# Eigenmode Wavefront Decoupling Algorithm for LC-DM Adaptive Optics Systems
## 3 Materials and Methods
### Decoupling of the LC-DM AO System  
There are three steps in the decoupling procedure. The first step is wavefront decomposition. For the proposed LC-DM cascaded AO system, the DM is required to correct low-order large amplitude aberrations to meet the imaging requirements of the infrared waveband. To make the DM correct only the first \(N\) eigenmodes, the coefficient matrix m is selected as follows:  
\[m^{\prime}=I_{N}\cdot m \tag{11}\]  
where \(I_{N}\) is an n \(\times\) n identity matrix, the first \(N\) diagonal elements are 1, and the rest of the diagonal elements are 0, that is  
\[I_{N}= \begin{bmatrix}1&&&&0\\ &\ddots&&\\ &&1&&\\ &&&0&&\\ &&&&\ddots&\\ 0&&&&0\end{bmatrix}  \tag{12}\]  
Figure 4: Simulation results of DM eigenmode correction (**a**) before and (**b**) after correction. Substituting Formulas (11) and (12) into Formula (10), the control voltage vector \(v^{\prime}\) can be used to calculate for the correction of the first N eigenmodes:  
\[v^{\prime}=v_{M}\cdot m^{\prime}=v_{M}\cdot I_{N}\cdot M^{-1}\cdot\overline{\varphi} \tag{13}\]  
After voltage calculation, the actual surface shape of the DM is obtained by employing the voltage and the DM response matrix. After the subtraction of the actual DM surface shape from the aberration to be corrected, the residual wave surface is sent to the LC:  
\[\overline{\varphi}_{LC}=\overline{\varphi}-\overline{\varphi}_{DM}=\overline{ \varphi}-\overline{R}\cdot v^{\prime} \tag{14}\]  
where \(\overline{\varphi}_{LC}\) is the wave surface to be corrected, which is sent to the LC and \(\overline{\varphi}_{DM}\) is the surface shape generated with the DM. Because the LC has thousands of pixels, if individual pixels are driven one by one, the calculation will be highly complicated and time-consuming; hence, the LC is driven with the Zernike mode surface. The wavefront correction process of the LC is as follows [12; 13]  
\[C=T^{-1}\cdot\overline{\varphi}_{LC} \tag{15}\]  
where \(C\) and \(T\) are the control signal and response matrix of the LC, respectively. So far, we have obtained the wavefront that needs to be corrected by the LC. This part of the wavefront is the residual wavefront after the DM correction. The second step is to obtain the coupling term between the LC corrector and the DM. In the LC-DM cascade AO system, the DM uses the eigenmode method for wavefront decomposition and reconstruction, which is different from the LC with the Zernike modes. The DM eigenmodes are obtained from the DM response, thus the DM can accurately fit its own eigenmodes without affecting the LC correction. The DM control voltage is calculated using Formula (13). For the LC corrector, the residual aberration is corrected using the Zernike modes. As there is no orthogonal relationship between the Zernike modes and the eigenmodes, the LC will generate the eigenmodes that have been corrected by the DM. That is to say, there are aberrations that have been corrected by the DM, which causes a reduction in the final correction performance. This part of the DM-corrected aberration \(\overline{\varphi}_{coup}\) is the coupling term to be calculated and eliminated. We first calculate the wavefront generated by the LC without decoupling. The control matrix \(C\) for the LC can be calculated using Formulas (13)-(15). The LC response wavefront can be expressed as follows:  
\[\varphi_{LC\_response}=T\cdot C \tag{16}\]  
There are aberrations that have been corrected by the DM in \(\varphi_{LC\_response}\). To avoid the repetitive generation of the coupling aberration \(\overline{\varphi}_{coup}\) by the LC, the DM eigenmodes \(M\) is used to reconstruct the LC wavefront \(\varphi_{LC\_response}\)  
\[\varphi_{LC\_response}=M\cdot m_{LC} \tag{17}\]  
where \(m_{LC}\) are the eigenmode coefficients corresponding to \(\varphi_{LC\_response}\).