## Abstract

A combined analytical approach to classify soliton dynamics from dissipative soliton to dissipative soliton resonance (DSR) is developed based on the established laser models. The approach, derived from two compatible analytical solutions to the complex cubic-quintic Ginzburg-Landau equation (CQGLE), characterizes the pulse evolution process from both algebraic and physical points of view. The proposed theory is proved to be valid in real world laser oscillators according to numerical simulations, and potentially offers guideline on the design of DSR cavity configurations.

© 2015 Optical Society of America

## 1. Introduction

Passively mode-locked fiber lasers, as ultrafast light sources, are more compact and user-friendly than the solid-state mode-locked lasers. However, as the key property of a mode-locked laser, the pulse energy is easily limited by the overdriven nonlinear effect because of the relatively high nonlinearity of the fiber. It has been demonstrated that, when the pulse energy reaches to a specific value, for example, 0.1 nJ in the anomalous-dispersion regime, the mode-locked pulse would tend to break up and is manifested as multiple pulses in fiber lasers [1]. So far, various approaches have been proposed to avoid multi-pulse generation in fiber lasers [2–4]. Among them, the dispersion engineering method was regard as an effective way to circumvent the multipulsing instability (MPI). Shifting the cavity dispersion from anomalous to an all-normal one enables the output pulse energy from a fiber laser to be enhanced to a much higher level, i.e., 20 nJ [5], which greatly stimulates the exploration of a new soliton family — dissipative soliton [6]. The conventional soliton concept is extended when taking into account energy exchange, which indicates that the dissipative soliton is balanced through the interaction between gain, loss, dispersion and nonlinearity. Theoretically speaking, dissipative soliton solution is an attractor of the Ginzburg-Landau equation with the inclusion of dissipative gain (loss) terms [7]. However, if the pump power level is further increased, the dissipative solitons still suffer from MPI owing to the over-accumulated nonlinear phase shift [5, 8]. From the viewpoint of some practical applications requiring large pulse energy, it is of great significance to develop other methods to achieve wave-breaking-free pulses in ultrafast fiber lasers.

Recently, a novel concept — dissipative soliton resonance, was theoretically proposed to boost the single-pulse energy [9]. DSR phenomenon features the generation of flat-top pulse without wave-breaking while almost keeping the amplitude constant with the increasing pump power under certain parameter selections in the framework of CQGLE. Therefore, the pulse energy operating in DSR region could be infinitely increased in theory, which provides an effective way to obtain wave-breaking-free high-energy pulse. Although the origin of DSR phenomenon could be traced back to about 20 years ago [10], the DSR concept was formally proposed in 2008 by N. Akhmediev’s group. Up to now, DSR phenomenon has been numerically proven to be existed not only in the normal dispersion regime but also in the anomalous dispersion regime, which severely counters the conventional idea of soliton dynamics [11]. In addition, to avoid the tedious numerical work searching for the CQGLE parameters that matched DSR phenomenon, roadmaps illustrated by analytical resonance curves in the parameter space were also given throughout the dispersion region [9, 12].

Passively mode-locked fiber lasers, which are actually nonlinear dissipative systems characterized by the generalized Ginzburg-Landau equation, are suitable for observing versatile soliton nonlinear phenomena. From another point of view, the oscillator characteristics will be easy to trace if the solution (especially the analytical one) of the corresponding master equation is achieved, for instance, Kalashnikov proposed a purely analytical approach to analysis the pulse energy scalability [13]. Here, we focus on the dynamic courses of the pulse evolution in the realistic fiber lasers, e.g., convergences to the normal dissipative soliton and DSR with pumping power. In particular, although long square-wave pulse, approximate state of DSR in the practical operating condition of virtually available pump power, has been frequently observed in passively mode-locked fiber lasers in despite of the dispersion regimes [14–16] or mode-locking techniques [17–19], a unified analytical viewpoint has not been built yet [20].

The purpose of the present paper is to explore an analytical identification of soliton dynamics covering from dissipative soliton to DSR in the real world fiber lasers. We start from handling with the problem in a completely algebraic scheme demonstrated as master curve, and subsequently propose the notions of limiting intensity and reference intensity stemming from an analytical solution to physically understand the distinct pulse dynamics. The combined approach is verified and embodied by numerical simulations of three typical laser oscillators, which is capable of offering guideline of desired cavity designs. Table 1 includes primary abbreviations and some symbols corresponding to the definitions that are not usually adopted in the previous work but are essential in the present article.

## 2. Combined analytical approach to the analysis of pulse dynamics in the CQGLE

The universal (1 + 1)-dimensional form of CQGLE reads [6]:

*ψ*is the field envelope,

*z*is the propagation distance,

*t*is the retarded time,

*D*is the second-order dispersion and restricted to negative value here,

*γ*is the nonlinearity,

*δ*is the linear gain or loss,

*β*describes the coefficient related to gain bandwidth,

*ε*and

*μ*account for nonlinear gain-abosrption process,

*v*accounts for the quintic nonlinearity.

The high chirp approximation that underlies the analytical approach is assumed in the present paper, which is justified when the pulse durations are in the picosecond time scale.

#### 2.1 Algebraic approach: master curve

An analytical solution of Eq. (1) for *v* = 0, which represents the dissipative soliton family with a single composite parameter, is written in spectral domain as [21, 22]

*H*(

*x*) is the Heaviside function,

*Δ*implies the positions of two steep edges in spectrum,

_{f}*I*

_{s}for

*I*

_{s}= -

*ε*/2

*μ*physically represents the switching intensity that is given detailed description in the following section. As seen in Eq. (2), –

*βγ*/

*Dε*and –

*δ*/

*εI*

_{s}are the only variables of the governing parameter

*R*, thereby constituting a master diagram of the dissipative soliton (–

*βγ*/

*Dε*as

*x*-axis and –

*δ*/

*εI*

_{s}as

*y*-axis) [23]. The minus sign added ensures that the values of the two terms are positive.

Since the value of *υ* is usually 3 orders of magnitude smaller than the nonlinearity *γ* (see Sec. 3 below), it is rational enough to use Eq. (2) in our study for |*ψ*|^{2} <<1000. The relevant master diagram is illustrated in Fig. 1, where the gray area is the existence region and the DSR curve (*R* = 0) is colored green. We add the curve *I*_{max} = *I*_{s}, and the peak power of the pulse solution *I*_{max} is given by

The composite linear loss (negative) *δ* expressed as *g* + *a*_{0} (*g* denotes gain coefficient and *a*_{0} accounts for intrinsic cavity loss), as it is known, is energy dependent due to variable gain strength *g* in an oscillator. As to the fiber lasers, particularly the gain guided erbium-doped ones [24], the gain dispersion *β* = *g*/*Ω*^{2} (*Ω* is the gain bandwidth) leads to the underlying relationship of the plots. Namely, the master curve representing the soliton dynamics in the master diagram is in the form of

*x*= –

*βγ*/

*Dε*with energy independent

*β*, which is additionally controlled by the slope aside from the intercept.

The modification shall not be underestimated, take the resonance phenomenon for example, the vertical type starts to approach the resonance curve by simply reducing the value of –*βγ*/*Dε*. The transition from non-DSR convergence to DSR convergence is denoted as DSR transition I and is shown in Fig. 1, where the two master curves (red lines) are plotted according to the parameters extracted from [25]. On the other hand, the oblique type determined by intercept and slope makes the approaching scheme twofold. First, DSR transition II, in analogy with DSR transition I, describes a lateral translation of the master curve. In this case, the DSR master curve is characterized by large slope *DΩ*^{2}/*I*_{s}*γ* and small intercept *a*_{0}*γ*/*εDΩ*^{2}, meanwhile lies in the unstable regime. Actually, we find the negative branch of the solution Eq. (2) stabilized when taking into account the gain saturation, in other words, the DSR master curve is indeed reasonable. We call it the conventional DSR master curve (CDMC) for the similarity with the one for fixed *β*. Second, in contrast, DSR transition III exhibits a rotate of the master curve to achieve DSR. The relevant master curve demands more delicate selection of the parameter set and is always in the stable regime, accordingly, we call it the stable DSR master curve (SDMC).

#### 2.2 Physical approach: an intuitive viewpoint

Another solution to Eq. (1), proposed by Akhmediev et al., is given by [26]

To establish a reduced frame, we introduce two new ingredients that are physically interpreted as limiting intensity and reference intensity. The limiting intensity *I _{l}*, mathematically defined by

*I*= −2

_{l}*d*

_{0}/

*d*

_{2}, is the maximum peak intensity among the available pulselike solutions when −2

*d*

_{0}/

*d*

_{2}>0. Although the condition −2

*d*

_{0}/

*d*

_{2}<0 can also support Akhmediev solution as mentioned in [27] (e.g., ones with M-shaped optical spectra), hereafter it is fair enough to only consider the case −2

*d*

_{0}/

*d*

_{2}>0 on the assumption of the forementioned high chirp approximation. The reference intensity

*I*

_{r}incorporates two branches ${I}_{r}^{\pm}=\left(-\epsilon \pm \sqrt{{\epsilon}^{2}-4\mu \delta}\right)/2\mu $, which are virtually the two distinct roots of the quadratic equation represented by $\delta +\epsilon {I}_{r}+\mu {I}_{r}^{2}=0$. The quadratic equation initially arises from the CQGLE in the continuous wave approximation [12], and by substituting

*I*for

_{l}*I*

_{r}, it is rewritten as

*δ*from

*I*= −2

_{l}*d*

_{0}/

*d*

_{2}and Eq. (6) to obtain the relation as follow

*d*

_{2}and

*d*

_{4},

The term ${d}_{2}^{2}-4{d}_{0}{d}_{4}$, similar to parameter *R* in Eq. (2), plays a dominant role in determining the temporal profile of Eq. (15). Using the substitution given by Eq. (8), the decisive term ${d}_{2}^{2}-4{d}_{0}{d}_{4}$ transforms to

*Δ*= 0, namely, DSR is obtained once the limiting intensity equals to the reference intensity (either of the branches).

One more step has been made by utilizing a general limit -*D*>>*β* that is well grounded in the previous investigations of solid-state laser oscillators [28]. In this case, the lengthy algebraic term denoted by *p* in the following form can be greatly simplified since *Dd* affects *p* in the leading order,

*I*= −3

_{l}*δ*/

*ε*. In what follows, we consider the nonlinear gain in the CQGLE defined by

*g*

_{NL}=

*δ*+

*εI*+

*μI*

^{2}and the prior switching intensity

*I*

_{s}= -

*ε*/2

*μ*is corresponding to the vertex of

*g*

_{NL}. From physical point of view, the switching intensity demonstrates the critical point that overdrives the saturable absorber effect (SAE). To associate with the SAE, the reference intensity is rewritten aswith the underlying relation $\delta =\mu \left({I}_{s}^{2}-\Delta {I}_{r}^{2}\right)$, the positive sign of

*ΔI*

_{r}is regulated. Consequently, ${I}_{l}/{I}_{r}^{\pm}$ can be put into the form,

Equation (12) clarifies our intuitive viewpoint: when limiting intensity is below the switching point (*I _{l}*<

*I*

_{s}), the negative branch of the reference intensity ${I}_{r}^{-}$ at play dominates the pulse shaping process, thus the criterion of the dissipative solution existence (

*Δ*>0) is always satisfied, while the criterion of the DSR formation (

*Δ*= 0) can never be attained for ${I}_{l}/{I}_{r}^{-}>1$; as the limiting intensity climbs over the switching point (

*I*>

_{l}*I*

_{s}), positive branch of the reference power ${I}_{r}^{+}$ begins to take over, then DSR is achieved (

*Δ*= 0) when ${I}_{l}={I}_{r}^{+}=4{I}_{s}/3$; the limiting intensity is not allowed to develop because Akhmediev solution is truncated for

*Δ*<0.

## 3. Numerical simulations of the nonlinear polarization evolution based mode-locked fiber laser

Nonlinear polarization evolution (NPE), acted as artificial saturable absorber, is a satisfying technique to offer the overdriven SAE in the laser cavity [29]. To relate the analytical approach with the real world NPE-based oscillator configuration, the bridge between the distributed CQGLE and the full lumped model is required.

#### 3.1 Averaging procedure

The illustration of the cavity configuration is presented in Fig. 2. It is reduced in contrast with the one including five or more discrete elements in Ding’s work and verified to be explicit enough to describe the NPE mode-locking process [30]. We set the position right after the polarization beam splitter (PBS) as the beginning of the round trip (terminal of the last round trip). By making use of the well-developed averaging, split-step procedure [31, 32], one obtains the iterative equation when propagation loss in the fiber is negligible,

*g*is the saturable gain strength,

*g*

_{0}is the small-signal gain and

*E*

_{s}represents the gain saturation energy.

*J*

_{Analyzer},

*J*

_{PBS}, and

*J*

_{PC}are Jones matrixes of the analyzer, PBS and polarization controller (PC), respectively.

*α*

_{1},

*α*

_{2}are orientation angles of the PC and analyzer with respect to the fast axis of the SM gain fiber respectively,

*α*

_{3}is the phase delay induced by the PC.

*K*is the birefringence matrix associated with the fiber birefringence

*k*,

*Γ*is the dispersion operator obtained by the Taylor expansion approximately, and

*Ω*is the gain bandwidth.

*ψ*

_{n}is the field envelope at the terminal of the

*n*

^{th}round trip,

*L*is the cavity length,

*γ*,

*D*are the nonlinearity and dispersion of the SM gain fiber,

*J*= |

*ψ*

_{n}|

^{2}cos

*α*

_{1}sin

*α*

_{1}sin

*α*

_{3}

*L*. The scalar form of Eq. (14) is shown as

*T*(|

*ψ*

_{n}|

^{2}) is the only function resulting in the intensity discrimination according to NPE technique and can be put into the following partial differential equation (PDE) via taking a continuous limit of Eq. (15).

*T*

_{s}(|

*ψ*

_{n}|

^{2}) is the real part of the logarithmic term, namely

We tackle the transmission function *T*_{s} via the quadratic fit to transform the sinusoidal Ginzburg-Landau type Eq. (16) to the universal CQGLE [33]. For clearness, we write Eq. (1) in another way

*a*

_{0},

*a*

_{1},

*a*

_{2}are the coefficients of the fitted quadratic function,

*b*

_{1}and

*b*

_{2}are the calculated coefficients of the corresponding Taylor series for the imaginary part of the logarithmic term in the vicinity of |

*ψ*|

^{2}=

*I*

_{s}.

#### 3.2 Full lumped scheme

As for the piecewise dynamics, PC, PBS and analyzer are treated as discrete elements and modeled by Jones matrixes. The initial field is linear polarized at position A in Fig. 2, after passing through the PC the light transforms to an elliptical polarization state and subsequently propagates in the single-mode gain fiber. The coupled nonlinear Schrödinger (CNLS) equations are used to characterize the pulse dynamics in the fiber:

*ψ*

_{x},

*ψ*

_{y}are light fields of two artificial directions. Note that Eq. (19) is only an approximation to the real evolution process of the pulse in the fiber referring to the leading order effect, i.e., second-order dispersion, self and cross phase modulation, gain and gain bandwidth. When entering the analyzer, the two orthogonal fields undergo a coordinate transformation attributed to the change of the orientation angle. Finally one polarized field is retained for the next round trip while the other is output. More specific process of the numerical simulation can be found in [34]. The full model accounts closely to the realistic fiber laser, and underlies the master CQGLE.

#### 3.3 Cavity parameters and numerical comparisons

We demonstrate with three typical oscillators I, II, III and intend to shift the soliton dynamics primarily by changing the cavity length [implied by Eq. (17)] meanwhile slightly adjusting other parameters. The main simulation parameters are listed as: for oscillator I, *L* = 5 m, *D* = 20 ps^{2}/km, *γ* = 4 W^{−1}km^{−1}, *g*_{0} = 0.5 m^{−1}, *Ω* = 12 THz, *α*_{1} = *π*/5, *α*_{2} = *π*/5 + *π*/2, *α*_{3} = 0.4, *k* = 0.6 (*α*_{1}, *α*_{2}, and *α*_{3} are used in lumped scheme), *a*_{0} = −0.259, *a*_{1} = 7.93 × 10^{−4}, *a*_{2} = −6.15 × 10^{−7}, *b*_{1} = 0.0015, *b*_{2} = 1.45 × 10^{−6} (*a*_{0}, *a*_{1}, *a*_{2}, *b*_{1}, and *b*_{2} are used in distributed equation); for oscillator II, *L* = 10 m, *D* = 12 ps^{2}/km, *γ* = 4 W^{−1}km^{−1}, *g*_{0} = 1 m^{−1}, *Ω* = 12 THz, *α*_{1} = *π*/6, *α*_{2} = *π*/6 + *π*/2, *α*_{3} = 0.3, *k* = 0.6 (*α*_{1}, *α*_{2}, and *α*_{3} are used in lumped scheme), *a*_{0} = −0.1113, *a*_{1} = 4.899 × 10^{−4}, *a*_{2} = −5.432 × 10^{−7}, *b*_{1} = −3.4 × 10^{−4}, *b*_{2} = 2.89 × 10^{−7} (*a*_{0}, *a*_{1}, *a*_{2}, *b*_{1}, and *b*_{2} are used in distributed equation); for oscillator III, *L* = 25 m, *D* = 30 ps^{2}/km, *γ* = 4 W^{−1}km^{−1}, *g*_{0} = 3 m^{−1}, *Ω* = 12 THz, *α*_{1} = *π*/10, *α*_{2} = *π*/10 + *π*/2, *α*_{3} = 1.55, *k* = 0.8 (*α*_{1}, *α*_{2}, and *α*_{3} are used in lumped scheme), *a*_{0} = −0.085, *a*_{1} = 0.0019, *a*_{2} = −1.09 × 10^{−5}, *b*_{1} = 4.698 × 10^{−4}, *b*_{2} = −2.21 × 10^{−6} (*a*_{0}, *a*_{1}, *a*_{2}, *b*_{1}, and *b*_{2} are used in distributed equation).

To verify the averaging process, comparisons between the simulation results calculated from both approaches are made (*E*_{s} = 300 pJ for oscillator I, *E*_{s} = 1000 pJ for oscillator III). As shown in Fig. 3, square-wave pulse is well reproduced and the spectra are all in good qualitative agreement. The explanation to the quantitative error are mainly threefold: first, averaging scheme itself may induce error which is closely related with *a*_{0} [31]; second, rough assessments of the parameters *b*_{1} and *b*_{2} affect the simulations of the master equation to some extent; third, the quadratic form is not that favourable to the transmission function *T*_{s}, in contrast, polynomial or sinusoidal function can give a better description. In other words, a certain error is induced by the defect in the form of the master equation. Nevertheless, the error is quite admissible and the mode-locking dynamics that we trace is qualitative. Hence, by means of the averaging procedure, the soliton dynamics of the mode-locked fiber laser model can be reflected in the master diagram and intuitive framework.

## 4. Results and discussion

To investigate the pulse dynamics for oscillators I, II and III, we raise the gain saturation energy continuously till the unstable operation occurs and simultaneously obtain the corresponding gain coefficient *g*. Following the algebraic approach, the parameter sets extracted from the averaged models of the three instances are mapped into the master diagram, denoted by pentagrams, diamonds and dots, respectively, as demonstrated in Fig. 4. Color scales the values of *R*^{2} [see in Eq. (2)] from zero (blue) to high (red) levels, manifesting the pulse shapes changing from flat-top (blue) to hyperbolic-secant (red). As a result, Master curves of the oscillators directly reflect the transforming processes of the pulses.

By combining with the physical approach, we classify the soliton dynamics into three main groups: conventional dissipative soliton (CDS), transition state (TS) and DSR. It is worth mentioning that the classification here is limited to a specific group of dissipative solitons for the high chirp approximation and prerequisite *I _{l}*>0 (i.e., −2

*d*

_{0}/

*d*

_{2}>0). Nevertheless, the identification is fairly rational in the laser oscillators that generating relatively long pulses (several picosecond to several nanosecond).

#### 4.1 Conventional dissipative soliton

According to the numerical simulation of oscillator I, curve of the linear loss *δ* bends over in response to the increasing gain saturation as shown in Fig. 5(a). It is easily understood because the saturation of the gain is constrained by the finite gain bandwidth [25]. Correspondingly, the limiting intensities are trapped in quite a limited scope below the switching intensity, whereas the peak intensities tend to increase monotonously to the switching point owing to the SAE.

Consider the physical definition of the limiting intensity, it seems to me that after exceeding the limiting intensity the peak power further raises by “over-compression” of the pulse and finally collapses due to the finite tolerance. This can be partially supported by the pulse evolution process illustrated in the inset of Fig. 5(a). Since the mechanism of the pulse destruction is similar to that leading to multisolton in the anomalous regime [30], we call this variety of soliton dynamics conventional dissipative soliton. Besides, the negative branch of the reference power retains below the limiting power, which reasonably matches the prediction of the physical approach.

In the master diagram as shown in Fig. 4, plots experience a lateral translation from the positive feedback region owing to the non-negligible effect of *v* on Eq. (2), which is implied by the truncated master curve.

#### 4.2 Transition state

The master curve of oscillator II in Fig. 4 indicates that when entering into the negative feedback region the dots can get close enough to the resonance curve but finally deviate from it due to the improper selection of slope or intercept. It is obvious that, under a certain perturbation, master curve of the TS can develop to SDMC.

Turn to the physical viewpoint, it is found in Fig. 6 that limiting intensity releases from the constraint and gets over the switching point at a certain pump power, which reveal the intracavity negative feedback. See in the inset of Fig. 6(a), as long as SAE is overdriven the energy flowing in tends to broaden the pulse instead of over-compressing it. Limiting intensity ceases from getting closer to the reference level when further increasing gain saturation energy. According to our simulation, MPI appears owing to the enhanced amplification of the background noise caused by the artificial reverse saturable absorber effect [34]. The failure to DSR, indicated by the characteristic of the master curve, results from a composite effect rather than the spectral filter effect.

#### 4.3 Dissipative soliton resonance

Pulse dynamics of oscillator III in the master diagram belongs to the CDMCs, as illustrated in Fig. 4. Therefore, in the process of approaching resonance curve, the linear loss *δ* converges to a limit corresponding to the point where the master curve and resonance curve cross. In other words, gain strength won’t get saturated. It is mainly because pumping power only amplifies and compresses the bell-shaped top of the spectrum while remaining the rectangular edges unchanged to prevent the drop of gain efficiency, as illustrated in the inset of Fig. 7(a).

In this case, revealed by Fig. 7(b), the limiting intensity is approaching the positive branch of the reference intensity with growing energy and eventually exceeds it for $\Delta /{I}_{l}^{2}=-0.07$ in the limiting case. The restraint of the exact approach between the reference intensity and limiting intensity (Δ = 0) results from the two algebraic equations used in the analytical procedure, namely,

Equations (20) and (21) establish the relationship among*D*,

*β*,

*ε*,

*γ*,

*μ*,

*ν*while only one free parameter

*d*is contained, which indicates one of the CQGLE coefficients, e.g. quintic nonlinearity

*ν*, can’t be set arbitrarily. As a result, the deviation from the analytical prediction comes from the Akhmediev solution that is not exactly fit the master CQGLE for the oscillator III. To further confirm our point, a delicately designed set of parameters [26] for

*D*= −1,

*β*= 0.5,

*γ*= 1,

*υ*= −0.1,

*ε*= 0.362,

*μ*= −0.061,

*g*

_{0}= 1 and

*a*

_{0}= −0.54 is exploited and satisfying results are achieved. Now that the relation ${I}_{l}={I}_{r}^{+}$ is ensured, it is reasonable enough to describe the well known peak power clamping effect through the comparison between the limiting intensity and reference intensity.

In what follows, we estimate the forementioned limit -*D*>>*β* in the oscillator III. It seems that the limit can be well generalized to the fiber laser systems as illustrated in Fig. 8, however, it is actually not the case for the oscillator I. The two opposite assessments stem from the difference in the gain dispersion *β*. Considerable saturation of the gain in the situation of DSR leads to a significantly deceased *β*, which supports the limit. By the way, gain dispersion management implemented by employing passive fiber may also reduce the *β* value via the averaging process. In a word, the validity of the limit -*D*>>*β* depends on either soliton dynamics or gain dispersion managed design.

Furthermore, we make an attempt to relate the specific simulation results with the analytical solution. Take the DSR condition for example, the approximate form Eq. (13) is chose as the analytical solution where *δ* extends to a *E*_{s}-dependent function achieved by exponential fit to the simulated *δ* values, namely,

The comparison between the simulated *δ* dots and the relevant fitting curve is made in the inset of Fig. 9. Figure 9 reveals a linear increase in pulse width towards the growing gain saturation energy. Refer to a work of Liu [34], gain saturation energy in the numerical work can well represent the pump power, that is to say, the linear tuning behavior of DSR can be analytically predicted.

To summarize the arguments, the schematic is given as shown in Fig. 10. We have discussed three main classes of soliton dynamics. (1) CDS. With the pumping power, positive feedback dominates the pulse shaping mechanism because the limiting intensities are always lower than the switching point. Correspondingly, master curve is restricted in the positive feedback part of the master diagram. (2) TS. In this condition, the parameter set enters into the negative feedback region of the master diagram since limiting intensity can get over the switching power. However, master curve cannot reach DSR curve due to the improper selection of cavity parameters. (3) DSR. In order to achieve DSR, as a first step, negative feedback with the increasing pump power is required. Then, the master curve in the parameter plane should be either CDMC or SDMC.

## 5. Insights into the DSR formation in mode-locked fiber lasers

We attempt to consider the prerequisites for DSR by taking account of the relation between the established real world oscillator and the corresponding master governing equation. Negative feedback implies that the switching intensity should be low enough to make sure the limiting intensity can pass through before spectral filter effect takes effect. In the present model, cavity length and fiber nonlinearity are two decisive factors determining the switching level. As demonstrated in Fig. 11(a), lengthening the cavity can significantly reduce the switching intensity, if the switching level is fairly low further increase in the cavity length produces little effect. The influence of the cavity length on the switching intensity results from the increasing linear phase delay. In Fig. 11(b), switching intensity decreases monotonously when the fiber nonlinearity becomes stronger, it is obvious that lower light intensity is demanded to attain a constant nonlinear phase delay while propagating in the fiber with larger nonlinear coefficient. In short, first, lengthening the cavity is an efficient way allowing the negative feedback; Second, the stronger nonlinearity is, the shorter cavity can be. Note that once the laser configuration is decided, states of the polarization components and fiber birefringence which can be simultaneously adjusted by polarization controller also affect switching power to some extent.

CDMC features large slope *DΩ*^{2}/*I*_{s}*γ* and small intercept *a*_{0}*γ*/*εDΩ*^{2}, which directly indicates that larger *D*/*γ* and lower switching intensity promote DSR. Both *a*_{0} and *ε* are obtained from the quadratic fit to Eq. (17), and the effect of cavity length eliminates in *a*_{0}/*ε*. In other word, the term *a*_{0}/*ε* depends only on the shape of the transmission function. On the other hand, SDMC, attained by properly selecting the cavity parameters including dispersion, nonlinearity and cavity length, is more sensitive to the perturbation induced by simply rotating PC.

## 6. Conclusion

In this paper, we developed a combined analytical approach to identify the soliton dynamics in the normal-dispersion fiber lasers. The theory is in the basis of high chirp approximation. In the algebraic approach, the concept of master curve for energy- dependent *β* is proposed to characterize the pulse dynamics in the master diagram. As to the physical approach derived from Akhmediev solution, the notions of limiting intensity and reference intensity are presented to understand the pulse shaping processes of different soliton varieties.

After establishing the reduced NPE-based mode-locked fiber laser model, soliton dynamics in the fiber laser is classified into three categories by means of the combined approach. First, conventional dissipative soliton, corresponding to the master curve lying in the positive feedback part of the master diagram, is distorted for the constraint of the limiting intensity. Second, transition state, operating in the negative feedback regime, deviates from the dissipative soliton resonance illustrated by the master curve and is broken up via the excessive amplification of the background. Third, dissipative soliton resonance is arisen due to indefinite approach between limiting intensity and reference intensity, which is reflected as CDMC or SDMC in the master diagram.

Since the master curve is especially convenient to trace the mode-locking dynamics in fiber lasers, insights into DSR formation are provided. It is revealed that lengthening the cavity and employing high nonlinearity fiber can stimulate the resonance phenomenon in the oscillator for introducing intracavity negative feedback, then *D*/*γ* begins to take effect and large values of *D*/*γ* promote the generation of wave-breaking-free square pulses.

## Acknowledgments

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).

## References and links

**1. **L. E. Nelson, D. J. Jones, K. Tamura, H. A. Haus, and E. P. Ippen, “Ultrashort-pulse fiber ring lasers,” Appl. Phys. B **65**(2), 277–294 (1997). [CrossRef]

**2. **M. E. Fermann, A. Galvanauskas, and M. Hofer, “Ultrafast pulse sources based on multi-mode optical fibers,” Appl. Phys. B **70**(1), S13–S23 (2000). [CrossRef]

**3. **K. Tamura, E. P. Ippen, H. A. Haus, and L. E. Nelson, “77-fs pulse generation from a stretched-pulse mode-locked all-fiber ring laser,” Opt. Lett. **18**(13), 1080–1082 (1993). [CrossRef] [PubMed]

**4. **M. H. Ober, M. Hofer, and M. E. Fermann, “42-fs pulse generation from a mode-locked fiber laser started with a moving mirror,” Opt. Lett. **18**(5), 367–369 (1993). [CrossRef] [PubMed]

**5. **A. Chong, W. H. Renninger, and F. W. Wise, “All-normal-dispersion femtosecond fiber laser with pulse energy above 20 nJ,” Opt. Lett. **32**(16), 2408–2410 (2007). [CrossRef] [PubMed]

**6. **P. Grelu and N. N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photonics **6**(2), 84–92 (2012). [CrossRef]

**7. **T. Schreiber, B. Ortaç, J. Limpert, and A. Tünnermann, “On the study of pulse evolution in ultra-short pulse mode-locked fiber lasers by numerical simulations,” Opt. Express **15**(13), 8252–8262 (2007). [CrossRef] [PubMed]

**8. **D. N. Schimpf, J. Limpert, and A. Tünnermann, “Controlling the influence of SPM in fiber-based chirped-pulse amplification systems by using an actively shaped parabolic spectrum,” Opt. Express **15**(25), 16945–16953 (2007). [CrossRef] [PubMed]

**9. **W. Chang, A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative soliton resonances,” Phys. Rev. A **78**(2), 023830 (2008). [CrossRef]

**10. **D. J. Richardson, R. I. Laming, D. N. Payne, V. Matsas, and M. W. Phillips, “Selfstarting, passively mode-locked erbium fibre ring laser based on the amplifying Sagnac switch,” Electron. Lett. **27**(6), 542–544 (1991). [CrossRef]

**11. **W. Chang, J. M. Soto-Crespo, A. Ankiewicz, and N. Akhmediev, “Dissipative soliton resonances in the anomalous dispersion regime,” Phys. Rev. A **79**(3), 033840 (2009). [CrossRef]

**12. **P. Grelu, W. Chang, A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative soliton resonance as a guideline for high-energy pulse laser oscillators,” J. Opt. Soc. Am. B **27**(11), 2336–2341 (2010). [CrossRef]

**13. **V. L. Kalashnikov and A. Apolonski, “Energy scalability of mode-locked oscillators: a completely analytical approach to analysis,” Opt. Express **18**(25), 25757–25770 (2010). [CrossRef] [PubMed]

**14. **X. Wu, D. Y. Tang, H. Zhang, and L. M. Zhao, “Dissipative soliton resonance in an all-normal-dispersion erbium-doped fiber laser,” Opt. Express **17**(7), 5580–5584 (2009). [CrossRef] [PubMed]

**15. **Z. C. Luo, W. J. Cao, Z. B. Lin, Z. R. Cai, A. P. Luo, and W. C. Xu, “Pulse dynamics of dissipative soliton resonance with large duration-tuning range in a fiber ring laser,” Opt. Lett. **37**(22), 4777–4779 (2012). [CrossRef] [PubMed]

**16. **L. Duan, X. Liu, D. Mao, L. Wang, and G. Wang, “Experimental observation of dissipative soliton resonance in an anomalous-dispersion fiber laser,” Opt. Express **20**(1), 265–270 (2012). [CrossRef] [PubMed]

**17. **S. K. Wang, Q. Y. Ning, A. P. Luo, Z. B. Lin, Z. C. Luo, and W. C. Xu, “Dissipative soliton resonance in a passively mode-locked figure-eight fiber laser,” Opt. Express **21**(2), 2402–2407 (2013). [CrossRef] [PubMed]

**18. **N. Zhao, M. Liu, H. Liu, X. W. Zheng, Q. Y. Ning, A. P. Luo, Z. C. Luo, and W. C. Xu, “Dual-wavelength rectangular pulse Yb-doped fiber laser using a microfiber-based graphene saturable absorber,” Opt. Express **22**(9), 10906–10913 (2014). [CrossRef] [PubMed]

**19. **Y. Liu, W. Li, X. Shen, C. Wang, Z. Yang, and H. Zeng, “Square nanosecond mode-locked laser based on nonlinear amplifying loop mirror,” IEEE Photonics Technol. Lett. **26**(19), 1932–1935 (2014). [CrossRef]

**20. **E. Ding, P. Grelu, and J. N. Kutz, “Dissipative soliton resonance in a passively mode-locked fiber laser,” Opt. Lett. **36**(7), 1146–1148 (2011). [CrossRef] [PubMed]

**21. **E. Podivilov and V. L. Kalashnikov, “Heavily-chirped solitary pulses in the normal dispersion region: new solutions of the cubic-quintic complex Ginzburg-Landau equation,” J. Exp. Theor. Phys. Lett. **82**(8), 467–471 (2005). [CrossRef]

**22. **V. L. Kalashnikov, “Chirped dissipative solitons of the complex cubic-quintic nonlinear Ginzburg-Landau equation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **80**(4), 046606 (2009). [CrossRef] [PubMed]

**23. **D. S. Kharenko, O. V. Shtyrina, I. A. Yarutkina, E. V. Podivilov, M. P. Fedoruk, and S. A. Babin, “Highly chirped dissipative solitons as a one-parameter family of stable solutions of the cubic–quintic Ginzburg–Landau equation,” J. Opt. Soc. Am. B **28**(10), 2314–2319 (2011). [CrossRef]

**24. **L. M. Zhao, D. Y. Tang, and J. Wu, “Gain-guided soliton in a positive group-dispersion fiber laser,” Opt. Lett. **31**(12), 1788–1790 (2006). [CrossRef] [PubMed]

**25. **A. Komarov, F. Amrani, A. Dmitriev, K. Komarov, and F. Sanchez, “Competition and coexistence of ultrashort pulses in passive mode-locked lasers under dissipative-soliton-resonance conditions,” Phys. Rev. A **87**(2), 023838 (2013). [CrossRef]

**26. **J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **55**(4), 4783–4796 (1997). [CrossRef]

**27. **W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A **77**(2), 023814 (2008). [CrossRef]

**28. **V. L. Kalashnikov, E. Podivilov, A. Chernykh, S. Naumov, A. Fernandez, R. Graf, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators: theory and comparison with experiment,” New J. Phys. **7**(1), 217 (2005). [CrossRef]

**29. **Z. Cheng, H. Li, and P. Wang, “Simulation of generation of dissipative soliton, dissipative soliton resonance and noise-like pulse in Yb-doped mode-locked fiber lasers,” Opt. Express **23**(5), 5972–5981 (2015). [CrossRef] [PubMed]

**30. **D. Y. Tang, L. M. Zhao, B. Zhao, and A. Q. Liu, “Mechanism of multisoliton formation and soliton energy quantization in passively mode-locked fiber lasers,” Phys. Rev. A **72**(4), 043816 (2005). [CrossRef]

**31. **E. Ding and J. N. Kutz, “Operating regimes, split-step modeling, and the Haus master mode-locking model,” J. Opt. Soc. Am. B **26**(12), 2290–2300 (2009). [CrossRef]

**32. **E. Ding, E. Shlizerman, and J. N. Kutz, “Generalized master equation for high-energy passive mode-locking: the sinusoidal Ginzburg–Landau equation,” IEEE J. Quantum Electron. **47**(5), 705–714 (2011). [CrossRef] [PubMed]

**33. **F. I. Khatri, J. D. Moores, G. Lenz, and H. A. Haus, “Models for self-limited additive pulse mode-locking,” Opt. Commun. **114**(5-6), 447–452 (1995). [CrossRef]

**34. **X. Liu, “Hysteresis phenomena and multipulse formation of a dissipative system in a passively mode-locked fiber laser,” Phys. Rev. A **81**(2), 023811 (2010). [CrossRef]