We numerically and experimentally investigate the pulse evolution to the edge of destabilization against pumping powers in a strongly dissipative-dispersive laser configuration mode locked by nonlinear polarization evolution (NPE) technique. Two distinct dynamic processes are indicated by numerical results and further evidenced by experimental observations, where one depicts the monotonous increase in peak power and slight narrowing of duration, the other is different in exhibiting obvious broadening in temporal domain. Correspondingly, it is demonstrated in the simulation of cavity dynamics that the artificial saturable absorber plays two opposite roles in pulse shaping, which implies the switch of cavity feedback. Mechanisms with respect to different cavity feedbacks are analyzed based on a newly-proposed theoretical viewpoint, for positive feedback single pulse operation is restricted by the limit of peak power mainly dependent of the gain bandwidth; for negative feedback the breakup is attributed to the limited strength of clamping effect determined by multiple ingredients.
© 2015 Optical Society of America
Pulse splitting in mode-locked lasers has been investigated from both experimental and theoretical points of view in the past two decades. It is generally considered as a ubiquitous phenomenon in mode-locking , which is justified except for some special circumstances, e.g. dissipative soliton resonance (DSR) . Multi-pulsing, as one of the most universal attractors after pulse splitting, has been extensively studied. Several relevant features including soliton energy quantization, pump hysteresis and multistability are recognized [3–5 ]. To understand the underlying instability mechanism resulting in multi-pulse formation, the detailed transition dynamics between single pulsing and double pulsing was revealed by Bale and Kutz [6, 7 ]. A stable breather was experimentally observed before a more chaotic oscillatory state occurred, which was subsequently reproduced via waveguide array architecture. Besides, there were also other phenomena related to the breakup of pulse, for instance, noise-like emission and soliton explosion. Noise-like emission is indeed chaotic pulse bunching and is shown as a quasi-stationary structure characterized by a narrow peak riding a wide pedestal by averaged autocorrelation measurement . It is recently proved to be an ideal state generating optical rogue wave, thereby becoming a hot issue . Soliton explosion is one of the most exotic phenomena in nonlinear system, where the soliton solution switches back and forth between the quasi-stable pulse and the collapsed temporal structure. The concept was firstly proposed in theory in the basis of numerical solution to the complex cubic-quintic Ginzburg-Landau equation (CQGLE) . The corresponding experimental evidence was lately found in a fiber laser oscillator operating in a transition zone between stable mode-locking and noise-like emission . Despite the criterions of the convergence to multiple pulse formation, soliton explosion and noise-like pulse are elusive at present; they are literally linked with the specific cavity parameters (e.g. gain bandwidth) .
Besides the fundamental exploration of the aforementioned nonlinear dynamics, the study on the pulse splitting is essential in practical operating condition to acquire pulse with better performances. Early at 1998, Kärtner et al. proposed a theory of pulse splitting and attributed the phenomenon to the continuous soliton narrowing in the process of enhancing amplification . During the evolution course reported by Kärtner, every pulse was almost transform-limited; however, it was usually not the case in mode-locked fiber lasers. By virtue of unapparent pulse narrowing, Tang et al. preferred to peak-power limiting effect arising from NPE .
Dissipative soliton, an extended concept in contrast with the conventional soliton, motivates the development of fiber laser technology. Dissipative soliton fiber laser is a promising alternative to the solid state mode-locked laser for its desirable pulse energy and compact structure . Hence, more attention has been paid to the mechanism of the dissipative soliton destabilization so as to boost higher pulse energy. Wise et al. phenomenologically considered the problem via evaluating the nonlinear phase shift. Compared to the theory of convention soliton, it is fairly rational to believe that the dissipative soliton is inherently subjected to a certain value of nonlinear phase shift . By imposing strong dissipation, i.e., spectral filtering and saturable absorber, accumulated chirps are possibly linearized and excessive nonlinear phase shifts are estimated as large as 10π . So far the available mechanism seems to be quite integrated. When tackling with the dissipative soliton implemented by gain spectral filtering (gain-guided type) and NPE, however, the mutual interaction makes the question more complex. Referring to the numerical calculation, Komarov pointed out that in this situation the reverse saturable absorption provided by NPE would play a dominative role in inducing pulse splitting . The exact switching point from positive to negative feedback and critical point of retaining stabilization in cavity were identified for the first time by exploiting a split-step averaging procedure. However, the well-accepted standpoint given by Komarov is challenged by the experiments that showed no continuous wave (CW) state after the breakup (theoretically, negative feedback finally leads to the CW operation) . Liu introduced an additive constraint of excessive pulse chirps to explain such phenomenon, implying that the effects of accumulated nonlinear phase shifts and cavity feedback switching might compete with each other. Nevertheless, there is currently no clear identification of the predominant effect at play when this type of soliton is about to split.
In this paper, we present numerical simulation and experimental observation focused on the pump-power-dependent pulse evolution process in a large normal-dispersion NPE-based mode locked fiber ring laser. It is shown that the pulse, no matter in simulation or in experiment, evolves obviously in two different manners before splitting in response to different states of polarization controllers (PCs). The intuitive understanding inferred from the features of the distinct evolution processes – switch of the cavity feedback, is corroborated through the simulated cavity dynamics. The mechanism corresponding to different cavity feedback is analyzed in detail in terms of the framework derived from an analytical approach. Consequently, we give a relatively universal explanation to the wave breaking phenomenon in fiber lasers by NPE mode-locking.
2. Numerical simulation
2.1 Model of the laser configuration
The numerical model is built based on the experimental setup. As schematically shown in Fig. 1(a) , the realistic fiber laser consists of two polarization-sensitive isolators (PS-ISOs), two pairs of PCs, optical coupler (OC) with 10% output, two sets of wavelength-division multiplexing (WDM) couplers and a segment of 19.5 m erbium-doped fiber (EDF) with absorption of 6.5 dB/m at 980 nm (Nufern EDFC-980-HP). The overall length of pigtailed single-mode fiber (SMF) of the optical components is estimated at 1.48 m, which leads to a 20.98 m long ring configuration. Two 976nm laser diodes (LDs) offer enough power to trace full evolution process of dissipative soliton by forward and backward pumping. The PS-ISO together with two PCs provides an intra-cavity intensity discrimination, thereby acting as additive pulse shaper (APS). We attempt to acquire a laser cavity possessing more flexible nonlinear dynamics by employing two sets of APS.
To specify the idea, the reduced models mode locked by one APS and two APSs are proposed, respectively, when only a segment of 19.5 m-long gain fiber with the nonlinear parameter of 4.2 W−1km−1 is included [see in the insets of Fig. 1(b)]. The split-step averaging procedure, developed from the initial work by Leblond et al. [18, 19 ], is exploited. Thus, the nonlinear loss functions T 1 and T 2 corresponding to the fiber lasers with one APS and two APSs are described as following :Eq. (1) are illustrated in Fig. 1(b). To compare the scope of nonlinear dynamics obtained by one APS and two APSs in a more intuitive way, kurtosis parameters of the given nonlinear loss curves are also plotted. It is found that the values of kurtosis parameters cover a much wider range when two APSs are utilized, which suggests diverse types of dissipation in the laser cavity.
After theoretically verifying the effect of the versatile combined-APSs architecture, we carry out numerical simulation of the real world laser structure through a full lumped scheme; see in Fig. 1(a) (under the arrow). The propagation of light field in the fiber is characterized by the coupled nonlinear Schrödinger equations,5]; Consider that the cavity birefringence can be adjusted in some extent by tuning PCs, k = 4, corresponding to the beat length of ~0.8 m, is used in the simulation of the EDF-dominant configuration; δ = kλ c/(2πc) is the inverse group velocity leading to an additional intra-cavity spectral filtering effect and the chosen wavelength λ c = 1563 nm. β and γ are dispersion and nonlinear parameter of the fibers, in which β EDF = 19.2 ps2km−1, γ EDF = 4.2 W−1km−1 for EDF; β SMF = −22 ps2km−1, γ SMF = 1.3 W−1km−1 for SMF. g = g 0exp[-(|ψ x|2 + |ψ y|2)/E s] is the saturable gain coefficient for EDF (apparently g = 0 for SMF) in which g 0 is the small-signal gain and E s is the gain saturation energy, g 0 = 1 is assumed in the calculation. Ω = 9.5 THz accounts for the spectral gain bandwidth.
The matrices characterizing the analyzer, PC and polarizer are presented as follows, respectively,
2.2 Simulation results
In a certain pump level, by properly setting the angles θ and φ i as well as the phase delay ξ i (i = 1, 2) stable and robust mode-locking is numerically achieved as a localized solution. Figure 2 shows that, when increasing the gain saturation energy various evolution processes are manifested corresponding to different positions in the parameter space. Instead of giving a more detailed analysis of every parameter, the influences of φ 2 and ξ 1 are demonstrated in Fig. 2. Consequently, the decrease of φ 2 and the increase of ξ 1, typically relevant to the growth of nonlinear and linear phase shifts respectively, lead to the enhancement of bell-shaped top in the spectrum. Two categories named as type I and II, typically represented by two extreme situations in Fig. 2, are realized by the feature of the evolution process.
We start from studying type I for θ = 0.75 + π/2, φ 1 = 0.75, φ 2 = π/4, ξ 1 = 0.1 and ξ 2 = 0. Figures 3(a) and 3(b) depict a sequence of autocorrelation traces and spectra against different gain saturation energies, respectively. Autocorrelation traces are adopted here instead of the temporal profiles because they are more convenient to compare with the direct experimental measurement. Once saturation energy E s reaches the threshold of destabilization, the spectrum is perturbed (especially near the central part) while the temporal structure exhibits no prominent fluctuation, as illustrated by the gray lines in Figs. 3(a) and 3(b). Before the breakup takes place, one can observe from the inset of Fig. 3(a) that the pulse narrows with the boost in gain saturation energy, albeit inapparent (drop from 15.2 ps to 14.9 ps). The change in pulse duration at a full width at half maximum (FWHM) is similar to one acquired in the conventional soliton region , despite the dissipative solitons here are much longer. Correspondingly, the spectrum varies with nearly the self-similar shape and the width ranging from one steep edge to another broadens to 26.6 nm when E s increases to 650 pJ. Since the assumed gain bandwidth Ωλ = Ωλ c 2/πc is ~24.5 nm, it is implied that the destabilization may be connected with the constraint imposed by the spectral gain filtering.
Type II is demonstrated with the example illustrated by Figs. 3(c) and 3(d), the selected values of the parameters are: θ = 0.45 + π/2, φ 1 = 0.45, φ 2 = 0.35, ξ 1 = 1.2 and ξ 2 = 0. When entering the unstable regime, the emission turns out to be noise-like but sustains quite stable averaged properties, for instance, the averaged autocorrelation trace and spectrum represented by the gray lines in Figs. 3(c) and 3(d). As to the steady-state operation, the pulse has a clear tendency to broaden, contrary to the prior slight narrowing; as shown in Fig. 3(c), the FWHM pulse width can reach as long as ~70 ps on the condition that E s = 1000 pJ. In response to the broadening in the temporal window, the spectra in Fig. 3(d) display a pronounced variation characterized by a growing and compressing bell-shaped top. In this case, the edge-to-edge width of the spectrum is always less than the gain bandwidth (e.g., ~15 nm for E s = 1000 pJ), indicating that a new mechanism substitutes for the aforementioned spectral gain filtering.
Note that the asymmetric profiles of the spectra, numerically implemented by introducing inverse group-velocity difference in Eq. (2), results from the wavelength-dependent transmission property of the laser structure. Aside from the nonlinear phase shift frequency-dependent linear phase shift also plays an important role in determining transmission characteristic of the system. Because of significant cavity birefringence, an invisible birefringent filter is formed. Furthermore, it is worth nothing that, although the performances of the laser configuration might differ from the previous examples in the unstable regime, the two categories are universal.
In NPE-based laser systems, it is probable to overdrive the saturable absorber effect owing to sinusoidal structure of the extracted transmission curve [21, 22 ]. Mathematically speaking, the switch of the cavity feedback is corresponding to the intensity that obtains maximum transmission. Hence, when increasing cavity energy before the switching point is attained, a strong trend towards pulse with higher peak amplitude is driven by positive feedback; on the other hand, in the situation of negative feedback, the growth of pulse peak power is suppressed due to the degraded transmission and the pulse tends to broaden because of the increasing transmission of the lower amplitude. Then, take the aforementioned distinct in the temporal window into consideration, the intuition might suggest that the two cases result from different cavity feedbacks: type I for the positive feedback, type II for the negative feedback. We numerically analyze the intra-cavity dynamics of the two cases (E s = 650 pJ for type I and E s = 1000 pJ for type II) for trying to verify the intuition. See in Fig. 4 , significant contribution to pulse shaping is made by APS1. As expected, after passing through APS1 pulse narrowing and broadening are witnessed for type I and type II, respectively. The bifurcation of the pulse shaping caused by NPE is interesting and can partially confirm the shift of cavity feedback.
3. Experimental results and discussions
3.1 Experimental results
The schematic of the experimental setup has already been given in Sec. 2.1, an autocorrelator (FR-103XL) and optical spectrum analyzer (Yokogawa AQ6375) are used to simultaneously measure the autocorrelation trace and spectrum of the output pulse, respectively. In addition, a 1 GHz oscilloscope (Agilent DSO-X 3102A) paired with 7 GHz photodetector (Newport 818- BB-51F) is utilized to monitor the pulse train for identifying the single (multiple) pulse operation. By adjusting PCs, pulses tolerant to different scales of the forward pump power P f are identified: (i) pulses that break up at P f ≈50 mW; (ii) pulses that split at P f >200 mW.
A representative example belongs to the prior kind is demonstrated in Figs. 5(a) and 5(b) , the pulse evolves in the way qualitatively consistent with type I and is destabilized at P f ≈55 mW by the amplification of some frequencies near the central wavelength [shown as the gray line in Fig. 5(b)]. See in the inset of Fig. 5(a), pulse narrowing is well produced while the difference between the experimental data and the numerical result in the pulse width probably comes from the relatively reduced model used in the simulation and deviation in the assessment of fiber dispersion. Nevertheless, the spectra agree well in width and show no obvious change in shape. With respect to the spectrum shape, Chong suggests that it is corresponding to the nonlinear phase shift that varies with the bandwidth of the spectral filter . One can imply that the invariable spectrum shape might result from the intrinsic gain bandwidth or equally a certain tolerance of nonlinear phase shift. Namely, such spectral gain filtering hampers further development of the pulse energy. A similar viewpoint was previously proposed in , but the reported spectral width was far less than the common gain bandwidth so that Liu didn’t relate the over-accumulated pulse chirps with the gain filtering. Here, instead of separating the two factors we attribute the incompatible spectral width to the extra effect of high-order nonlinearity (e.g., quintic nonlinearity) induced by NPE technique itself . It seems that the limited gain bandwidth, coupled with the intra-cavity positive feedback that is numerically identified, leads to the pulse destabilization.
The latter kind of the pulses is typically illustrated in Figs. 5(c) and 5(d), where the evolution process is in good accordance with type II. Moreover, the pulse duration and spectral width coincide with the corresponding numerical results as well. In contrast to the pulses of the first variety, latter ones exhibit good resistance to breakup even at the maximum forward pump power P f ≈270 mW (the pump power is measured at the common port of WDM, so it is a bit lower than the standard level). Once the backward pump is activated, stable single-pulsing is disrupted and transforms to a noise-like emission which is quasi-stationary in the sense that stable averaged measurements, shown in Figs. 5(d) (gray curve) and 5(e). Although whether the notion “noise-like pulse” can be used here is remained to be further discussed, the chaotic nature of the noise-like emission is undoubted . Such onset of destabilization is common in gain-guided soliton fiber lasers . As to the increasing bell-shaped top in spectra – one notable feature for type II, it has been frequently referred when concerning DSR [27, 28 ]. Whereas we note that type II apparently differs from DSR  due to quite limited energy scalability. With regard to the negative feedback, since DSR is known as the phenomenon caused by the peak-power clamping effect , breakup of the pulse is presumably due to the failure in fully “peak-power clamping”.
3.2 discussions by a combined analytical approach
According to the numerical intra-cavity dynamics shown in Fig. 4, APS2 has a minor effect upon the pulse shaping; in these cases, the laser configuration can be assumed to be governed by the CQGLE because the kurtosis parameters of the nonlinear loss in one-APS model are in the vicinity of the parabolic level. Analytical solution to the CQGLE, especially when the equation coefficients are linked with the specific cavity parameters, provides an easy way to assess the performances of the laser architectures. By far, two prime approaches have been utilized to look for the exact solution of the CQGLE: directly solving the equation by assuming an ansatz [31, 32 ]; tackling the equation by considering a high-chirp approximation . Subsequently, in the basis of the prior works, analytical models of the dissipative soliton fiber lasers have been built up [34–36 ]; however, most preferred to deal with the problem of energy scalability instead of the pulse dynamics. Recently, a combined analytical approach, incorporating physical and algebraic approach respectively derived from Akhemediev solution (AS) and Podivilov solution (PS), was proposed and aimed at identifying the soliton dynamics . It is somewhat different from the aforementioned theories in the purpose, which is, by contrast, desirable to further explore the mechanisms of soliton fission.
Here, we outline two aspects to quickly review the approach. The scalar master CQGLE takes the form ofEq. (2), coefficients σ, ε, α, μ, and υ are derived from the contributed effect of the artificial saturable absorber and gain medium. σ = g-a 0 is the net linear gain (loss) while a 0 accounts for intrinsic cavity loss; ε and μ represent the nonlinear gain-absorption process; α accounts for the gain spectral bandwidth; υ is the quintic nonlinearity.
First, among the combined approach the physical one based on the AS contributes an intuitive standpoint to understand the pulse dynamics. The form of the AS is given byEq. (5) is rewritten as:Eq. (6) as the limiting intensity among the available ASs, which potentially constrains further increase in peak power of the pulse; simultaneously, pulse shape is controlled by the term and transforms from hyperbolic-secant to super-Guassian for . Namely, Ir (either branch of ) represents a reference level because relative distance toward Il is directly related to the pulse profiles.
The dynamical characteristic of limiting and reference intensity, schematically demonstrated in Fig. 6(a) (numerically validated in ), bifurcates in response to different levels of switching intensity I s = -ε/2μ. For high switching intensity the curve of limiting intensity bends over before reaching I s due to finite gain bandwidth, implying an upper limit to pulse stabilization in the positive feedback regime; lowering the switching intensity promotes the shift of cavity feedback before the drop of gain efficiency, meanwhile limiting intensity progressively approaches the reference intensity, corresponding to the profile evolvement: from sech2 to super-Gaussian.
Secondly, the algebraic approach fulfills the concise connection between pulse dynamics and cavity parameter by exploiting PS with the formEq. (7) is only the solution to Eq. (4) in the case of v = 0, the accuracy (particularly for type II) is inferred by the averaged model and is further assured via numerical calculation. One can see from Eq. (7) that αγ/βε and –σ/εI s are the only variables of the governing parameter R; accordingly, show in Fig. 6(b), master diagram illustrated as contour plot of the values of R 2 is set up in the two-dimensional space for x = αγ/βε and y = –σ/εI s. When the relationship α = g/Ω 2 (Ω is the gain bandwidth) is involved in the gain-guided situation, dots charactering pulse dynamics in the master diagram produce a straight line expressed as37], is convenient to trace the soliton evolution.
As suggested by the physical approach, the fact that the peak power of the pulse is driven to increase by notable positive feedback will result in two stages in the evolution: (i) as long as the peak power is well below the limiting intensity normal sech2 pulse maintains; (ii) once the peak power exceeds the limiting intensity sech2 pulse is distorted and becomes a over-compressed profile that is beyond the prediction of AS. It is evidenced by the experiment results of type I as illustrated in Figs. 7(a) and 7(b) . At a low forward pump power P f = 28.7 mW, the calculated autocorrelation trace with respect to sech2 pulse perfectly fit to the measured curve; in contrast, at a higher pump power P f = 50.4 mW near the threshold of destabilization, higher kurtosis and fatter tail of the measured autocorrelation trace indicates over-compression of the pulse. Therefore, by taking advantage of the physical approach, the leading effect of spectral gain filtering on the pulse splitting is further confirmed. From another perspective, the mechanism of destabilization can also be regarded as the constraint of the peak power. Remarkably, for the experiments performing as type II autocorrelation traces at P f = 88 and 243 mW manifest clear transformation of pulse shape in Figs. 7(c) and 7(d), which are in good agreement with the deduction arisen from the physical approach. The asymmetric structure of the autocorrelation trace inherits from the pulse. The asymmetry in the pulse shape, well reproduced by simulation as shown in the inset of Fig. 7(c), stems from relatively high cavity birefringence induced by a long segment of EDF .
Since the existence region of stable pulse-like solutions in the master diagram is known, master curve is capable of monitoring the whole pulse evolution under variable gain coefficient [37, 39 ]. As the slope –βΩ 2/I s γ in Eq. (8) is easy to identify according to the measurements, master curve of the realistic laser configuration could be picked from a set of parallel lines. Take the experimental evidence of type II for example; we assume the experimental laser design is free from dispersion and gain dispersion management by overlooking the influence introduced by the short segment of pigtailed fiber. Apart from the fiber parameters β, γ and Ω, the switching intensity I s that can be tuned by the polarization controllers is unclear. A method to estimate the value of I s from the peak power of DSR is given in , whereas the DSR phenomenon is inaccessible in the current experiment; so we roughly regard I s as the peak power of the pulse which is on the edge of destabilization. Corresponding to the output power of 19.06 mW and pulse width of 56.3 ps, the obtained critical pulse owns the peak power of 340 W (assumed Gaussian shape), then the slope –βΩ 2/I s γ ≈ −1.21. A chosen master curve is shown in Fig. 6(b), where the dots 1, 2, 3 and 4 are corresponding to the spectra plotted in the inset. The comparable result implies that the use of master curve may open a window to investigate the pulse evolution. Furthermore, strength of the peak power clamping effect is quantitatively described in the master diagram: minimum value of R 2 along the master curve, R 2 = 0 represents the complete peak power clamping effect. In accordance with the viewpoint in , steeper slope |–βΩ 2/I s γ| of the master curve enhances the resistance to the multiple pulse generation, which coincides with the conclusion proposed by Haboucha .
In this paper, we systematically analyze the mechanism of pulse breakup in gain-guided soliton fiber laser mode locked by NPE technique. It is numerically found and experimentally observed that the patterns of pulse evolving before destabilization bifurcate into two categories: type I for monotonous increase in peak power and slight narrowing of duration; type II for apparent broadening of the pulse width. The distinction arises from the difference in cavity feedback which is partially verified through investigating the intra-cavity dynamics. In positive feedback regime, the critical spectral width close to the gain bandwidth, no matter in simulation or in experiment, implies that spectral gain filtering is a dominant effect on pulse splitting. According to an analytical approach, the mechanism is further confirmed and is physically understood as the constraint of pulse peak power. With regard to negative feedback, we suggest that the resistance to pulse destabilization is determined by the strength of peak power clamping effect which can be intuitively represented in a master diagram. In consequence, steeper slope |–βΩ 2/I s γ| of the master curve might lead to an increased pulse-energy performance in negative feedback regime.
This research was supported by the China State 863 Hi-tech Program (2013AA031502 and 2014AA041902), NSFC (11174085, 51132004, and 51302086), Guangdong Natural Science Foundation (S20120011380), and China National Funds for Distinguished Young Scientists (61325024). W. C. Chen acknowledges the financial support from NSFC (No.11204037).
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