# A Multi-Layer Erbium-Doped Air-Hole-Assisted Few-Mode Fiber with Ultra-Low Differential Modal Gain
## 1 Introduction  
Mode-division multiplexing (MDM) transmission over few-mode fibers (FMF) is a promising candidate to overcome the capacity limitation of single-mode fibers (SMF) [1; 2; 3; 4]. It utilizes multiple modes on a single core at the same time, contributing to an efficient capacity expansion. The few-mode erbium-doped fiber amplifiers (FM-EDFAs) are required for long-haul transmission signal amplification [5; 6; 7]. Due to the different modal loss and mode field distribution, a large differential modal gain (DMG) is probably aroused after amplification. The power discrepancy between signals hinders the stability of the MDM system and the bit error rate may be unacceptable after long-haul transmission or multi-stage amplification. Therefore, researchers try to obtain the gain equalization in FM-EDFA by refractive index (RI) [8; 9; 10] or doping profile design [8; 11; 12; 13] of the FM-EDF as well as the pump optimization [1; 14; 15; 16]. In the FM-EDFAs, a large DMG may arise due to the obvious different modal profile between the convergent Gaussian-like LP\({}_{01}\) and divergent LP\({}_{mn}\) (\(m\geq 0,n>1\)) modes in the step-index FM-EDF. In this case, FM-EDFs were designed to balance the modal patterns by introducing a dip in the fiber center [17]. [PERSON] first demonstrated an FM-EDFA by using an FM-EDF with a tailored central dip [18], where the DMG is lowered by 2 dB more than the traditional step-index fiber. With the depending dip, the ring-core fiber (RCFs) is formed when the RI of the central region is equal to the clad. In RCF, a high- RI ring core is sandwiched between the two low- RI regions. RCF has been proved to have much potential in gain equalization. In 2015, a 2-LP ring-core FM-EDF is fabricated and a small DMG (\(\sim\)1 dB) is realized [19]. With the aim of increasing the supporting mode number, a trench-assisted five-mode ring-core erbium-doped fiber amplifier was proposed with a DMG of \(\sim\)1 dB, while the gain is only 10 dB [20]. However, the RCF may not be satisfactoryin mode modification due to its poor fabrication tolerance. Usually, in RCF, both the model number and sequence are sensitive to the thickness of the central dip and the RI difference between the core and cladding. The air-hole-assisted few-mode fiber (AH-FMF) has a higher RI depression in the central than the RCF, thus enabling a stronger capability in modal intensity modification [21; 22; 23; 24]. In the air-hole-assisted few-mode fiber, the high-refractive-index annular core (HRI-AC) is sandwiched between the air hole and the cladding. Due to the weak light confinement ability of the air hole, all the modes tend to be confined in the high-refractive-index annular core. It benefits for balancing the value of the overlap integral factors between different signal and pump mode groups, contributing to a low DMG. For instance, in a traditional step-index FMF (SI-FMF), the LP\({}_{01}\) mode distributes as a convergence Gaussian shape and its modal gain is much higher than other divergence high-order modes [8; 16]. While in the air-hole-assisted few-mode fiber, the LP\({}_{01}\) mode is restricted into the annual core region and the modal intensity is similar to other high-order modes. This contributes to the semblable overlap integral factors and benefits the FM-EDFA for gain equalization. In addition, the central air hole changes the mode cutoff condition. With the introduction of the central air hole, the fiber only supports single-radial-order modes (i.e., LP\({}_{\rm m1}\) modes where m is an integer), enlarging the effective RI between modes, which significantly reduces the modal crosstalk and coupling. In the terms of modal amplification, all of the guided modes possess similar modal intensities. On that basis of appropriate pump selection and an adaptive erbium-doping profile, the air-hole-assisted few-mode fiber is bound to provide an excellent ability in gain equalization. In 2015, [PERSON] et al. proposed an air-core erbium-doped fiber (AC-EDF).