# Relative Intensity Noise of Hybrid Mode-Locked Bound Soliton Fiber Laser: Theory and Experiment
## 2 Materials and Methods
By introducing the adjoint operator of \(S\) according to the following Equation (4),  
\[\left\langle u^{A}\middle|S\cdot u\right\rangle=\left\langle S^{A}\cdot u^{A} \middle|u\right\rangle \tag{4}\]  
the adjoint equation is defined by Equation (5) as below:  
\[\frac{\partial u^{A}(t,T)}{\partial T}=-S(t,T)\cdot u^{A}(t,T) \tag{5}\]  
Here, the inner product of two functions is defined as \(\left\langle f(t)\middle|g(t)\right\rangle=\frac{1}{2}\int\left(f(t)^{*}g(t)+ f(t)g(t)^{*}\right)dt\). The complete form of the adjoint equation is given in Equation (6), where \(E_{0}=\int u_{0}^{*}u_{0}dt\) represents the mean field pulse energy. \[ \begin{array}{c}\frac{\partial u^{A}}{\partial T}=-\left(\frac{g_{0}}{1+ \frac{g_{0}}{k_{s}}}-l_{0}\right)u^{A}-(d_{r}-jd_{i})\frac{\partial^{2}u^{A}} {\partial t^{2}}-(k_{r}-jk_{i})(2u_{0}^{*}u_{0})u^{A}-(k_{r}+jk_{i})(u_{0}u_{0 })u^{A^{*}}\\ \hskip 113.811024 pt+jMcos(w_{m}t)u^{A}+\frac{g_{0}u_{0}}{\left(1+\frac{g_{0}}{ k_{s}}\right)^{2}}\frac{1}{E_{s}}\int\left(u_{0}u^{A^{*}}+u_{0}^{*}u^{A} \right)dt\end{array}  \tag{6}\]  
One can easily prove that the solutions of the linearized equation and its adjoint will satisfy:  
\[\left\langle u^{A}(t,T)\middle|u(t,T)\right\rangle=\left\langle u^{A}(t,0) \middle|u(t,0)\right\rangle+\int_{0}^{T}\langle u^{A}\big{(}t,T^{\prime}\big{)} \middle|N\big{(}t,T^{\prime}\big{)}\rangle dT^{\prime} \tag{7}\]Equation (7) is main result of the backpropagation method, and is very physically meaningful. The final noise parameter defined by the projection in the left-hand side of Equation (7) comes from two contributions: the initial noise (the first term in the right-hand side) and the integrated added noises (the second term in the right-hand side). Thus, one can obtain information about the final noise parameter after numerically solving the adjoint system backwardly (from \(T\) to 0) with the initial condition given by \(u^{A}(t,T)\), which is the projection function for the noise parameter to be analyzed. By choosing a suitable projection function, the inner product in the left-hand side of Equation (7) can correspond to the perturbation of the chosen pulse parameter. From the definition of the pulse energy \(E=\int U^{*}Udt\), for calculating the pulse energy noise of the optical pulse, the projection function is simply 2 times the mean field, as shown in Equation (8). \[\Delta n(T)=\langle 2u_{0}(t,T)|u(t,T)\rangle \tag{8}\]  
The noise variance \(\text{Var}[\Delta n(T)]\) and the noise auto-correlation function \(\langle\Delta n(T)\Delta n(T+\Delta T)\rangle\) can also be derived after knowing the statistical properties (i.e., white noises) of the noise source term \(N(t,T)\). In the following, we will simply assume the noise is delta-function correlated, as given by Equation (9) [1; 2; 3; 4], where \(g=g_{0}/(1+\frac{E_{0}}{E_{s}})\) is the steady state gain coefficient of the laser and \(\theta\) is the noise enhancement factor compared to the ideal quantum noise level. \[ \begin{split}\langle N^{*}(t_{1},T_{1})N(t_{2},T_{2})\rangle=& \langle N(t_{1},T_{1})N^{*}(t_{2},T_{2})\rangle=\theta*2g\hbar v \;\delta(t_{1}-t_{2})\delta(T_{1}-T_{2})\\ &\langle N(t_{1},T_{1})N(t_{2},T_{2})\rangle=\langle N^{*}(t_{1},T_{1})N^{*}(t_{2},T_{2})\rangle=0\end{split}  \tag{9}\]  
The final formula for the intensity noise variance and the noise auto-correlation function are listed in Equations (10) and (11) below. Here we have assumed the contribution from the initial noise can be ignored, and only the integrated added noises are taken into account. For the mode-locked laser problems considered in the present work, this is a valid assumption. \[\text{Var}[\Delta n(T)]=\text{Var}\bigg{[}\int_{0}^{T}\langle u^{A}\big{(}t,T^ {\prime}\big{)}|N\big{(}t,T^{\prime}\big{)}\rangle dT^{\prime}\bigg{]}=\theta g \hbar v\int_{0}^{T}\int Re[u^{A^{*}}(t,T)u^{A}(t,T)]dt\;dT \tag{10}\]  
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