## 2 Adaptive Optics Enabled Free-Space Optical Communication
### Results and Analysis  
We begin by investigating the atmospheric turbulence-induced mode crosstalk and the merits of AO assisted wavefront correction. The power ratio between the target OAM modes (OAM states \(\pm 2\), \(\pm 6\)) and their adjacent modes are used here as a metric to evaluate the effect of mode crosstalk. As illustrated in Figure 2a, the power of target OAM mode measured was similar to the adjacent OAM modes, indicating that the data originating from mode crosstalk will severely interfere with the desirable data after mode detection. Data in Figure 2b shows that the average extinction ratio (ER) after AO assisted wavefront correction reaches a 6-dB improvement. The blue bars in Figure 2a and navy bars in Figure 2b represent the desirable transmitted OAM modes, and the green bars represents the unwanted OAM modes caused by OAM mode crosstalk.  
Figure 3a shows the average bit-error rate (BER) vs. OSNR performance with or without atmospheric turbulence effects. The data clearly demonstrates that the BER curves do not drop quickly, even with the increasing OSNR values. It is caused by the unperfect mode generation and detection patterns, which will also introduce unwanted inter-mode crosstalk effects. Note that the worse BER performance of OAM states \(\pm 6\) compared to OAM state \(\pm 2\) is due to high-order OAM mode sensitivity to the boundary effect.  
Figure 3: (**a**) Average BER vs. OSNR performance in cases with or without atmospheric turbulence effects. (**b**) Average BER vs. OSNR performance after AO assisted wavefront correction and low-density parity-check (LDPC) coding.  
Figure 2: Power distributions of orbital angular momentum (OAM) modes: (**a**) without AO assisted wavefront correction and (**b**) with AO assisted wavefront correction.  
Figure 3b shows that distinct average OSNR gain can be reached after AO assisted wavefront correction and LDPC coding. We observe that the BER curves after AO assisted wavefront correction can be lower than 0.04, which is the error correction threshold of the used LDPC code. Furthermore, the performance of the AO assisted wavefront correction on OAM states \(\pm 2\) are better than that of OAM states \(\pm 6\). This is caused not only by the boundary effect brought by the limited sizes of the used optical components, but also the features of the used probe beam. This beam is able to fully cover the small-size OAM modes (OAM states \(\pm 2\)) after the turbulent FSO transmission, rather than the large-size OAM modes (OAM states \(\pm 6\)). After the AO correction is performed, LDPC coding/decoding can then be applied to efficiently eliminate the post-FEC error floor. Figure 3b shows that the BER curves of all OAM modes can drop quickly as long as the OSNR is higher than 8 dB. More specifically, when the BER is \(2\times 10^{-2}\) the coding gains of 3.9, 4.1, 5.2, and 5 dB are reached for OAM states \(2\), \(-2\), \(6\), and \(-6\), respectively.  
In this study, we implement pre-compensation algorithms at the transmitter side and post-equalization algorithms at the receiver side to minimize the implementation penalty. Turning off the turbulence emulator by sending blank phase patterns to the SLMs in the turbulence emulator does not eliminate all OAM mode distortions caused by the imperfect SLM screens, especially at the OAM mode generation and detection steps.