With increasing demand on a laser source in the gigahertz pulse repetition rate regime, clarification on the mechanism of instability in high repetition rate fiber lasers – a promising alternative to solid state lasers – is of great importance and can potentially offer guideline for continuous wave (CW) mode locking. Here we present a theoretical approach together with relevant experimental corroboration to analyze the instabilities. By means of appropriate approximations, regimes from Q-switched mode locking, CW mode locking and pulsation are theoretically identified. Meanwhile, a critical curve that characterizes pump level for triggering Q-switched mode locking and pulsation for different repetition rates is given by virtue of both analytical and numerical procedures. In experiment, a passively mode-locked fiber laser with 1.6 GHz fundamental repetition rate is realized. The three regimes and corresponding pump power intervals are revealed, which are in consistence with theoretical prediction. Pulsation, as a relatively exotic phenomenon in GHz fiber laser, is well reproduced by the present model, which further verifies the accuracy of the approach as well as enriches the nonlinear dynamics.
© 2016 Optical Society of America
Lasers those are capable of generating gigahertz-repetition-rate pulse have attracted more and more attention in the past decade. Aside from the direct usage of the light source, e.g. bio-optical imaging , frequency combs achieved by multi-GHz lasers benefits a great number of applications including optical arbitrary waveform generation , high resolution astronomical spectrographs [3,4], precision spectroscopy , high-speed analog-to-digital conversion [6,7], low-noise microwave signal generation  and photonic radars . Despite multi-gigahertz solid-state lasers have demonstrated good performance [10,11], there is no doubt that fiber lasers possess unique advantages such as extremely compact structure, excellent thermal management and stable pulse generation. With the development of the optical fiber technology, the availability of high fundamental pulse repetition rate in fiber lasers becomes possible. Hence, a large number of reports on mode-locked fiber lasers performing operation with GHz repetition rate in 1.0 and 1.5 μm wavelength ranges have sprung up [12–16]. As a rapidly evolving field, GHz fiber lasers will be promising sources which are comparable with the best Ti: sapphire mode-locked lasers in the near future . However, mode-locked fiber lasers in 2 μm with a repetition rate up to gigahertz have merely been brought into sight [18–20]. Recently, with the help of strong optoacoustic interaction in photonic crystal fiber gigahertz-repetition-rate Tm3+-doped fiber laser has been successfully achieved , whereas, fundamental mode locking is always an option. Owing to the experimental difficulties in obtaining high fundamental repetition rate in Tm3+-doped fiber lasers, analysis of the instability inside, e.g. Q-switched mode-locking and pulse splitting, is of great importance.
Aiming at acquiring CW passive mode locking, the relevant theoretical work has been motivated since the early 1970s. After Haus had delicately built up the master equation for the mode-locked oscillator [22,23], the stability issue was able to be depicted in a complete mathematical frame. Subsequently, criterions for CW passive mode locking had been theoretically investigated in detail [24–26]. Due to the growing demand of the gigahertz pulse repetition rate, the concern of Q-switching instabilities raised attentions once again and an all-optical Q-switching limiter has been designed in purpose by means of inducing reverse saturable absorption (RSA) [27,28]. The averaging model as well as the governing equation of the mode-locked fiber laser has been hitherto developed in spite of fast and slow saturable absorber [29–32], which paves the road for extending the available theory in solid-state oscillator to fiber configuration. Therefore, with respect to the multi-GHz fiber lasers, such roadmap to CW mode locking is just in hand.
On the other hand, mode locked fiber lasers as an ideal platform for the fundamental exploration of complex nonlinear dynamics, manifest novel characteristics particularly in the unstable regime. By far, through shifting the fiber lasers to the previously unwanted regime like partially mode locking [33,34], noise-like mode locking  especially in relatively long cavity (typically from several tens of to a hundred of meters), scalar, vector rogue wave [36–38] as well as soliton explosion [39,40] have been experimentally verified. Whether soliton dynamics exhibits exotic behavior in an extremely short fiber oscillator is still remained to be find out, which makes the analysis of instability meaningful also from the theoretical point of view.
In this paper, we attempt to identify different operating regimes in a high repetition rate (from sub-GHz to multi-GHz) fiber laser mode locked by a semiconductor saturable absorber mirrors (SESAM). In theory, the method with inclusion of both numerical and analytical procedures incorporates two distinct approximations: one is slow gain relaxation and fast saturable absorption for analyzing the Q-switched mode locking; the other is slow saturable absorption with steady-state gain for studying the onset of pulsation – a phenomenon that replaces the occurrence of stable multi-pulsing. In experiment, an ultra-concise linear resonator with repetition rate up to 1.6 GHz is employed to display a continuous transition from Q-switched mode locking to pulsating. The experimental results agree rather well with the theoretical prediction, particularly in the pumping interval of the three regimes and pulsating characteristics. Moreover, it is revealed that bifurcation in energy is also realized in the nonlinear dynamical system without direct RSA.
2. Theoretical analysis of the instability within high-repetition-rate fiber lasers
The linear cavity scheme, constituting a segment of gain fiber, a mirror and SESAM used as cavity end-reflectors, is handled equally to a ring configuration when it is modeled. Since the standing-wave effect is neglected, the exploited saturation energies of gain medium (fiber) and SESAM are averaged to Eg = EG/2, Ea = EA/2, as illustrated in Fig. 1. EG and EA are inherent saturation energies of the materials. For convenience, the parameters used in the calculation are shown in Table 1, the corresponding values are rationally evaluated by referring to the data of Tm3+-doped barium gallo-germanate (BGG) glass fiber  and commercial SESAM products.
The relaxation time Tg of the upper level in Tm3+ is always much longer than that of the SEASM as well as the pulsewidth, hence the multiscale analysis is used to demonstrate the gain dynamics. Before specific master equations characterizing intra-cavity soliton dynamics are given, evolvement of gain in slow time scale is required. As is known, the rate equation for gain dynamics reads [22,24]:42], thereby obtaining the form of Eq. (1) varied by slow time τ
2.1 Analysis of the formation of Q-switched mode locking with gain dynamics
2.1.1 Numerical analysis
As to the numerical calculation, the laser configuration is tackled by a lumped scheme. The light field when propagating in the fiber is characterized by 43]). To deal with Eq. (2) in the numerical calculation, the window of the fast time T is chosen to be [-Tr/2, Tr/2]; the step in the domain of slow variable is Tr, namely, τ = i·Tr. In this case Eq. (2) is well fit in the recurrence and is rewritten asEq. (5) greatly enhances the convergence of the numerical computation. The iterative procedure is implemented by .
In the case of Tr = 630 ps (corresponding to a 1.6 GHz repetition rate in the experiment below), without the presence of the gain relaxation a wide range of pump-dependent parameter g0, covering from g0 = 10 to 385, results in robust, stable single pulse operation. With the addition of gain dynamics represented by Eq. (4), it is found in Figs. 2(a) and 2(b) that the initial stationary solutions (unperturbed by the gain dynamics) acquired from g0≤190 begin to undergo growing perturbation in pulse energy. On the other hand, when g0 is above the threshold value of 190, e.g., for g0 = 250, the energy of the initial pulse solution conserves as illustrated in Fig. 2(c). Relevant temporal profile of the pulse is given in Fig. 2(d). By referring to , we can naturally associate this instability with the Q-switched mode locking: the fixed point (initial condition) is the CW mode-locking; the perturbation is induced by rate equation of the gain coefficient g. In analogous to the circumstances in solid-state lasers, the stationary solution goes unstable under a certain pump threshold. Interestingly, the changes of pulse energy exhibit different behaviors in Figs. 2(a) and 2(b), whose profiles are bell-shaped and flat-top, respectively. It might be related with the eigenvalues of the nonlinear system, which is implied by the early stage of energy variation. According to Fig. 2(a), the energy increases exponentially (marked by the dashed curve) to a fairly high level within only 300 roundtrips; by contrast, in Fig. 2(b) energy experiences slowly varying perturbation in the vicinity of the initial stationary solution for almost 8000 roundtrips (shown in the inset).
Calculating threshold for each repetition rate, even in a narrow region, is really a hard work; therefore, we try to plot the threshold curve analytically and subsequently corroborate it by a few numerical points.
2.1.2 Analytical analysis
Proposed by Haus and Kärtner et al. [24,25], the analytical procedure characterizing the stabilization process of the pulse energy in the slow time scale is governed by coupled rate equations as follows:Eq. (6), the Assuming that the energy is perturbed by δ‖u‖, the ordinary differential equations (ODEs) transform to:
Equation (8) physically represents the criterion of CW mode locking. Consequently, the corresponding threshold curve against different roundtrip time Tr (or equivalently the linear cavity length Lc) is analytically achieved, as demonstrated in Fig. 3. It is revealed that, as expected, with the increasing repetition rate the regime of Q-switched mode-locking expands dramatically. Stable CW mode locking would not be reached when the gain of the active fiber (being fully pumped) is insufficient. To intuitively clarify this point, two fictitious fibers: the maximum gain coefficient of the one is 10 times as large as the other, are compared by the critical repetition rate. In consequence, the performance of gigahertz CW mode-locking is easily attained by virtue of the fiber with high gain while Q-switched mode locking turns out to be the only choice for the lower one. Comparison between analytical the numerical results will be given afterwards together with the one of pulsating threshold curves.
2.2 Analysis of the formation of pulsating with absorption dynamics
2.2.1 Numerical analysis
Since the attention here is focused on the absorption dynamics, it is assumed that the system can retain a steady-state condition (i.e. CW mode locking) within a long period. In accordance with the assumption, the stationary solution of g substitutes for Eq. (4) in the lumped model. Hence, the master generalized nonlinear Schrodinger equation is at present in the form ofFig. 4(a). The detailed evolution processes of the pulse in temporal and spectral domain are illustrated in Figs. 4(b) and 4(c), respectively, indicating how the pulse explicitly acts when experiencing energy oscillation. Namely, instead of switching to a multi-pulsing stationary state the attractor of the current nonlinear system evolves to a limit cycle. So far, diverse types of the pulse dynamics have been reported from both theoretical and practical point of view, including pulsation , soliton molecule (or crystal) [45,46], soliton explosion [47,48], and etc.. Previously, pulsation has been predicted in the framework of complex cubic-quintic Ginzburg-Landau equation (CQGLE) and has been experimentally observed in a NPE-based system, indicating the importance of the latent RSA. However, there is no direct over-saturation phenomenon in our configuration (despite the two photon absorption will result in RSA in practice , it is not considered in our present model). Since the pulsation never exhibits when one substitutes Eq. (5) for Eq. (10), it is justified to say that the relaxation process of the slow saturable absorber is the decisive ingredient. In contrast to the gain relaxation aforementioned, the recovery of the absorption limits further saturation of the absorber and imposes an upper limit to the CW mode locking. Likewise, analytical trial of seeking the relationship between pulsating threshold and repetition rate is made prior to the numerical verification.
2.2.2 Analytical analysis
It is obvious that the extensively used approximation of weak absorption is invalid in our circumstance because the saturation intensity Ia = Ea/Ta of the SESAM is rather low for Ia = 10 W. However, as to the slow saturable absorber , the prerequisite for expansion may hold. In the fiber laser, averaging procedure might inevitably introduce a certain error in reconstructing the pulse solution even in the scalar model. Nevertheless, guided by the well-developed iterative method, the master equation characterizing the laser oscillator reads:29,31], we use the variable Z instead of z to avoid confusion.
A subtle ansatz proposed by Akhmediev is used ,Eq. (12) into Eq. (11) and separate the real and imaginary part, the parameters d, C, τ, k, w and v defined in the ansatz are determined,
By plotting the pulse energy ‖u‖ versus pump-dependent parameter g0 in the case of Tr = 630, upper and lower branches of the solution are illustrated in Fig. 5(a). Obviously, the lower branch coincides with the practical situation: increasing in g0 results in the enhanced pulse energy. To validate the ansatz form Eq. (12) in the first place, we compare the analytical solution at the transition point A (corresponding to the maximum value of g0) with the numerical one at the edge of pulsation for g0 = 280, as shown in Fig. 5(b-d). With approximately the same pulse duration and admissible deviation in peak power, the analytical solution is reliable in drawing the pulsating threshold line for different repetition rates. Note that the negative values of the saturable loss in Fig. 5(d) (indicating gain instead of loss) is not surprising so as to guarantee the assumed symmetric hyperbolic- secant solution.
Maximum values of g0 are extracted in terms of Eqs. (13) and (14) in the extent of repetition rate from 0.5 to 2.2 GHz, which define the CW mode locking regime combined with the former threshold curve. Here, the critical values of g0 for Q-switched mode locking (g0p) and pulsation (g0pl) at 0.5 GHz repetition rate is assumed to be equal and uniformly denoted by g0c. Consequently, the analytically defined area for CW mode locking is colored cyan in Fig. 6. In the meantime, the numerically defined one is shown as gray shade, suggesting acceptable result of the analytical approach. Unexpectedly, the assumption g0p = g0pl is indeed the case according to the numerical calculation, which means that the CW mode locking region shrinks with the decreasing repetition rate and almost vanishes at 0.5 GHz.
3. Experimental evidence and discussions
To implement a GHz mode-locked fiber laser in 2.0 μm, a heavily Tm3+-doped BGG fiber and two artificial mirrors with high reflectivity are employed as schematically illustrated in Fig. 7(a). The 5.9 cm long Tm3+-doped BGG gain fiber is sealed in a 125 μm zirconia ferrule and both ends are perpendicularly polished. To ensure adequate energy within the resonator as well as optimal output power, a coated ferrule spliced to the wavelength division multiplexer (WDM) is optimized to be a ~90% reflector and is connected to the gain fiber ferrule in a mating sleeve. The SESAM with the chip area of 1.0 mm × 1.0 mm, used as the other end-reflector, is characterized by the parameters shown in Table 1 and is sandwiched between the gain fiber and another fiber ferrule. The oscillator scheme, with the cavity length equivalent to the gain fiber, is extremely compact and low-loss. A 793 nm laser diode (LD) with maximum power up to 250 mW is exploited as the pump source and can efficiently couple into the resonator through the coated ferrule with <10% loss. The optical spectrum of the output laser is monitored by an optical spectrum analyzer (YOKOGAWA AQ6375). The characteristics of the pulse signal are measured by a 13 GHz bandwidth real-time digital oscilloscope (Keysight DSA91304A) along with a 12.5 GHz photodetector (Newport 818-BB-1F), and a 3 GHz radio spectrum analyzer (Agilent N9320A).
By increasing the pump power, continuous transition from Q-switched mode locking, CW mode locking to pulsation is observed. Typical temporal features of the three regimes are demonstrated in Fig. 7(b). Besides the uniform combs in the oscilloscope for CW mode locking, strong and weak modulations of the pulse train feature the Q-switched mode locking and pulsating. Dependence of the output power on the launched pump power is shown in Fig. 8(a), the CW laser threshold is 46 mW and sudden change in output power implies the shift of the laser operation. In our experiment, CW state is always accompanied by Q-switched envelope in some time intervals, which is named as unstable Q-switched mode locking. With the pumping power, the train of the Q-switched envelopes tends to be stable, dense; and it evolves to a fully regular sequence without any modulation at the pump power of 105 mW. The CW mode locking maintains before the pump power exceeds 130 mW. Further increasing the pump power will disrupt the steady state by triggering weak, periodic modulation of the pulse train. The pump power is kept below 180 mW to prevent thermal damage caused by its residual components. We compare the intervals in pump axis with ones extracted from the numerical analysis towards the values of g0 (i.e. 10-190 for Q-switched mode locking, 190-282 for CW mode locking and 282-385 for pulsation), the results are in good agreement as shown in Fig. 8(a) if the value of gain coefficient g0 is regarded to be proportion to the pump power [51,52].
We have paid extra attention to the pulsating because a consistence between numerical and experimental results in an exotic regime might provide evidence in proving our present model. Numerical results for g0 = 350 are chose to make comparisons with the experimental data. Since optical spectrum analyzer is actually a stable averaged measurement, the relevant simulated curve given in Fig. 8(b) is obtained by averaging one thousand consecutive roundtrips. Despite the sampling rate of the real-time oscilloscope is high (up to 40 GSa/s for 100 ns/div), it is still not sufficient to represent the pulse details. Hence we approximately regard the envelope of the oscilloscope trace as the real-time record of the pulse energies. In Fig. 8(c), evolution of the pulse energy according to numerical simulation is shown as the reproduction of the oscilloscope data. Correspondingly, as seen in Fig. 8(d), the oscillation in oscilloscope is also visualized on the radio spectrum analyzer as sidebands to the frequency at repetition rate. Meanwhile, the simulated RF spectrum is computed by applying Fourier transform to numerically calculated energy oscillation within 1000 roundtrips. The results above, including averaged and real-time measurements, are all in perfect agreements with the theoretical prediction.
For now, the obstacle to the real-time spectral diagnostic of pulsation, taking dispersive Fourier transformation (DFT) technique for example , is to introduce sufficiently large dispersion when fiber delay line is not available for high propagation loss in 1950 μm (~10 dB/ km for SM1950). Besides, high enough sampling rate of the oscilloscope is required to ensure the spectral resolution owing to the short roundtrip time for GHz fiber lasers (e.g. 630 ps in our experiment). Note that shot-to-shot records of the spectra (or temporal profiles) are always preferred if the difficulties can be circumvented. Moreover, since previous studies of pulsation have primarily focused on the parameter space of CQGLE in averaged model and transmission curves of artificial saturable absorbers in discrete model [44,54], the specific effect of absorption dynamics in pulsating formation is worth further exploration.
Based on the proof in Fig. 8, the validity of the theoretical approach can be partially admitted; but still more data achieved from different pulse repetition rates is required to further support the theory. The explicit physical model, characterizing both gain fiber and saturable absorber by exact master equations, is rather difficult to numerically find a stationary solution. The present method is in fact based on a composite approach, which is a compromise between the explicit physical model and useful numerical solutions. That is why it has to be treated deliberately.
In conclusion, instabilities within high-repetition-rate mode-locked fiber lasers have been analyzed both in theory and experiment. Using the theoretical approach, the numerical and analytical analyses are performed to reveal the Q-switching and pulsation instabilities. Theoretical results show that: (1) for the fundamental pulse-repetition-rate varying from 0.5 to 2 GHz, the range of pump power for the Q-switching instability significantly increased, in addition that the CW mode-locking usually requires Tm3+-doped fibers with enough gain; (2) the appearance of pulsation, rather than stable multi-pulsing or associated harmonic mode- locking, is particularly verified even excited at high pump power, in the high-repetition rate oscillators using the slow saturable absorber. Three distinct regimes operation in a passively mode locked fiber laser with 1.6 GHz repetition rate is further experimentally displayed, and meanwhile the relevant intervals of pump are in good agreement with the theoretical prediction. The pulsating of the spectral and temporal domain data reinforce the present theoretical modeling, by indicating the period of pulsation. The analysis on instabilities described here is helpful to developing an ultra-stable Tm3+-doped mode-locked laser sources with a higher pulse-repetition-rate.
China National Funds for Distinguished Young Scientists (61325024), the High-level Personnel Special Support Program of Guangdong Province (2014TX01C087), Fundamental Research Funds for the Central Universities (2015ZP019), the China State 863 Hi-tech Program (2013AA031502 and 2014AA041902), NSFC (51472088, 61535014 and 51302086), the Fund of Guangdong Province Cooperation of Producing, Studying and Researching (2012B091100140), National Key Research and Development Program of China (2016YFB0402204), the Science and Technology Project of Guangdong (2016B090925004).
We thank Qi Qian for providing the platform for numerical calculation, Guowu Tang for useful discussions on fiber materials and Jinzhang Wang for discussions on pulsation and pulse splitting issue.
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