Abstract
Keywords
1 Introduction
During the last three decades, the study of flocking birds, supramolecular complexes, neurons in the cortex, modes synchronization in lasers, telecom, and sensing networks mainly focused on revealing the interactions between individual system components that produce large-scale collective patterns.1
Short-range interaction through the overlapping of solitons tails results in bound states (BSs) solitons which, in analogy to biochemical and biological supramolecules, are formed by strong covalent bonds, are frequently referred to as soliton molecules, soliton macromolecules, or soliton crystals.5,20,21 The strong, short-range interactions lead to the narrow spacing of the few pulse widths and locked phase differences between adjacent solitons, resulting in the highly challenging real-time characterization of their detailed temporal structure.5,20,21 Long-range interactions can be driven by Casimir-like,19,25 optoacoustic17,18,20,24 polarization instabilities,13
Breathing solitons, known as dynamic DSs, typically feature profile and energy oscillation (breath) of DSs with periods from 5 to 100 round trips (RTs).6
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A recently developed dispersive Fourier transform technique explores conventional soliton breathing regimes, revealing a range of dynamic vector waveforms with the periods of oscillations from tens to hundreds of round trips under the paradigm of coupled Ginzburg–Landau equations.34
To fill this lacuna, for the first time, we believe that we have demonstrated a new type of slow soliton breathing dynamics, showing double temporal scale behavior, caused by a vector mechanism allowing the transition from the zero-lag synchronization to phase difference entrainment and finally desynchronization of orthogonal SOPs. We modeled nonlinear polarization rotation (NPR) mode-locked Er fiber laser in the paradigm of coupled oscillators synchronization. Based on the vector model of MLL dynamics, we reveal that the phase desynchronization is connected to the emergence of the so-called spiral chaos in a heteroclinic system,37,38 from which the trajectory is periodically evolving nearby each of the orthogonal SOPs and is switching between SOPs.15,28
2 Results
2.1 Experimental Results on Vector Breathing Regimes
The MLL we used to conduct the experiment operated under a normal dispersion regime in the telecommunication C-band. Such a type of laser allows the generation of high-energy pulses on which most practical applications rely. The schematic configuration of the laser is shown in Fig. 1(a) (details are in Appendix A). Dispersion engineering is accomplished through a piece of erbium-doped fiber (EDF) with highly positive group velocity dispersion (GVD). The mode-locking mechanism is NPR, where the intrinsic birefringence of optical fiber is used to generate pulse intensity bias through the combination of a fiber polarizer and two polarization controllers (PCs). The NPR mechanism especially benefits routes to vector pulse shaping in the context of the SOPs. The vector features of DS are analyzed through a polarimeter and present dynamic SOP evolution on the surface of the Poincaré sphere in terms of normalized Stokes parameters, total power, degree of polarization (DOP), phase difference, and power for the orthogonal -polarization components.
Figure 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
Such orthogonal linear polarized fields of DS with correlated phase relation can be treated as a general phase-coupled oscillator, as shown in Fig. 1(b).34,37,40,41 The slow temporal behavior of coupled oscillators strongly depends on syn-/desyn-chronization of orthogonal fields through a phase change. Therefore, the DS oscillation can be well explained by the general Adler equation,42 describing the evolution of the phase difference between two coupled oscillators.15,29,42 The Adler equation is represented as
To obtain the slow vector breathing, we first operate the laser under the standard stable DS regime. In contrast to the laser with the scalar breathing dynamics, the reduction of pump power in our laser leads only to the elimination of stable pulses. Alternatively, we obtained a DS breathing by elevating the pump power under a stable DS mode-locking scenario followed by fine adjustment of the intracavity PC. The slow vector breathing can be accessed within a range of hundreds of milliwatts of pump power. Figure 2 shows the experimentally observed slow DS breathing dynamics for a normal dispersion laser. Figure 2(a) shows the averaged optical spectra of the breather with a dual-peak signature.8 Using a high-speed oscilloscope, the temporal intensity distribution of a single DS breather is clearly revealed in Fig. 2(b), illustrating a breather width of 200 RTs and a period of 10,000 RTs. Unlike the scalar DS characterization, through polarimetric measurement, both orthogonal polarization modes can be mapped into the temporal domain, as shown in Fig. 2(c). The disparity in oscillation intensity of orthogonal polarization modes manifests the vector feature of the soliton breathing. Focusing on a time scale, the damped oscillation-like behavior consolidates the breathing dynamics. The distinction of the pulses bunches’ shape [Figs. 2(b) and 2(c)] between single shot and polarimetric measurement is attributed to the time resolution of the polarimeter. In particular, polarimetric measurement gave us a unique opportunity to extract information about the phase difference between orthogonal SOPs and DOP of the slow breathing dynamics. Based on Adler’s equation, the phase information allows us to view the synchronization of orthogonal fields. In Fig. 2(d), one can clearly see that periodic phase difference slip in is related to the breathers’ power spike emergence and disappearance. The fast phase difference slip hence results in a fast change DOP from 60% to 90%, indicating strong polarization instability. Therefore, such slow breathing behavior originates from the longitudinal and orthogonal polarization modes synchronization (constant phase difference) and desynchronization (phase difference slip). The polarimetric measurement also provides a route to identify desynchronization by visualizing the SOP evolution of the DS breathing. We then evidenced the desynchronization of polarization modes on the surface of the polarization Poincaré sphere [Fig. 2(e)], from which the SOP takes the form of hops from a localized point.
Figure 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
The vector DS breathing waveform can survive hours under lab conditions. Further, increasing pump power would result in completely unstable mode locking. Nevertheless, the NPR mechanism offers control of the polarization dimension, thus providing access to various stable DS regimes such as DS molecules and noise-like pulses in our laser under fixed pump power (see Appendix B, Sec. 5.1). In a laser, the undamped relaxation oscillation mechanism of Q-switching intrinsically generates breathers at a longer time scale. By adjusting the PCs in our laser, we could obtain another breathing state that is typically named Q-switched mode locking (QML). To examine the vector feature of breathing, we again carried out optical spectral measurement, temporal trace record, and polarimetric measurement, as shown in Fig. 3. Figure 3(a) shows the typical optical spectrum of a breather with the signature dual-peak feature. The rectangular pedestal in the optical spectrum indicates spectral filtering induced strong dissipative effects within the normal dispersion MLL cavity. In Fig. 3(b), the double-scaled temporal pulses show Q-switching behavior with a kilohertz repetition rate, i.e., 200 kHz ( RTs). Uniquely, there appears another envelope of the Q-switching mode-locked pulses in addition to the general Q-switched pulse bunch. This slow envelope hence corresponds to the breathing QML pulses. The breathing period is (3000 RTs). Through polarimetric measurement, the intensity discrepancy between orthogonal polarization modes can be well identified, indicating the vector feature of QML pulses [Fig. 3(c)]. A zoomed-in pulse oscillation is shown with a period of 200 RTs corresponding well to the fast detected results of this new type of slow vector breather. Moreover, by investigating the DOP of QML, we found that the DOP maintained at a large time window of 5000 RTs, as shown in Fig. 3(d). Such high DOP implies that the dynamic SOP is indeed slow and can be finely resolved through our polarimetric measurement. Moreover, the phase change in Fig. 3(d) appears as a straight line with finite thickness. This depicts an observable change without sudden slip in the phase domain [inset in Fig. 3(d)]. The dynamic temporal oscillation is therefore caused by the phase difference entrainment of orthogonal fields. The resulting Poincaré sphere representation of SOP trajectory [Fig. 3(e)] hence forms a limited cycle, indicating phase difference entrainment synchronization of the vector breathers.15,28
Figure 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
2.2 Vector Model of Er-Doped Mode-Locked Fiber Laser
For modeling slow breathing dynamics, we developed a vector model for a mode-locked EDF laser (details are in Appendix A, Sec. 4.2). This model overcomes limitations of the previous scalar or even vector models such as coupled nonlinear Schrödinger or Ginzburg–Landau equations, where long-range pulse dynamics are missing. Our model mainly visits the orientation distribution of the dipole moments with absorption and emission, the temporal response of the active medium’s population inversion, linear, and circular birefringence of a laser cavity and pump light contribution. We adopt the slow evolution of the lasing field in terms of the Stokes vector , and orientation distribution of the active medium’s population inversion in terms of Fourier coefficients for angular distribution of the population inversion , initial equations were averaged over the pulse width. For emulating the contribution of NPR, we take the linear and circular birefringence into consideration, where is the linear (circular) birefringence beat length and is the cavity length. We also involve the pump light anisotropy in addition to the normalized pump power , hence reflecting a completely vectorial treatment of laser pulse [Eq. (4) in Appendix A, Sec. 4.2]. To investigate the phase synchronization scenarios in vector DS, we simply derive the equation for the phase difference between orthogonal polarization states using our vector model [Eq. (6) in Appendix A, Sec. 4.2]. The slow-varying phase difference relationship is similar to the Adler equation. Moreover, the presence of dynamically evolving Stokes parameters and orientation distribution of the population inversion in active medium manifests a new system in the MLL that is a coupled oscillator with dynamically evolving frequency difference and coupling.
We then map the breather’s emergence in the phase space with the variables mentioned above. To identify the range parameters, i.e., the normalized pump power , pump anisotropy , and the linear birefringence , we linearized Eq. (4) in Appendix A, Sec. 4.2 in the vicinity of the steady-state solution (, , ) and found numerical eigenvalues for the parameters. Using the Shil’nikov theorem, eigenvalues and the second saddle parameter take the form41
Figure 4.Self-pulsing maps of laser operation conditions for
The simulated tunability of the breathing dynamics along with alternating the ellipticity of the pump wave is shown in Fig. 5. As shown in Fig. 5, the weak linear birefringence and low anisotropy of the pump wave within the range of parameters correspond to a case of from which the laser operates under the breathing regime. The breather polarization dynamics reproduced in Figs. 5(a)–5(c) is quite close to the experimental observation shown in Figs. 2(c) and 2(d) in the context of shape, breathing period ( RTs) and the phase difference slips in radian. The main deviation from the experimental observation is the Poincaré sphere SOP trajectories that can be caused by the simplified matter of the model that does not take into account the dual-wavelength lasing shown in Fig. 2(a) and slow polarimeter operation with a sampling of 50 RTs. Nevertheless, the SOP trajectories still follow a heteroclinic path.
Figure 5.Simulation of desynchronized vector breathing dynamics. (a) Breathing polarization dynamic in the form of complex oscillations of the total output power
As follows from Eq. (4) in Appendix A, Sec. 4.2, the increased pump anisotropy from to leads to a more anisotropic distribution of the population inversion ( function) and so to enhanced coupling of the polarization components and . So, the chaotic behavior is transformed into periodic oscillations of the total’s [Fig. 6(a)] and polarization components’ powers [Fig. 6(b)], along with the periodic oscillations of the phase difference [Fig. 6(c)]. The simulated polarization dynamics is also quite similar to the experimental results shown in Figs. 3(c) and 3(d). The heteroclinic SOP trajectories further confirm the validity of our model. The main difference is that the trajectories on the Poincaré sphere can be caused by the dual-wavelength lasing shown in Fig. 3(a) and the low polarimeter sampling rate.
Figure 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
Figure 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.
3 Discussion and Conclusion
We have experimentally demonstrated the observation of slow vector DS breathing waveforms in a normal dispersion mode-locked fiber laser. Such a dynamic pulse regime is obtained in the laser cavity well above the lasing threshold. We have shown numerically that such a type of breathing dynamics exists not only through the balance of gain, loss, dispersion, and nonlinearity, but also more to an equilibrium based on pump wave ellipticity, cavity anisotropy, and intracavity birefringence. The developed vector model reproduced our experimental observation well. We have identified the region for vector DS breathing emergence. The pump power and cavity birefringence can be controlled to alter the coupling strength between orthogonal SOPs. Uniquely, we have linked the appearance of vector DS breathing regime with the general synchronization concept in the phase domain through orthogonal polarization mode coupling. Weak coupling leads to the polarized modes’ desynchronization, with periodic switching SOPs trajectories evolving around a fixed point, which corresponds well to the heteroclinic dynamics.41 Compared with the existing soliton breathers, the vector breathing period (dwelling time near each SOP) in the scale of thousands of cavity RT times is approximately 2 orders of magnitude longer than that for scalar breathers. This is in stark contrast to the previous vector DS breathing waveforms (breathers and QS-mode locked) observed experimentally and theoretically.34
Our results demonstrate that the slow vector DS breathing universally exists in a normal dispersion mode-locked fiber laser. Our investigation can contribute to the thoroughgoing understanding of DS and laser physics. With the vector model, observations further illustrate that the mode-locked fiber laser serves as an effective test bed for the study of complex nonlinear dynamics relevant to a large variety of physical systems in a higher dimension. Likely, similar slow vector DS breathing waveforms may also exist in other types of MLLs. Further detailed breathing dynamics may be characterized through polarization-resolved dispersive Fourier transform in the future. Our findings could help in designing an ultrastable high-power mode-locked fiber laser, from which frequency comb, machining, and metrology would find important applications. It is also anticipated that the observed breathing dynamics is not only limited to nonlinear optics but also benefits in discovering new dynamics in general nonlinear physics, plasmonics, biology, and materials science.
4 Appendix A
4.1 Experimental Design
The configuration of the NPR mode-locked fiber laser is shown in Fig. 1. A 1.48 m EDF with a normal GVD of is used in this cavity. In addition, the cavity also contains 0.9 m of OFS980 fiber with a normal GVD of and 2.34 m of a single-mode fiber with an anomalous GVD of . The total length of the cavity is 4.72 m, corresponding to the fundamental frequency of 44.18 MHz and the net dispersion of the cavity is , and so the laser operates in the normal dispersion. The pump light is coupled to the laser cavity through a wavelength division multiplexer (WDM). A 91:9 coupler is used to direct out 9% of the pulse energy outside the cavity. The polarization-insensitive isolator (PI-ISO) in the cavity makes the unidirectional pulse train propagation. A fiber polarizer and two PCs are used to support the NPR mechanism for passive mode locking. By using the fast photodetector and oscilloscope, we record the dynamics of waveforms. By utilizing a commercial polarimeter (THORLABS, IPM5300) with resolution, we observed the evolution of the polarization attractors at the Poincaré sphere in terms of the normalized Stoke parameters , , and , the power for the orthogonal and polarization components , the total power , the phase difference , and the DOP. The definition of raw/normalized Stokes parameters and DOP is given below,
4.2 Vector Model of Erbium-Doped Fiber Laser
Here, time and length are normalized to the RT and cavity length, respectively. Coefficient is the EDF absorption at the lasing wavelength, represents the normalized losses, () are the Stokes parameters; and are the output lasing and the pump power normalized to the corresponding saturation powers and , respectively. Here, , ( is the Planck’s constant, , are pump and lasing frequencies), , , are absorption and emission cross sections at the lasing wavelength and absorption cross section at the pump wavelength, and are the confinement factors of the EDF fiber at the lasing and pump wavelengths, is the concentration of erbium ions, and is the fiber core cross-section area. Coefficient is normalized to the cavity length and the saturation power; the Kerr constant, , is the ratio of the RT time to the lifetime of erbium ions at the first excited level ; is the detuning of the lasing wavelength with respect to the maximum of the gain spectrum (normalized to the gain spectral width); . To mimic the contribution of the two PCs and polarizer shown in Fig. 1, we account for the linear , and the circular birefringence is the linear (circular) birefringence beat length and is the cavity length and the pump anisotropy , where is the ellipticity of the pump wave.28
Equation (4) has been derived under the approximation that the dipole moments of the absorption and emission transitions for erbium-doped silica are located in the plane that is orthogonal to the direction of the light propagation. This results in the angular distribution of the excited ions , which can be expanded into a Fourier series as follows:28
To explore the concept of the phase-coupled oscillators, by using Eqs. (4) and (5), we derive the equation for the phase difference between two orthogonal SOPs as follows:
The simulation for stable DS and another example of breather is shown in Appendix A, Sec. 4.2.
As follows from Eqs. (6) and (1), increased birefringence strength and nonlinearity coefficient lead to increased frequency of oscillations. The dependence of the coupling coefficient in Eq. (6) on the dynamically evolving Stokes parameters results in alteration of the synchronization and desynchronization shown in Fig. 5(c) and so in breathing dynamics in Figs. 5(a) and 5(b). Also, the linear stability analysis [Eq. (2) and Fig. 4] demonstrates that the breathing dynamics exists for very narrow range of values of the birefringence strength and the pump anisotropy . For an example, increasing the pump anisotropy leads to modification of breathing dynamics toward oscillations shown in Fig. 6.
5 Appendix B
5.1 Additional Experimental Results
In the experiment, we can achieve different mode-locking states by increasing the pump power or adjusting the PCs individually at the same pump power. Figure 7 shows continuous wave (CW) mode-locking state of DSs at a pump power of 260 mW. When the pump power is increased from 260 to 450 mW, the DS pulse splits into an unstable multi-pulse state in Fig. 8. After that, under this pump power, only adjusting the PCs can obtain unstable NLP in Fig. 9, stable NLP in Fig. 10, and stable DS in Fig. 11. Figures 12 and 13 show two different types of breathing waveforms from those in the article.
Figure 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.
Figure 9.Unstable mode-locking state of NLP at the same pump power as in
Figure 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.
Figure 11.Stable mode-locking state of DS molecules at the same pump power as in
Figure 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
Figure 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
5.2 Additional Theoretical Results
In addition to the simulation of vector breathing dynamics mentioned above, the breathing dynamics can also emerge for different circular birefringence and anisotropy of the pump wave as shown in Fig. 14. Figure 15 shows the simulation of polarization-locked regime which corresponds to the experimentally observed case shown in Figs. 7 and 10.
Figure 14.Breathing polarization dynamics in the form of complex oscillations of the output power total power
Figure 15.Steady state (polarization-locked regime) in the form of the constant output powers. When the pump power is reduced from
Biographies of the authors are not available.
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