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Abstract
Generation of dissipative soliton resonance (DSR) is numerically investigated in an all-normal-dispersion Yb-doped fiber laser mode-locked by a real saturable absorber (SA). In the simulation model, the SA includes both the saturable absorption and reverse saturable absorption (RSA) effects. It is found that the RSA effect induced by the SA material itself plays a dominant role in generating the DSR pulses. We also systematically analyze the influence of key SA parameters on the evolution of DSR pulses in the cavity. Our simulation results not only offer insight into the underlying mechanism of DSR generation in mode-locked fiber lasers by means of real SAs, but also provide a guideline for engineering SA parameters to generate optical pulses with the highest possible energy.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
High-energy ultrashort-pulse fiber lasers have been extensively investigated due to their widespread applications in material processing, micromachining, and supercontinuum generation. To date, various mode-locking techniques, such as nonlinear polarization rotation (NPR), nonlinear optical loop mirror (NOLM), real saturable absorbers (SAs) (e.g., semiconductor saturable absorption mirrors (SESAMs), single wall carbon nanotubes (SWNTs), graphene, transition metal dichalcogenides and black phosphorous), have been exploited for realizing ultrashort pulses. When a mode-locked fiber laser consists of purely anomalous group-velocity dispersion (GVD) fibers, conventional solitons can be formed as a result of the combination of negative GVD and self-phase-modulation (SPM) effects. However, the pulse energy for a single soliton is restricted to ~0.1 nJ due to the soliton area theorem [1,2]. By shifting the cavity dispersion to large-positive or even all-normal dispersion, dissipative solitons (DSs) have been obtained [3–5]. Different from conventional solitons, the formation of DSs is a composite balance involving dissipative and dispersive effects [6]. Pulse shaping in normal-dispersion fiber lasers is attributed to spectral filtering of highly-chirped pulses [7] and the output spectrum shows characteristic steep spectral edges [4]. All-normal-dispersion fiber lasers can generate the highly chirped pulses with pulse energy over 20 nJ [8]. Attempts to further enhancement of the pulse energy always lead to wave breaking, which is caused by the accumulation of excessive nonlinear phase shift [9,10]. One approach that could be used to limit nonlinearities in fiber lasers for pulse energy scaling is to utilize large-mode-area (LMA) microstructure fibers [11]. Nevertheless, due to the structural difference between LMA fiber and conventional single-mode fiber, their direct fusion splicing is difficult, and the bulk components are needed in the cavity, leading to a non-all-fiber laser system. Therefore, there is always a strong motivation to exploit new operation regimes for circumventing wave breaking.
A common approach to modeling passively mode-locked lasers is an averaged model in which the effects of discrete laser components in the cavity are averaged over one round trip [12-14]. By using the complex cubic-quintic Ginzburg-Landau equation (CQGLE), a novel concept of pulse formation known as dissipative soliton resonance (DSR) was proposed to achieve high-energy wave-breaking-free pulses [15]. The DSR pulse features the flat-top pulse profile. With the increase of pump power, the pulse width can be arbitrarily broad while maintaining its amplitude constant. DSR has been demonstrated and characterized in many theoretical models [16–19]. Within the framework of the CQGLE model, the cubic and quintic nonlinear loss terms are introduced. The former describes nonlinear loss that decreases inversely with the pulse intensity, and the latter represents the saturation of nonlinear loss, which is necessary for suppressing the pulse intensity. As a result, for certain values of system parameters, these two terms may cause a rectangular pulse and the pulse energy can be increased indefinitely without pulse break-up. In general, these two terms are introduced by the mode-locking devices. Numerical simulations based on the lumped models have shown that the strong peak-power-clamping effect induced by a sinusoidal SA transmission function is key for DSR generation in fiber lasers mode-locked by NPR or NOLM techniques [20,21]. DSR phenomena have also been experimentally demonstrated in fiber lasers by employing NPR [22–24] or NOLM [25–27] as a mode-locker.
Recently, real SAs have been exploited for DSR generation in mode-locked fiber lasers. Zhao et al. demonstrated the generation of dual-wavelength DSR pulses from an Yb-doped fiber laser using a microfiber-based graphene SA, which emitted rectangular pulses with duration tunable from 1.41 ns to 4.23 ns [28]. Cheng et al. observed DSR phenomenon in a graphene oxide mode-locked Yb-doped fiber laser [29], where the DSR-pulse generation was attributed to the etalon effect induced by a pair of non-touched fiber connectors and latent NPR effect in the cavity. However, the influence of SA parameters on the formation of DSR pulses has not been reported. A natural question arises as to whether the nonlinear optical properties of SA materials are responsible for the DSR generation in fiber lasers. The exploration of this issue would be beneficial for generating high-energy pulses in mode-locked fiber lasers with real SAs.
In this paper, we report numerical simulations on dynamics of DSR generation in an all-normal-dispersion (ANDi) Yb-doped fiber laser mode-locked by a real SA. In our model, both the saturable absorption and reverse saturable absorption (RSA) effects are included in the SA. It was found that the RSA effect induced by the SA material itself is required for achieving the DSR pulses in the cavity. The influence of key SA parameters such as modulation depth, saturation power, and RSA coefficient, on DSR-pulse properties is discussed.
2. Numerical model
The proposed ANDi Yb-doped fiber ring laser is schematically shown in Fig. 1. The cavity consists of a piece of 1-m ytterbium-doped fiber (YDF), two pieces of single mode fiber (SMF) with a total length of 5 m, a Gaussian-shaped spectral filter with a bandwidth of 6 nm, a 20% output coupler (OC) and a real SA. The real SA is used as a mode locker. The dispersion parameters and nonlinear coefficients of the fibers are β2,YDF = 36 ps2/km, β2,SMF = 23 ps2/km, γYDF = 5.8 W−1km−1 and γSMF = 3 W−1km−1. The net cavity dispersion is 0.151 ps2.
We numerically simulated the pulse formation and evolution in the laser cavity based on a cavity round-trip model [30]. Briefly, the action of each cavity component on the optical pulses was taken into account. The simulation started from an arbitrary weak pulse, and eventually converged into a stable solution with appropriate parameter settings after a limited number of circulations in the cavity. Pulse propagation in fiber section can be described by the scalar nonlinear Ginzburg-Landau equation:
where u denotes the normalized electric field envelope; β2 refers to the fiber dispersion; γ represents the nonlinearity of fiber; the variables t and z indicate the pulse local time and the propagation distance, respectively; g describes the gain function of the YDF and Ωg is the gain bandwidth assuming a parabolic gain shape with a bandwidth of 40 nm. For the SMF g = 0 and the last two terms on the right side of Eq. (1) are ignored. For the YDF, the gain saturation effect is considered as:where g0 is the small signal gain, which is related to the doping concentration; Esat is the gain saturation energy and the increase of Esat is equivalent to increasing the pump power. In this paper, g0 is fixed at 7 m−1 and Esat is variable. The pulse energy Ep is given by:where TR is the cavity round trip time.The real SA used in the cavity is treated as a lumped SA. Different from a simple two-level SA model [31], the SA used in our simulation includes both the saturable absorption and RSA effects. The total power-dependent transmission coefficient can be expressed by:
where α0 represents the modulation depth; αns denotes the nonsaturable absorption; I is the instantaneous pulse power and Isat is saturation power; β refers to the RSA coefficient. Generally, typical SA materials reveal a monotonically increasing transmissivity with the input pulse power, as shown by the blue curve in Fig. 2, indicating a saturable absorption. However, it has been found that some SA materials (e.g., SESAM [32], SWNTs [33] and MoS2 nanoflake [34]) exhibit different nonlinear absorption properties depending on the input pulse power. In the case of low input pulse power, they show saturable absorption. With further increase of the pulse power, a changeover from saturable absorption to RSA can be observed. The corresponding transmission curve is not monotonic and the significant feature is the rapid roll-off in transmissivity under sufficiently high power (see the red curve in Fig. 2). The RSA effect might be attributed to two-photon absorption (TPA) or exited-state absorption (ESA). This transition from saturable absorption to RSA could affect the intra-cavity pulse-shaping dynamics and give rise to different mode-locking states.
Fig. 2 Different transmission curves of SAs. (blue solid curve for β = 0 kW−1 and red dashed curve for β = 3 kW−1).
3. Simulation results and discussion
3.1 Mechanism of DSR-pulse generation
The numerical model was solved by using the split-step Fourier method. We firstly simulated the intra-cavity pulse evolution in the absence of RSA effect in the SA, and the SA parameters were set as α0 = 0.25, αns = 0.3, Isat = 20 W and β = 0 kW−1. As can be seen in Fig. 2 (blue curve), the transmissivity of SA increases monotonically with the input pulse power. The simulation results show the stable DSs generated with increasing pump, as presented in Figs. 3(a) and 3(b). The pulse spectra exhibit the steep edges, which is a typical characteristic of DSs. When the pump strength Esat is increased from 0.1 nJ to 1.2 nJ, the pulse peak power increases continuously, leading to the SPM-broadened spectra.
Fig. 3 Simulation results in the absence of RSA effect in the SA. (a) Pulse temporal profiles with the increasing Esat. (b) Pulse spectral profiles with the increasing Esat. (c) NLP profile with a pump Esat of 6 nJ. (d) Autocorrelation trace with a pump Esat of 6 nJ.
Further increasing Esat to 6 nJ, the noise-like pulse (NLP) state occurs in the cavity, as observed from Figs. 3(c) and 3(d). The generated NLP is essentially an optical wave packet that contains a bunch of randomly separated sub-pulses with varying amplitudes and pulse widths [35]. In NLP operation regime, the pulse profiles change over successive round trips. The corresponding calculated autocorrelation (AC) trace has a narrow spike riding on a broad shoulder, which is a unique characteristic of the NLP [36]. Since the monotonic SA transmission curve has no limitation on the peak power of the DS, the NLP generation may be attributed to the spectral-filtering-effect induced by the filter in the cavity. The transition from the DS operation to the NLP generation can be explained as follows: when the pump power is increased continuously, the peak power of the DS keeps growing, leading to the SPM-induced spectrum broadening. Once the spectrum width reaches a certain level, which depends on the filter bandwidth, the soliton peak clamping will occur. Further increase in the pump power will not enlarge the soliton energy but amplify the dispersive waves. Meanwhile, the normal cavity GVD makes the dispersive waves and the soliton pulse almost co-propagate in the time domain due to a small group-velocity difference. Eventually, the interaction between the dispersive waves and the soliton can result in the occurrence of NLP [37]. Based on our simulations, it was also found that through increasing the filter bandwidth or reducing the pump power, the NLP can be always returned to a DS pulse. It should be emphasized that no DSR operation state can be achieved in the case of β = 0 kW−1 as is evident from the numerical results.
We also simulated the mode-locked pulse evolution in the cavity when both the saturable absorption and RSA effects are included in the SA. For the sake of comparing with the aforementioned case, we only altered the value of β from 0 kW−1 to 3 kW−1, and the corresponding transmission curve is shown in Fig. 2 (red curve). The simulation results are illustrated in Fig. 4.
Fig. 4 DSR-pulse generation. (a) Pulse temporal profiles with the increasing Esat. (b) Pulse spectral profiles with the increasing Esat. (c) The pulse energy and pulse duration versus Esat. (d) The pulse peak power versus Esat.
As presented in Fig. 4(a), when the pump Esat is increased from 0.1 nJ to 1.6 nJ, the pulse peak power initially increases and then clamps at a certain value while its temporal profile transfers from a Gaussian shape to a rectangular one. Figure 4(b) shows the change in the corresponding pulse spectra. The intensity in the central region of pulse spectrum increases continuously while the 3-dB bandwidth is significantly narrowed at first and then keeps almost constant. The evolution process from the DS to the rectangular pulse is in agreement with the DSR theory, indicating that stable DSR pulses are generated in the cavity. Under DSR operation conditions, the pulse duration and pulse energy increase linearly with Esat, while the pulse peak power maintains approximately unchanged, as depicted in Figs. 4(c) and 4(d). In this case, only a stable single DSR pulse exists with increasing pump and the NLP operation state is avoided.
The characteristics of the intra-cavity DSR-pulse evolution at Esat = 1.2 nJ are illustrated in Fig. 5. One can see from Fig. 5(a) that the relative fluctuations of the pulse duration and spectrum width are very low as the pulse propagates through the cavity, which is quite different from those of the stretched-pulses and self-similar pulses. It is interesting that the DSR pulse with a rectangular temporal profile can exist throughout the laser cavity, as displayed in Fig. 5(b).
Fig. 5 Intra-cavity DSR-pulse evolution. (a) Variations of the pulse duration and spectrum width along the cavity. (b) Three-dimension evolution.
Our simulation results indicate that the RSA effect in the SA plays a dominant role in generating the DSR pulses. We explain the mechanism of DSR generation as follows: when the pump power is high enough, the RSA effect is activated, namely the transmissivity of SA decreases as the pulse peak power increases, as shown in Fig. 2 (red curve). The higher the pulse peak power increases, the larger loss the pulse encounters in the SA. When the pulse peak power is increased to a certain level, the effective gain in the central region of the pulse is equivalent to the dynamical loss that it experiences, and at this point the pulse peak power will be clamped. Further increase in the pump power will lead to enlarging pulse duration rather than its peak power. Meanwhile, the frequency chirp is moderately low in the central region of the rectangular pulse, which is a direct consequence of the small pulse power gradient variations near the center. Most pulse energy is accumulated near the central wavelength of pulse spectrum. As a result, the spectral width of the pulse is considerably narrowed and far less than the bandwidth of spectral filter. Hence, the NLP operation state induced by the spectral filtering effect is circumvented and the DSR pulses are generated. More details are discussed in Subsection 3.4.
3.2 Effect of modulation depth
The influence of key SA parameters on the evolution of DSR pulses was numerically analyzed. We firstly varied the modulation depth α0 while fixing the other variables (Esat = 1.2 nJ, αns = 0.3, Isat = 20 W, β = 3 kW−1). It was found that no stable DSR pulses could be obtained if the modulation depth α0 was lower than 0.14, manifesting a threshold of modulation depth for DSR-pulse formation. Figure 6 shows DSR pulse duration and peak power as a function of α0. It can be seen that with increasing α0 from 0.15 to 0.5, the pulse duration decreases from 180 ps to 70 ps whereas the peak power increases from 22 W to 53 W. Our simulation results indicate that in DSR regime, larger modulation depth could give rise to a narrower pulse duration and higher peak power at fixed pump power.
3.3 Effect of saturation power
The impact of saturation power Isat on DSR-pulse properties was investigated. By changing the Isat value while maintaining the other parameters constant (Esat = 1.2 nJ, α0 = 0.25, αns = 0.3, β = 3 kW−1), we calculated numerically the dependence of DSR-pulse duration and peak power on Isat of SA, as illustrated in Fig. 7. It can be seen that when the Isat increases from 25 W to 55 W, the pulse duration increases from 115 ps to 148 ps while the peak power decreases from 33 W to 25 W. This means that in DSR regime, larger saturation power could lead to a wider pulse duration and lower peak power at fixed pump power. It should be noted that under DSR operation condition, the pulse duration increases linearly with the pump power while the pulse peak power maintains constant, as illustrated in Fig. 4. The maximum pule duration available for the DSR operation state is limited by the cavity round trip time [38]. In order to further scale up the DSR-pulse energy, another effective way is to raise the clamping level of pulse peak power. As shown in Figs. 6 and 7, the clamped pulse peak power in DSR regime can be increased by choosing appropriate SA with large modulation depth and small saturation power, resulting in a higher DSR-pulse energy.
3.4 Effect of RSA coefficient
We further simulated the effect of the RSA coefficient β on the DSR-pulse evolution. It was found that the β value is key for generating DSR pulses in the cavity. When it exceeds a critical value, DSR can be always obtained in the cavity, otherwise the NLP operation is observed. Under our current SA parameters settings, i.e., α0 = 0.25, αns = 0.3 and Isat = 20 W, the critical value is about 1 kW−1. At this critical point, the pulse has a rectangular envelope with obvious peak power fluctuations, as shown in Fig. 8(a). Correspondingly, the calculated AC trace has a small spike riding on a triangular pedestal [see Fig. 8(b)]. The resultant pulse contains the combined characteristics of DSR and NLP, indicating that there exists a mixture state in the cavity. Our results are in consistent with the previous study [20], where the authors utilize NPR as an artificial SA.
The interesting pulse dynamics observed can be attributed to the competition between the RSA effect and spectral filtering effect. When the RSA effect dominates, namely the β is larger than the critical value, the pulse peak power is clamped at a relatively low level and its spectrum width is thus confined within the filter bandwidth. In this way, the NLP operation state induced by the spectral filtering effect is avoided and the fiber laser operates in DSR regime. Conversely, when the RSA effect is weak, the pulse peak power cannot be effectively restricted, leading to the SPM-induced spectrum broadening. Eventually, the interaction between the dispersive waves and the DS can result in the occurrence of NLP, as analyzed in Subsection 3.1.
Additionally, we demonstrate the variations of temporal pulse shape and peak power with β, as shown in Fig. 9. When the β increases from 2.5 kW−1 to 4 kW−1, the pulse temporal profiles maintain a rectangular shape, and the peak power decreases from 41 W to 23 W. This means that in DSR regime, smaller β value will lead to the pulse peak power clamped at a higher level, which is beneficial for achieving higher pulse energy.
4. Conclusion
We have numerically demonstrated DSR-pulse generation in an ANDi Yb-doped fiber laser mode-locked by a real SA. The SA includes both the saturable absorption and RSA effects. Theoretical analysis shows that the RSA effect induced by the SA material itself plays a dominant role in generating the DSR pulses. When the RSA coefficient exceeds a critical value, DSR can be obtained in the cavity and otherwise the laser is prone to operating in the NLP state. Furthermore, the impact of key SA parameters (including modulation depth, saturation power and RSA coefficient) on DSR pulse properties has been presented. Our simulation results indicate that through choosing proper SA parameters, high-energy pulses can be directly obtained from a single fiber laser.
Funding
National Natural Science Foundation of China (NSFC) (61775031, 61435003, and 61421002); National Key R&D Program of China (2016YFF0102003).
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