# Eigenmode Wavefront Decoupling Algorithm for LC-DM Adaptive Optics Systems
## 3 Materials and Methods  
### Construction of the DM Eigenmode Matrix  
In most AO systems with dual (multi) DMs, the Zernike mode method is used to decompose wavefront aberrations. Then, DMs are used to produce the conjugated Zernike mode to correct distortion and aberrations. However, due to their parameters and other factors, DMs cannot achieve a perfect fit to the Zernike mode. The \(i\)th response function of the DM \(R_{i}\) was described as a Gaussian function in Formula (1). The LC was described by 209 Zernike modes, and the dimension of the wavefront is 256 \(\times\) 256 pixels. \[R_{i}=\exp\left[\ln\omega\bigg{(}\sqrt{(x-x_{i})^{2}+(y-y_{i})^{2}}/d\bigg{)} ^{a}\right] \tag{1}\]  
where \(\alpha\),\(\omega\), and \(d\) are the Gaussian exponent, the coupling value, and the spacing between the actuators in the DM, respectively. In this model, \(\alpha\) was 1.73, \(\omega\) was 0.23, and \(d\) was set to 0.083. These parameters are normalized values from the DM that was purchased from ALPAO. Figure 2 presents the residual aberration histogram of the top 54 Zernike modes with the 145 unit-DM, when the DM actuator stroke is not considered. Correction residual aberrations exhibit an overall upward trend with an increase in the number of modes mainly because when the number of modes increases, the spatial frequency of the wavefront is high, and the wavefront is considerably complicated. DMs cannot satisfy the requirements of high spatial resolution; the correction effect can be poor. Therefore, when the Zernike mode is used for wavefront decomposition and correction, fitting errors inevitably occur. Next, we give the derivation procedure of the DM eigenmodes. The response matrix, a two-dimensional matrix, is measured using the WFS. The response function of each actuator represents the response surface of each actuator. The final surface shape of the DM can be expressed through a linear combination of the response functions of each actuator:  
\[\varphi=\sum_{i=1}^{n}v_{i}R_{i} \tag{2}\]  
Figure 2: Fitting error of the top 54 Zernike modes with 145-unit _DM_. where n represents the number of actuators and \(v_{i}\) denotes the voltage applied using the \(i\)th actuator. The coupling relationship between the response functions can be expressed with a coupling matrix \(C\), where the coupling values \(C(i,j)\) of the \(i\)th and \(j\)th actuators are calculated as follows:  
\[C(i,j)=C(j,i)=D^{-1}\int_{D}R_{i}R_{j}dxdy \tag{3}\]  
where \(D\) is the DM aperture. \(C\) describes the correlation of the influence functions of different actuators and may be decomposed using the singular value decomposition method:  
\[C=USU^{T} \tag{4}\]  
where \(S\) is the diagonal matrix whose diagonal elements are the eigenvalue of matrix \(C\), \(U\) is the unitary matrix comprising the eigenvector of \(C\), and \(U-1=UT\). The linear combination of the response function of each actuator and the coupling matrix eigenvector constitutes a new two-dimensional matrix, the DM eigenmodes:  
\[M_{i}=\sum_{j=1}^{n}U(i,j)R_{j},i=1,2,\cdots,n \tag{5}\]  
where \(M_{i}\) is the \(i\)th DM eigenmodes. As \(U(i,:)\) is the eigenvector of the \(i\)th actuator, the eigenvectors are orthogonal, so the eigenmodes are also orthogonal. There are n eigenmodes, which is the same as the number of actuators. A random wavefront may be described with the following eigenmodes:  
\[\varphi=\sum_{i=1}^{n}m_{i}M_{i} \tag{6}\]  
where \(m_{i}\) is the \(i\)th coefficient of \(M_{i}\). Considering a 145-element DM from ALPAO in the laboratory as a prototype, Figure 3 shows the surface shapes of the top 27 eigenmodes. From the derivation procedure of the DM eigenmodes, it can be seen that the eigenmodes have the following properties: (1) The number of eigenmodes of a DM is equal to the number of its actuators. (2) The eigenmodes of the DM are orthogonal to each other. Figure 3: Top 27 eigenmodes’ profiles of the 145-unit _DM._  
(3) As the number of modes increases, the spatial frequency of the eigenmodes increases gradually.