Mode-locking via dissipative Faraday instability

Emergence of coherent structures and patterns at the nonlinear stage of modulation instability of a uniform state is an inherent feature of many biological, physical and engineering systems. There are several well-studied classical modulation instabilities, such as Benjamin–Feir, Turing and Faraday instability, which play a critical role in the self-organization of energy and matter in non-equilibrium physical, chemical and biological systems. Here we experimentally demonstrate the dissipative Faraday instability induced by spatially periodic zig-zag modulation of a dissipative parameter of the system—spectrally dependent losses—achieving generation of temporal patterns and high-harmonic mode-locking in a fibre laser. We demonstrate features of this instability that distinguish it from both the Benjamin–Feir and the purely dispersive Faraday instability. Our results open the possibilities for new designs of mode-locked lasers and can be extended to other fields of physics and engineering.

ey were spectrally shi ed by approximately . nm and stabilised by Peltier elements. Pump radiation at nm was coupled into the cavity with the help of wavelength division multiplexing coupler (WDM). Optical spectrum and intensity dynamics were monitored from the % rejection port immediately a er re ection from the output grating.
Fast GHz DC-coupled photodetector and GHz real-time oscilloscope were used to register the temporal intensity dynamics. Optical spectra were measured using . nm resolution bandwidth optical spectrum analyser Yokogawa AQ C. Intensity autocorrelator Femtochrome -XL was used for auto-correlation function (ACF) measurements. RF spectra were obtained with . GHz electrical spectrum analyser.
With the gratings spectral shi set by the Peltrier elements, the laser readily mode-locks as soon as the pump level is above the lasing threshold of . W, and operates stably up to the pump powers . W. e detuning of the gratings a ects both the pulse repetition rate and pulse timing jitter, which agrees well with the prediction given by the linear stability analysis and numerical simulations.
Other combinations of normal dispersion bres, such as OFS IDF, and FBGs with di erent width and chromatic dispersion values were also studied both numerically and experimentally, with the results that agree well with stability analysis predictions.
To provide another reference point, the radiation coming from the cavity was registered at the point before the output grating. As expected, the pulses had close to parabolic shape Supplementary Figure a, with a characteristic dip in the middle, which, again, was reproduced in the numerical simulations. e e ect of spectral broadening that pulses experience during the propagation in the cavity is demonstrated on the Supplementary Figure b, where the spectrum of laser radiation coming from the cavity, and incident on the FBGs (see Supplementary  Figure ), is shown together with the spectrum of the optical pulses registered immediately a er re ection by the FBG. Depending on the pump power, the spectrum can broaden by a factor > , maintaining nearly a at top pro le, typical for parabolic pulses.

Supplementary Note The scaling of the Faraday Instability
In order to provide an estimation for the functional dependence of the maximally growing mode in the dissipative Faraday instability, we considered the nonlinear Schrödinger equation for the electric eld envelope ψ propagating in a bre with normal group velocity dispersion β > and Kerr nonlinearity γ > : ( ) e Bogoliubov modes, oscillatory perturbations on top of the homogeneous eld background ψ , are stable for focussing non-linearity and normal group velocity dispersion and obey the following dispersion relation: In presence of the periodic forcing of a system parameter with spatial frequency k, we expect the Bogoliubov modes having wavenumber which is an integer multiple of half of the forcing frequency, to synchronize with the forcing and hence to be excited. From the parametric resonance condition the rst excited temporal mode has frequency ω which satis es the dispersion relation ω(k ). For a dissipation modulation having spatial period equal to Λ, we have k = π Λ and with help of Supplementary Equation in the long wavelength limit we obtain: Supplementary Equation shows that the instability frequency depends on the inverse of the squared root of the eld intensity and provides a phenomenological scaling formula for the systems with parametric modulations. e Raman laser used in our experiment is described by a much more complicated model based on coupled generalised nonlinear Schrödinger equations, where dissipation plays a key role and the in uence of cross-phase modulation, spatial nonuniformity of the gain and group velocity mismatch between pump and Stokes elds are not negligible. Nevertheless, Supplementary Equation provides the basic functional dependence on the laser pump power. Such a scaling can be obtained for dissipative systems described by the Complex Ginzburg-Landau Equation as well [ ].

Supplementary Note Instability diagrams and dispersion management
We stress the fact that the dispersive Faraday instability does not introduce substantial modi cations to the main tongue of the instability spectrum. In the case of pure dissipative modulation (unchirped gratings) the instability spectrum Supplementary It is important to comment that, even though the periodicity of the dissipation and of the pump depletion induced periodic power variation are the same, the tongue induced by the zig-zag modulation of the dissipation has a di erent and characteristic shape "attached" to the "zero" frequency as it is clearly depicted in Figure a of the main paper and in Supplementary Figure a; and it is the only one that can explain the periodic pulsation observed in the system's dynamics.
We emphasize what already mentioned in the main paper: dispersion management is necessary in the experiment in order to stabilise the pulse train that otherwise will be less regular and not easily tunable. Such an extra degree of freedom makes the systems much more exible and versatile and provides a key tool for mode-locked lasers design. e compensation of the accumulated dispersion by the gratings also provides a convenient mechanism of pulse compression before the pulses are outcoupled from the cavity. e impact of the dispersion modulation on the instability frequency and pulse repetition rate, however, is not very large. On the Supplementary Figure the frequency scaling for two di erent sets of gratings is shown. e bre dispersion at the Stokes wavelength was the same in both cases β s = . ps , while the dispersion of the gratings di ered by a factor of .
We have checked both numerically and experimentally that the purely dispersive Faraday instability was not able to excite any regular pulsation in our system most likely due to the highly noisy nature of the Raman laser, and the absence of a suitable saturable absorber or other pulse reshaping mechanism.