# Cascaded Nonlinear-Optical Loop Mirror-Based All-Optical PAM Regenerator
## 2 Regeneration Characteristics of a Single NOLM-Based Regenerator  
### Normalized Power Transfer Function of Single NOLM  
Nonlinear optical loop mirror (NOLM) is a Sagnac interference structure composed of highly nonlinear optical fiber (HNLF) and optical coupler, as shown in Figure 1. The \(2\times 2\) optical coupler (OC) divides the input optical field into two beams, which propagate in forward (\(E_{\mathrm{f}}^{(1)}\)) and backward (\(E_{\mathrm{b}}^{(1)}\)) directions, respectively. The optical fields after the HNLF, \(E_{\mathrm{f}}^{(2)}\) and \(E_{\mathrm{b}}^{(2)}\), interfere at a \(2\times 2\) optical coupler, resulting into the optical transmitted field \(E_{out}\) and the reflected field \(E_{r}\). In the HNLF, the SPM and cross-phase modulation (XPM) effects occur contributing to the nonlinear phase shift of the counter-propagating fields [14; 15], and the former effect is involved in the following discussion, performing the SPM-NOLM. Figure 1: Nonlinear optical loop mirror (NOLM). The expressions of the above-mentioned optical fields can be derived as follows [16; 17; 18]:  
\[\left\{ \begin{array}{l}E_{f}^{(1)}=\sqrt{\rho}E_{in}\\ E_{b}^{(1)}=i\sqrt{1-\rho}E_{in}\end{array} \right. \tag{1}\]  
\[\left\{ \begin{array}{l}E_{f}^{(2)}=e^{-\frac{1}{2}a\Lambda\Gamma}\sqrt{\rho}E _{in}e^{i\varphi_{f}}\\ E_{b}^{(2)}=ie^{-\frac{1}{2}a\Lambda}\sqrt{1-\rho}E_{in}e^{i\varphi_{b}}\end{array} \right. \tag{2}\]  
where \(\rho\) is the splitting ratio of the \(2\times 2\) coupler, \(\varphi_{b}=\beta L+\gamma L_{eff}/\left(\left|E_{b}^{(1)}\right|^{2}+2\overline {\left|E_{f}^{(1)}\right|^{2}}\right)\) and \(\varphi_{f}=\beta L+\gamma L_{eff}\left(\left|E_{f}^{(1)}\right|^{2}+2 \overline{\left|E_{b}^{(1)}\right|^{2}}\right)\) are the nonlinear phase shifts introduced by SPM and XPM in the HNLE, \(\beta\), \(\gamma\), \(L\), and \(\alpha\) are the propagation constant, the nonlinear coefficient, the fiber length, and the loss coefficient of the HNLF, respectively, and the effective length is given as \(L_{eff}=\left(1-e^{-a\Lambda t}\right)/\alpha\). After the interfering between \(E_{f}^{(2)}\) and \(E_{b}^{(2)}\) at the OC, the transmitted and reflected optical fields for a single NOLM can be expressed by:  
\[\left( \begin{array}{c}E_{out}\\ E_{r}\end{array} \right) = \left( \begin{array}{cc}\sqrt{\rho}&i\sqrt{1-\rho}\\ i\sqrt{1-\rho}&\sqrt{\rho}\end{array} \right)\left( \begin{array}{c}E_{f}^{(2 )}\\ E_{b}^{(2)}\end{array} \right)\] \[= e^{-\frac{1}{2}a\Lambda t}e^{i\varphi_{f}}E_{in}\left( \begin{array} []{c}\rho-(1-\rho)e^{\beta\Delta\varphi}\\ i\sqrt{\rho(1-\rho)}\left[1+e^{i\Delta\varphi}\right]\end{array} \right) \tag{3}\]  
where the nonlinear phase shift is \(\Delta\varphi=\varphi_{b}-\varphi_{f}=\gamma L_{eff}(2\rho-1)\left(\left|E_{ in}\right|^{2}-2\overline{\left|E_{in}\right|^{2}}\right)\). From Equation (3) we can get the PTF relation between the transmitted power \(P_{out}=\left|E_{out}\right|^{2}\) and the input power \(P_{in}=\left|E_{in}\right|^{2}\):  
\[P_{out}=e^{-a\Lambda t}\left[\rho^{2}+(1-\rho)^{2}\right]\left\{\left\{1- \frac{2\rho(1-\rho)}{\rho^{2}+(1-\rho)^{2}}\cos\left[\gamma L_{eff}\left(P_{in }-2\overline{P_{in}}\right)(2\rho-1)\right]\right\}\times P_{in}\right. \tag{4}\]  
where \(\overline{P_{in}}=\overline{\left|E_{in}\right|^{2}}\) is the average input power. From Equation (4), the output power is dependent on the splitting ratio \(\rho\). For the case with \(\rho=0.5\), the output power is equal to zero (\(P_{out}=0\)) and the SPM-NOLM acts as a mirror. Equation (4) can also be rewritten as follows:  
\[P_{out}=G_{0}P_{in}\left[1+\eta\sin^{2}\left[\kappa\left(P_{in}-2\overline{P_ {in}}\right)\right]\right] \tag{5}\]  
where \(\eta=\frac{4\rho(1-\rho)}{(2\rho-1)^{2}}\), \(\kappa=\frac{1}{2}\gamma L_{eff}(2\rho-1)\), \(G_{0}=e^{-a\Lambda}(2\rho-1)^{2}\).