Dissipative soliton evolution in passively mode-locked fiber lasers with large net-normal-dispersion and high nonlinearity is investigated numerically and confirmed experimentally. I have proposed a theoretical model including the nonlinear polarization evolution and spectral filtering effect. This model successfully predicts the pulse behaviors of the proposed laser, such as the multi-soliton evolution, quasi-rectangle-spectrum profile, trapezoid-spectrum profile, and unstable state. Numerical results show that, in contrast to the typical net- or all-normal-dispersion fiber lasers with the slight variation of the pulse breathing, the breathing ratios of the pulse duration and spectral width of our laser are more than three and two during the intra-cavity propagation, respectively. The nonlinear polarization rotation mechanism together with spectral filtering effect plays the key roles on the pulse evolution. The experimental observations confirm the theoretical predictions.
©2009 Optical Society of America
Recently, the fiber lasers based on the gain medium of erbium- and ytterbium-doped fiber have attracted a lot of interest in the past two decades due to their potential applications –. Especially, passively mode-locked fiber lasers have been investigated extensively -, because they have the capacity of producing ultra-short pulses that can help us directly observe some of the fastest processes in nature. When a fiber laser is made of fibers with purely anomalous group-velocity dispersion (GVD) (or large negative GVD together with small positive GVD), conventional soliton-like pulses are produced by the balance between the fiber nonlinearity (i.e., self-phase modulation) and the fiber linear dispersion (i.e., GVD) . Optical conventional-soliton pulses exist in conservative systems and hence they are used to describe nonlinear solitary wave solutions of integrable equations.
The dissipative soliton (DS) concept is a fundamental extension of the concept of solitons in conservative and integrable systems . DSs exist in non-conservative systems and thus their dynamics is very different from that of conventional solitons -. A fiber laser with purely normal GVD (or large normal GVD together with small anomalous GVD) would presumably have to exploit dissipative processes in the mode-locked pulse shaping  . DSs have attracted great interest in the development of fiber lasers because they are able to significantly improve the deliverable energy of pulse, approaching or even exceeding 100 nJ -.
Spectral filtering mechanism enables us to successfully explain the experiential results of DSs observed in net- and all-normal-dispersion fiber lasers   -. Microjoule-level pulse energy has been achieved from the all-normal-dispersion fiber lasers  . By using the coupled extended Ginzburg-Landau equations, the nonlinear pulse propagation in the cavity is described -. However, the reported DS schemes usually have a smaller net intra-cavity dispersion and/or a lower nonlinearity -. What happens in a laser cavity when its net intra-cavity dispersion is very large with a high nonlinearity? The current work answers these questions from simulations and experiments. In this paper, I have proposed a theoretical model including the nonlinear polarization rotation (NPR) mechanism and spectral filtering effect. DSs in passively mode-locked fiber laser with large net-normal-dispersion and high nonlinearity are investigated numerically by solving the extended nonlinear Schrödinger equations (NLSE) and are confirmed experimentally. The numerical simulations predict the pulse characteristics of the proposed laser, which are very different from the typical net- and all-normal-dispersion fiber lasers. The experimental observations are in good agreement to the theoretical predictions.
2. Numerical model and simulation
2.1. Laser cavity and model
To study the feature and dynamic evolution of DSs in a passively mode-locked fiber laser, we use a numerical model that incorporates the most important physical effects such as the NPR, spectral filtering, and Kerr effect. The propagation model is sketched in Fig. 1 . The laser cavity is made up of three segments. The fiber parameters for each segment are listed in Table 1 . The total length of laser cavity is 23.8 m with the net cavity dispersion of ~1 ps2. One can see from Fig. 1 and Table 1 that the long erbium-doped fiber (EDF) provides the large net-normal-dispersion and the high nonlinearity for the proposed laser (the nonlinear coefficient of EDF is about four times larger than that of SMF).
The polarization additive-pulse mode-locking (PAPM) system is made of a polarization-sensitive isolator and two sets of polarization controllers. The PAPM system is used to produce the NPR effect, which relies on the intensity-dependent rotation of an elliptical polarization state in a length of optical fiber. The nonlinear polarization evolution serves as the saturable absorber that cleans up both the leading and trailing edges of the pulse . The propagation of polarization field components (i.e., horizontal and vertical components) in the fibers is modeled by the two coupled equations. The intensity transmission of light through PAPM is determined by the polarization of polarizer and analyzer together with the linear and nonlinear phase delay   . However, a scalar model is used in this paper. The PAPM function is then assumed to be intensity-dependent transmittance with a second-order power and a transmittivity of 0.94, as shown in Fig. 2 . The intensity-dependent transmittance function is given by
The gain medium of EDF has a gain bandwidth of 35 nm, whose profile is illustrated in Fig. 3(a) . The PAPM element is assumed to have a super-Gaussian transmission function with 70-nm bandwidth, as shown in Fig. 3(b). Both serve as the spectral filtering elements, which can cut off the temporal wings of a pulse   .
2.2. Equations and simulations
Usually, since the PAPM mechanism acts on the polarization state, the propagation is modeled by the two coupled equations that involve a vector electric field. To simplify the propagation model with the PAPM function, a model as shown in Fig. 2 is proposed in this paper. To describe better the features of the laser of delivering DSs, we numerically simulated the laser operation by solving the extended NLSE, which includes the effects of GVD, self-phase modulation, and saturated gain with a finite bandwidth. For this context, this equation is given by 23] 
Here g 0 is the small signal gain coefficient, related to the doping concentration. Es is the gain saturation energy, which is pump-power dependent and varies along the fiber position z. The pulse energy Ep is given by
The light transmission through fibers and other elements can be simulated by solving Eq. (2) and by using the suitable temporal and spectral filters (i.e., Figs. 2, 3(a) and 3(b) for PAPM and EDF). To numerically simulate the feature and behavior of this dissipative system, the simulation has started from an arbitrary signal and converged into a stable solution after approximately 50 round trips, as shown by transient temporal evolution in Fig. 4 . Numerical results show that the signal eventually converges toward a steady-state solution, independent of the initial conditions. The intra-cavity pulse evolution and the pulse profile at the output position are illustrated in Figs. 5 and 6 , respectively. In simulations, the gain saturation energy at the incident position of EDF is Es 0 = 1.6 nJ and other parameters are shown in Table 1.
The pulse characteristics of one cavity round-trip are illustrated in Fig. 5. Figures 5(a) and (b) show the intra-cavity pulse evolutions for the spectral width, temporal duration, energy, and peak power over one cavity round-trip, respectively. One can see from Fig. 5(a) that the pulse duration and the spectral width decrease in the beginning of the EDF and then increases monotonically in the remaining EDF part. The strong chirp of pulse causes the anomalous behavior of evolution of the pulse duration and spectral width in the beginning EDF part. The pulse duration reaches the maximum at the end of the gain fiber (i.e., EDF) because EDF has normal dispersion and other fibers have anomalous dispersion (the normal and anomalous dispersions induce the pulses to be broadened and narrowed, respectively). In contrast to the typical pulse behavior of net- or all-normal-dispersion fiber lasers that the pulse duration and spectral bandwidth changes only slightly during the intra-cavity propagation    , the temporal and spectral breathing ratios in this laser are more than 3 and 2, respectively. As a result, the traditional theory (i.e., the balance between the spectral broadening resulting from self-phase modulation and the spectral filtering resulting from the gain fiber and other elements could achieve the self-consistency) fails to explain our results. The new mechanism is covered in the proposed laser.
Figure 5(b) shows that the pulse energy is maximum at the intra-cavity position z = 20 m and the peak power increases in the beginning of the gain fiber and then decreases. We can observe from Figs. 5(a) and 5(b) that, through the output coupler and PAPM element, the spectral width and peak power of pulse vary slightly whereas the pulse duration and pulse energy decrease remarkably. We estimate that, besides the balance between the spectral filtering resulting from EDF and the spectral broadening via the self-phase modulation, the balance between the temporal filtering by PAPM element and the broadening by the normal dispersion plays a key role on the pulse evolution. We name the former as the spectral balance and the latter as the temporal balance. Two kinds of balances together lead to a self-consistent intra-cavity pulse evolution. The detailed discussions are shown in the following section.
Figure 5 exhibits that the spectral width, pulse duration, pulse energy, and peak power at z = 20.5 m are about 22.3 nm, 20.8 ps, 3.4 nJ, and 164 W, respectively. The pulse profiles in the temporal and spectral domains at this position are shown in Figs. 6(a) and 6(b) in detail. Surprisingly, the spectral profile approximately is the trapezoid shape, instead of the quasi-rectangle shape with steep edges that is numerically and experimentally demonstrated in the typical net- and all-normal-dispersion fiber lasers -. As shown on Fig. 6(a), the pulse duration is 20.8 ps and the pulse chirp is nearly linear across the pulse. The pulses are numerically dechirped with a linear dispersive delay down to 274 fs leading to a time-bandwidth product of ~0.76 (Fig. 7 ). One can see from Fig. 7 that the small satellites exist on the dechirped pulse and they contain about 5% of pulse energy. The satellites result from the nonlinear chirp of the pulse edges (Fig. 6(a)).
The evolution of the output spectrum versus the initial gain saturation energy Es 0 is shown in Fig. 8 . For low Es 0 (e.g., Es 0 = 0.8 nJ), the spectrum approximately shows a flat top with steep edges and the sharp peaks on the edges (Fig. 8(a)). This characteristic is as the typical spectrum of net- and all-normal-dispersion fiber lasers. As Es 0 is increased (e.g., Es 0 = 0.9 nJ), the spectrum broadens while its edges fluctuate remarkably (Fig. 8(b)). With further increasing Es 0, the spectral profile is stable again. Surprisingly, the spectral profile at this case is similar to the trapezoid-spectrum shape (Fig. 8(c)), instead of the quasi-rectangle-spectrum shape as shown in Fig. 8(a) for Es 0 = 0.8 pJ. By comparing to Fig. 8(a), the pedestal of spectrum in Fig. 8(c) significantly broadens. These numerical results (Figs. 8(a)-8(c)) are consistent with the experimental observations (Figs. 5(a)-5(c) in Ref .). When Es 0 is enhanced to ~2 nJ, the spectral profile is very unstable and the numerical solution is divergent (Fig. 8(d)). Numerical simulation shows that, when Es 0 is less than 2 nJ, the laser only emits single pulse and the spectral width (especially, the pedestal of spectrum) increases with increasing Es 0.
As Es 0 is increased further, the numerical solution is convergent again whereas the laser emits two pulses over one cavity round-trip. Figures 8(e) and 8(f) illustrate two examples at Es 0 = 2.2 and 3.4 nJ, respectively. When Es 0 is increased to about 4 nJ, the numerical solution is divergent and the pulse is extremely unstable. Successively, the triple solitons are stably produced from the laser cavity over one cavity round-trip for Es 0>4.2 nJ. The pulse evolution repeats again, as predicted in Ref . These numerical results well agree to our previous experiment observations (Fig. 4 in Ref .), where the unstable and stable mode-locking states alternately evolve along the gain saturation energy (corresponding to the pump power). Figure 8 shows that the characteristics of DSs drastically depend on the system parameters and the given external conditions.
To further demonstrate the characteristics of pulses at low Es 0 (e.g., Fig. 8(a)), the temporal profiles of the pulses before and after extra-cavity dechirping are shown in Figs. 9(a) and 9(b), respectively. The pulse approximately has a hyperbolic tangent instantaneous frequency variation and its duration is 19.1 ps. The pulses are numerically dechirped with a linear dispersive delay down to 547 fs leading to a time-bandwidth product of ~0.85. Although the pulses have the quasi-rectangle-spectrum shape for the case of Es 0 = 0.8 nJ (Fig. 8(a)) and the trapezoid-spectrum shape for the case of Es 0 = 1.6 nJ (Fig. 6(b)), respectively, both are strongly chirped. Even after dechirping, their time-bandwidth productions are still about two times larger than the Fourier transform limit.
3. Experiments and comparisons
A polarization-sensitive isolator (PS-ISO) together with two polarization controllers (PC) forms a PAPM element (the green dashed-frame). A 20-m-long erbium-doped fiber (EDF) provides the gain amplification for the laser system. The fiber pigtail of wavelength-division-multiplexed (WDM) coupler is Hi1060 with the length of 1 m. Other fibers in cavity are standard single-mode fiber (SMF) with the length of 2.8 m. The 977-nm laser diode (LD) can provide the pump power of up to 500 mW. The polarization-sensitive isolator provides unidirectional operation and polarization selectivity in a ring-cavity configuration. The saturable absorber is implemented via NPR. An autocorrelator, optical spectrum analyzer (OSA), and 11-GHz oscilloscope together with a 12-GHz photodetector are used to simultaneously monitor the laser output.
The proposed laser cavity has dispersion map with the total normal and anomalous GVD of about + 1.07 and −0.07 ps2, respectively. The normal GVD is much larger than the anomalous GVD so that our laser cavity has very large net-normal GVD. The typical spectrum characteristics of the all-normal-dispersion pulses have the steep spectral edges and the sharp peaks on the edges of the spectrum  . Obviously, our laser cavity with the special dispersion map is different from that of the typical all-normal-dispersion laser cavities. As a result, the proposed fiber laser constitutes a new type of pulse shaping in mode-locked lasers and covers a new mechanism for the pulse evolution.
By adjusting two polarization controllers with appropriate orientation and pressure settings, self-started mode locking of the laser is achieved at the threshold pump power P≈70 mW. After mode locking, the fiber laser produces stable pulses with the typical all-normal-dispersion laser spectrum, i.e., the steep edges and the sharp peaks near its edges. When the pump power is about 122 mW, the laser emits the unstable pulses. However, when the pump power P is more than 123 mW, the fiber laser stably produces a type of pulse exhibited as the trapezoid-spectrum profile. The experimental results are shown in our previous reports, i.e., Figs. 5(a)-5(c) in Ref . These experimental observations well agree with the theoretical predictions as shown in Figs. 8(a)-8(c).
With further increasing the pump power, our fiber laser stably emits the pulses with the increase of the pulse duration and spectral width. An experimental example for P≈180 mW is shown in Fig. 11, where (a)-(c) show the optical spectrum of the pulses, the corresponding oscilloscope and autocorrelation traces, respectively. It is seen from Fig. 11 that (1) the pulse separation is about 114.7 ns, corresponding to the fundamental cavity frequency ~8.7 MHz; (2) the 3-dB spectral width Δλ of solitons is about 22.5 nm; (3) the autocorrelation trace has a full width at half maximum (FWHM) of about 31 ps. If a Gaussian pulse profile is assumed, the pulse width Δτ is about 22 ps. A time–bandwidth product is about 61 so that the pulses are very strongly chirped. The pulse energy in this case is ~0.25 nJ, corresponding to 2.5 nJ for the intra-cavity energy.
When P is increased to about 300 mW, the pulses become very unstable (the intra-cavity pulse energy is about 4 nJ). Figure 12 shows a typical result of the optical spectrum at this case. The unstable fluctuation of the top and edges of the spectrum are very strong. This experimental result is consistent with the theoretical prediction as shown in Fig. 8(d). When the pump power is about 320 mW with the appropriate settings of polarization controllers, two solitons emerge from the laser over a cavity round-trip. The oscilloscope trace for this case is demonstrated in Fig. 13 .
To confirm the proposed theoretical model, either the temporal balance via the temporal filtering (Fig. 2) or the spectral balance via the spectral filtering (Fig. 3) is used to simulate the laser system. Firstly, taking into account the temporal filtering and skimming the spectral filtering, we simulate the laser cavity and the numerical results are shown in Fig. 14(a) . In the simulations, all parameters are the same as those used in Fig. 6, except the spectral filtering effect via EDF and PAPM. One can obviously see from Fig. 14(a) that the numerical solutions are unstable. Secondly, when only spectral filtering is used, the numerical results are shown in Fig. 14(b). Similar to the previous step (Fig. 14(a)), it is also unstable for the numerical solutions. By comparing Fig. 6 to Fig. 14, we can conclude that the temporal filtering effect together with the spectral filtering effect leads to the self-consistent intra-cavity pulse evolution.
In experiments, the temporal filtering function via PAPM is very sensitive to the polarization of polarizer and analyzer. The simulation results show that the dynamic behavior and evolution of pulses are also sensitive to the intensity-dependent transmittance function of PAPM. In contrast, the evolution of pulses is hardly sensitive to the gain profile of EDF. Although an asymmetric EDF gain profile is assumed (Fig. 3(a)), the simulations show that the characteristic of pulses is determined by the width of the gain spectrum rather than its profile.
Both theoretical and experimental results show that infinitely increasing the net dispersion of laser cavity is not a reasonable solution for power scalability of mode-locked fiber lasers. Recently, many types of pulses have been reported, e.g., gain-guided solitons, stretched pulses or dispersion managed solitons, similaritons, conventional solitons, etc. From the theoretical and experimental results (Figs. 6, 8, and 11) and the laser cavity structure (Fig. 1) the pulse here does not belong to one of such above families. For example, the spectrum of gain-guided solitons is as the quasi-rectangle-spectrum shape with steep edges , whereas the spectral profile of our laser is similar to the trapezoid-spectrum shape. The breathing ratio as reported here is more than three, which approximately is the same order of magnitude of breathing ratio as shown in the stretched-pulse laser. But the proposed laser is very different from the stretched-pulse lasers, because the net cavity dispersion is near zero for the stretched-pulse lasers whereas more than zero for our laser. Furthermore, the pulses of our laser are strongly chirped and even they are dechirped to be far from the Fourier transform limit (Figs. 6, 7, and 9). Therefore the proposed laser here emits a new type of pulses.
To show the power scalability of laser, Fig. 15 demonstrates the extracted energy per pulse with respect to the output coupler ratio. In simulations, except that Es 0 = 1.2 nJ and the coupler ratio is variable, the other parameters are the same as those used in Fig. 8. Figures 15(a)-15(c) show the relationships of the pulse duration, the intra-cavity pulse energy, and the extracted pulse energy versus the output coupler ratio, respectively. One can observe from Fig. 5 that, with the increase of output coupler ratio, the pulse duration and the intra-cavity pulse energy decrease. Whereas the extracted pulse energy increases in the beginning of the output coupler ratio and then decreases. When the output coupler ratio is about 0.9 (i.e., 90%), the maximum of the extracted pulse energy is as high as ~1.9 nJ.
By solving the extended NLSE together with the temporal and spectral filtering effects, a passively mode-locked fiber laser with strong normal dispersion and large nonlinearity is numerically investigated. Numerical results show that the proposed laser can operate on the multi-soliton evolution, quasi-rectangle-spectrum shape, trapezoid-spectrum shape, and unstable state with the variation of the gain saturation energy Es (Es is a pump-power dependent variable). The experimental observations are in good agreement to the theoretical predictions. In contrast to the typical net- or all-normal-dispersion fiber lasers with the slight variation of the pulse breathing    , the breathing ratios of the pulse duration and spectral width of our laser are more than 3 and 2 during the intra-cavity propagation, respectively.
The proposed laser system covers the temporal and spectral balances. The former is from the balance between the NPR effect (i.e., the temporal filtering) via the PAPM element and the broadening via the self-phase modulation. The latter is from the balance between the spectral filtering effect (resulting from the gain bandwidth of EDF and the spectral bandwidth of other elements) and the broadening (resulting from the net-normal-dispersion effect). Both the temporal and spectral balances lead to the self-consistent intra-cavity pulse evolution.
This work was supported by the “Hundreds of Talents Programs” of the Chinese Academy of Sciences and by the National Natural Science Foundation of China under Grants 10874239, 10604066, and 60537060. The author would especially like to thank Leiran Wang, Yongkang Gong, and Xiaohong Hu for help with the experiments.
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