We present a comprehensive numerical investigation of the polarization dynamics of dissipative-soliton-resonance (DSR) pulses in a passively mode-locked Yb-doped fiber laser with a nonlinear optical loop mirror (NOLM). In our simulations, the NOLM’s saturable absorption effect is modeled by its transmission function. Depending on the strength of cavity birefringence, various types of vector DSR pulses, including the polarization-locked DSR (PL-DSR), the polarization rotation-locked DSR (PRL-DSR), and the group velocity-locked DSR (GVL-DSR) states are obtained in the laser cavity. The characteristics of these vector DSR pulses are shown. We also analyze the polarization evolution and internal polarization dynamics of PRL-DSR pulses within the cavity. Our simulation results offer insight into the vector nature of DSR pulses in polarization-insensitive mode-locked fiber lasers.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Solitons, which refer to the localized wave packets that remain intact even after mutual collisions, have been extensively investigated over the past decades. In the context of optics, Mollenauer et al. firstly demonstrated optical solitons in single-mode fibers (SMFs) . The formation of optical solitons is interpreted as a result of the interplay between anomalous group-velocity dispersion (GVD) and self-phase modulation (SPM) effects. The basic equation that governs the propagation of optical pulses in conservative systems is the nonlinear Schrӧdinger equation (NLSE). However, it is found that optical solitons can also exist in non-conservative or dissipative systems. The ultrafast fiber laser is a typical nonlinear dissipative system, which has been regarded as a convenient platform for the fundamental exploration of complex soliton dynamics. In contrast to the conservative solitons (CSs) observed in SMFs, solitons formed in fiber lasers are dissipative in nature. Such solitons are thus referred to as dissipative solitons (DSs). A DS is a localized structure which requires continuous exchange of energy with the nonlinear system. The dynamics of DSs can be well described by the complex Ginzburg-Landau equation (CGLE), which takes account not only of the cavity dispersion and nonlinearity but also the gain and loss. DSs have characteristics that are remarkably different from those of CSs. To date, versatile soliton dynamics such as multiple pulses , pulsating solitons , rogue waves , and soliton explosions  have been reported in ultrafast fiber lasers.
A novel type of DS known as dissipative soliton resonance (DSR) has been proposed in mode-locked lasers . The DSR pulse features the flat-top temporal profile. With the increase of pump power, the pulse duration can be arbitrarily broad while maintaining its peak power constant. This phenomenon was originally identified as a new class of localized solutions of the CGLE, and considerable theoretical studies have subsequently been reported in this framework [7–10]. Based on the lumped models, the mechanism of DSR-pulse generation in mode-locked fiber lasers has been investigated numerically [11–13]. DSR phenomena have also been experimentally demonstrated in fiber lasers by employing various mode-locking techniques, such as nonlinear polarization rotation (NPR) [14,15], nonlinear optical loop mirror (NOLM) [16,17], and real saturable absorbers (SAs) [18–20]. However, emphasis has been given to scalar DSR, neglecting the vector nature of DSR pulses. In reality, due to the manufacturing imperfections, externally applied stress or bending, a practical SMF is always weakly birefringent. The presence of birefringence lifts the degeneracy between the two orthogonal polarization modes, resulting in differences in both group and phase velocities. Owing to the nonlinearity coupling, two polarization components of the soliton can propagate as a unit. The multi-dimensional entity of the coupled soliton is referred to as vector soliton (VS). So far, various types of VSs, including the group velocity-locked (GVL) state [21,22], the polarization rotation-locked (PRL) state [23,24], and the polarization-locked (PL) state [25,26], have been confirmed in mode-locked fiber lasers without polarization discrimination devices. Recently, Luo et al. demonstrated the generation of GVL-DSR pulses in a figure-of-eight fiber laser with the net anomalous dispersion . Li et al. investigated the internal polarization dynamics of DSR pulses in an Yb-doped fiber laser mode-locked by NOLM . Though the vector properties of DSR pulses have been demonstrated in these two works, DSR polarization dynamics have not yet been fully explored. A natural question arises as to whether the GVL-DSR pulses can be formed in normal dispersion fiber lasers. Moreover, to the best of our knowledge, no PRL-DSR phenomenon has been reported and its polarization evolution and internal polarization dynamics remain unclear. The exploration of these issues would enrich the VS dynamics in polarization-insensitive mode-locked fiber lasers.
In this paper, we report numerical simulations on the polarization dynamics of DSR pulses in a passively mode-locked Yb-doped fiber laser based on a NOLM. In the simulation model, the saturable absorption of the NOLM is described by its transmission function. By varying the cavity birefringence, different types of vector DSR pulses, including the PL-DSR state, the PRL-DSR state, and the GVL-DSR state, are obtained in the laser cavity. The polarization evolution of these vector DSR pulses in the cavity is analyzed. Furthermore, the internal polarization dynamics of PRL-DSR pulses are discussed.
2. Numerical model
The configuration of an all-normal-dispersion mode-locked Yb-doped fiber laser is schematically shown in Fig. 1. The cavity contains a piece of 3-m ytterbium-doped fiber (YDF), a lumped polarization-insensitive SA, a Gaussian-shaped spectral filter with a bandwidth of 12 nm, two pieces of SMF with a total length of 1 m, and a 50:50 output coupler (OC). The dispersion parameters of the fibers are β2,YDF = 36 ps2/km and β2,SMF = 23 ps2/km, respectively. All the fibers in the cavity are assumed to have the same nonlinear coefficient of γ = 3 W−1km−1. The net cavity dispersion is 0.131 ps2.
We numerically simulated the pulse formation and evolution in the laser cavity based on a cavity round-trip model . Briefly, the action of each cavity component on the optical pulses was taken into account. The simulation started from an arbitrary weak pulse, and eventually converged into a stable solution with appropriate parameter settings after a limited number of circulations in the cavity. Pulse propagation in a linearly birefringent fiber can be described by the following coupled Ginzburg-Landau equationEq. (1) are ignored. For the YDF, the gain saturation effect is considered as
In our simulations, a simple model of NOLM is adopted. The saturable absorption of the lumped SA can be expressed by the transmission function of NOLM 
A convenient approach to analyzing the polarization dynamics of the pulses is based on the rotation of the Stokes vector on the Poincaré sphere. Four variables known as the Stokes parameters are introduced and defined asEq. (4) that s02 = s12 + s22 + s32 = 1. As s0 is invariant with time and space, the Stokes vector with components s1, s2, and s3 moves on the surface of a sphere of radius s0 as the pulse propagates in the cavity, which provides a visual description of the evolution of the polarization state.
3. Simulation results and discussion
The numerical model was solved by using the split-step Fourier method. Depending on the strength of cavity birefringence, three types of vector DSR pulses, including the PL-DSR state, the PRL-DSR state, and the GVL-DSR state, were obtained in the laser cavity. Their polarization dynamics were analyzed. The simulation results are presented as follows.
3.1 Generation of PL-DSR state
We firstly simulated the intra-cavity pulse evolution in the case of low cavity birefringence. When the cavity beat length Lb and the saturation power Isat were selected as 50 m and 150 W, a stable vector DSR pulse was obtained in the cavity. As shown in Fig. 2(a), the pulse consists of components along both principal axes of the birefringent fiber. The peak powers of the orthogonally polarized components are unequal and the strong polarization mode is along the fast axis. Figure 2(b) shows the corresponding pulse spectra, which have the same central wavelength centered at 1060 nm. As illustrated in Fig. 2(c), the phase difference between the two components is fixed at π/2 during the propagation, indicating that their phase velocities are locked. The polarization evolution of the vector DSR pulse throughout the cavity is presented by using the Poincaré sphere, as depicted in Fig. 2(d). It shows a fixed point on the Poincaré sphere, which means that the pulse has a stationary polarization state when propagating in the cavity. Specifically, as observed in Fig. 2(e), the pulse is elliptically polarized with its larger axis of polarization along the fast axis. Figure 2(f) illustrates the evolution of output pulses with cavity roundtrips after passing through an external polarizer. The pulse peak power is unchanged for each cavity roundtrip. These results are in agreement with the theory of PL-VSs, manifesting that stable PL-DSR pulses are generated in the cavity. The formation of the PL-DSR state can be explained as follows: when the peak-power-clamping effect induced by NOLM is strong enough, the pulse peak power will be confined at a relatively low level. In this way, the fiber laser is prone to operating in DSR regime . Meanwhile, owing to the linear birefringence, the refractive index of the fast axis is less than that of the slow axis. However, the magnitude of the nonlinear birefringence depends on an asymmetric peak power distribution between the two principal axes. If the fast axis component has a larger peak power, the nonlinear contribution to the refractive index of the fast axis is greater relative to the slow one, which thereby results in the reduction of the difference in total indices between axes. With the proper balance, the linear birefringence can be compensated by the nonlinear birefringence, enabling the dynamic equalization of the phase velocities between the two components. The vector DSR pulse can thus propagate without change of its elliptic polarization state.
The stability of the obtained PL-DSR pulses was considered. It was found that the applied perturbations could be suppressed after a limited number of circulations in the cavity. It should be emphasized that the PL-VSs formed in birefringent SMFs are weakly unstable owing to the fragile balance between the linear and nonlinear birefringence . However, the PL-DSR pulses generated in the fiber laser are highly stable as is evident from the numerical results. We conjecture that the dissipative features of the fiber laser are prone to damping the perturbations.
The influence of the cavity birefringence on the properties of PL-DSR pulses was analyzed. Figure 3 shows the difference between the peak powers of the two components and their relative phase as a function of the beat length Lb. In this case, we varied the beat length Lb while fixing the other variables. It can be seen that with increasing Lb from 30 m to 62 m, the peak power difference between the two components decreases from 53 W to 26 W. However, the phase difference is approximately fixed at π/2. Our simulation results indicate that in PL-DSR regime, the ellipticity of the polarization ellipse becomes smaller at higher cavity birefringence, while its orientation angle remains unchanged. This can be understood as a natural consequence of the balance between linear and nonlinear birefringence.
We also investigated the impact of the pulse peak power on the range of cavity birefringence where the PL-DSR pulse could be formed. It should be noted that under DSR condition, the generated pulse peak power can be easily controlled by varying the saturation power of NOLM . As illustrated in Fig. 4, when the pulse peak power increases from 32 W to 64 W, the minimum Lb for preserving a stable PL-DSR state is reduced from 62 m to 28 m. This means that the PL-DSR pulse with higher peak power could exist in a larger domain of cavity birefringence values.
3.2 Generation of PRL-DSR state
When the beat length Lb was decreased to 8 m (i.e., Lb/L = 2) and the saturation power Isat was selected as 150 W, PRL-DSR pulses were obtained in the cavity. As shown in Fig. 5(a), the pulse contains two polarization modes. The pulse peak power of the fast axis component is slightly greater than that of the slow one. Figure 5(b) shows the corresponding pulse spectra. Similar to the case of PL-DSR state, two components of the PRL-DSR pulse locate at the same central wavelength. As illustrated in Fig. 5(c), the phase difference between the two components increases linearly from −0.06π to 0.94π during the propagation, and the relative phase accumulated in one cavity roundtrip is thus π. The polarization evolution of the PRL-DSR pulse inside the cavity is presented in Fig. 5(d). The Stokes vector on the Poincaré sphere has the form of a close loop, which could be related to a periodic polarization attractor. The polarization state of the pulse rotates back to its original one after every two cavity roundtrips. Figure 2(e) presents the polarization evolution of output pulses with cavity roundtrips. It shows two fixed points on the Poincaré sphere, implying that the pulse has two alternating polarization states at a fixed cavity position. Specifically, as depicted in Fig. 5(f), there are two clear polarization ellipses at the position of OC in the cavity, and their orientations are orthogonal. Figures 5(g) and 5(h) illustrate the evolution of output pulses with cavity roundtrips and their evolution after passing through an external polarizer, respectively. As observed in Fig. 5(g), the pulse peak power is identical for each cavity roundtrip. However, one can see from Fig. 5(h) that it alternates between two values after the output pulses passing through the polarizer.
As shown in Fig. 6, we also explicitly calculated the polarization ellipses of the pulse at various locations inside the cavity during two consecutive roundtrips. Obviously, both the orientation angle and ellipticity of the polarization ellipse vary during the propagation, i.e., the polarization state changes. After every two cavity roundtrips, the polarization ellipse returns to its original state. We further investigated the polarization evolution of PRL-DSR pulses in the laser cavity when the beat length was varied to Lb/L = 1, Lb/L = 3, and Lb/L = 4, respectively. Similar results were obtained in these cases except that the number of the roundtrip required for the polarization ellipse returning to its original one was different, which was equal to the ratio of the beat length over the cavity length.
Actually, the aforementioned polarization evolution of the PRL-DSR pulses is represented by the polarization ellipses of the central point in the pulse envelope. Whether the polarization states across the pulse are identical remain unclear. Thus, we further investigated the internal polarization dynamics of the PRL-DSR pulses when the beat length Lb/L = 2. As shown in Fig. 7(a), the phase difference between the two components is not uniform across the pulse. However, the relative phase of any point in the pulse envelope accumulated in one cavity roundtrip is fixed at π. Figure 7(b) presents the internal polarization evolution of the pulse. It can be seen that the leading edge (red curve) and trailing edge (blue curve) of the pulse show a long trajectory on the Poincaré sphere, while its flat-top part converges to a fixed point (green dot). It should be mentioned that Li et al. investigated the internal polarization evolution of PL-DSR pulses , which exhibits a nonuniform polarization distribution across the pulse. Our simulation results clearly suggest that the PRL-DSR pulse does not acquire a single state of polarization either. Specifically, although the flat-top part of the pulse has a nearly fixed polarization state, the leading and trailing edges feature polarization states that vary with time. However, it should be noted that the polarization evolution of the entire pulse exhibits the same dynamics, i.e., the polarization state of any point in the pulse envelope rotates back to its own original one after every two cavity roundtrips.
The influence of pump power on the behavior of PRL-DSR pulses was also analyzed. We varied the pump Esat while fixing the other variables. It was found that under DSR operation conditions, the pulse duration and pulse energy increase linearly with Esat while the pulse peak power maintains approximately unchanged. On the other hand, the unique polarization-state nonconformity between the edges and flat-top part of the pulse is maintained with changes in the pump power.
3.3 Generation of GVL-DSR state
The GVL-DSR state could be formed in the laser with high cavity birefringence. As an example, when the cavity beat length Lb and the saturation power Isat were set to 0.4 m and 150 W, a stable GVL-DSR pulse was obtained in the cavity. As shown in Figs. 8(a) and 8(b), the temporal profiles of the two orthogonally polarized components almost overlap while their spectra center at different wavelengths, which is a typical characteristic of GVL-VSs .
Our simulation results unambiguously confirm the existence of the GVL-DSR state in normal dispersion fiber lasers. Its formation can be explained by the cross-phase-modulation (XPM) induced incoherent coupling between the two components. Owing to the linear birefringence, the group velocity of the fast axis component is greater than that of the slow one. In the absence of XPM effect, the two polarization modes evolve independently and separate from each other due to their different group velocities. However, the XPM-induced nonlinear coupling causes the central wavelengths of the two components to shift in opposite directions. In conjunction with a wavelength-dependent group velocity, these shifts cancel their group-velocity differences. Specifically, as the laser is operated in the normal-dispersion regime, the central wavelength of the slow axis component increases while the fast axis component decreases. Eventually, two DSR components formed along the principal axes of the birefringent fiber mutually trap each other and move at a common group velocity in the cavity.
We have presented a comprehensive numerical investigation on the polarization dynamics of vector DSR pulses in a passively mode-locked Yb-doped fiber laser with a NOLM. In the case of low cavity birefringence, coherent coupling between the two components polarized along both principal axes occurs. Either the PL-DSR or the PRL-DSR states could be formed in the cavity. The impact of the pulse peak power on the range of cavity birefringence where the PL-DSR pulse exists has been analyzed. The polarization evolution and internal polarization dynamics of the PRL-DSR pulses have been revealed. In the case of high cavity birefringence, incoherent coupling between the two components causes their spectra to have different central wavelengths. The two components can capture each other and eventually the GLV-DSR state is obtained. Our simulation results give an insight into the vector nature of DSR pulses in mode-locked fiber lasers without polarization discrimination devices.
National Natural Science Foundation of China (NSFC) (61775031, 61435003, and 61421002) and the National Key R&D Program of China (2016YFF0102003).
1. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45(13), 1095–1098 (1980). [CrossRef]
4. A. Klein, G. Masri, H. Duadi, K. Sulimany, O. Lib, H. Steinberg, S. A. Kolpakov, and M. Fridman, “Ultrafast rogue wave patterns in fiber lasers,” Optica 5(7), 774–778 (2018). [CrossRef]
5. A. F. J. Runge, N. G. R. Broderick, and M. Erkintalo, “Observation of soliton explosions in a passively mode-locked fiber laser,” Optica 2(1), 36–39 (2015). [CrossRef]
6. W. Chang, A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative soliton resonances,” Phys. Rev. A 78(2), 23830 (2008). [CrossRef]
7. W. Chang, A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative soliton resonances in laser models with parameter management,” J. Opt. Soc. Am. B 25(12), 1972–1977 (2008). [CrossRef]
8. W. Chang, J. M. Soto-Crespo, A. Ankiewicz, and N. Akhmediev, “Dissipative soliton resonances in the anomalous dispersion regime,” Phys. Rev. A 79(3), 33840 (2009). [CrossRef]
10. 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]
11. W. Du, H. Li, J. Li, P. Wang, S. Zhang, and Y. Liu, “Mechanism of dissipative-soliton-resonance generation in fiber laser mode-locked by real saturable absorber,” Opt. Express 26(16), 21314–21323 (2018). [CrossRef] [PubMed]
12. 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]
13. D. Li, D. Tang, L. Zhao, and D. Shen, “Mechanism of dissipative-soliton-resonance generation in passively mode-locked all-normal-dispersion fiber lasers,” J. Lightwave Technol. 33(18), 3781–3787 (2015). [CrossRef]
15. 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]
16. J. Zhao, D. Ouyang, Z. Zheng, M. Liu, X. Ren, C. Li, S. Ruan, and W. Xie, “100 W dissipative soliton resonances from a thulium-doped double-clad all-fiber-format MOPA system,” Opt. Express 24(11), 12072–12081 (2016). [CrossRef] [PubMed]
17. K. Krzempek and K. Abramski, “6.5 µ J pulses from a compact dissipative soliton resonance mode-locked erbium–ytterbium double clad (DC) laser,” Laser Phys. Lett. 14(1), 015101 (2017). [CrossRef]
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. Z. Cheng, H. Li, H. Shi, J. Ren, Q. H. Yang, and P. Wang, “Dissipative soliton resonance and reverse saturable absorption in graphene oxide mode-locked all-normal-dispersion Yb-doped fiber laser,” Opt. Express 23(6), 7000–7006 (2015). [CrossRef] [PubMed]
20. H. Yang and X. Liu, “WS2-clad microfiber saturable absorber for high-energy rectangular pulse fiber laser,” IEEE J. Sel. Top. Quant. 24(3), 1 (2018). [CrossRef]
26. L. M. Zhao, D. Y. Tang, X. Wu, H. Zhang, and H. Y. Tam, “Coexistence of polarization-locked and polarization-rotating vector solitons in a fiber laser with SESAM,” Opt. Lett. 34(20), 3059–3061 (2009). [CrossRef] [PubMed]
27. Z. C. Luo, Q. Y. Ning, H. L. Mo, H. Cui, J. Liu, L. J. Wu, A. P. Luo, and W. C. Xu, “Vector dissipative soliton resonance in a fiber laser,” Opt. Express 21(8), 10199–10204 (2013). [CrossRef] [PubMed]
28. D. Li, D. Shen, L. Li, D. Tang, L. Su, and L. Zhao, “Internal polarization dynamics of vector dissipative-soliton-resonance pulses in normal dispersion fiber lasers,” Opt. Lett. 43(6), 1222–1225 (2018). [CrossRef] [PubMed]
29. J. Wu, D. Y. Tang, L. M. Zhao, and C. C. Chan, “Soliton polarization dynamics in fiber lasers passively mode-locked by the nonlinear polarization rotation technique,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(4), 46605 (2006). [CrossRef] [PubMed]
30. N. N. Akhmediev, A. V. Buryak, J. M. Soto-Crespo, and D. R. Andersen, “Phase-locked stationary soliton states in birefringent nonlinear optical fibers,” J. Opt. Soc. Am. B 12(3), 434–439 (1995). [CrossRef]