## 3 Experiments  
To verify our model, we simulated different types of fiber systems with FRA, employing our modified NLSE in Equation (4) and the simplified NLSE in Equation (3). In the simulations, we employed the split-step Fourier method (SSFM) to simulate the performance of the fiber system, and the number of steps was always 1000. The computation cost of the SSFM was obtained from the FFT/IFFT operation. With the same number of steps, both our model and the simplified NLSE required 1000 pairs of FFT/IFFT operations. The computation cost of our model and the simplified NLSE in the SSFM were almost the same. The increased computation was due to the noise calculation by Equations (7) to (12). Therefore, the computation cost was acceptable. Then, we compared the simulating results with the experiments. The system architecture of our experiments and simulations is shown in Figure 2.  
At the transmitter, a 1550 nm wavelength laser with 100 kHz linewidth and an IQ modulator were used for optical modulation. A pseudo-random binary sequence (PRBS) of length \(2^{24}\) was generated by the software. After serial-to-parallel conversion and constellation mapping, the data of the I/Q channels were written to an arbitrary waveform generator (AWG). The raw data were  
Figure 2: Schematic of the experiment and simulation. our experiments and simulations. The baud rate of the symbol was 10 GBaud/s, and the working AWG was at a 10 GS/s sampling rate with 8-bit resolution. The output signal of the AWG in the two channels was sent to the IQ modulator through radio-frequency amplifiers. The modulated optical signal was fed into an EDFA to adjust the optical launch power. Then, the optical signal was transported through the fiber link and reached the receiver. At the receiving end, we employed a variable optical attenuator (VOA) to control the input optical power of the balanced photonics detector (BPD). The two output electrical signals of the BPD were acquired by a digital sampling oscilloscope with a 40 GS/s sampling rate. Subsequently, the offline digital signal processing (DSP) was employed to recover the data carried by the light. The offline DSP included electrical dispersion compensation (EDC), IQ orthogonalization, carrier recovery, and error-vector magnitude (EVM) calculation. The EVM was defined as follows:  
\[EVM=\frac{\sqrt{\frac{1}{M}\sum\limits_{i=1}^{M}\left(E_{r,i}-E_{t,i}\right)^ {2}}}{\left|E_{t,m}\right|} \tag{13}\]  
where \(E_{r,i}\) is the normalized symbol in the stream of received symbols, \(E_{t,i}\) is the corresponding ideal normalized constellation point, \(M\) is the number of symbols in the received stream, and \(\left|E_{t,m}\right|\) is the absolute value of the longest ideal normalized constellation vector. We measured the signal under different circumstances, including the 180 km link with only backward-pumped FRA and the 250 km link with both forward- and backward-pumped FRA. The efficient gains of the forward- and backward-pumped FRA were 10 dB and 16 dB, respectively. We also measured the 6-span 480 km fiber link, in which each span included an 80 km fiber, a backward-pumped FRA with 16 dB gain, and a VOA to adjust the optical launch power. The received optical power at the BPD was fixed at \(-25\) dBm by adjusting the VOA. In the three circumstances, we changed the optical launch power to obtain the variation of EVM. Figure 3a illustrates the simulating results and the experimental results of the 180 km fiber link with only backward-pumped FRA. The black diamonds are the experimental results, and the red and blue curves are the simulating results of the simplified NLSE and our model, respectively. In the single-span system with only backward-pumped FRA, both the simplified NLSE and our model could simulate the system precisely. The root mean squared error (RMSE) between the EVM of the simplified NLSE and the experimental results was 0.490%; the RMSE between the EVM of our model and the experimental results was 0.431%.