# Gain Equalization for Few-Mode Erbium-Doped Fiber Amplifiers via Strong Mode Coupling
## 1 Introduction  
With the increasing demand for optical fiber transmission capacity, space division multiplexing (SDM) has been explored, in depth, as an effective way to overcome the capacity limit of conventional single-mode fiber transmission systems by using spatially multiplexed channels in few-mode fibers (FMRs) and/or multi-core fibers (MCFs) [1; 2; 3]. Their matching optical fiber amplifiers are the key building blocks in long-haul SDM transmission systems, and various types of few-mode erbium-doped fiber amplifiers (FM-EDFAs) [4; 5; 6; 7; 8; 9; 10] and multi-core EDFAs [11; 12] have been developed over the past 10 years in order to simultaneously amplify all the spatial channels. These SDM amplifiers can bring significant advantages in terms of cost and space saving and efficient pump utilization compared to multiple single-mode EDFAs [4].  
One of the most important aspects of SDM amplifier design is to equalize the amplifier gain between all the spatial channels. In principle, the differential modal gain (DMG) is dependent on the spatial overlap between signal, pump intensity profile, and erbium-dopant distribution and can be effectively improved by optimizing these three aspects. Various advanced amplifier fiber designs (e.g., confinement doping or multi-layer doping) have been investigated to minimize the DMG [5; 6; 7], and it has been reported that the DMGs of FM-EDFAs supporting 6, 10, and 21 spatial modes can be reduced to 2-3 dB by adopting these approaches [8; 9; 10]. However, it is still very challenging to achieve a very low DMG (less than 1 dB) and the fiber design complexity gradually increases as the number of spatial modes increases. It is also important to point out that conventional single-mode EDFAs need gain equalization with respect to wavelength--i.e., in one dimension--while FM-EDFAs require two-dimensional gain equalization for both multiple wavelengths andspatial channels. This causes tremendously increased complexity in the fiber design and fabrication of FM-EDFAs as the number of modes increase.  
Intentionally enhancing the mode coupling (MC) within an FM-EDFA can be considered as an effective way to further reduce DMG. It has obvious advantages in two main aspects: (1) reduced DMG via completely mixing the signal modes and (2) reduced fiber design complexity with a large number of modes. Please note that a strong MC scheme has already been employed in long-haul FMF transmission systems and is regarded as an effective method to greatly reduce the complexity of MIMO signal processing [13, 14, 15, 16]. Therefore, the use of strong coupling onto FM-EDFAs can naturally give transmission fiber links improved DMG performance. Recently, mode scramblers have been investigated as a means to reduce the DMG in FM-EDFAs and a 1.8 dB DMG reduction was successfully achieved in a 10-mode fiber amplifier [17]. Therefore, it is highly desirable to systematically investigate the effect of strong mode coupling inside FM-EDFAs and explore its feasibility in practice.  
In this paper, we propose a strongly coupled FM-EDFA for the first time with greatly reduced DMG that can be controlled by choosing the appropriate coupling strength (CS) and number of coupling points (NCPs). In this way, the challenge of two-dimensional gain equalization can be reduced to a one-dimensional task. Using a fiber transfer matrix method model, we analyzed two exemplary six-mode EDFAs and found that strong mode coupling has a great effect on DMG mitigation in both of the FM-EDFAs. An 8.2 dB DMG reduction was achieved in a uniformly doped FM-EDFA via strong coupling, and this could be further reduced to 0.5 dB with an optimized fiber doping profile. The proposed strongly coupled FM-EDFAs were very tolerant to the mode-dependent loss in SDM transmission systems and the spectral gain flatness and differential noise figure (NF) could also be kept low.