# Cascaded Nonlinear-Optical Loop Mirror-Based All-Optical PAM Regenerator
## 1 Introduction  
Optical fiber communications are required for higher data rate and transmission capacity to satisfy the demands of new services such as mobile Internet, big data and Internet of things. Optical signals with advanced modulation formats such as pulse amplitude modulation (PAM) or quadrature amplitude modulation (QAM) have been implemented in the real high-capacity optical networks. However, when moving to the high constellation orders the signal is more sensitive to the distortion from the fiber nonlinearity or the amplified spontaneous emission noise (ASE) noise. All-optical signal processing is a promising technology to directly deal with these high-spectra-efficiency signals in the optical domain, bypassing the so-called \"electronic bottleneck\". In particular, optical signal regeneration also evolves from the traditional optical-electrical-optical (O/E/O) approach to the all-optical regeneration [1, 2]. All-optical PAM regeneration technologies begin to attract more attention because of the widespread use of PAM signals in data centers [3] and the need for medium- or long-distance communication between data centers [4]. For the all-optical amplitude regeneration, the nonlinear optical loop mirror (NOLM) structure has been intensively investigated. In 2004, References [5, 6] presented the self-phase modulation (SPM)-based three-stage NOLM scheme for all-optical return-to-zero (RZ) signal regeneration. In 2014, the simultaneous multi-level amplitude and phase regeneration was numerically demonstrated by using nonlinear amplifying loop mirror (NALM) with a phase-sensitive amplifier [7]. The design rule of a multi-level amplitude regenerator in a single NOLM has also been discussed by [PERSON] [8]. For the experimental investigation of the PAM regeneration, [PERSON] et al. improved thesignal linearity through degenerate four-wave mixing in a silicon waveguide [9]. Our group has also demonstrated the multi-level amplitude regenerations in a single NOLM [10; 11; 12] or a modified NOLM structure- polarization-orthogonal continuous-wave-light-assisted NOLM (PC-NOLM) [13]. The above studies show that the power transfer function (PTF) achieved in a single NOLM- regenerator satisfies the expression of \(P_{out}\propto P_{in}\sin^{2}[\varphi(P_{in})]\), where \(P_{out}\) is the output power of the NOLM regenerator and \(\varphi(P_{in})\) is the function of the input power \(P_{in}\)[10]. It can be seen that as the input power increases, the oscillatory strength of this kind of PTF curves becomes much stronger, and consequently, narrows down the regenerative regions, especially for the high amplitude level. Therefore, we propose a novel cascaded two-NOLMs regenerator for the PAM signal. We carefully design the splitting ratio of the two NOLMs and utilize the power-dependent regenerative performance of the NOLM unit. By optimizing the gain coefficient of the Erbium-doped fiber amplifier (EDFA) connecting the two NOLMs, the proposed regenerator offers the wider regenerative region on each level and supports more regenerative levels. In this paper, we analyze the matching relation between the working points (WPs) of the cascaded NOLMs through the normalized PTF curve relative to the step power of a PAM signal. The detailed design rule is also presented in order to equalize each regenerative level to perform the optimal reshaping behavior. By appropriately setting the optimal WPs of two stages of NOLMs, a PTF curve close to the ideal staircase like function is obtained for PAM-4 regeneration and the improvement of Q-factor is up to 25.32 dB for the input signal-to-noise ratio (SNR) of 20.5 dB. Compared with the traditional single NOLM structure, the improvement of SNR increases by 1.69 dB at lg(BER) = \(-\)3 [12]. A PAM-8 regenerator is also restructured by cascading two NOLMs with an inter-stage optical amplifier.