## 4 Results and Discussion
Figure 8 illustrates the SER performance as a function of the optical signal-to-noise ratio (OSNR/0.1 nm dB) with different \(\phi_{\text{IQ}}\). The shot noise (SN) is included, and the thermal noise (TN) is 10 \(\times\) 10\({}^{-12}\) A/H\({}^{(1/2)}\). The linewidth is set to 100 kHz. The 15\({}^{\circ}\) of the IQ phase mismatch is already a lot for a commercial device, and in practice, the commercially available coherent receivers have a phase mismatch less than 30\({}^{\circ}\). Thus, we only simulated the range less than 30\({}^{\circ}\) in this paper. It is evident that the ASOP and SYOP schemes outperform the GSOP. Here, for the convenience of comparison, we consider the worst case for the GSOP that it always selects the wrong initial reference vector. As shown in Figure 8a, we should note that the results after the GSOP become worse because of the wrong initial reference vector, so the original signal (I) is destroyed undesirably. The green line represents the ideal GSOP, which means that the right initial reference vector is selected. Figure 8b,c shows that the SYOP and ASOP can both improve the SER \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Wavelength & 1550 nm \\ Aperture diameter \(D\) & 5 cm \\ \(\text{C}_{n}{}^{2}\) & \(1.5\times 10^{-15}\) m\({}^{-2/3}\) \\ Transmission distance \(L\) & 1000 m \\ Phase variance \(\sigma_{\varphi}{}^{2}\) & 0.07 rad \\ Transmitted power & 10 dBm \\ Received power & \(-30\) dBm \\ Linewidth & 100 kHz \\ LO power & 0 dBm \\ DSP sampling rate & 20 Gsample/s \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters used in the numerical simulation. performance compared with no compensation of phase mismatch. When the \(\phi_{\text{IQ}}\) is less than \(5^{\circ}\), the SYOP outperforms the ideal GSOP, so the ASOP scheme applies the SYOP algorithm. This is because of the symmetry and resemblance properties of the SYOP so the noise can be evenly distributed to both the I and Q branches. However, when the \(\phi_{\text{IQ}}\) is greater than \(5^{\circ}\), the ideal GSOP with the right initial reference vector outperforms the SYOP, so the ASOP scheme applies the GSOP algorithm. Figure 9 illustrates the SER performance as a function of \(\phi_{\text{IQ}}\) with a frequency offset of 400 MHz. The phase drift in the receiver caused by the frequency offset between the transmitter and the local oscillator could be compensated with the carrier frequency estimation algorithm. The carrier frequency estimation was completed by a frequency offset estimation module in VPI with the FFT-based frequency offset estimation algorithm. As we can see, without nonorthogonality compensation, the SER performance of \(10\,\times\,10^{-4}\) can be obtained only in the rage of \(7^{\circ}\). However, a stable SER performance of \(10\,\times\,10^{-6}\) can be obtained after the ASOP in the range of \(20^{\circ}\), and in fact, it can reach a range of \(40^{\circ}\). The simulation results also show that the phase mismatch deteriorates the frequency offset estimation effect seriously, so the ASOP also contributes to the frequency offset estimation, thus the system SER performance can be much improved. Figure 8: Symbol error rate (SER) versus the optical signal-to-noise ratio (OSNR) with different \(\phi_{\text{IQ}}\) of \(5^{\circ}\), \(10^{\circ}\), and \(15^{\circ}\) for the (a) GSOP, (b) the SYOP, (c) and the ASOP; W/O: without. Figure 7: (**a**) Initial constellation; (**b**) after the GSOP; (**c**) after the SYOP; (**d**) after the ASOP. In order to verify the impact for the phase mismatch and the compensation effect of the proposed ASOP algorithm, we consider the effects of log-normal amplitude fluctuations and Gaussian phase fluctuations induced by turbulence. The atmospheric structure constant and phase variance are set as \({C_{n}}^{2}\) = 1.5 \(\times\) 10\({}^{-15}\) m\({}^{-2/3}\) and \({\sigma_{\varphi}}^{2}\) = 0.07 using Equation (5) in the simulation, which stand for weak atmospheric turbulence condition.