Objective  Multi-core supermode fiber (MCSMF) with small inter-core spacing enable multiple cores to form a core region that supports the transmission of multiple supermodes. In comparison to conventional single-core few modes fiber, MCSMF has a larger effective mode field area and lower mode crosstalk, making it highly attractive. When used for long-distance transmission, novel gain equalization amplifier that is compatible with MCSMF is a necessary device to achieve signal relaying and maintain stable signal transmission. Current research on MCSMF mainly focuses on increasing the number of spatially multiplexed channels, optimizing pumping methods, and adjusting the length of the erbium-doped fiber (EDF) to expand communication capacity and reduce differential mode gain (DMG). However, there are few reports on the structural design of MCSMF. Therefore, it is of great significance to optimize the fiber's structural parameters and erbium ion distribution to further reduce DMG. In this study, the particle swarm optimization algorithm was employed to flexibly control the erbium doping concentration in each fiber core, determining the optimal doping structure of the EDF. This approach reduces the overlap integral factors of different supermodes and achieves gain equalization for \rmLP_01 , \rmLP_11\texta , \rmLP_11\textb , \rmLP_21\texta , \rmLP_21\textb , \rmLP_02 , \rmLP_31\texta , \rmLP_31\textb , \rmLP_12\texta and \rmLP_12\textb in the MC-SM-EDFA.

  Methods  The designed MCSMF in this study consists of 19 fiber cores (Fig.1), including a central core (1), the first layer of cores (2-7), and the second layer of cores (8-19). These 19 cores are uniformly distributed in a hexagonal pattern. In the MCSMF, erbium ions are uniformly doped within a single layer of each fiber core. The doping concentration (volume fraction) in each core is denoted as N_1 , N_2 , ··· N_19 according to the core numbering. The particle swarm optimization algorithm is utilized to optimize the erbium doping concentration in each fiber core, aiming to reduce the overlap integral factors of different supermodes and further minimize DMG. This optimization process enables the achievement of gain equalization for various signal modes.

  Results and Discussions  After optimization, the MC-SM-EDFA achieved an average gain of 27.79 dB, DMG of only 0.20 dB, and NF below 3.79 dB at a signal wavelength of 1 550 nm. Furthermore, the MC-SM-EDFA exhibited gains higher than 25 dB and gain flatness below 1 dB for different signal wavelengths in the C-band (Fig.7). The noise figure ranged from 3.4 dB to 4.4 dB, and the DMG showed minimal variation with signal wavelength. Additionally, using the Monte Carlo method, this study conducted simulations to analyze the impact of erbium ion doping concentration deviations on the balancing performance of the MC-SM-EDFA. The results demonstrated that the proposed MC-SM-EDFA structure exhibits good robustness (Fig.8).

  Conclusions  The proposed MC-SM-EDFA in this study supports simultaneous amplification and gain equalization of 10 modes. Simulation results demonstrate that when erbium ions are flexibly doped at different concentrations in each fiber core of the MC-SM-EDFA, the DMG at a signal wavelength of 1 550 nm for the 10 modes is 0.20 dB. In the C-band (1 530-1 565 nm), all signal modes achieve gains exceeding 26.99 dB, with DMG below 0.26 dB and NF below 4.37 dB. Additionally, the gain flatness in the C-band is below 1 dB. Furthermore, the tolerance analysis of DMG to fiber manufacturing deviations indicates stable gain performance of the proposed MC-SM-EDFA. Moreover, the MC-SM-EDFA achieves gain equalization by uniformly doping erbium ions in a single layer within each fiber core, eliminating the need for a layered doping design, as required in few-mode erbium-doped fiber amplifier (FM-EDFA). Therefore, the MC-SM-EDFA offers certain advantages in terms of design and manufacturing.