Fig. 1. Vector breathing DS generation in a mode-locked fiber laser. (a) Schematic setup of the NPR mode-locked normal dispersion fiber laser. EDF, erbium-doped fiber; LD, laser diode; PC1 and PC2, polarization controllers; PI-ISO, polarization-insensitive isolator; Polarizer, 45 deg-tilted fiber grating based polarizer; WDM, wavelength division multiplexer; OC, 91:9% output coupler. The opaque yellow area represents the NPR mechanism. (b) A general schematic of orthogonal polarization modes resembled coupled oscillator systems. The two pendula are connected with a common beam indicating the coupling mechanism. Each pendulum has an initial phase angle ${\phi }_{1,2}$, initial frequency $f_{1,2}$, and intensity $I_{x,y}$. (c)–(e) The temporal behavior of coupled oscillator model for emitting vector breathing/stable DS. (c) The constant phase difference between orthogonal polarization fields, showing $d\mathrm{\Delta }\varphi /dt=0$. In this case, the output DS is synchronized and only stable DS will be generated. (d) Phase difference slip between orthogonal polarization modes, showing $d\mathrm{\Delta }\varphi /dt\ne 0$ with $|\mathrm{\Delta }\mathrm{\Omega }|>|K|$. In this case, the phase difference quickly jumps within two fixed points in the vertical axis (blue points). (e) Phase difference entrainment between orthogonal polarization fields, showing $d\mathrm{\Delta }\varphi /dt\ne 0$ with $|\mathrm{\Delta }\mathrm{\Omega }|>|K|$. In this case, $\mathrm{\Delta }\varphi$ oscillates within the region defined by two fixed phase points (blue points in the vertical axis), and the synchronization tends to occur but is never reached.

Fig. 2. Experimental observation of slow vector DS breathing dynamics. (a) Optical spectrum with dual-peak feature at 1570 nm; (b) single shot temporal trace of vector DS breather. The period of breather is 10,000 RTs. Inset: zoomed in temporal structure of vector DS breather with a width of $\sim 200$ RTs; (c) slow polarimetric trace of powers of the orthogonally polarized components $I_x$ (blue) $I_y$ (orange) at a scale of 1 ms. The lower figure is the total power $I=I_x+I_y$ (black). Inset: zoomed-in temporal trace of shaded area in (c) within $250\mu \mathrm{s}$ (10,000 RTs), and the pulse period is $\sim 200$ RTs. (d) Slow polarimetric measurement retrieved breather phase difference $\mathrm{\Delta }\varphi$ (blue) and DOP (red) within 1 ms. When breathing appears, DOP oscillates between 60% and 90%, and $\mathrm{\Delta }\varphi$ oscillates within $\pi$. Inset: zoomed-in phase difference transition and DOP of shaded area in (d) within $250\mu \mathrm{s}$. (e) SOP evolution trajectories of vector DS breathing on the surface of Poincaré sphere within 1 ms. All stokes parameters are normalized. Note: The laser pump power is $I_p=500\mathrm{mW}$.

Fig. 3. Experimentally observed vectorial breathing dynamics of QML in a normal dispersion fiber laser. (a) Optical spectrum with a dual-peak feature for vector breathers. (b) Single-shot temporal trace of Q-switched mode-locked DS for 10,000 RTs. The period of the breather is 3000 RTs. Inset: zoomed-in temporal structure of the breather within $250\mu \mathrm{s}$ in shaded areas of (b). (c) Polarimeter measured averaged powers of the orthogonally polarized components $I_x$ (blue) and $I_y$ (orange). The lower figure is the total output power and total $S_0=I_x+I_y$ (black). The period of the breathing waveform is $\sim 4000$ RTs. Inset: zoomed-in temporal trace within $250\mu \mathrm{s}$ in the shaded area of (c). (d) Retrieved phase difference $\mathrm{\Delta }\phi$ (blue) and DOP (red) through polarimetric measurement along 1 ms (44,000 RTs). Inset: zoomed-in phase difference evolution and DOP within $250\mu \mathrm{s}$ in the shaded area of (d). (e) SOP trajectories of Q-switched mode-locked breathers on polarization Poincaré sphere. All Stokes parameters are normalized. Note: The laser pump power is $I_p=360\mathrm{mW}$.

Fig. 4. Self-pulsing maps of laser operation conditions for $\rho >0$, $\omega \ne 0$. (a) The pump wave is isotropic where $\xi =0$. (b) The laser cavity is isotropic where ${\beta }_L=0$. (c) The pump power is fixed. (d) Shil’nikov chaos conditions for the second saddle parameter ${\sigma }_2$ at $I_p=55$.

Fig. 5. Simulation of desynchronized vector breathing dynamics. (a) Breathing polarization dynamic in the form of complex oscillations of the total output power $I=I_x+I_y$. The breathing period is 104 RTs. (b) The simulated powers of the individual polarization components $I_x$ (blue) and $I_y$ (orange). (c) The calculated phase difference $\mathrm{\Delta }\phi$ between orthogonal polarization modes. The phase slip in $\pi$ accords to the intensity breathing in (a). (d) SOP trajectories shown on the Poincaré sphere in normalized Stokes parameters. Note: Main parameters used in the model are ${\beta }_L=2\pi ·0.001$, ${\beta }_c=0$, $I_p=55$, $\sigma =0.9$ ($\xi =0.11$). The other parameters $\mathrm{\Delta }=0.025$, $\epsilon =0.22·{10}^{-5}$, ${\alpha }_1=12.9$, ${\alpha }_2=2.3$, ${\chi }_s=2.3$, ${\chi }_p=1$, and $\gamma =2·{10}^{-6}$.

Fig. 6. Simulation of phase difference entrainment vector breathing dynamics. (a) Simulated temporal trance in the form of complex oscillations (QML) of the output power total power $I=I_x+I_y$. The breathing period is 3000 RTs. (b) Simulated temporal trace of the powers of polarization components $I_x$ and $I_y$. The breathing period is 3000 RTs. (c) Simulated phase difference $\mathrm{\Delta }\varphi$ evolution along 10,000 RTs. (d) SOP trajectories in the Poincaré sphere. The main parameters used in the model are $\sigma =0.8$ ($\xi =0.22$), ${\beta }_L=0$, ${\beta }_C=2\pi ·0.001$, $I_p=55$. The other parameters: $\mathrm{\Delta }=0.025$, $\epsilon =0.22·{10}^{-5}$, ${\alpha }_1=12.9$, ${\alpha }_2=2.3$, ${\chi }_s=2.3$, ${\chi }_p=1$, $\gamma =2·{10}^{-6}$.

Fig. 7. CW mode-locking state of DSs with a locked state of polarization (SOP) at a pump power of 260 mW. The CW mode-locking case at a pump power of 260 mW of DS is shown in (a)–(e). The wide optical spectrum typical for the normal dispersion operation is shown in (a). (b) A stable mode-locking pulse train with a repetition rate of 44.18 MHz accords well with the length of the cavity, and the pulse train has a stable amplitude with the small variation of the peak power at the fast and slow time scales. (c) The autocorrelation trajectory. (d) The output power of two orthogonal polarization components giving stable evolving power. (e) The fixed phase difference and SOP locking with high DOP above 90%, indicating the soliton is polarization locked vector dissipative soliton (PLVDS) caused by strong coupling between two orthogonal polarization components. (f) The averaged SOP on the Poincaré sphere within 1 ms in the form of a fixed point.

Fig. 8. Unstable multi-pulse states of DSs when the pump power is increased from 260 to 450 mW. When the pump power is increased from 260 to 450 mW, the stable fundamental frequency DSs become unstable multi-pulse states of DSs shown in (a). The pulse train has an unstable amplitude with a repetition rate of 88.36 MHz and pulse duration of ~10 ps, as shown in (b) and (c). (d), (e) The polarization state of the pulse, indicating the soliton is PLVDS so we cab increase the pump power to obtain an unstable mode-locking state. (f) The averaged SOP on the Poincare sphere within 1 ms in the form of a fixed point.

Fig. 9. Unstable mode-locking state of NLP at the same pump power as in Fig. 8. At the same pump power as Fig. 8, we can obtain NLP with polarization instability mode-locking state only by adjusting the PCs. (a) A typical spectrum of NLP. (b) The pulse with a repetition rate of 44.18 MHz accords well with the length of the cavity. (c) The autocorrelation trajectory, which is very consistent with the typical characteristics of NLP with a large energy base and a very narrow peak. As shown in (d) and (f), the output power of the cavity varies within a certain range, and the averaged SOP on the Poincaré sphere is not a fixed point, which indicates that the laser is not operating in a stable state, mainly due to the weak coupling of the strengths $I_x$ and $I_y$. (e) The fixed phase difference and SOP locking with high DOP above 90%.

Fig. 10. (a) A typical spectrum of NLP. (b)A stable mode-locking pulse train with a repetition rate of 44.18 MHz accords well to the length of cavity and the pulse train has a stable amplitude. (c) The autocorrelation trajectory which is very consistent with the typical characteristics of NLP with a large energy base and a very narrow peak. As shown in (d) and (e), the output powers of two orthogonal polarization components keep unchanged with fixed phase difference and high DOP above 90% indicating that the soliton is polarization locked vector soliton (PLVS). (f) The averaged SOP on the Poincaré sphere within 1 ms in the form of a fixed point.

Fig. 11. Stable mode-locking state of DS molecules at the same pump power as in Fig. 9. At the same pump power as in Fig. 9, the CW mode-locking state of NLP can be turned into a stable mode-locking state of DS molecules with a locked SOP only by adjusting the PCs. (a) Significant regular periodic modulation, and the overall shape of the modulation spectrum has the typical characteristics of a single-pulse DS spectrum. As shown in (b), the bound state solitons as a whole propagate in the laser cavity with a repetition rate of 44.18 MHz. (c) An autocorrelation trajectory with three peaks. (d) The output power of two orthogonal polarization components varying within a certain range. (e) and (f) The fixed phase difference, a fixed point on the Poincaré sphere, and high DOP above 90%, which indicates that the soliton is PLVDS.

Fig. 12. Experimental observation of desynchronized vector breather dynamics. In addition to the breathers in the article, it can also be obtained desynchronized vector breather at the pump power of 500 mW. The optical spectrum, pulse trains, and autocorrelation trajectory are shown in (a)–(c), respectively. As shown in (a), the optical spectrum has two maxima that reflects the breathing spectral dynamics. (d) A breather width of $200\mu \mathrm{s}$. (e) The breather’s power spikes emergence and disappearance is related to the periodic phase difference slip in $\pi$ and DOP hops. As shown in (f), the trajectories in the Poincaré sphere also take the form of hops from the localized SOPs.

Fig. 13. Experimental observation of phase difference entrainment vector breather dynamics. In addition to the breathers in the article, a phase difference entrainment vector breather at the pump power of 460 mW can also be obtained. The optical spectrum, pulse trains, and autocorrelation trajectory are shown in (a)–(c), respectively. Similar to the previous case shown in Fig. 13, the optical spectrum in (a) also exhibits two maxima. However, the breathing dynamics takes the form of two-scale oscillations (QML) with the periods of $125\mu \mathrm{s}$ in (e). As follows from (d), the dynamics is caused by the phase difference entrainment (oscillations). The DOP above 90% indicates that the dynamic is slow at the time scale from $1\mu \mathrm{s}$ to 1 ms. As shown in (f), the trajectory in the Poincaré sphere is a cycle, and so the vector soliton breathing dynamics takes the form of a phase difference entrainment synchronization scenario.

Fig. 14. Breathing polarization dynamics in the form of complex oscillations of the output power total power $I=I_x+I_y$. Though breathing regimes are not mapped in terms of the circular birefringence, the dynamics can also emerge for the weak circular birefringence and low anisotropy of the pump wave, as shown in (a)–(d). The main parameters: $\sigma =0.95$ ($\xi =0.05$), ${\beta }_L=0$, ${\beta }_c=2\pi ·0.001$, $I_p=55$. The other parameters: $\mathrm{\Delta }=0.025$, $\epsilon =0.22·{10}^{-5}$, ${\alpha }_1=12.9$, ${\alpha }_2=2.3$, ${\chi }_s=2.3$, ${\chi }_p=1$, $\gamma =2·{10}^{-6}$. (a) and (b) The powers of the polarization components and the total power. As shown in (c), the phase difference slips in $\pi$ radian. (d) SOP trajectories in the Poincaré sphere.

Fig. 15. Steady state (polarization-locked regime) in the form of the constant output powers. When the pump power is reduced from $I_p=55$ to $I_p=29$ and high pump anisotropy $\xi =0.2$, it leads to the increase coupling between $x$ and $y$ components and so to the steady-state operation (CW or polarization-locked regime) shown in (a) and (b). The other parameters: $\mathrm{\Delta }=0.025$, $\epsilon =0.22·{10}^{-5}$, ${\alpha }_1=12.9$, ${\alpha }_2=2.3$, ${\chi }_s=2.3$, ${\chi }_p=1$, and $\gamma =2·{10}^{-6}$.