## 3 High-Dynamic Atmospheric Turbulence Compensation Algorithm
### Capturing Aberrations
According to [32], consider the 2 nd and 3 rd order of Zernike polynomials in Cartesian coordinate system marked as \(\mathbf{Zernike}_{2}\) and \(\mathbf{Zernike}_{3}\), as shown in Equation (18). \[ \begin{split}\mathbf{Zernike}_{2}&=2\mathbf{x},\\ \mathbf{Zernike}_{3}&=2\mathbf{y},\end{split}  \tag{18}\]  
where \(\mathbf{x}\) and \(\mathbf{y}\) are coordinate vectors within the normalized area. The formulas above reveal that the 2 nd and 3 rd of the Zernike polynomials vary monotonically with \(\mathbf{x}\) and \(\mathbf{y}\). They depict the horizontal and vertical tilt of the phase screen which demands barely any capability of local deformation. This fits the deformation capability of FSM shown in Figure 3c which forms a tilt shape with a spatial resolution of two. In contrast, the phase screens described by the fourth and above Zernike polynomials are not a plane, for example, the 4 th\(\sim\)6 th order of Zernike polynomials in Equation (19). There are local fluctuations in the phase screen, which require a higher spatial resolution to be expressed. Therefore, the workload allocated to FSM is to compensate the 2 nd and 3 rd order Zernike aberrations while the rest of the wavefront aberrations belong to the DM. \[ \begin{split}\mathbf{Zernike}_{4}&=2\sqrt{3}(\mathbf{x}^{ 2}+\mathbf{y}^{2})-\sqrt{3},\\ \mathbf{Zernike}_{5}&=2\sqrt{6}\mathbf{x}\mathbf{y},\\ \mathbf{Zernike}_{6}&=\sqrt{6}(\mathbf{x}^{2}-\mathbf{y}^{2}) \end{split}  \tag{19}\]  
Assuming the coefficients of Zernike polynomials are,  
\[\mathbf{a}=[a_{2},a_{3},\cdots,a_{15}]^{T} \tag{20}\]  
where subscripts symbolise order. Decoupled offsets expressed in \(offset_{Decoupled,low}\) and \(offset_{Decoupled,high}\) (tilts aberrations and high-order aberrations, respectively) are gained according to the following procedure. \[\mathbf{offset}_{Decoupled,Low}=Z\times I_{Low}\times\mathbf{a}\div\frac{2\pi}{ \lambda f} \tag{21}\]  
\[\mathbf{offset}_{Decoupled,High}=Z\times I_{High}\times\mathbf{a}\div\frac{2\pi}{ \lambda f} \tag{22}\]  
where, \(Z\) is the wavefront restoration matrix which corresponds to 2 nd\(\sim\)15 th order Zernike polynomials, while \(I_{Low}\) and \(I_{High}\) are shown as follows  
\[I_{Low}= \begin{bmatrix}1&&&\\ &1&&\\ &&0&&\\ &&\ddots&\\ &&&0\end{bmatrix} ,I_{High}= \begin{bmatrix}0&&&\\ &0&&\\ &&1&&\\ &&\ddots&\\ &&&1\end{bmatrix}  \tag{23}\]  
Denoting \(\mathbf{offset}\) with a subscript containing '\(low\)' as spots offsets caused by low-order aberrations while \(\mathbf{offset}\) with a subscript containing '\(high\)' as spots offsets caused by high-order aberrations. Representing iterative target as \(\mathbf{offset}_{Decoupled,Low,Target}\) and \(\mathbf{offset}_{Decoupled,High,Target}\), they are set as follows  
\[ \begin{split}\mathbf{offset}_{Decoupled,Low,Target}=f_{FSM}(\mathbf{s} _{FSM})&+(-\mathbf{offset}_{Decoupled,Low,Residual})\\ &+(-\mathbf{offset}_{Decoupled,Low,Detected})\\ \mathbf{offset}_{Decoupled,High,Target}=f_{DM}(\mathbf{s}_{DM})&+(-\mathbf{offset}_{ Decoupled,High,Residual})\\ &+(-\mathbf{offset}_{Decoupled,High,Detected})\end{split}  \tag{24}\]  
where \(f_{FSM}(\mathbf{s}_{FSM})\) and \(f_{DM}(\mathbf{s}_{DM})\) illustrate model values, \(\mathbf{offset}_{Decoupled,Low,Residual}\) and \(\mathbf{offset}_{Decoupled,High,Residual}\) are the decoupled wavefront residual of the previous turn, while \(\mathbf{offset}_{Decoupled,Low,Detected}\) and \(\mathbf{offset}_{Decoupled,High,Detected}\) are decoupled current spots offsets. As we are expected to drive the wavefront correctors eliminating the detected offsets, the target should be set opposite to the decoupled current spots or decoupled wavefront residual. The above method effectively suppresses the wavefront residual and distributes the workload. Meanwhile, it saves the stroke of DM by passing the pressure of the stroke to FSM.