# Research on a Decoupling Algorithm for the Dual-Deformable-Mirrors Correction System
## 1 Introduction  
The Woofer-Tweeter system is a typical dual-deformable-mirrors wavefront correction system [1], consisting of a large stroke, low spatial-frequency deformable mirror (Woofer), and a small stroke, high spatial-frequency deformable mirror (Tweeter). These two deformable mirrors work together to improve the wavefront correction effect. However, when both deformable mirrors are working simultaneously, coupling effects occur, which can lead to the wastage of correction ability by some deformable mirrors. If this phenomenon cannot be effectively suppressed, coupling errors will gradually accumulate, eventually leading to the complete consumption of the stroke of the deformable mirror by coupling. When the aberration to be corrected in the system changes, the deformable mirror affected by coupling will have difficulty in effectively compensating for the aberration, leading to a decrease in the system's correction ability [2]. Therefore, coupling compensation between deformable mirrors has become an important research direction for dual-deformable-mirrors wavefront correction systems in terms of aberration compensation ability and stability. To fully utilize the advantages of adaptive optics dual-deformable-mirrors wavefront correction systems, it is crucial to further improve their performance and eliminate potential coupling between the deformable mirrors. Numerous scholars have addressedthis issue, with different approaches proposed over the years. In 1998, [PERSON] analyzed combining low and high spatial-resolution systems to enhance their spatial correction capability, but this approach required two expensive adaptive optical systems [3]. In 2013, [PERSON] proposed a decoupling control algorithm based on Zernike mode decomposition that greatly reduced costs but could not independently correct high-order aberrations [4]. In 2016, [PERSON] proposed a Zonal decoupling algorithm that utilizes the traditional direct slope method to control both deformable mirrors [5]. This approach effectively suppressed the coupling effect between the deformable mirrors, but was not suitable for engineering applications [2]. In 2018, [PERSON] proposed a slope-based decoupling algorithm that showed better performance in suppressing coupling errors but was also unsuitable for dual-deformable-mirrors wavefront correction systems with FSM and DM combined [6]. In 2021, [PERSON] introduced a decoupling control algorithm based on Numerical Orthogonal Polynomials (NOP), which aimed to eliminate cross coupling between dual wavefront correctors. However, the limitation of this algorithm is that each order of aberration cannot be independently corrected [7]. In 2021, [PERSON] used dual-deformable-mirrors combination control for fiber power coupling, enhancing the efficiency of spatial optical fiber coupling [8]. In 2023, [PERSON] proposed a dual-deformable-mirrors decoupling control method based on pattern projection suppression that enabled the two deformable mirrors to work efficiently together while eliminating low-order correction components in Tweeter [9]. These decoupling algorithms have been successfully applied to various adaptive optical dual-deformable-mirrors wavefront correction systems, significantly reducing coupling effects between deformable mirrors. Nonetheless, few decoupling algorithms are available for dual-deformable-mirrors wavefront correction systems combined with FSM and DM, where each order of aberration cannot be independently controlled or corrected. Different application scenarios have varying requirements for different types of aberrations. Hence, specific aberrations can be corrected according to actual needs to meet the diverse needs of different application scenarios, improving system performance and laying the foundation for large aperture telescope application. In the field of astronomical observation, as the aperture of telescopes continues to increase, so does their sensitivity to atmospheric turbulence.