# Eigenmode Wavefront Decoupling Algorithm for LC-DM Adaptive Optics Systems
## 4 Results
### Results of Eigenmode Correction  
First, the initial 55 eigenmodes were experimentally corrected according to Section 3.4. The turbulence simulator was placed at the system aperture diaphragm. The system aberrations detected through the WFS were PV = 4.32 \(\upmu\)m and RMS = 0.71 \(\upmu\)m (Figure 7). After the correction of the top 55 aberrations through the DM, the PV and RMS decreased to 1.21 and 0.08 \(\upmu\)m, respectively. Figure 7b presents the wavefront of the residual aberration. According to the relationship between the residual RMS and the number of eigenmodes reported in [24], after the correction of 55 eigenmodes, the wavefront residual RMS can be lowered to approximately 0.47 rad, that is 0.07 \(\upmu\)m, for the 950 nm waveband. Hence, the correction results are consistent with the theoretical results, and better than the finding of an RMS of 0.12 \(\upmu\)m with the Zernike modes presented in [13].  
After we obtain the wavefront aberration of the system, we can simulate the point spread function (PSF) of the system and calculate the optical transfer function (OTF) of the system by Fourier transform of the PSF, and then take the absolute value of the OTF, that is, the modulation transfer function (MTF) of the system. The MTF reflects the relationship between spatial resolution and signal intensity. Since the RMS of the wavefront residual aberration is 0.07 \(\upmu\)m, which is better than 1/14 \(\lambda\) in the 0.95-1.7 \(\upmu\)m waveband, the MTF curve will be close to the diffraction limit. In order to experimentally prove that the resolution can reach the diffraction limit, the CG-USAF-1951-0 standard resolution plate was used as the observation target, and the imaging experiment was performed. The USAF target was placed near the light source, and the image was captured with the infrared camera before and after AO correction. Figure 8a shows the USAF target captured using the infrared camera in the 1.5-1.7 \(\upmu\)m waveband before and after the DM correction. Before the DM correction, the camera was blurred, and the image details could not be distinguished. After the DM correction of the top 55 eigenmodes, the fifth element of the fifth group in the USAF target was distinguished, and the corresponding element spatial frequency was 50.8 line pairs/mm, that is, the resolution was 19.7 \(\upmu\)m. The diffraction limit resolution of an optical system can be calculated by 1.22 \(\frac{\lambda}{D}f\), where \(\lambda\) is the wavelength, \(D\) is the pupil diameter, and \(f\) is the effective focus length. In this AO system, the aperture of the entrance pupil was 20 mm, the effective focus length was 200 mm, and the diffrac  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Lens** & **Focus Length (mm)** & **Aperture (mm)** \\ \hline L1 & 200 & 20 \\ L2 & 548 & 20 \\ L3 & 300 & 20 \\ L4 & \(-\)43.5 & 2.9 \\ L5 & 100 & 2.9 \\ L6 & 200 & 5.8 \\ L7 & 150 & 2.9 \\ L8 & 300 & 5.8 \\ L9 & 300 & 5.8 \\ L10 & 272 & 5.8 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of each lens in the LC–DM adaptive optics system.  
tion limit at the central wavelength of 1.6 \(\upmu\)m was 19.5 \(\upmu\)m. Therefore, the image of the 1.5-1.7 \(\upmu\)m waveband attained diffraction-limited resolution after the DM correction. Figure 8b shows the 950-1500 nm waveband before and after correction. After correction, the first element of the sixth group was resolved. The corresponding spatial frequency was 64 line pairs/mm, that is, the resolution was 15.6 \(\upmu\)m. The diffraction-limited resolution of the central wavelength at 1.2 \(\upmu\)m was 14.6 \(\upmu\)m. Therefore, the image in the 950-1500 nm waveband achieved diffraction-limited resolution.  
Figure 7: Wavefront detected with the WFS: (**a**) original and (**b**) residual wavefront after correction with the DM.