# Gain Equalization for Few-Mode Erbium-Doped Fiber Amplifiers via Strong Mode Coupling
## 4 Implementation of Strongly Coupled FM-EDFAs  
The feasibility of our proposed strongly coupled FM-EDFAs was also examined. Various fiber-type mode scramblers can be considered as a strong MC, but we particularly focused on the long-period gratings (LPGs) inscription technique during the fiber drawing process by CO\({}_{2}\) laser annealing [25,26] because it can provide highly controllable MC along the entire length of active fiber. According to [27], the grating angle \(\Psi_{0}\) and grating strength \(\Delta n_{g}\) are the most essential parameters for LPG inscription and directly determine CS. With \(\Psi_{0}=0\), the refractive index change is described as \(\Delta n_{g}(x,y,z)=\Delta n_{g}exp\Big{[}-\rho\Big{(}\sqrt{a^{2}-y^{2}}-x \Big{)}\Big{]}[1+cos(2\pi z/\Lambda)]\), where \(\Lambda\) is the grating period, \(\rho\) represents the asymmetric shape of the transverse index profile, and a is the core radius. Specifically, taking into consideration the dimension of our fibers, a 10% power transfer (i.e., CS = \(-10\) dB) occurred after a 4 cm-long fiber when \(\Delta n_{g}\) equaled 7.5 \(\times\) 10\({}^{-5}\). For the abovementioned 6M-EDFAs, we chose the length of EDF to be 12 m, which can accommodate three hundred 4 cm-long LPGs in total, meaning NCP = 300. Moreover, according to [28], the mode-dependent loss (MDL) for the device is assumed to be 4 dB and uniformly distributed along the EDF. Table 2 summarizes the gain characteristics with or without losses for two types of EDFs. Although the averaged MGs was reduced by about 1 dB due to the MDL, there was little change in both DMGs and their standard deviations (only 0.5-dB and 0.2-dB changes of DMGs for Type-A and Type-B EDFs, respectively).  
Note that conventional LPGs are strongly wavelength-dependent, which may limit the gain bandwidth of the proposed amplifiers. The mode coupling induced by LPGs varies with the wavelength, which will also cause the DMG to vary with the wavelength based  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Parameters** & **Type A** & **Type B** \\ \hline \multirow{3}{*}{Without Losses} & Averaged MG (dB) & 31.31 & 30.65 \\  & DMG (dB) & 3.40 & 1.59 \\  & Standard deviation of DMG (dB) & 0.65 & 0.24 \\ \hline \multirow{3}{*}{With Losses} & Averaged MG (dB) & 30.10 & 29.57 \\  & DMG (dB) & 3.91 & 1.38 \\ \cline{1-1}  & Standard deviation of DMG (dB) & 0.72 & 0.25 \\ \hline \hline \end{tabular}
\end{table}
Table 2: EDF gain characteristics w or w/o losses at CS = \(-10\) dB/pt and NCP = 300.  
Figure 6: (**a**) Pump power and (**b**) wavelength dependence of uncoupled and strongly coupled 6-mode EDFAs at NCP = 300 and CS = \(-10\) dB/pt.  
on the analysis in Section 3. Therefore, in order to obtain a low DMG over a wide band, wideband LPGs are needed. Many research works have recently reported that they can improve the bandwidth [29; 30] using chirped grating profiles [31] or cascading several gratings with different periods [32], and the widest bandwidth achieved so far with an LPFG is \(\sim\)300 nm (10-dB bandwidth) [33]. Therefore, we believe that the proposed strongly coupled FM-EDFAs can be achievable with currently available LPG fabrication techniques.