## 3 Results  
In this section, we will present the simulated retrieving performances of the OAM modes after transmitting in the different turbulences with the aids of GS algorithm, which are described by the power density (PD) defined as the ratio of the obtained target OAM mode power on the original incident OAM mode power. Based on the obtained PD spectra, we will discuss and interpret the results in details. In simulations, we set three different values of turbulence refractive-index structure parameters, including \(C_{n}^{2}=1\times 10^{-15}\)m\({}^{-2/3}\), \(C_{n}^{2}=1\times 10^{-14}\)m\({}^{-2/3}\) and \(C_{n}^{2}=1\times 10^{-13}\)m\({}^{-2/3}\), and the propagation distance \(z=1000\) m is divided into several parts with different number of random phase screen including \(N=20\), \(50\), and \(100\). The beam waist is \(\omega_{0}=5\) cm with the wavelength \(\lambda=1550\) nm. Firstly, we explore the effect of turbulent channel with a different refractive index structure parameter \(C_{n}^{2}\) and phase screen number \(N\) on the transmitted single OAM beam. Figure 3 shows the intensity and phase distributions of beams and the corresponding PD spectra without and with the pre-compensation of GS algorithm under simulated atmospheric turbulence with \(C_{n}^{2}\) valued as \(1\times 10^{-15}\)m\({}^{-2/3}\) but different number of random phase screens. Sub-figures (a1, c1, e1) and (b1, d1, f1) are the concrete intensity distributions of the OAM beams, without pre-compensation, after transmitting in simulated turbulence channels with random phase screens number of \(N=20\), \(50\) and \(100\) respectively, in which the topological charge of OAM beam are \(l=3\) and \(l=5\) respectively. Sub-figures (a3, c3, e3) and (b3, d3, f3) of Figure 3 are their phases distributions respectively. Sub-figures  
Figure 2: Iterative step of computing the compensation phase screen based on the Gerchberg-Saxton (GS) algorithm; (FFT: Fast Fourier Transform; IFFT: Inverse Fast Fourier Tranform). (a5, c5, e5) and (b5, d5, f5) of Figure 3 are their PD spectra respectively. Sub-figures (a2, c2, e2) and (b2, d2, f2), (a4, c4, e4) and (b4, d4, f4), and (a6, c6, e6) and (b6, d6, f6) of Figure 3 are the corresponding situations (intensity distributions, phases distributions, and PD spectra) with the pre-compensation of GS algorithm. The parts circled in green (a1\(\sim\)a6; b1-b6), blue (c1\(\sim\)c6; d1\(\sim\)d6) and red (e1\(\sim\)e6; f1\(\sim\)f6) dashed frames represent simulated turbulence channels with \(N=20\), 50 and 100 respectively. We can observe that when the refractive-index structure parameter is fixed, with increasing the number of inserted random phase screens, the damage effect of simulated turbulence on the wave-front of the transmitted OAM beam becomes more obvious. To further analyze quantitatively, we have measured the PD spectra of the obtained LG beams with initial topological charges of \(l=3\) and \(l=5\) under turbulence with refractive index structure parameter \(C_{n}^{2}=1\times 10^{-15}\)m\({}^{-2/3}\). For the transmitted OAM beams with topological charge of \(l=3\), without pre-compensation by GS algorithm, it shows that the PD spectra are 94.96% (Figure 3(a5)), 88.09% (Figure 4(c5)) and 67.94% (Figure 4(e5)) corresponding to different channels with \(N=20\), 50 and 100 respectively, and for \(l=5\), the corresponding PD spectra are 91.56% (Figure 3(b5)), 82.15% (Figure 3(d5)) and 64.33% (Figure 3(f5)) respectively. Then, with the aids of pre-compensation, the PDs reach 99.98% (Figure 3(d6)), 99.88% (Figure 3(c6)) and 99.46% (Figure 3(e6)) for \(l=3\), and 99.73% (Figure 3(b6)), 99.42% (Figure 3(d6)) and 98.35% (Figure 3(f6)) for \(l=5\).