Results and Discussions The experimental data and simulation results are compared to demonstrate the feasibility of the above simulation method. Two-mode amplification experiments of LP₀₁ and LP₁₁ under bidirectional pumping were carried out using the optically isolated wavelength-division multiplexers (IWDMs). In the experiment, the total pump power is fixed at 200 mW while the forward pump powers are varied. The optimal equivalent erbium doping concentrations for forward-pumping and backward-pumping cases are determined using the VPI simulation platform. The equivalent erbium doping concentration under the bidirectional-pumping case is obtained using a simple weighting method of Nˉ=(1-ηb)Nfˉ+ηbNbˉ and the simulation curve and experimental data of the mode gain are shown in Fig.3. The feasibility of the weighting method is shown by the 1.3 dB difference between the simulation and the experimental data. The weighting method is very useful for the realization of FM-EDFA’s automatic gain control. The maximum difference between the simulation and the experimental results increased to 2.1 dB when the fitting process is done by use of the conventional method with a fixed erbium ion doping concentration, which is greater than that of the equivalent concentration simulation method proposed in the paper. The simulation method of equivalent erbium doping concentration doesn't limit to the two-mode amplification. Figure 6 shows that our simulation curve coincides with the VPI reference data for the six-mode amplification. An equalized FM-EDFA with a mode gain of about 11 dB is designed by optimizing the power ratio of the forward and backward pumps. Our simulation shows that the DMG and optical signal-to-noise ratio (OSNR) after loop transmission with 10 FM-EDFAs are respectively 0.56 dB and 27 dB, exceeding the OSNR threshold of no bit error for 32 GBaud DP-32QAM MDM system.

Space-division multiplexing appears to be the only way to increase the capacity of optical transport networks since wavelength-division multiplexing is rapidly approaching its scaling limits. As a key relay device, the few-mode erbium-doped fiber amplifier (FM-EDFA) plays an important role in long-distance mode division-multiplexing (MDM) systems. The differential mode gain (DMG) and modal gain are important factors to evaluate the FM-EDFA’s performance. The intensity model of FM-EDFA can be described by the rate equation of erbium ion concentration and the power evolution equation of the signal (or pump) mode. The distributions of erbium ion doping concentration and signal (or pump) mode field should be known in advance for theoretical calculations, with application to compare with experimental results or to optimize the performance of FM-EDFA. For example, the refractive index distribution must be imported into the simulation system as files in many simulation platforms. However, the complicated erbium-doped concentration distribution and the refractive index profile are usually unknown in most experiments. This research aims to offer a new simulation method to bypass the erbium-doped concentration distribution and accomplish the consistency of FM-EDFA’s theoretical calculation (or simulation results) with the experimental data as possible.

According to the relationship between the parameters used in the FM-EDFA’s theoretical model shown in Fig.1, an equivalent erbium doping concentration parameter was introduced to replace the real erbium doping concentration N₀ in the theoretical model, and then the equivalent erbium doping concentration parameter was obtained by simulating and fitting the mode gain curve measured by the experiment. Regardless of whether the actual N₀ is uniform in the erbium-doped fibers, the gain of the signal modes depends on the pumping mode, pumping mode distribution, and its power; therefore, the equivalent erbium doping concentration also varies with the pumping parameters. Based on the equivalent erbium doping concentration, a new simulation method was proposed for FM-EDFAs, in which the equivalent erbium doping concentrations under the bidirectional pump case can be obtained from the two separate experiments with forward and backward pumping by weighting the pump power ratio. The simulation method is also used to efficiently simulate the cascading transmission performance of the FM-EDFAs.

The two-mode amplification experiment of bidirectionally pumped EDFA is carried out using few-mode IWDM devices, which shows the feasibility of the FM-EDFA simulation method based on the equivalent erbium doping concentration, and provides a simple method to calculate the equivalent doping of bidirectionally pumped FM-EDFA. The highest difference between the simulation results and the experimental data using the weighting method of erbium concentration in the VPI simulation platform was only 1.3 dB. A gain-balanced FM-EDFA with mode gain of about 11.2 dB is built, and its cascaded transmission properties are investigated using optical cycle simulation. This is accomplished by optimizing the bidirectional pump power distribution to compensate the mode-dependent loss of FM-IWDM. The simulation shows that after ten cyclic transmissions, the DMG and OSNR of the entire system are about 0.56 dB and 27 dB, respectively, which applies to the long-distance transmission of DP-32QAM high-order modulation format signals.