# Cascaded Nonlinear-Optical Loop Mirror-Based All-Optical PAM Regenerator
## 4 Cascaded NOLM-Based Regeneration Scheme  
According to the above analysis, a desirable PAM regenerator should satisfy \(g_{d}(x_{l})\to 0\) at every regenerative level as possible. For this reason, we investigate a regeneration scheme of cascaded NOLM and optimize the regenerative WPs of the NOLMs, as well as the inter-stage matching amplifier through the NPTF curves. The design rule is given as follows: the output WPs of the former NOLM stage is identical to the input WPs of the latter NOLM stage and consequently performs a one-to-one regenerative-level matching relation between two NOLMs. Therefore, we can regenerate the high levels from the first NOLM with the large splitting ratio, and the low levels from the second NOLM stage. with the small splitting ratio. The optimization of the splitting ratio helps to achieve the regenerative behavior on particular level and brings the less impact on other levels. The matching amplifier is another important device to match the two NOLM operation. The gain of this amplifier is determined by \(G_{12}=P_{0,2}/(G_{0,1}P_{0,1})\), in which \(P_{0,1}\) and \(P_{0,2}\) are the step powers of the two NOLMs and \(G_{0,1}\) is the linear gain of the former NOLM stage. Then, we discuss the NPTF and NDG results achieved by the cascaded NOLM regenerator. Following the definition in Section 2, the NPTF can be expressed as \(f(x)=f_{2}[f_{1}(x)]\) and the NDG is \(g_{\rm d}(x)=g_{\rm d}{}^{(1)}(x)\cdot g_{\rm d}{}^{(2)}[f_{1}(x)]\), where \(f_{1,2}(x)\) and \(g_{\rm d}{}^{(1,2)}(x)\) correspond to the results from the first and second NOLMs. Obviously, the most straightforward approach to \(g_{\rm d}(x_{l})\to 0\) is to optimize the NDG of the first NOLM stage \(g_{\rm d}{}^{(1)}(x)\), and to set its optimal WP at a higher regenerative level in view of the oscillation properties of the NPTF curve. Naturally, the optimal WP of the second NOLM stage should locate at a lower regenerative level by appropriately choosing a smaller splitting ratio of the coupler. After the thoughtful investigation, we perform the PAM-4 regeneration when the optimal operating levels of the first and second NOLM stages are, respectively, set at the third level (\(\rho=0.97\)) and the first level (\(\rho=0.92\)) for \(G_{12}=6.59\) dB. Figure 4 depicts the calculated NPTF and NDG results of the cascaded NOLM scheme. To investigate the regenerative performance we carried out the numerical simulation through the OptiSystem platform. The schematic diagram of the whole PAM regeneration system is shown in Figure 5. The signal is modulated by an intensity modulator 1(IM1) and then degraded by an ASE noise source. An optical isolator (ISO) is used to prevent unwanted reflected light. After an Erbium-doped fiber amplifier 1 (EDFA1), the signal power is boosted to match the optimal regenerative level. Figure 6 gives the level distribution of input and output signal and the simulated waveforms of the PAM signals before and after regeneration. When a degraded PAM-4 signal with the SNR of 20.5 dB is input to the cascaded NOLM-based regenerator, the numerical simulation shows that the NRRs of the four levels are, respectively, 27.11 dB, 18.10 dB, 25.14 dB and 16.65 dB, and the overall Q-factor can be improved by 25.32 dB. The relationship of NRR corresponding to different input SNR is analyzed as shown in Figure 6c. The signal quality improvement can be observed at every input SNR value. The maximal NRR over 30 dB is achieved demonstrating extremely high noise suppression capability by the proposed cascaded NOLM regenerator. The better performance is obtained when the input SNR is larger than 20 dB because the flat power response achieved around WPs, see in Figure 4. Although the noise suppression becomes weaker for the lower SNR, we can still get the NRR improvement of more than 10 dB. Figure 4: The NPTF and NDG curves for a PAM-4 regenerator with two NOLMs. We also test the bit error rate (BER) versus input SNR when the proposed regenerator located in the middle of the transmission link.