turitsyn, and C. Finot, “Amplifier similariton fiber laser with nonlinear spectral compression,” Opt. Lett. 37(21), 4531–4533 (2012).
    [Crossref] [PubMed]
  16. C. Finot, B. Barviau, G. Millot, A. Guryanov, A. Sysoliatin, and S. Wabnitz, “Parabolic pulse generation with active or passive dispersion decreasing optical fibers,” Opt. Express 15(24), 15824–15835 (2007).
    [Crossref] [PubMed]
  17. B. Oktem, C. Ülgüdür, and F. Ö. Ilday, “Soliton-similaritonfibre laser,” Nat. Photonics 4(5), 307–311 (2010).
    [Crossref]
  18. B. Nie, D. Pestov, F. W. Wise, and M. Dantus, “Generation of 42-fs and 10-nJ pulses from a fiber laser with self-similar evolution in the gain segment,” Opt. Express 19(13), 12074–12080 (2011).
    [Crossref] [PubMed]
  19. S. Lefrancois, C.-H. Liu, M. L. Stock, T. S. Sosnowski, A. Galvanauskas, and F. W. Wise, “High-energy similariton fiber laser using chirally coupled core fiber,” Opt. Lett. 38(1), 43–45 (2013).
    [Crossref] [PubMed]
  20. M. E. Fermann and I. Hartl, “Ultrafast fibre lasers,” Nat. Photonics 7(11), 868–874 (2013).
    [Crossref]
  21. W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A 82(2), 021805 (2010).
    [Crossref] [PubMed]
  22. N. N. Akhmediev, A. Ankiewicz, M. J. Lederer, and B. Luther-Davies, “Ultrashort pulses generated by mode-locked lasers with either a slow or a fast saturable-absorber response,” Opt. Lett. 23(4), 280–282 (1998).
    [Crossref] [PubMed]
  23. F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasersbased on pulse propagation at normal dispersion,” Laser Photonics Rev. 2(1–2), 58–73 (2008).
    [Crossref]
  24. C. Finot, F. Parmigiani, P. Petropoulos, and D. Richardson, “Parabolic pulse evolution in normally dispersive fiber amplifiers preceding the similariton formation regime,” Opt. Express 14(8), 3161–3170 (2006).
    [Crossref] [PubMed]
  25. M. Baumgartl, B. Ortaç, J. Limpert, and A. Tünnermann, “Impact of dispersion on pulse dynamics in chirped-pulse fiber lasers,” Appl. Phys. B 107(2), 263–274 (2012).
    [Crossref]

2014 (1)

2013 (3)

2012 (3)

M. Baumgartl, B. Ortaç, J. Limpert, and A. Tünnermann, “Impact of dispersion on pulse dynamics in chirped-pulse fiber lasers,” Appl. Phys. B 107(2), 263–274 (2012).
[Crossref]

S. Boscolo, S. K. Turitsyn, and C. Finot, “Amplifier similariton fiber laser with nonlinear spectral compression,” Opt. Lett. 37(21), 4531–4533 (2012).
[Crossref] [PubMed]

W. H. Renninger, A. Chong, and F. W. Wise, “Pulse shaping and evolution in normal-dispersion mode-locked fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 18(1), 389–398 (2012).
[Crossref] [PubMed]

2011 (2)

2010 (2)

W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A 82(2), 021805 (2010).
[Crossref] [PubMed]

B. Oktem, C. Ülgüdür, and F. Ö. Ilday, “Soliton-similaritonfibre laser,” Nat. Photonics 4(5), 307–311 (2010).
[Crossref]

2009 (1)

C. Finot, J. M. Dudley, B. Kibler, D. J. Richardson, and G. Millot, “Optical parabolic pulse generation and applications,” IEEE J. Quantum Electron. 45(11), 1482–1489 (2009).
[Crossref]

2008 (2)

W. H. Renninger, A. Chong, and F. W. Wise, “Giant-chirp oscillators for short-pulse fiber amplifiers,” Opt. Lett. 33(24), 3025–3027 (2008).
[Crossref] [PubMed]

F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasersbased on pulse propagation at normal dispersion,” Laser Photonics Rev. 2(1–2), 58–73 (2008).
[Crossref]

2007 (3)

2006 (1)

2004 (2)

G. Chang, A. Galvanauskas, H. G. Winful, and T. B. Norris, “Dependence of parabolic pulse amplification on stimulated Raman scattering and gain bandwidth,” Opt. Lett. 29(22), 2647–2649 (2004).
[Crossref] [PubMed]

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902 (2004).
[Crossref] [PubMed]

2002 (2)

2000 (1)

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(26), 6010–6013 (2000).
[Crossref] [PubMed]

1998 (1)

1996 (1)

Akhmediev, N. N.

Andrés, M.

Ankiewicz, A.

Barviau, B.

Baumgartl, M.

M. Baumgartl, B. Ortaç, J. Limpert, and A. Tünnermann, “Impact of dispersion on pulse dynamics in chirped-pulse fiber lasers,” Appl. Phys. B 107(2), 263–274 (2012).
[Crossref]

Boscolo, S.

Bucklew, V. G.

Buckley, J. R.

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902 (2004).
[Crossref] [PubMed]

Chang, G.

Chong, A.

W. H. Renninger, A. Chong, and F. W. Wise, “Pulse shaping and evolution in normal-dispersion mode-locked fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 18(1), 389–398 (2012).
[Crossref] [PubMed]

W. H. Renningar, A. Chong, and F. W. Wise, “Amplifier similaritons in a dispersion-mapped fiber lasers [Invited],” Opt. Express 19(13), 12074–12080 (2011).
[PubMed]

W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A 82(2), 021805 (2010).
[Crossref] [PubMed]

F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasersbased on pulse propagation at normal dispersion,” Laser Photonics Rev. 2(1–2), 58–73 (2008).
[Crossref]

W. H. Renninger, A. Chong, and F. W. Wise, “Giant-chirp oscillators for short-pulse fiber amplifiers,” Opt. Lett. 33(24), 3025–3027 (2008).
[Crossref] [PubMed]

Clark, W. G.

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902 (2004).
[Crossref] [PubMed]

Clausnitzer, T.

Cormier, E.

Dantus, M.

Díez, A.

Druon, F.

Dudley, J. M.

C. Finot, J. M. Dudley, B. Kibler, D. J. Richardson, and G. Millot, “Optical parabolic pulse generation and applications,” IEEE J. Quantum Electron. 45(11), 1482–1489 (2009).
[Crossref]

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3(9), 597–603 (2007).
[Crossref]

V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, “Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers,” J. Opt. Soc. Am. B 19(3), 461–469 (2002).
[Crossref]

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(26), 6010–6013 (2000).
[Crossref] [PubMed]

Fermann, M. E.

M. E. Fermann and I. Hartl, “Ultrafast fibre lasers,” Nat. Photonics 7(11), 868–874 (2013).
[Crossref]

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(26), 6010–6013 (2000).
[Crossref] [PubMed]

Finot, C.

Fuchs, H.

Galvanauskas, A.

Georges, P.

Guryanov, A.

Hanna, M.

Hartl, I.

M. E. Fermann and I. Hartl, “Ultrafast fibre lasers,” Nat. Photonics 7(11), 868–874 (2013).
[Crossref]

Harvey, J. D.

V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, “Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers,” J. Opt. Soc. Am. B 19(3), 461–469 (2002).
[Crossref]

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(26), 6010–6013 (2000).
[Crossref] [PubMed]

Iakushev, S. O.

Ilday, F. O.

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902 (2004).
[Crossref] [PubMed]

Ilday, F. Ö.

B. Oktem, C. Ülgüdür, and F. Ö. Ilday, “Soliton-similaritonfibre laser,” Nat. Photonics 4(5), 307–311 (2010).
[Crossref]

Kibler, B.

C. Finot, J. M. Dudley, B. Kibler, D. J. Richardson, and G. Millot, “Optical parabolic pulse generation and applications,” IEEE J. Quantum Electron. 45(11), 1482–1489 (2009).
[Crossref]

Kley, E.

Kruglov, V. I.

V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, “Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers,” J. Opt. Soc. Am. B 19(3), 461–469 (2002).
[Crossref]

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(26), 6010–6013 (2000).
[Crossref] [PubMed]

Lederer, M. J.

Lefrancois, S.

Limpert, J.

M. Baumgartl, B. Ortaç, J. Limpert, and A. Tünnermann, “Impact of dispersion on pulse dynamics in chirped-pulse fiber lasers,” Appl. Phys. B 107(2), 263–274 (2012).
[Crossref]

J. Limpert, T. Schreiber, T. Clausnitzer, K. Zöllner, H. Fuchs, E. Kley, H. Zellmer, and A. Tünnermann, “High-power femtosecond Yb-doped fiber amplifier,” Opt. Express 10(14), 628–638 (2002).
[Crossref] [PubMed]

Liu, C.-H.

Luther-Davies, B.

Millot, G.

C. Finot, J. M. Dudley, B. Kibler, D. J. Richardson, and G. Millot, “Optical parabolic pulse generation and applications,” IEEE J. Quantum Electron. 45(11), 1482–1489 (2009).
[Crossref]

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3(9), 597–603 (2007).
[Crossref]

C. Finot, B. Barviau, G. Millot, A. Guryanov, A. Sysoliatin, and S. Wabnitz, “Parabolic pulse generation with active or passive dispersion decreasing optical fibers,” Opt. Express 15(24), 15824–15835 (2007).
[Crossref] [PubMed]

Mottay, E.

Nakazawa, M.

Nie, B.

Norris, T. B.

Oktem, B.

B. Oktem, C. Ülgüdür, and F. Ö. Ilday, “Soliton-similaritonfibre laser,” Nat. Photonics 4(5), 307–311 (2010).
[Crossref]

Ortaç, B.

M. Baumgartl, B. Ortaç, J. Limpert, and A. Tünnermann, “Impact of dispersion on pulse dynamics in chirped-pulse fiber lasers,” Appl. Phys. B 107(2), 263–274 (2012).
[Crossref]

Papadopoulos, D. N.

Parmigiani, F.

Peacock, A. C.

Pestov, D.

Petropoulos, P.

Pollock, C. R.

Renningar, W. H.

Renninger, W. H.

V. G. Bucklew, W. H. Renninger, F. W. Wise, and C. R. Pollock, “Average cavity description of self-similar lasers,” J. Opt. Soc. Am. B 31(4), 842–850 (2014).
[Crossref]

W. H. Renninger, A. Chong, and F. W. Wise, “Pulse shaping and evolution in normal-dispersion mode-locked fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 18(1), 389–398 (2012).
[Crossref] [PubMed]

W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A 82(2), 021805 (2010).
[Crossref] [PubMed]

F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasersbased on pulse propagation at normal dispersion,” Laser Photonics Rev. 2(1–2), 58–73 (2008).
[Crossref]

W. H. Renninger, A. Chong, and F. W. Wise, “Giant-chirp oscillators for short-pulse fiber amplifiers,” Opt. Lett. 33(24), 3025–3027 (2008).
[Crossref] [PubMed]

Richardson, D.

Richardson, D. J.

C. Finot, J. M. Dudley, B. Kibler, D. J. Richardson, and G. Millot, “Optical parabolic pulse generation and applications,” IEEE J. Quantum Electron. 45(11), 1482–1489 (2009).
[Crossref]

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3(9), 597–603 (2007).
[Crossref]

Schreiber, T.

Shulika, O. V.

Sosnowski, T. S.

Stock, M. L.

Sukhoivanov, I. A.

Sysoliatin, A.

Tamura, K.

Thomsen, B. C.

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(26), 6010–6013 (2000).
[Crossref] [PubMed]

Tünnermann, A.

M. Baumgartl, B. Ortaç, J. Limpert, and A. Tünnermann, “Impact of dispersion on pulse dynamics in chirped-pulse fiber lasers,” Appl. Phys. B 107(2), 263–274 (2012).
[Crossref]

J. Limpert, T. Schreiber, T. Clausnitzer, K. Zöllner, H. Fuchs, E. Kley, H. Zellmer, and A. Tünnermann, “High-power femtosecond Yb-doped fiber amplifier,” Opt. Express 10(14), 628–638 (2002).
[Crossref] [PubMed]

Turitsyn, S. K.

Ülgüdür, C.

B. Oktem, C. Ülgüdür, and F. Ö. Ilday, “Soliton-similaritonfibre laser,” Nat. Photonics 4(5), 307–311 (2010).
[Crossref]

Wabnitz, S.

Winful, H. G.

Wise, F. W.

V. G. Bucklew, W. H. Renninger, F. W. Wise, and C. R. Pollock, “Average cavity description of self-similar lasers,” J. Opt. Soc. Am. B 31(4), 842–850 (2014).
[Crossref]

S. Lefrancois, C.-H. Liu, M. L. Stock, T. S. Sosnowski, A. Galvanauskas, and F. W. Wise, “High-energy similariton fiber laser using chirally coupled core fiber,” Opt. Lett. 38(1), 43–45 (2013).
[Crossref] [PubMed]

W. H. Renninger, A. Chong, and F. W. Wise, “Pulse shaping and evolution in normal-dispersion mode-locked fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 18(1), 389–398 (2012).
[Crossref] [PubMed]

B. Nie, D. Pestov, F. W. Wise, and M. Dantus, “Generation of 42-fs and 10-nJ pulses from a fiber laser with self-similar evolution in the gain segment,” Opt. Express 19(13), 12074–12080 (2011).
[Crossref] [PubMed]

W. H. Renningar, A. Chong, and F. W. Wise, “Amplifier similaritons in a dispersion-mapped fiber lasers [Invited],” Opt. Express 19(13), 12074–12080 (2011).
[PubMed]

W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A 82(2), 021805 (2010).
[Crossref] [PubMed]

F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasersbased on pulse propagation at normal dispersion,” Laser Photonics Rev. 2(1–2), 58–73 (2008).
[Crossref]

W. H. Renninger, A. Chong, and F. W. Wise, “Giant-chirp oscillators for short-pulse fiber amplifiers,” Opt. Lett. 33(24), 3025–3027 (2008).
[Crossref] [PubMed]

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902 (2004).
[Crossref] [PubMed]

Zaouter, Y.

Zellmer, H.

Zöllner, K.

Appl. Phys. B (1)

M. Baumgartl, B. Ortaç, J. Limpert, and A. Tünnermann, “Impact of dispersion on pulse dynamics in chirped-pulse fiber lasers,” Appl. Phys. B 107(2), 263–274 (2012).
[Crossref]

IEEE J. Quantum Electron. (1)

C. Finot, J. M. Dudley, B. Kibler, D. J. Richardson, and G. Millot, “Optical parabolic pulse generation and applications,” IEEE J. Quantum Electron. 45(11), 1482–1489 (2009).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

W. H. Renninger, A. Chong, and F. W. Wise, “Pulse shaping and evolution in normal-dispersion mode-locked fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 18(1), 389–398 (2012).
[Crossref] [PubMed]

J. Opt. Soc. Am. B (2)

Laser Photonics Rev. (1)

F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasersbased on pulse propagation at normal dispersion,” Laser Photonics Rev. 2(1–2), 58–73 (2008).
[Crossref]

Nat. Photonics (2)

M. E. Fermann and I. Hartl, “Ultrafast fibre lasers,” Nat. Photonics 7(11), 868–874 (2013).
[Crossref]

B. Oktem, C. Ülgüdür, and F. Ö. Ilday, “Soliton-similaritonfibre laser,” Nat. Photonics 4(5), 307–311 (2010).
[Crossref]

Nat. Phys. (1)

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3(9), 597–603 (2007).
[Crossref]

Opt. Express (6)

Opt. Lett. (7)

N. N. Akhmediev, A. Ankiewicz, M. J. Lederer, and B. Luther-Davies, “Ultrashort pulses generated by mode-locked lasers with either a slow or a fast saturable-absorber response,” Opt. Lett. 23(4), 280–282 (1998).
[Crossref] [PubMed]

S. Boscolo, S. K. Turitsyn, and C. Finot, “Amplifier similariton fiber laser with nonlinear spectral compression,” Opt. Lett. 37(21), 4531–4533 (2012).
[Crossref] [PubMed]

K. Tamura and M. Nakazawa, “Pulse compression by nonlinear pulse evolution with reduced optical wave breaking in erbium-doped fiber amplifiers,” Opt. Lett. 21(1), 68–70 (1996).
[Crossref] [PubMed]

S. Lefrancois, C.-H. Liu, M. L. Stock, T. S. Sosnowski, A. Galvanauskas, and F. W. Wise, “High-energy similariton fiber laser using chirally coupled core fiber,” Opt. Lett. 38(1), 43–45 (2013).
[Crossref] [PubMed]

D. N. Papadopoulos, Y. Zaouter, M. Hanna, F. Druon, E. Mottay, E. Cormier, and P. Georges, “Generation of 63 fs 4.1 MW peak power pulses from a parabolic fiber amplifier operated beyond the gain bandwidth limit,” Opt. Lett. 32(17), 2520–2522 (2007).
[Crossref] [PubMed]

G. Chang, A. Galvanauskas, H. G. Winful, and T. B. Norris, “Dependence of parabolic pulse amplification on stimulated Raman scattering and gain bandwidth,” Opt. Lett. 29(22), 2647–2649 (2004).
[Crossref] [PubMed]

W. H. Renninger, A. Chong, and F. W. Wise, “Giant-chirp oscillators for short-pulse fiber amplifiers,” Opt. Lett. 33(24), 3025–3027 (2008).
[Crossref] [PubMed]

Phys. Rev. A (1)

W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A 82(2), 021805 (2010).
[Crossref] [PubMed]

Phys. Rev. Lett. (2)

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902 (2004).
[Crossref] [PubMed]

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(26), 6010–6013 (2000).
[Crossref] [PubMed]

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Figures (7)

Fig. 1
Fig. 1 Experimental setup.AL: aspheric lens; DM: dichroic mirror; PBS: polarization beam splitter; HWP: half-wave plate; GR: grating; HR: high-reflectivity mirror; OC: output coupler; SL: slit
Fig. 2
Fig. 2 Simulated results: evolution of (a) pulse chirp and (b) M parameter within one round-trip in the cavity (SA: saturable absorber; OC: output coupler; DDL: dispersion delay line; SF: spectral filter); simulated pulse intensity and instantaneous frequency (c) at the OC and (d) at the position after SF.
Fig. 3
Fig. 3 Output power as a function of pump power.
Fig. 4
Fig. 4 (a) Autocorrelations and (b) spectra at different pump power.
Fig. 5
Fig. 5 (a) Measured pulse train and (b) its radio frequency spectrum under pump power of 12.76W.
Fig. 6
Fig. 6 (a) Pulse intensity (black solid line) and instantaneous frequency (blue solid line) retrieved by PICASO and (b) measured spectrum (black solid line) on log scale under the max output pulse energy, with the parabola fit (red dash line), hyperbolic secant fit (purple dash dot line) and Gauss fit (olive dot line).
Fig. 7
Fig. 7 Dechirped pulse intensity (black line) and the instantaneous frequency (blue line) retrieved by PICASO.

Tables (1)

Tables Icon

Table 1 Simulation parameters

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

A z + i β 2 2 2 A t 2 = G 2 A + i γ | A | 2 A
G = G 0 1 + E p u l s e E s a t / g a i n
E p u l s e = T R 2 T R 2 | A | 2 d t
R ( t ) = e δ s ( t ) δ n s
δ s ( t ) = Δ R f ( t ) [ 1 τ r e l f ( t ) d t + 1 ]
f ( t ) = exp [ ( 1 τ r e l + | A | 2 E s a t / a b s ) d t ]
M 2 = ( | A | 2 + | A p f i t | 2 ) 2 d t | A | 4 d t

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