Stability of the similariton mode-locked regime in Yb-doped fiber laser in the vicinity of zero cavity dispersion is studied by means of numerical simulations. It is shown that similariton pulses which initially arise from laser noise collapse into a continuous wave state. The mode-locked pulses are found to be stable after a certain cavity dispersion threshold is exceeded. From analysis of the instability development, we conclude that instability has parametric nature. We compare our results with stability analysis based on the Ginzburg-Landau approach. Analogies with instabilities found in the long-haul fiber communication systems are also discussed.
©2007 Optical Society of America
Similariton mode-locked regime has attracted a lot of attention due to its potential of generating femtosecond pulses with energy of microjoule level directly from a fiber laser cavity . Stability of similariton pulse relies on a property of a pulse of parabolic shape to propagate self-similarly in a nonlinear fiber with positive group velocity dispersion (GVD) . In the case of laser cavity, however, the self-similarity is strongly perturbed by the self-amplitude modulation as well as by the significant losses associated with light output from the cavity. Nevertheless, dominating role of self-similar mechanism in mode-locked pulse formation has made the use of the term “similariton“ well established [1, 3–8].
Historically, stability of mode-locked pulses has been studied in the framework of the Haus master model [9, 10] followed by development of more general versions of the Ginzburg-Landau (GL) equation [11–14]. Though validity of the GL approach seems to be questionable for the case of similariton dynamics, recent studies of the heavily-chirped mode-locked pulses in the normal dispersion domain have revealed striking resemblance of similaritons with the solutions of the cubic-quintic complex GL equation [13, 14]. In particular, in both cases the pulse stability is guaranteed by the strongly pronounced chirp while the spectrum demonstrates profile with truncated wings. Another common feature is the loss of stability when the cavity dispersion decreases as it has been demonstrated by very recent numerical studies of similaritons . The latter observation seems to be especially important because the shortest mode-locked pulses are achievable in the vicinity of zero cavity GVD. Although similariton dynamics in mode-locked lasers has been extensively studied, instability mechanism close to zero dispersion point has not been investigated in depth. Moreover, higher order dispersion has not been properly considered.
In the present work we numerically study the instability which prevents the generation of stable similariton pulses in the vicinity of zero cavity dispersion. We find that the similariton pulses, which initially establish from laser noise, are eventually unstable versus cw-regime once cavity dispersion is below a certain threshold value. We present details of the instability development and conclude that the instability is of parametric nature. We draw parallels with the mode-locked pulse instabilities demonstrated in the positive cavity dispersion domain in the framework of the Ginzburg-Landau equation. The analogy with the parametric instability known in long-distance optical fiber communication links is also discussed.
2. Laser model
We model Yb-doped fiber laser in ring cavity configuration schematically shown in Fig. 1. The cavity consists of relatively short section (30 cm) of Yb-doped fiber for light amplification, the single mode fiber (SMF) of length 2.5 m incorporating a WDM coupler, dispersion delay line typically using a diffraction grating, polarization beam splitter for light output from cavity as well as for providing mode-locking action through nonlinear polarization evolution mechanism, and wave plate for controlling light polarization. Light propagation in Yb-doped fiber is described by the equation for the field amplitude, A :
whereas for the SMF fiber the last two terms describing respectively gain and loss effects are omitted. In Eq. (1) β2 is the GVD parameter taken as 24 fs2/mm, β3 is the third order dispersion (TOD) parameter used as 30 fs3/mm, γ is the nonlinearity parameter equal to 0.005 W-1/m for both fibers, the gain parameter g(E) has saturated dependence  on the pulse energy Epulse: g=g0/[l+ Epulse/Esat] with the small signal gain g 0 equal to 6 m-1 and the saturation energy Esat equal to 3 nJ, the spectral width of the gain Ωg corresponds to 30 nm, and the linear loss is l=0.04 m-1. Mode-locking mechanism associated with the nonlinear polarization evolution is modeled as an intensity dependent transmission T=l-l 0/[l + |A|2/Psat, where l 0=0.5 is the unsaturated loss, Psat=12kW is the saturation power. A standard split-step Fourier algorithm has been used in simulations with a white noise as initial field.
3. Results and discussion
In simulations we find that once positive cavity dispersion exceeds certain value which is 0.004 ps2 for our particular laser parameters, the pulses with specific similariton features arise from noise. Let us consider first the similariton dynamics for the case of zero TOD and then the role of TOD will be elucidated. In Fig. 2(a) the power distribution of the similariton pulse taken close to the end of SMF section is presented and its spectrum is shown in Fig. 2(b) at cavity dispersion equal to 0.007 ps2. The parabolic fit shown in Fig. 2(a) by the dashed line assumes that the pulse demonstrates self-similar propagation features. The spectrum with the parabolic top and truncated wings also corresponds to the characteristics of the similariton [1,2]. It has been found in simulation that the similariton pulse becomes unstable with time and the cw component starts growing.
The snapshots of the pulse profile and the spectrum during instability development are shown in Figs. 2(c) and 2(d) respectively. The cw component is quite pronounced in Fig. 2(c) and the ripples arise on the pulse wings with oscillation frequency increasing further from the pulse center. It is instructive to consider the development of the instability in the frequency domain. In Fig. 2(d) the top part of the spectrum is shown at the moment corresponding to initial stage of instability. The whole spectrum is similar to the one presented in Fig. 2(b). Monitoring the spectrum we see that at first the zero frequency component becomes pronounced and then other low frequency harmonics become visible forming the ripple pattern similar to the one observed in time domain in Fig. 2(c). Finally after the instability is fully developed, the cw state with some phase defects establishes.
The transient dynamics of the similariton pulses at different cavity dispersion is presented in Fig. 3(a) while in Fig. 3(b) the energy of the mode-locked pulses is shown as a function of the total cavity dispersion with stable “S” and unstable “U” regions. The solid line in Fig. 3(a) corresponds to the transient similariton with energy close to 3.5 nJ obtained at a total cavity dispersion of 0.004 ps2. This is the lowest energy similariton observed in simulations. It is rather short-lived as the pulse collapses, and then the cw-state with energy Ecw=4.65 nJ is built up beyond about 80 cavity roundtrips.
The similariton with energy of 4 nJ, obtained at cavity dispersion around 0.007 ps2 lives about 200 roundtrips as illustrated by the dash-dotted line in Fig. 3(a). Afterwards, the switch to the cw state occurs. Finally, the dashed curve in Fig. 3(a) corresponds to the similariton circulating about 1000 roundtrips in the cavity then it collapses to the same cw state. It should be noted that instability develops faster for lower dispersion, i.e., the switching to the upper cw state is slower for the dashed curve than for the solid curve in Fig. 3(a).
It is depicted in Fig. 3(b) that as the dispersion increases, the energy of the similariton grows and saturates when approaching the cw state energy. Such saturation behavior with increasing dispersion is known from the early analytical studies of the sech-type solution for the mode-locked pulses in fiber lasers . In our simulations we observe that after some dispersion value depicted by the vertical dash-dotted line in Fig. 3(b), the mode-locking stabilizes and the pulse circulates for several tens thousands of roundtrips without any changes. We consider this as stable mode-locked state. Let us note that instability threshold for mode-locked pulses in the positive cavity dispersion range has been predicted by the early theoretical studies on basis of the Haus master model [9, 10] as well as by recent analysis of the heavily chirped pulses as solutions of the cubic-quintic GL equation .
When we trace the cw component amplitude with time, we observe dynamics illustrated in Fig. 4. Here the instability is presented at early stage, Fig. 4(a) and at further developed stage, Fig. 4(b). At the early stage we see multi-period pulsations, meaning that the initial period corresponds to the cavity roundtrip time. In the case of Fig. 4(a), at the early stage of instability development laser dynamics repeats after 4 roundtrips. It is seen from Fig. 4 that while growing in general, the cw component demonstrates a drop in intensity during some roundtrips. This implies that different frequency components nonlinearly interact in the way that energy redistributes over the spectrum.
Instabilities and dynamical chaos are well known in passive and active nonlinear optical cavities , including mode-locked fiber lasers . In our case, in particular, the instability arises in already pulsating laser cavity and multi-period pulsing corresponds to excitation of the frequency by its multiplier which points out to the parametric nature of instability. Similar parametric instabilities have been studied in the context of long-distance optical transmission systems [18, 19] where the signal perturbation by the periodically placed optical amplifiers corresponds to repeated transmission of light through the same Yb-doped fiber in the case of our laser cavity.
When the third order dispersion is included into consideration (β 3=30 fs3/mm for both SMF and Yb-doped fiber) the instability development does not change qualitatively. In Fig. 5, the detailed picture is presented for one roundtrip. Namely, instability development for the parameters of Fig. 1 is shown while pulse propagates through the SMF and Yb-doped fibers. The starting pulse profile in SMF in Fig. 5(a) corresponds to the upper spectrum in Fig. 5(b), while the pulse at the output from Yb-doped fiber is characterized by the lower spectrum in Fig. 5(b). It is seen that TOD brings asymmetry into temporal and spectral profiles, now more pronounced ripples develop on the leading front of the pulse. It should be noted that the normalized spectra in Fig. 5(b) all have maxima at zero frequency so that from SMF to Yb-doped fiber the zero component grows, whereas the amplitude modulation action of the polarizer suppresses the cw-component. Increasing TOD shifts instability boundary in the domain of larger cavity GVD like it has been discussed in the context of the transition between the stretched pulse and similariton-cubicon regimes for cavities containing photonic bandgap fibers .
It is worth also to emphasize the difference of the similariton instability considered here from the sideband instability  found for the case of the stretched-pulse regime in the region of positive cavity dispersion . Only parametric mechanism can provide gain for cw component as it has been discussed by Matera et al , while the sideband generation is always taking place at the frequencies with the offset from the spectrum center.
In summary, we have demonstrated parametric instability of the similariton mode-locked pulses in the vicinity of zero cavity dispersion by using relatively simple model of ring cavity fiber laser. The generic features of parametric instability persist when third order dispersion is included into consideration or/and the pulse shape deviates from similariton parabolic profile at variation of the model parameters. Our numerical data are in qualitative agreement with the analytical results obtained in the framework of the Ginzburg-Landau approach.
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