Laser turbulence is a phenomenon caused by nonlinear interactions between millions of longitudinal modes in the cavity, which are far from the equilibrium state. The presence of turbulence can significantly alter the coherence and statistical properties of the laser, thus hindering the practical application of high-performance fiber lasers. As a typical nonlinear system, the continuous-wave fiber laser serves as an excellent platform for studying the laser turbulence phenomenon. Many phenomenological statistical methods have been applied to study laser turbulence. However, most of these methods use only a single approach. In this paper, we employ simulation experiments to demonstrate the differences between the laminar and turbulent states and analyze the effects of pump power, cavity length, and reflection bandwidth of the fiber Bragg grating (FBG) on the transition between these states in the cavity. Our simulation results provide important insights for better controlling laser turbulence and optimizing the application of high-performance Raman lasers.

The optical turbulence of the continuous-wave Raman fiber laser is modeled using the coupled nonlinear Schr?dinger equations for the pump and Stokes waves. First, by varying the pump power, we observe the transition from laminar to turbulent states in our simulations. The differences between these states are characterized by the output energy, the auto-correlation function, the intensity probability distribution, and the phase evolution. After analyzing a large dataset from simulations, we define the criteria for the laminar state as an auto-correlation background value greater than 0.94 and a spectral width of the laser less than 1/5 of the FBG’s reflection bandwidth. All other cases are considered turbulent states. Next, using the control variables method, we examine how pump power, cavity length, and FBG reflection bandwidth affect the laser’s operating state in the cavity, revealing key details of the laminar-to-turbulent transition process.

By comparing and analyzing the differences between the laminar and turbulent states, we observe that the turbulent state exhibits unstable power distribution, weak time-domain coherence, and a probability density function close to a Gaussian distribution (Fig. 2). The spectral width of the turbulent state is significantly wider than that of the laminar state. As the pump power increases, the originally stable dark soliton structures are destroyed, and the creation and disappearance of unstable dark solitons give rise to a macroscopic turbulent structure (Fig. 4). By adjusting the cavity length of the resonant cavity, we find that shorter cavity lengths favor the generation of turbulence, while longer cavity lengths favor laminar states. However, overly long cavity lengths introduce greater background noise. In addition, increasing the cavity length results in significant undulations in the background of the dark soliton (Fig. 5). Adjusting the FBG reflection bandwidth reveals that too large a bandwidth reduces the modulation effect on the resonant cavity and introduces more spectral noise, while too small a bandwidth filters out the intrinsic modes of the resonant cavity, both of which lead to the formation of turbulence (Fig. 6).

In this paper, we provide mathematical conditions for delineating the laminar state and investigate the effects of pump power, cavity length, and FBG reflection bandwidth on the laminar-to-turbulent transition in the laser cavity. The results show that the realization of the laminar state is the result of the combined efforts of pump power, resonant cavity length, and FBG reflection bandwidth in maintaining equilibrium within the cavity. Our simulations demonstrate that larger pump power and shorter cavity length make the nonlinear effects in the cavity dominant, reducing the coherence of the laser and causing the transition from laminar to turbulent states. Only a very narrow FBG reflection bandwidth allows the laser system to maintain the laminar state, as the laser mode must strictly meet the requirements for steady laminar state evolution. To maintain the laminar state without disruption, factors such as low pump power, a resonant cavity length greater than the dispersion length, and a suitably narrow FBG reflection bandwidth are crucial. Our simulation results offer theoretical guidance for experimental studies on turbulence in fiber Raman lasers and contribute to improving the theory of laser turbulence. Lasers in the laminar state exhibit ultra-high time-domain coherence and extremely narrow spectral width, which may provide a reference for the future development of a new generation of high-performance fiber lasers.