The coexistence of weakly sech-shaped solitons and strongly dissipative solitons is experimentally observed in an ultra-large net-anomalous-dispersion mode-locked fiber laser for the first time to author’s best knowledge. Both sech-shaped and dissipative solitons appear to be the asymmetrically combined pulse state with one pulse component much smaller than the other. The energy of dissipative solitons is over three orders of magnitude larger than that of sech-shaped solitons. Two different types of pulse-shaping mechanisms coexist in the laser: one is the dissipative processes and the other is the balance between anomalous dispersion and nonlinear Kerr effect. Numerical simulations and analysis confirm the experimental observations.
©2011 Optical Society of America
Solitons as a kind of stable self-localized waves occur in various physical systems and are one of the most fascinating nonlinear phenomena [1–6]. Soliton formation and dynamics in optical fibers result from a balanced interaction between the anomalous dispersion of fibers and the nonlinear Kerr effect . However, solitons formed in passively mode-locked (PML) fiber lasers subject to the interplay of the laser cavity dispersion and nonlinearity and gain and losses [8–11]. When a PML fiber laser is made of fibers with purely anomalous dispersion, the fiber dispersion balances the fiber nonlinearity to produce conventional soliton-like pulses that are well described by the nonlinear Schrödinger equation (NLSE) and have the hyperbolic-secant profile. In this case, the spectral bandwidth of solitons is much narrower than the laser gain bandwidth , and the cavity pulse peak clamping effect limits soliton to less than 0.1 nJ of the pulse energy in standard fibers [10,12].
To increase the pulse energy, most modern ultra-short pulse cavities have the dispersion management in which a fiber laser can operate either in the positive or negative cavity dispersion regime. When the net dispersion of the laser cavity is anomalous enough, approaches zero, and is normal enough, sech-shaped solitons , dispersion-managed solitons (i.e., stretched pulses) , and self-similar parabolic pulses  can be formed, respectively. When the laser cavity contains components without anomalous dispersion or with small anomalous dispersion together with large normal dispersion, dissipative solitons (DSs) and DS molecules can be generated [15–18]. Pulse evolutions in the aforementioned lasers are qualitatively distinct from each other and they are controlled by the distinctive types of soliton-shaping mechanisms. The influence of the laser gain on the conservative sech-shaped solitons is very weak, whereas the gain and loss play an essential role in the formation of DSs [19,20].
Recently, the theoretical predictions and experimental observations show that with a appropriate cavity configuration a laser is expected to produce two distinctive types of pulses at the same cavity. When the net cavity dispersion βnet approaches zero, dispersion-managed solitons and sech-shaped solitons coexist in the same laser . With the moderate normal βnet, both stretched-pulse and self-similar operation can be observed in a laser . When βnet is normal enough, the fiber laser either works on a status that is similar to an all-normal-dispersion laser or generates a type of pulses exhibited as the trapezoid-spectrum profile according to the pump power . The numerical simulations show that the high-energy DSs can even be generated by lasers operating in the anomalous dispersion regime [22–25].
In this work, we experimentally observe that weakly sech-shaped solitons and strongly DSs coexist in ultra-large net-anomalous-dispersion PML lasers. Two different types of solitons attribute to two completely different soliton formation mechanisms and are governed by different theoretical modeling. The theoretical simulation and analysis successfully explain our experimental observations. Our investigation provides a better understanding of the pulse shaping in PML fiber lasers and a way to achieve high-energy nanosecond pulses.
2. Experimental setup and observations
2.1. Experimental setup
The proposed fiber laser is shown schematically in Fig. 1 . The laser cavity is a 720-m-long loop consisting of a polarization-sensitive isolator (PS-ISO), two sets of polarization controllers (PCs), a segment of standard single-mode fiber (SMF), a wavelength-division-multiplexed (WDM) coupler, a piece of erbium-doped fiber (EDF), and a fused coupler (10% output). EDF has a length of about 10 m with absorption of 6 dB/m at 980 nm, dispersion of about −42 ps/nm/km, and nonlinear coefficient of 4.5 /W/km at 1550 nm. The standard SMF has a length of about 700 m with dispersion of 17 ps/nm/km and nonlinear coefficient of 1.3 /W/km at 1550 nm. The total length of fiber pigtail of components is about 10 m. The polarization-sensitive isolator provides unidirectional operation and polarization selectivity in a ring-cavity configuration, and forms a polarization additive pulse mode-locking (PAPM) element by combining with two polarization controllers. The polarization state of lightwaves in the cavity can be controlled by adjusting two polarization controllers. The nonlinear polarization rotation technique is used for locking the laser. A 977-nm laser diode (LD) supplies the pump power of up to 500 mW. A polarization beam splitter (PBS) is used to separate the two orthogonal polarizations of the laser emission. An autocorrelator, an optical spectrum analyzer (OSA), and a 70-GHz oscilloscope (Tektronix TDS8200) together with a 50-GHz photodetector are used to simultaneously monitor the laser output.
The laser has the fundamental cavity frequency of about 278 kHz, corresponding to the round-trip time of about 3.6 μs. The laser cavity has the dispersion management with total anomalous and normal dispersion of about −15.4 ps2 and 0.5 ps2, respectively. The proposed laser system has extremely large net-anomalous dispersion so that it may exploit two soliton-shaping mechanisms. One is the balance between the anomalous dispersion and the nonlinear Kerr effect and thus the conventional soliton-like pulses are generated. The other is the balance between gain and loss processes that induce the DS pulses.
2.2. Weakly sech-shaped soliton pulses
The proposed laser starts to emit CW for the pump power P≈10 mW, as shown in Fig. 2(a) . With appropriate orientation and pressure setting of two polarization controllers, self-started mode locking can be achieved when P is beyond a threshold value of about 15 mW. After mode locking, the fiber laser emits stable pulses with the typical soliton-like spectrum, i.e., spectral sideband structure related to the linear dispersive waves . An example for P = 18 mW is shown in Figs. 2(b), 2(d) and 2(e), where they are optical spectra, autocorrelation trace, and oscilloscope trace of solitons, respectively. One can see from Figs. 2(b), 2(d), and 2(e) that (1) the spectral width of solitons is about 1 nm; (2) the autocorrelation trace has a full width at half maximum (FWHM) of 4.5 ps (corresponding to the pulse duration of about 2.9 ps); (3) the time–bandwidth product is 0.36, showing that it is sech-shaped pulses rather than Gaussian-shaped pulses; and (4) the laser is multiple pulse operation with the nonuniform temporal spacing [Fig. 2(e)]. The nonuniformity of the measured pulse height [Fig. 2(e)] is caused by the limit of the electronic detection system. Theoretically, solitons in Fig. 2(e) have the same pulse height due to the soliton energy quantization effect . By increasing P, the spectral sidebands of soliton are significantly enhanced whereas the spectral width varies slightly. Figure 2(c) shows an example for P = 21 mW.
Figure 3 shows the experimental observations for the optical spectra of u and v components of pulses at the pump power P = 18 mW. Note that the y-axis of Fig. 3 is the linear scale, instead of the logarithmic scale in Fig. 2(b). It is found from Fig. 3 that the ratio RP of peak power of u and v components of pulses is about 11.5. According to the theory , the ratio of pulse energy of u and v components also is 11.5.
The experimental observations show that the laser only produces conventional solitons with sech-shaped profile when the pump power is from about 15 mW to about 25 mW. In this operation regime, the soliton number in the cavity increases with the increase of pump power, whereas the peak and duration of solitons change slightly. The intracavity energy of each soliton is about 0.04 nJ. By comparing to the traditional fiber soliton laser , the proposed laser generates solitons with narrower spectral width Δλ and wider pulse duration Δτ, e.g., Δλ≈1 nm and Δτ≈2.9 ps here instead of Δλ≈9 nm and Δτ≈0.45 ps in Ref .
2.3. Coexistence of two types of weak and strong pulses
When the pump power P is beyond about 28 mW, a strange phenomenon occurs, i.e., the laser stably emits a type of pulses without spectral sidebands and with an order-of-magnitude increase in the pulse height. An experimental example for P = 29 mW is shown in Fig. 4 . Figure 4(a) is the oscilloscope trace at the range from 0 to 100 μs, and Figs. 4(b) and 4(c) are the local view of Fig. 4(a) from 0 to 2.5 μs and from 3.2 to 3.4 μs, respectively. Figure 4(d) is the optical spectra, in which the ratio RP of peak power of u and v components is about 11.1.
It is found from Figs. 4(a)–4(c) that two different types of pulses coexist in the laser cavity, where weak pulses [Fig. 4(b)] are the same with the sech-shaped pulses [Fig. 2(e)] and strong pulses [Fig. 4(c)] have quasi-rectangular temporal profile with the pulse duration of about 5 ns. The strong pulse increases the pulse height by a factor of about 10 in comparison with the weak pulse. Figure 4(a) shows that the temporal spacing of strong pulses is a round-trip time of cavity and a strong pulse together with multiple weak pulses coexists within the laser cavity. By comparing Fig. 4(b) to Fig. 2(e), one can see that the height of the weak pulses is approximately the same with that of sech-shaped solitons. The experimental observations show that the energy of weak pulse varies slightly with the increase of pump power. The intracavity energy of strong pulse for P = 29 mW is about 40 nJ, increasing three orders of magnitude in comparison with the weak pulse. Comparing Fig. 4(d) to Figs. 2(b) and 2(c), we can see that the optical spectrum does not have spectral sidebands for P>28 mW rather than have spectral sidebands at the pump power range from about 10 mW to about 25 mW. It originates from the fact that the strong pulses dominate the spectral property for P>28 mW.
The experiments show that with the increase of P the height of strong pulses hardly changes while their width broadens. Some experimental examples are shown in Fig. 5(a) , in which Δτ≈6.7 and 8.5 ns for P = 40 and 50 mW, respectively. No internal fine structures are observed within the pulses. We note that the trailing and leading parts of the pulses are different [three examples are shown in Figs. 4(c) and 5(a)]. The difference is caused by the response time of our measurement systems. The optical spectra for P = 50, 150, 300, 450 mW are shown in Fig. 5(b). One can see from Fig. 5(b) that the optical spectra approximately are the Gaussian profiles in the linear scale.
Based on net-normal-dispersion or all-normal-dispersion laser configuration, the fiber laser oscillator can generate high-energy sub-nanosecond and nanosecond DSs [10,28,29]. In this report, an ultra-large net-anomalous-dispersion PML laser is proposed to generate high-energy nanosecond pulses. We believe that the dissipative processes based on gain and loss in laser system dominate the pulse-shaping mechanism and the strong pulses are a class of DSs.
We can observe from Figs. 3 and 4(d) that the ratio RP of peak power of u and v components of pulses is more than 11. Then the ratio of pulse energy for u and v components of pulse is beyond 11 according to the theory , whether the laser delivers the weakly conventional sech-shaped solitons or the strong DSs. As a result, the pulse approximates a linearly polarized wave propagating along the fiber.
3. Theoretical simulation and analysis
3.1. Coexistence of two types of weak and strong pulses
The conventional solitons propagating in the standard fibers are usually simulated by a NLSE , where the birefringence of fiber is not taken into account. However, the formation and propagation of solitons in PML fiber lasers are modeled by the coupled complex NLSEs [10,30]. To find the physical mechanism of soliton formation in our laser, we numerically solve the coupled complex NLSEs to simulate the modeling of laser. The coupled complex NLSEs are expressed by 31]30].
The coupled complex NLSEs are solved with a predictor–corrector split-step Fourier method . The following parameters are employed for our simulations for possibly matching the experimental conditions: α = 0.2 dB/km, g 0 = 2 m−1, γ = 4.5 W−1km−1 for EDF and 1.3 W−1km−1 for SMF, Ωg = 30 nm, and β 2 = 53.5 × 10−3 ps2/m for EDF and −21.7 × 10−3 ps2/m for SMF. The simulation starts from an arbitrary signal and converges into a stable solution when the appropriate parameters are given.
When θ = π/3.5, φ = π/10, ϕ 1 = 0.9 + π/2, and Es = 0.24 nJ, the weakly sech-shaped soliton solution is obtained, as shown in Fig. 6 . Figures 6(a) and 6(b) illustrate the temporal and spectral domain profiles of solitons. The autocorrelation trace of soliton is shown in Fig. 6(c). It is found from Fig. 6 that there are significant spectral sidebands [Fig. 6(b)] and the pedestal of solitons has the oscillating structure [Fig. 6(a)]. It is the oscillating structure of the pedestal that causes the complex structural spectrum. The numerical results show that the pulse duration and the spectral width are 3.1 ps and 0.9 nm, respectively, corresponding to the time–bandwidth product of about 0.34. The intensity peak of soliton is about 16 W and the soliton energy is about 0.048 nJ, as are in good agreement with the experimental results. Figures 6(a) and 6(b) show that the ratio RP of peak intensity of u and v components of pulses is about 11.2. The theoretical predictions [Figs. 6(b) and 6(c)] agree well with the experimental observations (Figs. 2(b), 2(d), and 3).
Figure 6(d) exhibits that the instantaneous frequency δω of the pulses is nearly zero at the time range from −3 ps to 3 ps and hence the pulse chirp is about zero across the pulse width. So this kind of solitons is stable chirp-free pulses with low energy and solitons are static, chirp-free solutions of the coupled complex NLSEs [i.e., Eq. (1)].
3.2. Theoretical modeling and results for strongly dissipative solitons
Both the theoretical results and the experimental observations show that the weakly sech-shaped pulse approximates a linearly polarized wave propagating along the fiber (Figs. 3 and 6). The experimental results also demonstrate that the laser emits the strong DSs approximating the linearly polarized wave [Fig. 4(d)]. The sech-shaped pulse and DS appear to be the asymmetrically combined pulse state with one pulse component much smaller than the other. Then the v component can be dealt with a small perturbation on the u component. A pair of coupled complex NLSEs can be simplified to a single equation, e.g., Ginzburg-Landau equation.
We have numerically solved the coupled complex NLSEs to simulate the laser model by using the Fourier method. In simulations, the time interval must be much bigger than the width of the pulses. For the simulations based on the personal computer, usually, the time window is less than 1 ns and the spectral window is less than 150 nm (corresponding to about 19 THz). To meet the solution accuracy, the temporal and spectral step sizes are chosen to less than 50 fs and 1 GHz, respectively. Then the grid number is up to 215 = 32768. As a result, the numerical investigations of laser system are very time-consuming. When the pulse duration is more than 5 ns [Figs. 4(c) and 5(a)], the time interval should be more than 500 ns to meet the solution accuracy. So the numerical investigations for this case are hardly completed. As a result, we have to utilize the soliton amplitude and phase ansatzes to search the analytical stationary solution of Ginzburg-Landau equation modeling the dissipative processes.
By adding a real cubic (i. e., amplitude-modulating) saturable absorber term ε, a real quintic saturable absorber term μ, and a real quintic (reactive) nonlinearity term ν, the complex cubic-quintic Ginzburg-Landau equation (CGLE) replaces the NLSE to describe the laser oscillators with more complicated structures [33–37]. For instance, CGLE has the capacity of predicting the DS resonances and structural DSs in the anomalous dispersion regime [22,23]. When the quintic nonlinearity term v was ignored, Wise et al. had predicted the flat-top solitons solutions existing in the normal-dispersion lasers by an analytic theory . In a fiber laser, the reactive nonlinearity comes mainly from the nonlinear susceptibility of the fiber . We cannot ignore it when the lasers operate on the high intensity pulse regimes. In fact, the DS resonances were theoretically predicted in the anomalous dispersion regime by analyzing CGLE with the nonlinearity term ν . The generalized CGLE describing pulse propagation in a PML fiber is given by [38–41]39]), σ is the linear gain, and β describes spectral filtering. In this report, we focus on the anomalous dispersion regime and the dispersion coefficient in Eq. (4) is given by D = 1 . Then, we expect to search for the stationary solutions of Eq. (4) of the form [38,41]Eq. (5) into Eq. (4) and separating real and imaginary terms, we can achieve two coupled ordinary differential equations. We use a phase ansatz(6) into the intermediate differential equations and after some cumbersome transformations, one can obtain the reduced system of nonlinear equationsEquation (7) is an elliptic-type differential equation, and its solution for our purposes is given by Fig. 7 .
It is found from Fig. 7 that, with the increase of line gain (corresponding to increasing n), the pulse width broadens and the pulse height hardly varies. These theoretical predictions are in good agreement with the experimental observations. Therefore, both theoretical and experimental results prove that (1) with the increase of pumping strength the width of the quasi-rectangular DSs broadens with the fixed amplitude, (2) this type of DSs is a class of the wave-breaking-free pulses, (3) the ultra-large net-anomalous-dispersion PML fiber laser exploit a new way to generate high-energy nanosecond pulses, and (4) the dissipative processes based on the balance between the energy being supplied and lost in laser system governs the formation and evolution of DSs.
Although the profiles of solitons (especially two edges) in Fig. 7 have some difference from Fig. 4(a), the evolution of pulses in theory is consistent with the experimental results. Of course, investigating CGLE in detail is beyond the range of this report and CGLE only is an equation for approximately simulating the modeling of PML fiber lasers. In this work, the theoretical and experimental results show that two different types of pulse-shaping mechanisms coexist in the proposed laser. One is the dissipative processes that govern the generation of the high-energy nanosecond pulses. The other is the balance between anomalous dispersion and nonlinear Kerr effect, which determines the formation of the low-energy picosecond sech-shaped solitons.
The theoretical and experimental results show that the state of polarization controllers of PAPM determines the operation of laser oscillator. In experiments, the sech-shaped soliton pulses can be achieved with the appropriate orientation and pressure setting of polarization controllers in this report, whereas they are never observed in the previous report . The experimental observations show that the proposed laser here delivers the sech-shaped solitons coexisting with the DSs only when the polarization controllers are adjusted to the special state. On the other hand, it is found from the numerical simulations that the sech-shaped soliton solutions can be obtained from the coupled complex NLSEs in this report only when φ and θ are about 0.1π and 0.28π, respectively. But the sech-shaped soliton solution is never achieved numerically when the parameters of laser oscillator of Ref . are used.
The laser oscillator has a 10-m length of EDF with normal dispersion of about 0.5 ps2 in current work, instead of a 0.8-m length of EDF with normal dispersion of about 0.01 ps2 in Ref . The total length of laser cavity here is about 720 m rather than about 530 m in Ref . We can conclude that the sech-shaped solitons depend on not only the angles of polarizer and analyzer, φ and θ, but also the cavity parameters (e.g., the total normal and anomalous dispersions of cavity, the length of EDF and SMF, and even their location in cavity). The experimental results here show that, if the polarization controllers are not set to the appropriate orientation and pressure, the proposed laser does not deliver the weakly sech-shaped solitons and its operation is very similar to reports in Ref . To achieve the sech-shaped solitons, in fact, the length of EDF and SMF in the cavity here is adjusted again and again.
Based on the coupled complex NLSEs [i.e., Eq. (1)], we simulate the proposed laser and achieve the pulse solutions. Since the time–bandwidth product of pulses is about 0.34 and the pulse chirp is about zero across the pulse width [Fig. 6(d)], the chirp-free pulses are confirmed to be the sech-shaped solitons. The theoretical results (Fig. 6) are in excellent agreement with the experimental observations (Figs. 2 and 3). When the laser delivers the DSs, however, the pulse duration is more than 5 ns so that the accurately numerical simulations are hardly implemented. Then we have to search the analytical stationary solutions from other equation. Taking into account the fact that the weakly sech-shaped solitons and the strongly DSs approximate the linearly polarized wave propagating along the fiber (Figs. 3, 4(d), and 6), the two coupled NLSEs [Eq. (1)] that involve a vector electric field can be simplified to a scalar CGLE [Eq. (4)]. We then use CGLE to search the analytical solutions for explaining the experimental observations of DSs (Fig. 5). Figure 6 shows the accurate simulations whereas Fig. 7 only is the approximately analytical solutions.
By comparing our pulses to the vector solitons reported on Ref , their behaviors have very distinct difference. For instance, the u and v components along the two orthogonal polarization axes of the fiber almost have the same spectral and temporal profiles (see Figs. 3, 4(d), 6(a), and 6(b) in this paper), except the intensity difference of u and v. For the vector solitons, however, the spectral and temporal profiles for the horizontal axis are completely different from those for the vertical axis (see Figs. 4(c), 5, and 6(b) in Ref .). Therefore, we believe that the solitons here are not a sort of vector solitons.
We have experimentally observed the coexistence of weakly sech-shaped solitons and strongly DSs in an ultra-large net-anomalous-dispersion PML fiber laser for the first time to author’s best knowledge. The sech-shaped solitons here have the pulse duration Δτ of about 2.9 ps and the spectral width Δλ of about 1 nm, which are much wider and narrower than Δτ and Δλ of the typical sech-shaped solitons , respectively. The laser generates a new class of DSs exhibiting as the quasi-rectangular temporal and quasi-Gaussian spectral profiles in linear scale. The pulse energy of DSs increases more than three orders of magnitude by comparing to the pulse energy of sech-shaped solitons. The experimental observations show that two different types of pulse-shaping mechanisms exist simultaneously in our laser. One is the dissipative processes that play an essential role in the formation of nanosecond DSs. The other is the balance between anomalous dispersion and nonlinear Kerr effect, which governs the generation and evolution of picosecond sech-shaped solitons. Both sech-shaped and dissipative solitons appear to be the asymmetrically combined pulse state with one pulse component much smaller than the other. The theoretical predictions and experimental observations show that the energy of pulses along one of the two orthogonal polarization axes of the fiber is over 11 times lower than the energy along the other axis. The theoretical results are in good agreement with the experimental observations. The proposed laser provides a new way to generate high-energy nanosecond pulses and the investigation here provides a better understanding of the pulse shaping in PML fiber lasers.
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 Xiaohui Li and Dong Mao for help with the experiments.
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