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In recent years, the demand for network bandwidth has increased significantly, and the capacity of wave division multiplexing (WDM) systems has reached the nonlinear Shannon limit. In order to adapt to the development of future networks, space division multiplexing (SDM) technology attracts more and more attention. In this paper, we put forward a novel structure of pulse amplitude modulation(PAM) regenerator based on few-mode nonlinear optical fiber loop mirror (FM-NOLM) for the first time, and theoretically analyze the working principle for few-mode reshaping. It can regenerate degraded PAM signals and improve transmission performance in SDM system. The detailed design steps of the regenerator are given, in which the sulfide highly nonlinear fiber and multimode coupler are used to build up the FM-NOLM and their mode characteristics are simulated by COMSOL software. The parameters of the regenerator are determined by taking the few-mode optical fiber supporting LP01, LP11, and LP21 modes as an example, and then the power transfer function (PTF) curve of each mode for PAM signals is calculated. We simulate and analyze the noise reduction ratio (NRR) performance of the few-mode regenerator for PAM-4 signals, and compare with the case of single mode fiber. Our simulation shows that: (1) for each spatial mode of PAM signal, all regenerative levels have the same consistent power transfer performance; (2) for the input signal-to-noise ratio (SNR) greater than 20 dB, the NRR for each mode can exceed 3 dB, and increase with the input SNR at the slope of about 1.2; (3) the NRR difference between the three modes is less than 1.1 dB for the same input SNR. In order to illustrate the reshaping function of the regenerator, we also present the power distribution histograms for PAM-4 signals before and after regeneration when the input SNR is 25 dB. This scheme proposed here has some advantages over the existing regenerators in the applicability for the long-haul SDM system with high spectral efficiency and regeneration of any level number of PAM signals in theory due to its uniform multi-level regeneration function, but also is capable of being extended to the wavelength domain for higher transmission capacity.

In recent years, the transmission capacity of wavelength division multiplexing (WDM) communication systems has gradually approached to the nonlinear Shannon limit. To meet the increasing demand for communication capacity, space division multiplexing (SDM) has become one of the most concerned technologies. In this paper, the four-wave mixing process (FWM) in fibers is considered from the frequency domain to the mode division multiplexing (MDM) spatial domain under pump depletion and the exact analytical solution to the FWM coupled-mode equations in the space-frequency domain is in detail deduced. The analytical method is verified by numerically calculating the amplitude and phase evolution of the idler wave in non-degenerate four-wave mixing. We discuss three new applications of the analytical solution as follows. 1) Using the phase matching condition we select the terms in the multi-wave coupling equation, and only retain the coupling term that plays a major role. According to the analytical solution in this paper, the phase matching percentage parameter is introduced to determine the FWM coupling terms necessary for multi-wave coupling equations, thus simplifying the multi-wave coupling problem in the study. 2) Combining the analytical solution with the numerical calculation results, we find the initial phase relationship between the output idler and the input guided wave for phase-insensitive FWM, and we provide the analytical expression for a theoretical basis to efficiently design the FWM-based phase arithmetic devices in parallel operating at WDM and MDM systems. 3) We propose a nonlinear compensation algorithm based on analytical solution, which can be used in the few-mode transmission system. The algorithm can fast evaluate or compensate for the fiber nonlinearity by taking into account the pump depletion of the FWM effect. Compared with the traditional digital back propagation (DBP) algorithm, our algorithm has the advantage of lower complexity.