1. Introduction

Many practical as well as fundamental research applications request compact, reliable, efficient and low-cost laser sources capable of delivering highly energetic ultrashort pulses. Fiber-based lasers are promising systems for fulfilling these requirements in the very near future. Recent demonstration of such sources operating in the 1-µm range with deliverable energy per pulse approaching or even exceeding 100 nJ [1,2] have made them henceforth available now and fully competitive with their bulk-based counterparts like the more standard Ti-Sa based femtosecond oscillators.

The main concept that allows energy scaling in mode-locked fiber lasers depends on group velocity dispersion (GVD) management to operate with large net normal cavity GVD. In particular, it has been demonstrated that pulse shaping in normal GVD fibres is favourable for the generation of self-similar pulses in fibre oscillators [3, 4]. Indeed, the interplay between normal dispersion, self-phase modulation (SPM) and gain leads to the propagation of linearly chirped pulses which are more resistant to the restrictions imposed by the fibre nonlinearities [3,4]. These demonstrations have attracted much interest in the development of purely normal dispersion fibre lasers in the last few years leading to a significant improvement in the energy extracted from fibre oscillators [5-7]. Herda et al. [8] have reported a first demonstration of picoseconds pulse generation from a purely normal GVD fiber laser. This laser consisted in a short Yb-fiber with positive GVD and used a high-modulation depth saturable absorber mirror (SAM) for mode-locking. The second important contribution was the report of Zhao et al. [9-11] of the so-called gain-guided soliton formation in an all-normal dispersion fiber laser. The all-normal dispersion fiber laser concept has been significantly improved by Chong et al. [12, 13] by adding a spectral filter inside the cavity leading to the achievement of more than 20 nJ energy in step index fiber lasers [6]. Pulse shaping in these lasers arises from the spectral filter, which converts frequency chirp to self-amplitude modulation. A wide variety of pulse shapes are obtained with variation of the filter bandwidth [13] and have been interpreted in terms of highly chirped dissipative solitons in the frame of the cubic-quintic Ginzburg-Landau equation [14]. The spectral filter bandwidth being much narrower than the gain bandwidth, gain dispersion did not play a key role in pulse shaping in such lasers [13].

The situation seems different in the laser of Zhao et. al. where pulse shaping is dominated by the saturable absorber nonlinearity whereas the intra-cavity pulse dynamics and its output characteristics are highly dependent on the gain saturation and dispersion [11]. Such gain-guided solitons have been observed in purely positive dispersion fiber lasers as well as in dispersion managed fiber lasers [9, 10] and are often associated with gain-bandwidth limited pulses. In addition, such solitons seemed to suffer from low dechirped capabilities [10, 11]. Recently, it has been shown that these gain-guided solitons could be obtained in a dispersion managed erbium laser with nearly zero net cavity dispersion [15]. In particular, it has been shown that these gain-guided solitons can coexist with the dispersion-managed solitons in the same cavity configuration. The transition from one regime to another can be obtained by adjusting the intra-cavity pulse peak power. In addition, recent experimental and numerical studies of all-normal dispersion large mode area fiber lasers show that the gain dispersion effect could manifest even for mode-locking regimes with spectral widths much narrower than the gain bandwidth [1]. So, one wonders if gain-guided solitons with spectral widths much narrower than the gain bandwidth could exist.

In this contribution, we report on a passively mode-locked dispersion-managed erbium-fiber laser with large net cavity dispersion. The laser generates positively chirped 5.3 ps output pulses that are compressed down to 757 fs. The optical spectrum is characterized by steep edges and a narrow width of 8.3 nm, which is far below the gain-bandwidth of erbium. Numerical simulations show that in addition to the SAM nonlinearity, the gain and gain dispersion play a key role in the formation of these pulses. Then our results confirm the tendency that gain-guided solitons belong to a more general class of so-called ‘dissipative solitons’ as highlighted in recent publication [14] or book [16]. Such term is used hereafter in order to generalize discussion.

2. Experiments and results

The experimental set-up of the laser is drawn on Fig. 1. It consists in a short heavily doped erbium fiber (HDEF) laser presenting high normal GVD. The net cavity dispersion is highly positive thanks to the minimization in length of all other passive fibered components which are restricted, in our case, solely to the pigtails of a wavelength division multiplexer (WDM). The laser cavity is mounted in a sigma configuration using a polarization-sensitive optical isolator. The 1.15 m HDEF presents an unpumped absorption of 80 dB/m at 1530 nm. Its GVD has been estimated at -48 (ps/nm)/km at 1550 nm. Pigtails of the WDM are made up of Hi1060 fibers with a measured GVD of 8.7 (ps/nm)/km. The net positive cavity dispersion is β₂ ≈ +0.063 ps² around 1550 nm. Rest of the cavity comprises bulk isolator, wave-plates, coupling lenses and commercial SAM (Batop 1550-23). The total optical cavity length is about 3.26 m leading to a pulse repetition rate of 92 MHz. The SAM presents a low intensity reflectivity of 77%, modulation depth of 14% and saturation fluence of 25 µJ/cm². Temporal relaxation of the SAM is roughly estimated around 2 ps but clearly present a bi-temporal response time with a fast sub-picosecond component and a slower part approaching 10 ps at high fluence. One should note that up to a fluence of 458 µJ/cm², no reverse saturable absorption (free carrier absorption-based, FCA) has been observed for this SAM structure. A negligible positive dispersion (+1000 fs²) has been evaluated for the SAM at the laser central operating wavelength of 1560 nm.

 

Fig. 1. Experimental set-up. WDM : 980/1550 nm multiplexer; L₁, L₂, L₃: coupling lenses; λ/2, λ/4: half- and quarter- wave plates.

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For an optimized setting of the wave-plates, the laser starts in a CW regime for a pump power of 40 mW and self-starts in a pure mode-locking regime when pump power reaches 220 mW. It should be noted here that, even if we have carefully checked that mode-locking could not be experimentally obtained without the SAM, we expect that nonlinear polarization evolution (NPE) mechanism participates to pulse shaping. Indeed, the laser operation is very sensitive to the wave-plates orientation and mode-locking can not be obtained for all settings. Numerical simulations reported on next section will provide further information on such hypothesis. The single-pulse mode-locking operation is sustained up to the maximum of available pump power of 450 mW. At maximum pump level, the measured output average power is 31 mW. The positively chirped output pulses are well fitted with a Gaussian shape with pulse duration of 5.3 ps (Fig. 2). The optical spectrum of Fig. 2 is characterized by steep edges similarly to the dissipative solitons regime and more generally to mode-locked fiber lasers operating in the highly positive dispersion regime [14]. However, the narrow spectral width of 8.3 nm obtained in our case is different from that of the dissipative solitons reported by Zhao et al. which present a broad gain-bandwidth limited spectrum [9-11]. Interestingly, in our configuration the output pulses are extra-cavity dechirped down to 757 fs using bulk gratings (Fig. 3(a)). It is about 15 % higher than the pulse duration obtained from Fourier-transformation of the experimental optical spectrum. The average power after extra-cavity compression is 24 mW.

 

Fig. 2. Right: Typical output spectrum on a linear scale. Inset presents the spectrum on a logarithmic scale. Left: Autocorrelation trace of the output pulse on linear and logarithmic (inset) scales.

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To evaluate the quality of the mode-locked pulse-train, we performed amplitude noise measurements using the radio-frequency (RF) power spectrum obtained with a microwave spectrum analyzer via a high-speed photodetector (8-GHz bandwidth). Spectra were taken at different spectral ranges from 100 MHz to 10 kHz at the fundamental harmonic around 92 MHz. The different spectra presents only one noise-substructure at low-frequencies <40 kHz (see Fig. 3(b)). The noise level associated with this substructure is lower than 0.25% which highlights the good-stability of the output pulse train that did not suffer from Q-switch mode-locking instabilities. Indeed, Q-switching sidebands are at least ≈ 95 dB below the carrier, which is an extremely good value for a highly normal dispersion fiber laser.

 

Fig. 3. (a). Autocorrelation trace of the dechirped pulse. (b) rf spectrum recorded at the fundamental frequency. Resolution bandwidth is 300 Hz.

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3. Numerical results

To understand better the features of the above-reported laser of delivering quasi-linearly chirped pulses, we numerically simulated the laser operation using the extended Non Linear Schrödinger Equations (NLSE) based model as published previously [1]. However, in order to get additional insight on the fundamental effects governing the present regime, we focused only on a scalar NLSE. This means that no NPE mechanism (and so no pulse-shaping related effect) is taken into account in this case. Pulse propagation in the gain fiber is described by the scalar NLSE which includes the effects of dispersion, Kerr nonlinearity and saturated gain with a finite bandwidth :

where A(z,t) is the complex pulse envelope and t is the time in the co-moving frame. We denote by β₂ the group velocity dispersion, γ is the self phase modulation coefficient and g(z) is the saturated gain coefficient given by g(z)=g ₀/[1+E(z)/E_sat], where g ₀ is the small signal gain coefficient (related to dopant concentration), E(z) is the pulse energy and E _sat is the gain saturation energy which is pump-power dependant, [17]. A 25 nm width Gaussian profile is assumed for the second Er gain band centered around 1560 nm.

Table 1:. Intra-cavity fiber parameters used in the simulation of the laser dynamics

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Since a scalar model is used here, no value for the linear birefringence of the fibers is requested. Local losses are set around the experimentally estimated value of 85 % (taking both the polarizing output coupler ratio (70 %) and propagation along the bulk elements). For all settings, the pulse evolution in each segment was numerically solved using a split step Fourier method until it reached a steady state. The intra-cavity fiber parameters used in the simulation are given on Table 1.

 

Fig. 4. Predicted evolution of the spectral shape vs. pump power for (a) a real saturable absorber with 2 ps relaxation time and (b) a SAM with a monotonic saturation and an infinite fast response.

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In order to allow a stable pulsed regime to emerge from an initially injected noise, we first implemented a real saturable absorber with 2 ps time response [18]. The evolution of the output spectrum versus pump power is shown on Fig. 4(a). The output spectra present highly asymmetric shapes, which are very different from experiment. We attribute such asymmetry to the finite temporal response for the SAM acting on a highly chirped pulse with strong SPM-induced spectral components. Suggesting that the NPE mechanism plays a key role in pulse shaping, we assumed in a second step a SAM with a monotonic saturation and an infinite fast response, which is a good approximation for NPE mechanism, see e.g. [13], as long as precise information on the polarization state for the pulse is not required. In this case, our stable pulsed solutions are not achievable from noise. To access the pulsed solution, a pulse with a Gaussian shape is used as an initial condition. The simulations have been checked to converge exactly to the same attractor for different initial pulse shapes and chirps. This confirms the experimental observations that mode-locking operation is not achievable with the NPE mechanism alone. Figure 4(b) shows the evolution of the corresponding output spectra versus pump power. It appears that the spectral width always increases with increasing pump power and the output spectra display a dramatic variation. For low pump power, the spectra show a multi-peaked shape with steep edges as observed experimentally. As pump power is increased, the spectrum broadens while developing sharp peaks around its edges. Symmetric spectra contrast with those obtained on Fig. 4(a). With both SAM models, pulse duration always decreases with increasing pump power. Since the intracavity pulse dynamics is similar for both SAM models, subsequently we will detail only the regimes obtained with the fast SAM model.

 

Fig. 5. Results of numerical simulations for Esat=120 pJ : (a). intra-cavity pulse evolution in the temporal and spectral domains. OC: Output coupling and SA: Saturable absorber. (b). Output power spectrum (solid curve) and simulated gain profile (dashed curve). Temporal intensity profile (solid curves) and instantaneous frequency (dashed curves) of the output pulses before (c) and after extra-cavity dechirping (d).

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Numerical results close to experiments are obtained for E_sat=120 pJ where the output pulse energy is about 310 pJ. Figure 5(a) shows the corresponding evolution of the pulse width and its spectral bandwidth versus the position in the ring cavity. Intra-cavity evolution is essentially dominated by gain amplification in the doped fiber up to 1.15 m long. Then the pulse evolves in 0.6 m of passive fiber with negative dispersion and around 84% of the total power is extracted from the cavity just before the SAM.

As revealed by the simulation, the pulses are always positively chirped inside the cavity with only one minimum located inside the gain fiber. A weak gain filtering effect can be recognized even at such narrow spectral bandwidth of 8.7 nm (Fig. 5(b)). Pulse duration decreases slightly in the beginning of the gain fiber and then increases monotonically during the amplification. A relatively weak pulse shortening effect (35%) happens in the passive fiber but the main shortening part (60%) results from the SAM nonlinearity. Spectral broadening via self-phase modulation (SPM) can be observed after sufficient peak power is reached during the amplification. Finally, the balance between the SPM broadening and the spectral filtering resulting from the SAM nonlinearity could achieve self-consistency. The optical spectrum agrees as well in width (8.7 nm) and shape with the experimentally measured spectrum (see Fig. 2(b)). Indeed, the latter reproduces quite well the multi-peaks shape and steep edges observed experimentally. As shown on Fig. 5(c), the output pulse duration is 5.33 ps and the pulse has a hyperbolic tangent instantaneous frequency variation as predicted theoretically [14, 19]. The output pulses are numerically dechirped with a linear dispersive delay down to 745 fs leading to a theoretical time-bandwidth product of about 0.8 (Fig. 5(d)). We note the presence of small satellites on the dechirped pulse, which result from the steep edges of the spectrum. The lobes contain less than 7% of the pulse energy. These profiles are in very good accordance with the experimental autocorrelation trace of Fig. 3.

 

Fig. 6. Results of numerical simulations for Esat=350 pJ : (a) intra-cavity pulse evolution in the temporal and spectral domains. (b) Output power spectrum (solid curve) and simulated gain profile (dashed curve). Temporal intensity profile (solid curves) and instantaneous frequency (dashed curves) of the output pulses before (c) and after extra-cavity dechirping (d).

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Increasing pump power, we observe the diminution of the impact of the SAM nonlinearity on pulse shaping. This observation is illustrated on Fig. 6 which shows the results obtained for E_sat=350 pJ. It corresponds to an extracted energy of about 800 pJ. As illustrated on Fig. 6(a), pulse shaping is less affected by the SAM nonlinearity and the main shortening effects happen in the passive fiber (59%) and the beginning of the gain fiber (32%). The nonlinear phase accumulated along the cavity is around π/2 leading to broader spectrum. The output spectrum width is 18.5 nm and presents sharp peaks around its edges (see Fig. 6(b)). The output pulse duration is 3.31 ps (Fig 6(c)). These pulses can be dechirped with a linear dispersive delay to about 5% beyond the Fourier transform limit (Fig. 6(d)). The energy contained in the secondary structures of the dechirped pulse is about 17% of the total pulse energy.

Numerical simulations show that increasing the pump-power-dependant gain saturation energy leads to two different regimes of operation that correspond to different pulse shaping mechanisms. For low pump powers, the spectra present a multi-peaked shape with steep edges and the SAM nonlinearity dominates pulse shaping. Pulse duration increases monotonically within the gain fiber and then decreases slightly in the negative GVD fiber. It is important to note that even if the essential pulse shortening effect happens in the SAM, the gain spectral filtering is also necessary to get stable pulsed operation. The temporal and spectral evolutions along the cavity obtained at low power are qualitatively the same as for the wave-breaking free laser. The pulse breathing ratio is about 1 while it could be more than 10 in a wave-breaking free regime [3]. In addition, the phase accumulated by the pulse along the cavity is well below π while it could be of several times π in the wave-breaking-free regime [3]. At high pump powers, the spectra exhibit a more structured shape with sharp peaks around their edges. This is very similar to the ANDI fiber laser comprising a passive spectral filter [13]. Indeed, in the two configurations, the pulse breathing within the cavity is small and output spectra present similar shapes. However, the major difference relates to the amount of nonlinear phase shift of the pulse along the cavity, which is limited to low values (lower than π) in our configuration while it could be as high as 16π in the ANDI fiber laser. These differences arise mainly from the cavity design and fiber lengths (especially gain fiber) in between ANDI cavity and the one of Fig. 1. Dispersion-managed fiber laser with large normal cavity dispersion has been recently addressed in [20]. In our case, pulse narrowing in the gain fiber is clearly visible (see Fig. 5 and 6). Numerous simulations have revealed that it could be linked to the small-signal gain value g₀. Indeed, with smaller g₀ value we obtained intracavity pulse dynamics that resemble curves given on Fig. 4 of ref. [20]. However, in that case it was not possible to reproduce both our experimentally measured temporal and spectral widths simultaneously. Further prospects are currently under progress in order to explore energy scaling capabilities in this dissipative soliton laser for different laser configurations. Preliminary numerical results show that by increasing the positive net cavity GVD, several nanojoules of energy could be achieved.

4. Conclusion

In conclusion, we have demonstrated the generation of picosecond pulses from a highly normal heavily doped erbium fiber laser. The regime is fully self-starting with the help of a SAM with a high modulation depth. We extracted more than 31 mW average output power at a repetition of 92 MHz. The output pulse duration is 5.3 ps and spectral bandwidth around 8.3 nm with steep edges. Pulses have been dechirped externally to the cavity down to 757 fs. Numerical simulations taking into account all the main aspects of the cavity reproduced fully the experimental results and show that this laser operates in the dissipative soliton regime.