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Abstract
Transition from a gain-guided soliton (GGS) to a fully developed noise-like pulse (NLP) is numerically demonstrated in fiber lasers operated in the normal dispersion regime, which explains well the experimental observation of spectrum evolution that the bottom of the averaged spectrum gradually broadens with pump power increasing. Numerical results suggest that the transition could also happen under the condition of cavity linear phase delay bias change with fixed pump power. It is demonstrated that the peak power clamping effect and the normal dispersion are the key factors leading to the spectrum evolution. In addition, intermittent meta-stable states between GGS and NLP can be obtained when the cavity dispersion is chosen at small normal dispersion.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Compared with solid-state lasers, fiber lasers have the advantages of good beam quality, high energy conversion efficiency, compact structure, reliable operation and good heat dissipation. Passively mode-locked fiber lasers are an important source of ultra-short pulses and an ideal platform for investigating various nonlinear effects. Fiber lasers can work either in the anomalous dispersion regime [1,2], near zero dispersion regime [3] or normal dispersion regime [4]. The soliton generated in the normal dispersion regime can have much larger pulse energy, therefore, fiber lasers operated in the normal dispersion regime have become a hot research spot [5–8].
Noise-like pulses (NLPs) is a kind of pulse emission state commonly observed in fiber lasers. It is a pulse wave packet similar to noise, which composed of many small pulses with random variation of peak power and pulse width. NLPs was first reported by Horowitz et al. in a nonlinear polarization rotation (NPR) mode-locked erbium-doped fiber laser operating in the anomalous dispersion regime [1]. In the following investigations, Zhao et al. found that NLPs could also be generated in the normal dispersion regime [9]. Compared with gain-guided solitons (GGSs) generated in the same fiber laser, NLPs have wider averaged spectrum and larger pulse energy. NLPs have a good application prospect in fields such as laser micro-machining [10], generation of supercontinuum spectrum [11,12], low-coherence spectral interferometry [13,14] and so on. The investigation on the properties and formation mechanism of NLPs can help us to obtain NLPs with required performance. In the anomalous dispersion regime, the formation of NLPs is attributed to the soliton collapse effect [15], while the generation of NLPs in the normal dispersion regime is determined by the peak power clamping effect [9]. It is found that there exists a spectrum evolution that the bottom of the averaged spectrum observed on an Optical Spectrum Analyzer (OSA) gradually broadens during pump power increasing, which is different from the case happened in the anomalous dispersion regime where the NLP is abruptly generated with pump power increasing. Therefore, it is interesting to investigate the detailed evolution in fiber lasers operated in the normal dispersion regime.
In this paper, the evolution of NLPs in fiber lasers operated in the normal dispersion regime is numerically discussed. The averaged spectrum bottom of NLPs broadens with the increase of gain, which reproduces experimental details [9]. Similar phenomena can be achieved by increasing the linear cavity phase delay bias too. From the autocorrelation trace evolution, it can be found that this phenomenon is actually the development of a pulse from a GGS to a fully developed NLP. Numerous simulations suggest that the pulse peak power clamping effect results in the evolution of the averaged spectrum bottom of NLPs during pump power increase or the linear cavity phase delay bias change. Plenty of simulations suggest that the evolution only occurs in the normal dispersion regime. The smaller the total dispersion in the cavity is, the faster the evolution develops. When the net cavity dispersion is normal but small, an intermittent meta-stable state between a GGS and NLPs can be obtained.
2. Fiber laser setup and theoretical model
The investigation is based on a typical NPR mode-locked fiber laser, as shown in Fig. 1. The fiber laser contains a 2 m Erbium-doped fiber (EDF) and two dispersion-shifted fibers (DSF) totaling 3.5 m. The group velocity dispersion (GVD) parameter of the EDF and the DSF is $51.64\textrm{p}{\textrm{s}^2}/\textrm{km}$ and $0.258\textrm{p}{\textrm{s}^2}/\textrm{km}$ [9], respectively, to form an all-normal-dispersion fiber laser. Two polarization controllers and a polarizer are used for achieving mode-locking. A 10% output coupler is used to output the pulse. The initial condition is set as a small hyperbolic secant pulse. By circulating the pulse in the cavity, a steady pulse can be eventually formed with specific parameters.
The propagation of pulse in fibers can be described by the coupled Ginzburg–Landau equation (GLE) [16],
In an NPR mode-locked fiber laser, the NPR together with a polarization projection can introduce artificial saturable absorption effect to achieve mode-locking. This saturable absorption effect can be described as a sinusoidal function with a period of $2\pi $ of the cavity phase delay. The transmission function of an NPR cavity can be denoted as [16]:
Some parameters used in the simulations are given in Table 1.
Table 1. Main parameters used in the numerical simulations
3. Simulation results and discussions
3.1 Spectral evolutions of NLPs
Numerically we obtained various NLP generation in the fiber laser. When $\textrm{Ph} = 1.3\mathrm{\pi }$, $\theta = 0.152\pi $, $\varphi = 0.652\pi $, as shown in Fig. 2(a), after a GGS (${G_0}$=1000) is obtained, if the pump power continued to increase, the bottom of the pulse spectrum broadened, and the central part of the spectrum nearly maintained unchanged. The corresponding pulse profiles are shown in Fig. 2(b), where we purposedly shifted individual pulse to make them clearly visible. The pulse gradually changes from a GGS into an NLP with pump power increasing. We note that all the spectra in the simulation are an averaged one of 3000 roundtrips as the experimental results of optical spectrum measurement [9] is indeed an averaged one. A typical output pulse train of a NLP state when ${G_0}$=2000 is shown in Fig. 2(c). We note that here the fiber laser parameters and operation conditions are so selected that an NLP is more favourite than multiple soliton generation under pump power increasing.
Fig. 2. Pulse evolution with gain increasing: (a) averaged spectra; (b) pulse profiles (pulse profiles are shifted to avoid overlapping); (c) NLP pulse train when ${G_0}$=2000; (d) averaged autocorrelation traces.
Pump power and polarization are two easy-controlling factors in experiment. Generally, there exist two operation routes to explore pulse properties: one is to manipulate the polarization state while fixing the pump power, the other is to change the pump power while maintaining the polarization state. Numerically we followed these two operation routes. Figure 2 shows the spectral evolution with ${G_0}$ increasing when Ph is fixed at $1.3\pi $. Figure 3 shows the other case when ${G_0}$ is fixed at $1250$ with the increase of Ph. Under both operations, the extension and increasing of averaged spectrum bottom could be identified once a GGS is transformed into an NLP. In Fig. 2(d) and Fig. 3(b), the evolution of the averaged autocorrelation trace shows that the transition from soliton to NLP in normal dispersion regime is a development. As the gain increases, the normalized intensity of the base of the averaged autocorrelation trace decrease from 1 to ∼0.6, which indicates the coherence of the pulse decreases and the pulse becomes more “noise-like”. When the gain is large enough, the pulse can be considered as a fully developed NLP, and one of its typical characteristics is that the normalized intensity of the base of the averaged autocorrelation trace is about 0.5. The spectrum evolution in normal dispersion regime is actually a development of the pulse from a GGS to a fully developed NLP. The normalized pedestal became narrower with Ph increasing as shown in Fig. 3(b). That is because the pulse intensity limitation is gradually reduced with Ph increasing. Consequently, more energy was stored in intra-pulses instead of distributing in the wave packet. Therefore, the width of the wave packet became narrower.
Fig. 3. When ${G_0}$=1250 and Ph varying, the evolution of (a) the averaged spectrum of NLPs; (b) the averaged autocorrelation trace of NLPs.
Our simulation results reproduce the experimental observation of spectral evolution from a GGS to a fully developed NLP by pump increasing [9]. NLP generation is the result of the peak power clamping effect. The peak power clamping effect means there exists a switching point between the positive and negative feedback regimes in NPR mode-locking, which related to the pulse peak power. Consequently, the pulse peak power will be limited once it is increased to larger than a threshold [17]. It is straightforward if the cavity transmission trace is a sinusoidal function as denoted by Eq. (3). When the pulse intensity increases gradually, the nonlinear phase delay increases until the transmission reaching a switching point. Then, as the transmission reduces, the pulse peak power is limited.
According to Fig. 2 and Fig. 3, the peak power clamping effect is also one of the factors leading to the NLP spectrum evolution. To uncover it, we showed the cavity transmission function of Eq. (3) with different Ph in Fig. 4. The x-axis represents the nonlinear phase delay caused by the peak power of the pulse. The y-axis represents the cavity transmission. The area between two switching points is the positive feedback regime [16]. A stable mode-locked pulse usually exists in the positive feedback regime. We assume that the nonlinear phase delay corresponding to A1 or A2 is the initial operating point of the pulse (it corresponds to the nonlinear phase delay of the initial small pulse in the cavity). A1C1 represents the nonlinear phase delay corresponding to the maximum peak power the pulse can achieve if there is no peak power clamping effect. Then A1B1 and B1C1 represent the nonlinear phase delays corresponding to the upper limit of the soliton peak power and the power of the background (such as dispersive waves) amplified by the extra gain, respectively. When Ph is fixed, A1B1 will be fixed. If the gain increases, only B1C1 increases, that is, the spectrum bottom broadens while the central part of the spectrum nearly maintained unchanged, which explains the spectrum evolution in Fig. 2. When Ph increases, the transmission trace changes. If the gain is fixed, that is, A2C2 = A1C1, then we can see that as Ph increases, A2B2 keeps decreasing while B2C2 increasing, which corresponds to spectrum bottom extension and the central part of the spectrum decreasing in Fig. 3. Due to the peak power clamping effect, when the pulse peak power reaches the switching point, the extra gain in the cavity will not boost the soliton energy but amplify the background. With the increasing of gain, the soliton gradually develops into an NLP. And when the gain is large enough, the pulse becomes a fully developed NLP. Therefore, the spectral evolution of NLP is caused by the peak power clamping effect.
3.2 Impact of dispersion on NLP evolution
The NPR technique can be used in a fiber laser operated in the anomalous dispersion regime, and the peak power clamping effect also exists. But in anomalous dispersion regime, it usually leads to the generation of multiple solitons [16]. And different from the slow evolution from a GGS to a fully developed NLP when the fiber laser is operated in the normal dispersion regime as shown in Fig. 2, the change from a stable conventional soliton to an NLP is abrupt. When the operating parameters reach a threshold, the pulse state in the cavity changes from a soliton to NLP immediately. There is no obvious evolution can be obtained. Tang et al have demonstrated that NLPs generated in fiber lasers operated in the anomalous dispersion are resulted from periodic soliton collapse and amplification [15]. The abrupt change from a stable conventional soliton to an NLP is resulted from the soliton collapse.
Obviously, the operation in different dispersion regime determines the different NLP evolution. To confirm the impact of dispersion especially in the normal dispersion regime, by changing the total dispersion in the cavity, comparative analysis was carried out. Considering the feasibility of experiment, we use fibers of anomalous dispersion to reduce the normal dispersion. Dispersion management was adopted by replacing the DSF with single mode fiber (SMF), and the dispersion of SMF is set to $- 15\textrm{p}{\textrm{s}^2}/\textrm{km}$ and $- 22\textrm{p}{\textrm{s}^2}/\textrm{km}$ (the latter is the standard dispersion of SMF) respectively, so that the total dispersion in the cavity is reduced from ${\sim} 0.104p{s^2}$ to ${\sim} 0.051p{s^2}$ and ${\sim} 0.026p{s^2}$, respectively. And the other parameters are fixed. Pulse evolution when total dispersion is ${\sim} 0.051p{s^2}$ and ${\sim} 0.026p{s^2}$ is shown in Fig. 5 and Fig. 6 correspondingly.
Fig. 5. Pulse evolution when total dispersion is ${\sim} 0.051p{s^2}$: (a) averaged spectra; (b) pulse profiles (pulse profiles are shifted to avoid overlapping); (c) averaged autocorrelation traces.
Fig. 6. Pulse evolution when total dispersion is ${\sim} 0.026p{s^2}$: (a) averaged spectra; (b) pulse profiles (pulse profiles are shifted to avoid overlapping); (c) averaged autocorrelation traces.
We found that the pulse intensity of the NLP of all-normal-dispersion fiber laser (Fig. 2(b)) is much lower than that of dispersion-managed fiber lasers (Fig. 5(b) and Fig. 6(b)), which is resulted from the pulse compression when it propagates along the fiber segment of anomalous dispersion.
In addition, when the dispersion in the cavity is reduced to ${\sim} 0.026p{s^2}$, as shown in Fig. 6, when ${G_0}$=595, there is a stable mixture of GGS and small pulses. The wave packet has a more regular waveform and spectrum, and its autocorrelation trace is more like a soliton than an NLP. We called it “meta-stable state” or developing NLP. Figure 7 shows the output pulse in 200 roundtrips (after the pulse stabilizes) of the meta-stable state. The pulse repeats itself every ∼20 cycles. The intermittent meta-stable states exist from ${G_0}$=590 to ${G_0}$=630. More roundtrips will be needed for the pulse waveform returning for larger gain. Further increasing the gain destroys the meta-stable state and develops the pulse into a fully developed NLP as shown in Fig. 6(b). We note that the meta-stable state is similar to the performance of a pulsation state always observed in fiber lasers operated in the anomalous dispersion regime [18].
Fig. 7. The pulse evolution in 200 roundtrips of the meta-stable state (${G_0}$=595, after the pulse stabilizes).
4. Conclusion
In conclusion, we have investigated the evolution of NLPs in fiber lasers being operated in the normal dispersion regime, which actually is a development of the pulse from a soliton to a fully developed NLP. We found that the peak power clamping effect determines the development of NLP based on different operation conditions. Either pump power increase or linear phase delay bias increase can result in the averaged spectrum bottom expansion of pulses. Moreover, this evolution is dispersion-dependent, it does not exist in anomalous dispersion regime. The evolution is faster with smaller cavity dispersion. By reducing the cavity dispersion, a meta-stable state can be obtained, which is a mixture of the GGS and small pulses.
Funding
Fundamental Research Funds for the Central Universities (HUST 2020kfyXJJS007); the Protocol of the 9th Session of China-Croatia Scientific and Technological Cooperation Committee (Grant No. 9-28); National Undergraduate Training Projects for Innovation and Entrepreneurship (GD2022114).
Acknowledgments
This work was supported by Fundamental Research Funds for the Central Universities; the Protocol of the 9th Session of China-Croatia Scientific and Technological Cooperation Committee; National Undergraduate Training Projects for Innovation and Entrepreneurship.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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