# Relative Intensity Noise of Hybrid Mode-Locked Bound Soliton Fiber Laser: Theory and Experiment
## 3 Results  
The typical output characteristics of the experimental fiber laser are summarized in Table 1 and illustrated in Figure 2. The laser can be operated under the single-soliton state at lower power levels, and under the bound-soliton state at higher power levels. In the middle of the transition, the laser is not very stable, mainly due to the co-existence of the two states. The measured pulse widths are in the ps level for both states, and the maximum output averaged power is near 60 mW. Figure 2a shows the auto-correlation trace for the single-soliton state. The red dashed line is the fitting curve with an assumed hyperbolic secant pulse shape. The insets of the sub-figures indicate the corresponding optical spectra. In the inset of Figure 2b, one can observe an interference dip located at the center of the spectrum, which indicates that the two bound-pulses are with \(\pi\) phase difference. Figure 2c,d show the RF spectral measurement results through a high speed photo-detector for both the single- and bound-soliton states. The repetition rate is around 10 GHz and the laser is with a good super mode suppression ratio (SMSR). Figure 3 shows the measurement data for the laser intensity noise spectra with a span from 1 kHz to 500 kHz, and a resolution bandwidth of 10 Hz. In the measurement range from 1 kHz to 500 kHz, one can avoid the dominant pump-induced amplitude noises near direct current (DC) so as to more clearly observe the intrinsic pulse intensity fluctuations of the mode-locked fiber laser [13; 14; 15]. The measurement results approach the background floor at frequencies higher than 500 kHz. One can observe a flat intensity noise spectrum for both states from 1 kHz to the turning point near 100 kHz. The noise of the bound-soliton state is lower than that of the single-soliton state. The averaged noise power levels for the single- and bound- soliton states within the span from 1 kHz to 30 kHz are \begin{table}
\begin{tabular}{c c c} \hline \hline
**Operation States** & **Single Soliton** & **Bound Soliton** \\ \hline Output averaged power: & 25 mW & 55 mW \\ Pulse width/Timing distance & 2.4 ps/X & 1.95 ps/4.8 ps \\ Repetition rate & 10 GHz \\ SMSR: (Super mode suppression) & 50 dB under the REW 470 kHz \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main output characteristics of the fiber laser. Figure 2: Auto-correlation and optical spectrum measurements for (**a**) single- and (**b**) bound-soliton states respectively. The red dashed line in (**a**) shows the pulse intensity auto-correlation fitting under the sech shape assumption. RF spectrum measurement for (**c**) single- and (**d**) bound-soliton states within the 50 MHz span range with the resolution bandwidth of 470 kHz. \(-96\) dBm and \(-101\) dBm respectively, and the total integrated power within the span from 1 kHz to 500 kHz are \(-36.3\) dBm and \(-38.3\) dBm. One can observe some oscillation peaks near the turning point for both states, and the flat bandwidth of the bound-soliton state is wider than that of the single-soliton state. One possible cause for these peaks is the relaxation oscillation effect caused by the gain dynamics inside the cavity [6; 35]. However, as we will see in the theoretical results below, parts of these resonance peaks may also be attributed to the mode-locking dynamics. Note that, experimentally, we cannot operate the single- and bound-soliton states under the same power level, because the laser becomes unstable due to the co-existence of the two states. Therefore, for direct experimental 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. For numerically calculating the intensity noises, the physical normalization units and the various cavity parameters used in our simulation are given below. The normalization unit of \(t\) is 0.5 ps, and the unit of \(T\) is the cavity roundtrip time, which is about 200 ns, as determined from the cavity length.