# Relative Intensity Noise of Hybrid Mode-Locked Bound Soliton Fiber Laser: Theory and Experiment
## 1 Introduction  
Investigation of mode-locked laser noises is important both for studying delicate laser dynamics and for optimizing laser performance. Theoretically, by introducing proper noise sources into the Master equation model for mode-locked lasers, different laser noise properties, such as the relative intensity noises (RIN), as well as the timing jitter noises, can be calculated based on soliton perturbation theory [1; 2; 3; 4; 5; 6] or stochastic numerical simulation [7; 8; 9; 10; 11]. In the literature, both passive and active mode-locked lasers have been studied extensively, and the agreement between theoretical and experimental results are reasonable for the considered single-pulse mode-locking cases [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17]. It is well known that some mode-locked lasers can be operated under a bound-pulse mode-locking state, in which two or more adjacent bound-pulses are formed inside the laser cavity [18; 19; 20; 21; 22; 23; 24; 25]. Because of the sophisticated balancing effects among different optical linear/nonlinear processes, bound-pulse mode-locking states can exhibit more interesting appearances. The noise properties of bound-pulse mode-locked lasers have also been less investigated, but remain of particular research interest. Some preliminary experimental results have been reported in our early work [26]. Due to the more complicated pulse shape of the bound-pulses, analytic soliton perturbation theory is no longer applicable. The stochastic numerical simulation is applicable in principle, but in practice will beextremely time consuming, due to the required computation efforts. To overcome these difficulties, in the present work we develop a new theoretical method based on linearized backpropagation to calculate the laser RIN noise spectra of bound-soliton mode-locking. The linearized backpropagation approach was originally developed for modeling quantum nonlinear optical pulse propagation problems [27; 28; 29]. Here we generalize the method to calculate the laser noise spectra, as well as the total noise variance, for the first time. Due to the deterministic computational nature of the approach, the required computation is much less than that of the stochastic simulation method. Since no particular pulse shape is assumed, the developed new approach is a general method applicable to the considered bound-pulse mode-locking case. As a real example, we investigate an environmentally stable 10 GHz hybrid mode-locked Er-doped fiber laser with active phase modulation. The laser is experimentally built with the interesting property that its mode-locking state can transmit from the single-soliton to the bound-soliton mode-locking state by simply increasing pump power [25; 26]. The RIN noises of the laser are experimentally measured under both the bound- and single-soliton states for comparison. The observed experimental tendency agrees reasonably with the theoretical prediction based on the linearized backpropagation approach. One interesting finding is that the RIN noise of the bound-soliton state can be lower than that of the single-soliton state by following the laser power scaling tendency. The results also demonstrate that the developed theoretical method is very well-suited for studying general mode-locked laser noise problems.