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Abstract
We systematically study the stable-, period-N- and multiple-soliton regimes in an Erbium-doped fiber laser effectively mode-locked by nonlinear polarization rotation technique. In the stable mode-locked regime, an invariant soliton with 497 fs pulse duration and 6.9 nm optical spectrum are obtained. With a larger pump power of 180 mW, the period-N state (in which the pulse intensity returns to its original value after N cavity-roundtrips) emerges, accompanied by sub-sideband generation on the first Kelly sideband and spectrum shift. Considering the inconsistent central wavelengths between gain and polarization-dependent isolator (PD-ISO) firstly, to our knowledge, the numerical results are in good agreement with the experiment and reveal the composite filtering of gain and PD-ISO takes major responsibility for spectrum shift, which causes group velocity offset simultaneously. Further study shows the continued increase of pump power can lead to the laser operating in the unstable multi-pulse state and the narrow spectral width contributes to stabilizing the multi-pulse state. Our work can promote the understanding of soliton dynamics and filtering in ultrafast fiber lasers.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The study of operating states in ultrafast fiber lasers involves soliton localizations in complex circumstances consisting of dispersion, nonlinearity, gain and loss, etc., which promotes mode-locked fiber laser an excellent platform for revealing soliton behaviors in dissipative nonlinear systems [1]. Apart from various stable soliton regimes [e.g. conventional soliton, stretch soliton, self-similar soliton, dissipative soliton, dissipative soliton resonance and spatio-temporal soliton [2–7]] relating to the fiber lasers with different dispersion, there are tremendous researches on unstable soliton states especially the emerging of recently developed dispersive Fourier transform technology [8]. For example, soliton explosion, pulsating soliton, multi-pulse instability, shaking soliton molecules, and unstable soliton in transition states have been proposed theoretically and experimentally [9–13]. The relevant researches not only lead the instabilities to be circumvented, but also propel the progress of ultrafast optics as well as nonlinear science [14].
In fiber lasers, numerous effective saturable absorbers [e.g. SESAM, carbon nanotubes and two-dimensional materials [15–20] and mode-locking techniques [e.g. Nonlinear polarization rotation (NPR) [21], nonlinear optical/amplifying loop mirror (NOLM/NALM) [22,23]] have been used to achieve mode-locked regimes. Among these methods, NPR is an effective mode-locking technique that takes advantage of the combination of a polarizer, a Kerr medium and an analyzer as fast saturable absorption effect [24], possessing the ability to tolerate higher pulse energy. Therefore, NPR mode-locked fiber lasers have been used to achieve a variety of above-mentioned soliton regimes [25–31]. Among these solitons, conventional soliton in an anomalous dispersion fiber laser is of great attractiveness because the balance of dispersion and nonlinearity facilitates the formation of femtosecond solitons [32]. With the modulation of polarization settings or pump power in an anomalous dispersion fiber laser, periodic pulsating soliton, multi-soliton can be achieved [21,33,34]. However, the research difficulty of the relevant subject is increased due to the complicated mechanism in the cavity. As for the original mechanism of accompanied sub-sidebands generation in period-N soliton regime, there are numerous investigations in the past two decades and several key points appeared: dispersion wave modulation instability (MI), complex dynamical origin over MI, slow gain dynamics and incoherent interaction between the synchronized dispersive waves and soliton [35–39]. In the proposed researches, the sub-sidebands generation is investigated both in experiments and numerical simulations. By contrast, there are fewer explanations on the optical spectrum shift in simulation. Besides, excluding gain spectrum, the filtering of intracavity components was left out of consideration in the numerical models for period-N solitons, which is a ubiquitous effect in fiberized devices and affects significantly on the soliton regimes [40–42].
In this work, we establish a mode-locked fiber laser based on NPR mode-locking technique which is realized by the combination of intracavity fibers, a polarization controller (PC) and PD-ISO. By modulating the pump power, stable-, period-N- and multiple-soliton regimes can all be achieved in the laser. According to the experimental condition, a rigorous numerical model considering the transmittance of every individual component and filtering of PD-ISO is created to systematically study the pulse behaviors in the three soliton regimes, especially for spectrum shift. There is a high consistency between the experimental results and numerical simulations.
2. Experimental configuration and numerical model
To easily understand of our experiment, we erect a schematic of the mode-locked fiber laser, which is shown in Fig. 1. Initially, the amplified spontaneous radiation is generated in EDF due to the pump excitation through a WDM. Following the fibers, the signal encounters a coupler, an inline PC, a PD-ISO in sequence and finally back into the WDM. After one roundtrip, the light enters the next cycle. In the cavity, the light amplification is provided by the 0.5-m-long EDF (Liekki, Er110-4/125). The following 20/80 coupler splits the 20% of light energy out of the cavity for measurement. While the other part of light continues to transmit in the cavity. An inline PC and a fiber polarization beam splitter (PBS) are used after the output to measure the pulse trains along the two axes of the cavity. The PC and PD-ISO are utilized to modulate the polarization direction of the signal and eliminate the light energy perpendicular to the intrinsic transmission direction, respectively. Thus, the intensity-relevant polarization direction of light due to the fiber-provided nonlinear phase shift leads to the different transmittances after passing through the PD-ISO, which acts as the analogous behavior of saturable absorber. Besides, the PD-ISO (40 nm operating spectral bandwidth) ensures the unidirectional cavity configuration and spectral filtering. All the components are well spliced by standard single-mode fibers (SMF-28e). The whole cavity length is $L_c\approx 9.9$ m and the total group velocity dispersion (GVD) is −0.201 ps$^2$.
Fig. 1. Schematic of the mode-locked fiber laser. WDM: wavelength division multiplexer; EDF: Erbium-doped fiber; PC: polarization controller; PD-ISO: polarization-dependent insulator; PBS: polarization beam splitter.
According to the experimental setup in Fig. 1, a rigorous numerical model is established to investigate the remarkable features of pulse properties. A lumped model that marks every component in the cavity is employed, rather than a distributed model based on master equation or cubic-quintic complex Ginzburg-Landau equations which consider the average effect of cavity parameters. In our model, irrespective of the high-order dispersion and Raman scattering effect for reducing discussion complexity, the propagation of low-power pulse in every segment of fiber is depicted by the coupled Ginzburg-Landau equations [21]
In order to precisely express the devices in the cavity, the commonly used equation of NPR mode-locking in [21] is not employed in our model. When the vector field passing though the PC is the multiplication of the light before the PC by the following Jones matrix
To approach the experimental condition, the residual parameters are listed in our simulation: $\beta _2=29.53$ ps$^2$/km, $\gamma =3$ W$^{-1}$ km$^{-1}$, $g_0=7$, $\lambda _0=1560$ nm for the EDF; $\beta _2$=−23 ps$^2$/km, $\gamma$=1.3 W$^{-1}$ km$^{-1}$ for SMF; the central wavelength $\lambda _c=1570$ nm and the bandwidth of PD-ISO is $\omega _f=40$ nm; the length of the EDF and the five segments of SMFs are 0.5 m and 1.5, 2.3, 2, 2.6, 1 m, respectively.
3. Results and discussion
When the pump power is 100 mW, the laser operates in a stable mode-locked state after properly modulating the angle and screw of the PC. Once the mode-locking is formed, the laser can self-start if the PC settings are fixed and the pump power is raised to 100 mW. The corresponding pulse properties are shown in Fig. 2. One can see that a stable pulse train with a pulse-to-pulse interval of 49.8 ns is obtained by an oscilloscope (Agilent Technologies, DSO9104A) [see Fig. 2(a)]. To measure the pulse trains along the two axes of the cavity, the port1 and port2 are connected to the oscilloscope through two photoelectric detectors simultaneously. From Fig. 2(b), both the two pulse trains on the two axes are existing and stable, which means the polarization-relevant loss induced by PD-ISO is effective and there is a misalignment between the PD-ISO and both the two axes. Figure 2(c) depicts the optical spectra of the pulses, which is measured by an optical spectrum analyzer (YOKOGAWA, AQ6370C). The total spectrum is centered at 1567.5 nm and the 3dB spectral width is 6.9 nm. The obvious Kelly sidebands indicate the pulse belongs to conventional soliton. After sech $^2$ fitting the auto-correlation trace, the pulse duration is 497 fs [see Fig. 2(d)]. The time-bandwidth product is calculated as 0.419, indicating a slight frequency chirping is contained in the pulse. The radio frequency signal is monitored by a frequency spectrum analyzer (Agilent Technologies, N9000A), as shown in Figs. 2(e) and (f), the fundamental frequency centered at 20.1 MHz manifests a signal-to-noise ratio (SNR) of 53.2 dB. Besides, the high-order harmonic signal can extend to the range beyond 1 GHz.
Fig. 2. Pulse characteristics of stable soliton regime. (a) pulse train; (b) pulse trains from port1 and port 2; (c) optical spectra; (d) auto-correlation trace; (e) fundamental frequency signal; (f) frequency spectrum train.
As the pump power is increased, the pulse characteristics are changed, as shown in Fig. 3. There are spectrum shift and CW component generation in the description of Fig. 3(a). If the pump power is raised to 180 mW, the first order of Kelly sideband on the left (L1) broadens, as featured in Fig. 3(b). With a higher input power, sub-sidebands begin to emerge on L1 and become more obvious when the pump power is 220 mW. Besides, there is a CW signal on the main peak of the spectrum [see the yellow line in Fig. 3(b)]. From Fig. 3(c), the intensity of pulse train at the pump power of 220 mW is modulated periodically and the period is about 51 multiples of pulse interval, indicating the laser operating in the period-N soliton regime which emerges generally in the transition process of stable soliton to chaos [43]. When the pump power is large enough, the complex interaction of gain, loss, dispersion, nonlinearity and other effect causes the unstable dispersive wave [39], it combines the intracavity soliton results in the generation of period-N state [35,36]. Meanwhile, the peak intensity of a single pulse signal on the oscilloscope depicted in Fig. 3(d) reduces gradually with the raising pump power. Figure 3(e) describes the frequency properties, from which two weak satellite frequencies symmetrically distribute at the two sides of the fundamental frequency. And the separation of the two new frequencies enlarges corresponding to higher pump power. When the pump power is 220 mW, the separation of satellite frequencies and the fundamental frequency is $\Delta =390$ Hz, producing a period of pulse intensity modulation of $P=f_c/\Delta \times L_c=51.5 L_c$, which coincides with Fig. 3(c). Besides, the Kelly sidebands shift towards to long wavelength when the pump power is raised, as shown in Fig. 3(f). The shift has two properties: one is that the sidebands on the left of central wavelength shift longer than the right counterparts; the other is the shift distances become short with high pump power. Simultaneously, the CW signal keeps almost invariant.
Fig. 3. Pulse performances of period-N soliton regime. (a) optical spectra with different pump power; (b) enlarged optical spectra; (c) pulse train; (d) enlarged pulse train; (e) fundamental frequencies; (f) positions of Kelly sidebands versus pump power.
The spectrum shift can be easily explained by the effective gain spectra which are shown in Fig. 4. In our theoretical model, the gain spectrum and the transmittance of PD-ISO are simplified as Gaussian shape [Fig. 4(a)]. If the linear loss is not considered, there is nearly no affection on the effective gain spectrum duration the change of gain, as shown in Fig. 4(b). Once the loss is considered, there is an obvious shift of central gain spectrum towards the transmittance peak of the filter when the gain is raised [Fig. 4(c)]. The variation of effective gain is the key factor on spectrum shift. In fact, the gain spectrum is not a perfect Gaussian type, there is a narrow peak value on the effective gain spectrum corresponding to the spectral peak of CW signal. Due to the peak of CW signal is far higher than the mode-locking spectrum, the gain for CW is far higher than the loss. Therefore, according to Fig. 4(b), the effective gain of the narrow gain peak remains nearly constant. Figure 4(d) depicts the effective gain when the bandwidth of PD-ISO is changed to 15 nm, one can see that the spectral bandwidth of the effective gain becomes narrow significantly.
Fig. 4. The variation of the gain spectrum with the effect of filtering. (a) Spectra of gain and filter; (b) effective gain spectral without considering the linear loss; (c) effective gain spectra with linear loss; (d) effective gain spectra with enhanced filtering.
To better understand the process of solitons in the stable- and period-N regimes, the corresponding numerical simulation is employed. Only properly adjusting the angles of PC and PD-ISO, the mode-locked state can be achieved and the results are displayed in Fig. 5. Figures 5(a1) and (a2) are the evolutions of pulse peak power and pulse energy at $E_{sat}$ of 50 pJ, respectively. Clearly, both the peak power and pulse energy increase initially, and get stable after 400 roundtrips, manifesting a stable soliton state is formed. Besides, the peak powers and pulse energies on the two axes of the cavity remain unchanged due to the polarization-relevant loss of PD-ISO, which is the same as experimental results. If the $E_{sat}$ is set as 80 pJ, the period-N state occurs, as shown in Figs. 5(b1) and (b2), the intensity of peak power variates periodically and the pulse energy oscillates with weak magnitude. Simultaneously, the L1 Kelly sideband broadens [see Figs. 5(c2) and (d1)]. When the $E_{sat}$ is lifted to 120 and 160 pJ, the obvious sub-sidebands on L1 emerge. Figure 5(e) is the spatio-temporal evolution of pulse intensity at $E_{sat}$ of 120 pJ, the pulse acquires a GVD and shifts from central time of cavity roundtrip to negative time direction, which also means the effective cavity length is slightly reduced. Correspondingly, the optical spectrum moves to long wavelength direction [see Fig. 5(f)]. Clearly, the simulation results with respect to the movement principle of Kelly sidebands are highly consistent with the experimental results shown in Fig. 3(f). In our simulation, the spectrum shift is dominated by the deviation of the effective gain center due to the inconsistency of spectral centers between the gain and the PD-ISO. As a result, a larger gain gives rise to the effective gain shift towards the center of filtering. Therefore, the pulse spectrum moves with the effective gain. Also, the spectrum shift also causes the GVD offset of pulse intensity [44]. From the simulation, the inconsistently central wavelength between gain and PD-ISO shifts the effective gain which causes the spectrum shift, thereby altering the effective dispersion of the cavity (which is supported by the GVD offset). Due to the positions of Kelly sidebands is related to the dispersion [45], the Kelly sidebands move to the long wavelength. In the period-N regime, the soliton intensity returns to its origin after tens of cavity period, periodically radiating resonant dispersive waves which contain synchronized resonant dispersive waves and unsynchronized resonant dispersive waves [39]. The interaction of unsynchronized resonant dispersive wave and soliton cause the variation of pulse intensity, thereby inducing the oscillation of the soliton and synchronized resonant dispersive waves. The oscillation changes the spectrum on the Kelly sideband. Moreover, the higher the pump power, the stronger the dispersive wave energy [36]. Therefore, the sub-sidebands emerge at the maximal Kelly sideband, namely, the L1 Kelly sideband in the experiment.
Fig. 5. Numerical simulation of outputs in stable- and period-N soliton regimes. (a1, a2) and (b1, b2) Peak powers and pulse energies corresponding to the stable state and period-N state, respectively; (c) optical spectrums with different $E_{sat}$; (d) enlarged optical spectra; (e) spatio-temporal evolution of pulse intensity; (f) side-bands shift and GVD offset speed versus $E_{sat}$.
When the pump power is further increased, the laser operates in an unstable multi-pulse state. Figure 6 describes the non-stationary chaotic bunches comprising five pulses with the pump power of 250 mW. The pulses in four time shots distribute differently, as shown in Fig. 6(a). The pulse in one cycle measured by oscilloscope variates significantly, especially for pulse intervals. For a single-short, the consecutive four pulse clusters also variate slightly, manifesting the pulses change rapidly versus time. Noted that the multi-pulse state is different from soliton molecules whose pulses are bounded together stably with small intervals [1,46–48], the pulse-to-pulse interval in our multi-pulse regime is large enough to be distinguished by an oscilloscope. Under this circumstance, the soliton dynamics are influenced by direct soliton interaction and intense dispersive waves [11]. The large pulse-to-pulse interval means the repulsion between solitons is weaker than the attractive force originating from nonlinearity and thereby the collapse cannot be prevented. Simultaneously, the unstable intensity dispersive waves are generated, which in turn alter the intensity and phase of pulses, randomizing the direct interaction of solitons and adding chaos. As a result, the pulse-to-pulse interval is fluctuated. The corresponding simulation results in Figs. 7(a1-a3) manifests that the pulse peak power, pulse energy and pulse intervals are chaotic when the $E_{sat}$ is large enough for the laser operating in the unstable multi-pulse state. Even there are collisions between different solitons. Fortunately, if the bandwidth of the spectral filter is narrowed, the situation can be improved [49]. For example, with a filter bandwidth of PD-ISO of 15 nm, the simulation results are in Figs. 7(b1-b3). Initially, the pulse peak power jitter is violent and decreases after 220 cycles. Thereafter, the laser operates in a state more stable than that with PD-ISO spectral bandwidth of 40 nm. Meanwhile, from Fig. 7(b2), the pulse energy also gets stable after 200 roundtrips. Figure 7(b3) depicts the spatio-temporal evolution when the spectral bandwidth of PD-ISO is 15 nm. The number of solitons increases and there is no soliton collision. Through comparison of the two columns in Fig. 7, the peak power of solitons is reduced dramatically while the pulse energy is almost unchanged. This is because the enhanced filtering increases the number of solitons but the loss induced by filtering is small in this situation in which the filter bandwidth of PD-ISO is still larger than the spectral width of the output pulse. The reduction of pulse peak power decreases the nonlinearity induced attractive force between solitons. Besides, the spectrum of unstable dispersive wave is limited by strong spectral filtering. Therefore, the perturbation caused by unstable dispersive waves is reduced. The simulation results coincide with the previous investigation of the spectral filtering effect in [50,51] and manifest the potential of generating high-repetition pulses of the laser.
Fig. 6. Pulse characteristics of the unstable multi-pulse regime. (a) pulse intensity with different time shots; (b) pulse intensity with different roundtrips in one single-short.
Fig. 7. Numerical simulating the pulse performances of unstable multi-pulse state. (a1-a3) and (b1-b3) pulse evolutions with the PD-ISO spectral width of 40 nm and 15 nm, respectively.
4. Conclusion
We constructed an Erbium doped mode-locked fiber laser in which the coaction of intracavity fibers, a polarization controller and a PD-ISO acts as an equivalent saturable absorber. Based on the experimental setup, a lumped numerical model is established for simulating the pulse performances of the laser. When the pump power is 100 mW, the laser can emit a stable conventional soliton with the pulse duration and spectral bandwidth of 497 fs and 6.9 nm, respectively. With a higher pump power, the laser operates in a period-N state. The corresponding spectrum shift and sub-sidebands generation are experimentally observed and numerically realized. Besides, the spectrum shift is prominently caused by effective gain spectrum movement due to the inconsistently central wavelengths between gain and PD-ISO. Further raising the pump power results in the laser operating in the unstable multi-pulse regime which can be stabilized by enhancing the filtering in the simulation. In addition, the high consistency between the experiment and numerical simulation not only indicates the effective mode-locking via a PC and a PD-ISO in the fiber laser, but certifies the accuracy of our numerical model. Our work systematically investigates the pulse performances in different mode-locked regimes stemming from the increasing pump power and filtering, which is beneficial for studying soliton behaviors in dissipative systems.
Funding
National Natural Science Foundation of China (61905193); National Key Research and Development Program of China (2017YFB0405102); Key Laboratory of Photoelectron of Education Committee Shaanxi Province of China (18JS113); Key R&D project of Shaanxi Province-International Science and Technology Cooperation Programme (2020KW-018); Northwest University Innovation Fund for Postgraduate Student (YZZ17099).
Disclosures
The authors declare no conflicts of interest.
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