Bifurcations and spectral pulsations in ultrafast fiber lasers

. We employ real-time spectral measurements to investigate the transitions from the stable mode locking to period doubling and long-period pulsations. We highlight the role of self-phase modulation as a general mechanism triggering period-2 bifurcations. We analyze the new frequencies generated during the sequence of bifurcations and report an entrainment where the period of the long oscillation locks to multiples of the cavity roundtrip time.


Period-doubling as an initiating bifurcation
Ultrafast lasers are nonlinear dissipative oscillators with a virtually infinite number of degrees of freedom -namely, the field amplitude values as a function of time.They make a convenient experimental platform for studying period doubling as well as more complex bifurcations.Following a bifurcation that generates a periodic oscillation of the pulse parameters when monitored at a fixed location -such as the laser output port -the laser dynamics is qualified as a pulsating dissipative soliton regime.Pulsating solitons can involve short-period pulsations (SPP) whose periodicity remains comparable to the cavity roundtrip time, long-period pulsations (LPP), or even multiple-period pulsations that combine both (1) .
In this communication, we first highlight the role of the period-2 bifurcation as a universal warning signal of the pulse instability, investigating the onset of the bifurcation in several fiber laser cavities.The latter include fiber lasers operating at different average chromatic dispersion (either anomalous or normal) and different operating wavelengths (in the 1.5-micron or in the 1.9-micron regions).These observations are performed in real time in the spectral domain, using the time-stretch dispersive Fourier transform (DFT) method.Such real-time spectral monitoring allows an early detection of the instability, which could otherwise remain invisible if we merely recorded the pulse energy over successive cavity roundtrips (2) .We also compare our experimental observations with numerical simulations of the intracavity dynamics.One example of period-2 spectral pulsations of a dissipative soliton circulating round a fiber ring laser is featured in Fig. 1, corresponding here to an anomalous dispersion cavity and central wavelength around 1567 nm.During the conference, we will elaborate on similarities and differences reported when the laser parameters are varied, with an overall emphasis on the universal features of the period-2 bifurcations observed.

Long-period pulsations and entrainment phenomena
To understand the bifurcation cascades involved in the case of a larger parameter excursion, we record the transient dynamics of the bifurcation sequences starting from a single-period soliton and leading to period-N solitons, by ramping up (linearly with time) the pump power from 70 mW to 90 mW. Figure 2 (a) represents the DFT evolution map over 15000 cavity roundtrips as the pump power is increased, showing that oscillations quickly appear.The evolution of the intensity in the oscillating part of the spectrum at λi, namely the central wavelength, is plotted as a function of the roundtrip number in Fig. 2 (b).The first bifurcation after 4000 roundtrips leads to the period-2 pulsation, followed by a second bifurcation at 8000 roundtrips corresponding to the addition of a long-period modulation.Interestingly, between 10500 and 12500 roundtrips, a structure emerges in the oscillation pattern, revealing the organization of the oscillating patterns into an integer periodicity.For a better visualization, one out of every three data points of the same acquisition has been plotted in red, with 4 branches appearing clearly, hence an exact 12-periodicity.This is the signature of an entrainment of the pulsation to an integer multiple of the cavity roundtrip period.To follow more accurately the evolution of this modulation, a spectrogram of the intensity at λi is shown on Fig. 2 (c): for each round-trip, we calculate the Fourier transform of the next 1024 points and plot the evolution of the spectral density (in log scale) as the round-trip number increases.

Conclusions
We observed and simulated period-2 spectral pulsations in ultrafast fiber lasers under different dispersion regimes, confirming its universality.The period-2 spectral dynamics primarily manifest as peak-to-dip (dip-to-peak) transformation of the central intensity of the spectrum in the frequency domain, which is initiated by an excess of the Kerr nonlinearity.We have highlighted how the period-2 bifurcation can be followed by period-N (N>2) bifurcations, leading to doubly periodic pulsations.During the bifurcation cascade, we have observed experimentally and corroborated numerically several illustrations of the entrainment of the pulsation period to an integer multiple of the cavity roundtrip period.Whereas the self-synchronization of periodic pulsations was predicted quite a few years ago (1) , our experimental observations bring unprecedented information about spectral dynamics and confirm the universal reach of synchronization phenomena (5) .This echoes recent observations of the subharmonic entrainment of longperiod pulsations, also called breather solitons, in both driven Kerr microresonators and ultrafast lasers (3,4) .

Fig. 1 .
Fig. 1.Period doubling of dissipative solitons in an anomalous dispersion ultrafast fiber laser.(a) Time-averaged spectrum (blue curve) and DFT spectra for two successive roundtrips.(b) Time-averaged second-order autocorrelation (AC) trace of the pulse train (blue curve) and first-order AC trace obtained by Fourier transforming the single-shot DFT spectra.(c) Evolution map of the DFT measurement showing the stable period-2 spectral oscillations.(d) Evolution with the round-trip number of the intensity of the pulse (thin black) and of the 1567 nm component using the DFT measurements.The oscillations of the intensity at 1567 nm are stable all over the long recording time (> 4000 roundtrips).From Ref. 5.

Fig. 2 .
Fig. 2. Period doubling of dissipative solitons in an anomalous dispersion ultrafast fiber laser.(a) Time-averaged spectrum (blue curve) and DFT spectra for two successive roundtrips.(b) Time-averaged second-order autocorrelation (AC) trace of the pulse train (blue curve) and first-order AC trace obtained by Fourier transforming the single-shot DFT spectra.(c) Evolution map of the DFT measurement showing the stable period-2 spectral oscillations.(d) Evolution with the round-trip number of the intensity of the pulse (thin black) and of the 1567 nm component using the DFT measurements.The oscillations of the intensity at 1567 nm are stable all over the long recording time (> 4000 roundtrips).Adapted from Ref. 5.