Observation of Dissipative Bright Soliton and Dark Soliton in an All-Normal Dispersion Fiber Laser

This paper proposes a novel way for controlling the generation of the dissipative bright soliton and dark soliton operation of lasers. We observe the generation of dissipative bright and dark soliton in an all-normal dispersion fiber laser by employing the nonlinear polarization rotation (NPR) technique.Through adjusting the angle of the polarizer and analyzer, themode-locked and non-modelocked regions can be obtained in different polarization directions. Numerical simulation shows that, in an appropriate pump power range, the dissipative bright soliton and dark soliton can be generated simultaneously in the mode-locked and non-mode-locked regions, respectively. If the pump power exceeds the top limit of this range, only dissipative soliton will exist, whereas if it is below the lower bound of this range, only dark soliton will exist.


Introduction
With the rapid development of ultrafast optical field, a great many researchers have been focusing on mode-locked fiber laser recently [1][2][3][4][5][6][7].Passive mode-locked fiber lasers have shown many advantages over solid-state systems such as the compact design, low-cost, and stability [8][9][10][11].It has also some potential applications, like laser processing, optical communications, medical equipment, military, and so on [12][13][14].Ultrashort pulses are stabilized in an oscillator when the effects of optical nonlinearity are exactly balanced by other processes after one cycle around the cavity.The most widely used method to compensate nonlinearity is group velocity dispersion (GVD).The traditional soliton is formed by the balance between nonlinear and negative dispersion phase changes.The resulting soliton propagates indefinitely without change.However, the traditional soliton is limited to 0.1 nJ of the energy in standard fibers [15].When the GVD reaches zero, the dispersion managed soliton has been formed, which enables emitting the pulses with the pulse energy reaching the 1 nJ level [16].Instead of conventional soliton and dispersion managed soliton, self-similar soliton can tolerate strong nonlinearity without wave breaking.But due to the restricted gain bandwidth, self-similar soliton allows the energy to reach the 10 nJ level [17].Dissipative soliton (DS) has attracted great interest in the development of fiber lasers because it can significantly improve the deliverable energy of pulse, approaching or even exceeding 100 nJ [18,19], and the DS exists in nonconservative systems whose dynamics are extremely different from those of conventional soliton [20][21][22].A fiber laser with pure normal GVD (or large normal GVD together with small anomalous GVD) would presumably have to exploit dissipative process in the modelocked pulse shaping [23][24][25].
The research of dark soliton lags behind that of bright soliton due to the difficulty of dark soliton generation.However, dark soliton is more suitable than bright soliton when used in optical communications [26].Dark soliton is broadened during propagation at nearly half the rate of bright soliton, and dark soliton shows more resistance than bright soliton to perturbation during propagation.Furthermore, as the background noise mainly affects the background of the dark soliton, it is less sensitive to background noise [27][28][29].Recently, the dark soliton formation in fiber lasers has been reported in literature [30][31][32][33].Zhang et al. have experimentally observed dark soliton in a fiber ring laser with all-anomalous dispersion fibers [30] and with all-normal dispersion fibers [31].Tang et al. demonstrated the dark 2 International Journal of Optics soliton formation in an all-normal dispersion cavity fiber laser without an antisaturable absorber in cavity [33].
In this paper, we propose a novel way for controlling the generation of the dissipative bright soliton and dark soliton operation of lasers.The nonlinear polarization rotation technique is implemented for generating mode-locked and nonmode-locked region.The dissipative soliton and dark soliton are obtained in mode-locked and non-mode-locked regions, respectively.

Modeling
To study the feature and dynamic evolution of dissipative and dark soliton in an all-normal dispersion fiber laser, we implement a numerical model that incorporates the most important physical effects like the nonlinear polarization rotation (NPR), spectral filtering (SF), and so forth.The propagation model is shown schematically in Figure 1.It consists of a 4 m erbium-doped fiber (EDF) and 1.8 m dispersion compensating fiber (DCF), and the polarization additive-pulse modelocking (PAPM) system is made of a polarization-sensitive isolator and two sets of polarization controllers, two quarter waveplates (QWP), and a half waveplate (HWP) made of the polarizer; one QWP and a HWP constitute the analyzer.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.By setting the angle of the polarizer and the linear cavity phase delay of the cavity appropriately, we can obtain the mode-locked and the non-mode-locked area in PAPM where mode-locked area means that the light intensity is inversely proportional to the loss and the opposite non-mode-locked area.When the soliton propagates through the PAPM element and the intensity transmission,  is expressed as [11]  = sin 2 () sin 2 () + cos 2 () cos 2 () where  and  represent the angle between the fast axis of the birefringent fiber and polarizer (analyzer), Δ  is the linear cavity phase delay, Δ  = (2/)(  −   ) +   ,   is the phase difference between two polarization directions, and  and  represent the cavity length of the fiber and the wavelength of light, respectively.Δ  is the nonlinear phase delay caused by self-phase modulation (SPM) and cross-phase modulation (XPM).Δ  = −(2 2 /3 eff ) cos 2,  2 is the nonlinear coefficient,  eff is the effective mode area, and  is optical power.
The pulse propagation in the weak birefringent fiber can be modeled well by two coupled nonlinear Schrodinger equations (NLSE).However, the fiber lasers have components which cannot be modeled by several phase modulation terms in NLSEs.For example, a fiber laser has a gain with a limited gain bandwidth (BW) and a saturable absorber (SA).They do not induce the phase modulation, but they definitely cause the amplitude modulation in the frequency and time domain.To count the amplitude modulation effects in a laser cavity properly, more terms needed to be added to NLSEs.The resulting differential equations are called coupled cubic Ginzburg-Landau equations (CGLE) [34], shown as where   and   denote the envelopes of the optical pulses along the two orthogonal polarization axes of the fiber, and  is the loss coefficient of the fiber. = ( 1 − 1 )/2 is the group velocity difference between the two polarization modes,  2 represents the fiber dispersion, Ω  is the bandwidth of the laser gain,  =  − ( 1 +  1 )/2 and  indicate the pulse local time and the propagation distance, respectively, and  describes the gain function of EDF which is expressed by  =  0 exp(− pulse / sat ) [2], where  0 is the small signal gain coefficient, related to the doping concentration, and  sat is the gain saturation energy, which corresponds to the pumping strength [2,34].The pulse energy  pulse is given by , where   is the cavity round-trip time.

Simulation Results and Discussion
The model is solved with a standard symmetric split-step Fourier algorithm.The initial   denotes an arbitrary signal, while the initial   denotes a weak continuous wave (CW) light which contains a small segment dip on the top of it; by appropriately adjusting the angle of the polarizer (analyzer) and the linear cavity phase delay of the cavity, both dissipative and dark solitons can be achieved in different directions of the polarization.Since the saturation energy  sat is proportional to the pumping strength [35], this means increasing  sat corresponds to increasing the pump power in the practical system.The following parameters are applied for the simulations possibly matching the experimental conditions:  = 0.2 dB/km,  0 = 2m −1 ,  = 4.2 W −1 km −1 for EDF and  = 1.3 W −1 km −1 for DCF, Ω  = 30 nm, EDF = 4.0 m, DCF = 1.8 m,  2 = +31×10 −3 ps 2 /m for EDF and  2 = +20× 10 −3 ps 2 /m for DCF, and net cavity GVD  net ≈ 0.16 ps 2 ,  = /8,  = 5/8,   = 0.96, and when  sat = 460 pJ, it is capable of achieving both dissipative and dark solitons in fiber laser.
The formation and evolution of dissipative and dark solitons at  sat = 460 pJ are illustrated in Figure 2. We can find that the dissipative soliton pulse duration, pulse energy, and peak power are about 2.96 ps, 229.19 pJ, and 89.16 W in one polarization, respectively.In the other polarization, we can generate the dark soliton with 960 fs.Figures 2(c) and 2(d) show that dissipative and dark solitons remain stable when they circled 500 times in the cavity separately.According to Figure 2(d), we found that the dip exist not only in the pulse central but also in the edge, because in the simulation we used the finite background pulse instead of infinite background pulse to generate dark soliton.Numerical simulation shows that if the background pulse width is 10 times the soliton width, the transmission characteristics of finite and infinite backgrounds of dark solitons are basically the same, so we can ignore the edge dip.From Figures 3(a) and 3(b) we can observe the impact of the various parts of the components for peak power and pulse duration.After propagating the EDF, both the dissipative soliton and the dark soliton peak power show an increase trend.However, as for the pulse duration, the dissipative soliton shows an increase while the dark soliton remains nearly stable.When soliton transmitted to PAPM, both dissipative and dark soliton pulse durations are rapidly compressed while their peak power remains the same.In the DCF, the dissipative soliton peak power related to the previous EDF has been increased rapidly and then decreased slowly, but it has no effect on the pulse duration.As for the dark soliton, the peak power in DCF increases rapidly and then remains unchanged, and the same basic has no effect on dark soliton pulse width.These results are obtained when  sat = 460 pJ.Next, we analyze the various  sat effects for our system.
Figure 4 shows the change of dissipative and dark solitons with the increase of  sat .Figure 4(a) reflects the notion that dissipative soliton pulse duration and peak power increase with  sat .When  sat increases to about 980 pJ, peak power is nearly unchanged, and when  sat reaches 1560 pJ, dissipative soliton appears split in Figure 4(c).If we continue to increase  sat , the dissipative soliton will generate an unstable pulse which is different from the phenomenon adopted in Liu [2].However, Figure 4(b) shows that dark soliton varies with different dissipative soliton; firstly the peak power increases with  sat , but the dark soliton pulse duration gradually decreases.When  sat increases to 550 pJ, the peak power reduces gradually and eventually in  sat = 640 pJ, the dark soliton disappears in Figure 4(d).
Figures 5(a) and 5(b) show that with the decrease of  sat , both the dissipative soliton and the dark soliton peak power show a decrease trend.However, as for the pulse duration, the dark soliton shows an increase while the dissipative soliton remains nearly stable.When  sat is reduced to 335 pJ, the dissipative soliton is unstable.When  sat is below 318 pJ, the dissipative soliton cannot be generated.Then we conclude that the generation of dissipative soliton requires a pump power exceeding a threshold.When  sat is reduced to 58 pJ, the dark soliton is unstable, but it is still capable of forming a dark soliton.It can be found that obtaining a dark soliton does not require high  sat , which means the dark soliton does not require high pump power.

Conclusion
We propose a novel way for controlling the generation of the dissipative bright soliton and dark soliton operation of lasers.Numerical simulation shows that, in an appropriate pump power range, the dissipative bright soliton and dark soliton can be generated simultaneously in the mode-locked and non-mode-locked regions, respectively.When  sat is up to 640 pJ, only dissipative soliton can be achieved, and the dissipative soliton is unstable when  sat is close to 1560 pJ, whereas it does not generate multipulse phenomenon [2].Then if we decrease  sat to 335 pJ, only dark soliton can be obtained.Hence, we conclude that the dissipative soliton can tolerate higher pump power but its generation process requires a high threshold, whereas dark soliton can be obtained at a low pump power threshold.