# Research on a Decoupling Algorithm for the Dual-Deformable-Mirrors Correction System
## 2 Principle of Wavefront Correction with Dual Deformable Mirrors
### Principle of Separation Coefficient Algorithm  
The distorted wavefront generated by the interference of atmospheric turbulence can be expanded using Zernike polynomials, and the expression for wavefront \(\phi(\rho)\) is [11; 12]:  
\[\phi(\rho)=\sum_{i=1}^{\infty}a_{i}Z_{i}(\rho) \tag{1}\]  
Figure 1: Schematic diagram of dual-deformable-mirrors wavefront correction system. In the equation, \(a_{i}\) represents the \(i\)-th order Zernike coefficient and \(Z_{i}(\rho)\) represents the \(i\)-\(th\) order Zernike polynomial [12]. This study uses the Zernike polynomial defined by Noll in polar coordinates, whose expression is [13]:  
\[Z_{i}= \begin{cases}\sqrt{n+1}R_{n}^{m}(\rho)\sqrt{2}\cos(m\theta)&m\
eq 0\text{ \emph{and} }m\text{ \emph{is even}}\\ \sqrt{n+1}R_{n}^{m}(\rho)\sqrt{2}\sin(m\theta)&m\
eq 0\text{ \emph{and} }m\text{ \emph{is odd}}\\ \sqrt{n+1}R_{n}^{m}(\rho)&m=0\end{cases}  \tag{2}\]  
Among them, \((\rho,\theta)\) is the polar coordinate. The integers \(m\) and \(n\) represent the angular and radial levels, respectively, and there is a relationship between the two as follows:  
\[ \begin{cases}m\leq n\\ n-|m|=k\quad k\in N\end{cases}  \tag{3}\]  
And the radial polynomial \(R_{n}^{m}(\rho)\) is defined as [14]:  
\[R_{n}^{m}(\rho)=\sum_{s=0}^{\frac{n-m}{2}}\frac{(-1)^{s}(n-s)!}{s![(n+m/2)-s]! [(n-m/2)-s]!}\rho^{n-2s} \tag{4}\]  
The lower-order modes of Zernike polynomials in the circular domain are consistent with Seidel aberrations, and the specific expressions for the aberrations represented by the first 15 orders of Zernike polynomials are shown in Table 1.  
Assuming the influence function of the \(i\)-th actuator of the DM is \(f_{i}\), it can generally be measured using a Zygo interferometer or wavefront sensor. The \(j\)-th order Zernike mode is denoted as \(Z_{j}\). The cross-correlation matrix \(C_{f_{2}}\) between the influence function of the DM and the Zernike modes can be represented as [9]:  
\[C_{f_{2}}(i,j)=\int f_{i}(r)Z_{j}(r)d^{2}r \tag{5}\]  
Similarly, the expression for calculating the autocorrelation matrix \(C_{ff}\) between the influence functions of the DM actuator is as follows [9]:  
\[C_{ff}(i,j)=\int f_{i}(r)f_{j}(r)d^{2}r \tag{6}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**i** & **(n, m)** & **Polar Coordinates** & **Cartesian Coordinates** & **Aberration Type** \\ \hline
1 & \((0,0)\) & \(1\) & \(1\) & Piston \\
2 & \((1,1)\) & \(2\rho\cos\theta\) & \(2x\) & X-direction Tilt \\
3 & \((1,-1)\) & \(2\rho\sin\theta\) & \(2y\) & Y-direction Tilt \\
4 & \((2,0)\) & \(\sqrt{3}(2\rho^{2}-1)\) & \(\sqrt{3}(2x^{2}+2y^{2}-1)\) & Defocus \\
5 & \((2,-2)\) & \(\sqrt{6}\rho^{2}\sin 2\theta\) & \(2\sqrt{6}xy\) & Y-direction Coma \\
6 & \((2,2)\) & \(\sqrt{6}\rho^{2}\cos 2\theta\) & \(y^{2}-x^{2}\) & X-direction Coma \\
7 & \((3,-1)\) & \(2\sqrt{2}(3y^{2}-2\rho)\sin 2\theta\) & \(2\sqrt{2}(3x^{3}+3 xy^{2}-2x)\) & Y-direction Astigmatism \\
8 & \((3,1)\) & \(2\sqrt{2}(3y^{3}-2\rho)\cos 2\theta\) & \(2\sqrt{2}(3x^{3}+3 xy^{2}-2y)\) & X-direction Astigmatism \\
9 & \((3,-3)\) & \(2\sqrt{2}\rho^{3}\sin 3\theta\) & \(2\sqrt{2}(3y^{2}-x^{3})\) & Trefoil Aberration \\
10 & \((3,3)\) & \(2\sqrt{2}\rho^{3}\cos 3\theta\) & \(2\sqrt{2}(y^{3}-3x^{2}y)\) & Trefoil Aberration \\
11 & \((4,0)\) & \(\sqrt{5}(6\rho^{4}-6\rho^{2}+1)\) & \(\sqrt{5}(6y^{4}+12x^{2}y^{2}+6x^{4}-6x^{2}-6y^{2}+1)\) & Spherical Aberration \\
12 & \((4,2)\) & \(\sqrt{10}(4\rho^{4}-3\rho^{2})\cos 2\theta\) & \(\sqrt{10}(4y^{4}-4x^{4}+3x^{2}-3y^{2})\) & \\
13 & \((4,-2)\) & \(\sqrt{10}(4\rho^{4}-3\rho^{2})\sin 2\theta\) & \(\sqrt{10}(8 xy^{3}+8x^{3}y-6 xy)\) & \\
14 & \((4,4)\) & \(\sqrt{10}\rho^{4}\cos 4\theta\) & \(\sqrt{10}(4x^{4}-6x^{2}y^{2}+y^{4})\) & Quatrefoil Aberration \\
15 & \((4,-4)\) & \(\sqrt{10}\rho^{4}\sin 4\theta\) & \(\sqrt{10}(4 xy^{3}-4x^{3}y)\) & Quatrefoil Aberration \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Aberration expressions for the first 15 orders Zernike polynomials.