# Relative Intensity Noise of Hybrid Mode-Locked Bound Soliton Fiber Laser: Theory and Experiment
## 3 Results
The normalization unit of the pole energy is 10 pJ, which is estimated from the experimental results. The group velocity dispersion parameter \(d_{i}=0.4\) is estimated from the experimental condition. We set the SPM parameter at \(k_{i}=0.7\) according to the soliton assumption, and finely adjust it to make the numerical and experimental results better matched. The saturable absorption parameter \(k_{r}\) is set to 0 because there was no passive saturable absorption in our laser due to the sigma-type cavity. The phase modulation frequency is set around 10 GHz, with a modulation depth around 1. The cavity linear gain is set to be variable from \(g_{0}=2.5\) to 5 to accord with the lower and higher power level cases according to the relation \(\int\lvert u_{0}\rvert^{2}dt\approx(g_{0}/l_{0}-1)E_{s}\)[25]. The noise enhancement factor \(\theta\) is set to be 10. Other parameters are also reasonably determined so that the simulation results can be closely matched with the experimental results. The initial condition of the single-soliton case is a symmetrically single secant-shape pulse, and the initial condition of the bound-soliton case is two anti-symmetrical secant-shape pulses with \(\pi\) phase difference. The total propagation distance in the simulation is 1000 roundtrips with the step size \(dT=0.005\). The small-scale time step is \(dt=0.01\) in the normalization units, and the total simulation window is 25 ps. We have checked that these step sizes are small enough to ensure the accuracy of the calculation. Figure 4 shows the steady state propagation solutions of the single- and bound-soliton states in the case of \(g_{0}=2.5\) and 5. The pulse width for the single-soliton is around 2.41 ps, while the pulse width and timing separation of the bound-soliton state are around 1.72 and 4.58 ps. We gradually increase \(g_{0}\) from 2.5 to 5 to investigate the pulse properties under different pulse energies. Figure 5a indicates  
Figure 3: Measured intensity fluctuations of (**a**) single-soliton state with output averaged power of 25 mW; (**b**) bound-soliton state with output averaged power of 55 mW. The span range is from 1 kHz to 500 kHz. For direct comparison, the measurement results for both states are performed with the same averaged optical power incident on the photo-detector through the use of an optical attenuator. the simulated pulse energy for both states versus \(g_{0}\). Basically, a linear relation can be obtained to match with the relation \(\int\abs{u}^{2}dt\approx(g_{0}/l_{0}-1)E_{s}\). Figure 5b is the pulse width and bound-pulse timing separation as a function of \(g_{0}\). The pulse width and timing separation are both decreased when \(g_{0}\) is increased, because of larger nonlinearity. The pulse width reduction is extremely sensitive in the single-soliton state; this is because the single-soliton is with a larger peak power to induce larger nonlinearity. Figure 5c shows the required steady-state lasing gain for the considered mode-locking states by analyzing the Master equation as an eigenvalue problem [25; 36]. The negative real part of the solved eigenvalue exactly corresponds to the required steady-state lasing gain for the corresponding mode-locking state in the presence of all the cavity effects. Physically, the state with the minimum steady-state lasing gain is the most stable solution that will emerge as the final steady state of the system. This is because once the state emerges, other states will see net losses, and thus, decay away. This is why the steady state lasing gain can be used as a meaningful physical quantity for determining the stability of the lasing states. Nevertheless, if there are several states with enough close steady-state lasing gains, the perturbations in the laser can cause state jumping, and the differential losses may be too small to make the perturbations decay quickly. Under this situation the laser will become unstable.