Mode-locking via dissipative Faraday instability

Abstract

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.

Introduction

Understanding the mechanisms underlying generation of coherent structures from noise or uniform field distribution is a fundamental problem of nonlinear science and its numerous practical applications ranging from biology to astrophysics. Formation of coherent patterns originates in nonlinear instabilities that can often be described within generic mathematical models1, and, as a result, are very similar across different fields of science. Modulation instability is responsible for the symmetry breaking of homogeneous spatio-temporal states or wave envelopes and the formation of stable patterns in a variety of physical media. There are three major and well-known classes of instabilities. First, Benjamin–Feir (BF) instability, originally introduced in fluid dynamics2,3, and later demonstrated in a variety of physical systems2,4,5. Second, Turing instability6, where the combined action of local self-enhancement and lateral inhibition due to the interplay between nonlinearity and diffusion in coupled equations leads to the pattern formation in chemical, biological systems7,8, as well as in nonlinear optics where diffraction or dispersion substitute diffusion9,10,11. Third and finally, Faraday instability, which results from the periodic in time modulation of a dispersive parameter of the system12 and was studied in different systems, ranging from vertically shaken granular media13 to periodically driven spatially extended chemical systems14, repulsive (defocusing-type) Bose–Einstein condensates15,16 and nonlinear optics17,18. Recently, a new dissipative type of Faraday instability was demonstrated theoretically19, in the framework of the complex Ginzburg–Landau equation, where a suitable parametric modulation of spectral losses can lead to pattern formation. The dissipative Faraday instability, induced by periodic modulation of spectral losses, differs substantially from the usual Faraday instability where a dispersive parameter, diffraction, dispersion or nonlinearity, is periodically modulated. Although the traditional Faraday instability can also occur in a dissipative system, such as an externally driven optical resonator, as it has been predicted theoretically18,20, and recently demonstrated experimentally21, both the excitation method and the sidebands growth process differ from the case where a dissipative element is modulated.

In nonlinear science, instability of a uniform state in non-equilibrium systems triggers a transition to new states with a rich variety of spatio-temporal structures. In engineering, instabilities are often associated with somewhat undesirable effects and problems that ‘should be avoided’. However, they can also play a constructive role in technology, defining device design and control methods in non-equilibrium systems. For instance, instabilities might be important for seeding, enhancing certain frequencies, dumping others and leading to the formation of stable patterns, with the characteristics of emerging structures determined at the nonlinear stage18,20,22,23.

There is a great practical demand for devices and physical mechanisms, which break the symmetry of the uniform or continuous wave state of the laser radiation leading to the formation of temporal structures—optical pulses. Usually, in laser systems, the symmetry breaking is achieved through modulation instability22,23, the introduction of a modulator or a saturable absorber. The latter can be either material, such as for instance carbon nanotubes24 and SESAM25, or an effective one based on physical propagation effects, such as, for example, nonlinear polarization evolution26, Kerr lens27, nonlinear optical loop mirror28 and others. The demand for a new controllable and stable all fibre mode-locking mechanism is driven by the field of high power mode-locked fibre lasers, where it has a great practical value.

Here we experimentally demonstrate the recently theoretically predicted dissipative Faraday instability19 in a simple configuration Raman fibre laser. The induced instability leads to high-order harmonic mode-locking with tunable repetition rate and high environmental stability. The experimental results are in a good agreement with theoretical predictions and numerical simulations.

Results

Dissipative Faraday instability

The instability is initiated by the introduction of a periodic spatial modulation of a dissipative parameter of the system (Fig. 1a). When parameters of a system are modulated with the longitudinal period Λ, corresponding to the spatial frequency k=2π/Λ, then the Faraday instability is initiated, with the first unstable mode oscillating with the wavenumber k/2, that is, the double period. The corresponding pattern forms in the temporal domain, with the characteristic frequency ωrelated to the wavenumber k/2, via the nonlinear dispersion relationω(k). In the proposed configuration, the light travelling in the cavity experiences periodic spectral losses after reflection from the spectrally shifted mirrors at each end of the cavity (zig-zag-like spectral filtering). The reflection profiles of mirrors were shifted in spectral domain by +Δω and −Δω.

Figure 1: Dissipative Faraday instability in a fibre laser.

(a) The light propagating in a linear laser cavity experiences periodic modulation of group velocity dispersion and spectrally dependent losses. In the particular example of dispersion modulation, the normal dispersion accumulated along the propagation over the fibre length is partially compensated at the cavity mirrors. The zig-zag spatial modulation of the dissipation with spatial frequency k=2π/Λ, where Λ=2L, excites the dissipative Faraday instability. The instability frequency is related to half of the spatial forcing frequency, k/2 (parametric resonance condition), via the dispersion relation ω(k). (b) The Faraday instability gain developed in the system couples the phases of each optical cavity mode ωnand cavity modes separated by the instability frequency f=ω(k)/2π. (c) Coupling of modes separated by frequency f leads to the harmonic mode-locking and pattern or pulse train formation in the temporal domain. At the later stages, the shape of the pulses is defined by the combination of self-similar propagation and spectral filtering. FBG, fibre Bragg gratings.