Simulation setup

Numerical modeling of laser dynamics serves two primary objectives. First, it provides valuable insights into the physics and operational dynamics of the lasers. Second, it offers a straightforward and rapid tool for investigating various operating regimes. Due to the large number of variable parameters and the complexity of laser systems, it is virtually impossible to experimentally explore the full parameter space. Therefore, simulations are indispensable. Successful computations necessitate a model that accurately captures the complex dynamics of realistic experiments with minimal approximations. In this paper, we employed the nonlinear Schrödinger equation to describe pulse propagation through each fiber segment:

here, z and T are the propagation distance and local time; γ is the Kerr nonlinear parameter; α represents the linear loss. Ω is the 3 dB bandwidth of the gain fiber (doped-fiber), g=g0exp(−∫(|A|2)dtEsat) is the gain of fibers, where g₀ is the small-signal gain, which is taken to be non-zero only in the intra-cavity gain fiber (doped-fiber), and the saturation energy E_sat can be adjusted to simulate changes in the pump power and the intra-cavity loss. Considering the finite gain bandwidth of EDF, we added a Lorentzian profile filter with a bandwidth of 25 nm to the gain model. The saturable absorber is represented by the transmission function of the intensity T=α0−α/(1+|A|2Psat), where α₀ is saturation absorption, α presents the modulation depth of a saturable absorber, and P_sat is saturable power. The simulation parameters are consistent with their experimental values, that is, 0.45, 0.17, and 25 W. In the simulation, the laser initially propagates through the pigtail fiber (HI1060) of WDM. Subsequently, the intra-cavity solitons are amplified by the 1.2 m EDF due to its saturable amplification property and exhibit an almost linear increase in the initial part, while the enlargement rate slows down in the latter part. Following the EDF, the solitons traverse through the SMF28e and are outputted by a 50% output coupler. Then, the solitons propagate further through the circulator and the SESAM, where the duration and intensity decrease due to the saturable absorption effect. In the final section, spectral pulse shaping is modeled by multiplying the electric field by a phase following the expression in Eq. (4) in the spectrum domain to realize high-order dispersion management.