# Relative Intensity Noise of Hybrid Mode-Locked Bound Soliton Fiber Laser: Theory and Experiment
## 2 Materials and Methods  
Our experimental mode-locked fiber laser configuration is constructed as a sigma-shape cavity which is able to achieve environmentally-stable operation, as indicated in Figure 1[30; 31; 32; 33; 34]. The cavity is composed of a polarization-maintaining (PM) reflection loop, and a non-PM linear fiber section with a Farady rotator mirror for automatic birefringence cancellation. In the PM-loop, a LiNbO\({}_{3}\) phase modulator is inserted to provide an active mode-locking mechanism, and the phase modulation frequency is set at 10 GHz so that the laser can be harmonically mode-locked under a high repetition rate. In the non-PM linear section, a piece of 4 m dispersion-shifted high nonlinearity fiber (DS-HNLF) is used to enhance the cavity Kerr nonlinearity, and a piece of bent dispersion-shifted fiber (DSF) is inserted to adjust the lasing wavelength through its wavelength-dependent loss. The Er-doped gain laser is bi-directionally pumped so as to achieve enough lasing power for observing the bound-soliton mode-locking. Notice that in our mode-locked fiber laser, there is no equivalent saturable absorption effect [33; 34]. The bound-soliton state can be formed in the higher pump power levels in comparison to the single-soliton state [25]. The RIN noises of both states are measured with a fast photo-detector and a radio frequency (RF) spectrum analyzer (Agilent E4440A, Santa Clara, CA, USA) for detailed comparison. Figure 1: Configuration of the environmentally stable 10 GHz hybrid mode-locked Er-doped fiber laser with a sigma-type fiber cavity. Our theoretical model is based on the Master equation, as shown in Equation (1), which assumes all the cavity parameters are averagely distributed. The physical meanings of the symbols are described as follow. \(U(t,T)\): complex pulse envelope, \(t\): small time scale, \(T\): large time scale in terms of the cavity roundtrip number, \(g_{0}\): small signal gain, \(E_{s}\): gain saturation energy, \(l_{0}\): linear loss, \(d_{l}\): cavity dispersion, \(d_{r}\) : optical filtering, \(k_{r}\): equivalent nonlinear saturable absorption (=0 for the studied laser), \(k_{i}\): self-phase modulation (SPM), \(M\) : active phase modulation depth, \(w_{m}\): modulation frequency, and \(N(t,T)\): noise source. \[\frac{\partial U}{\partial T}=\left(\frac{g_{0}}{1+\frac{f[|U|^{2}dt}{E_{s}}}- l_{0}\right)U+(d_{r}+jd_{i})\frac{\partial^{2}U}{\partial t^{2}}+(k_{r}+jk_{i})U^{*}UU \ +jMcos(w_{m}t)U+N(t,T) \tag{1}\]  
The backpropagation method starts from the linearization of the above nonlinear equation. We assume a small perturbation field \(u(t,T)\) around the mean field solution \(u_{0}(t,T)\) and substitute \(U(t,T)=u_{0}(t,T)+u(t,T)\) into Equation (1) to derive the linearized equation for \(u(t,T)\). The result is shown in Equation (2). \[ \begin{array}{c}\frac{\partial u}{\partial T}=\left(\frac{g_{0}}{1+\frac{g_{ 0}}{k_{s}}}-l_{0}\right)u+(d_{r}+jd_{i})\frac{\partial^{2}u}{\partial t^{2}}+( k_{r}+jk_{i})(2u_{0}^{*}u_{0})u+(k_{r}+jk_{i})(u_{0}u_{0})u^{*}\\ \hskip 113.811024 pt+jMcos(w_{m}t)u-\frac{g_{0}u_{0}}{\left(1+\frac{g_{0}}{k_{ s}}\right)}\frac{1}{E_{s}}\int(u_{0}u^{*}+u_{0}^{*}u)dt+N(t,T)\end{array}  \tag{2}\]  
One can rewrite Equation (2) in a compact form, as in Equation (3). Note that the operator \(S\) now denotes a special operator that operates both on \(u\) and \(u^{*}\) as defined in Equation (2). \[\frac{\partial u(t,T)}{\partial T}=S(t,T)\cdot u(t,T)+N(t,T) \tag{3}\]  
Corresponding to the above linearized perturbation equation, we then define an adjoint equation such that the inner-product conservation between two systems can be ensured.