# Eigenmode Wavefront Decoupling Algorithm for LC-DM Adaptive Optics Systems
###### Abstract  
To realize the decoupling control of the dual corrector of liquid crystal (LC)-deformable mirror (DM) adaptive optics (AO) systems and prevent the loss of correction ability induced by coupling aberrations, a wavefront decoupling method based on DM eigenmodes was proposed. The coupling relationship of two correctors was studied, and the distribution of aberrations corrected using the DM and LC was investigated. First, a DM eigenmode matrix was derived according to the DM influence function. Then, aberrations were decomposed according to the Strehl ratio, and the projection of the LC response matrix on the DM eigenmode matrix, that is, the coupling term, was derived. Before using the LC corrector, the coupling item was eliminated for the decoupling of the two correctors, and the decoupling effect was verified through simulations. Finally, a 2 m telescope LC-DM AO system was developed to validate the proposed method. The simulation and experimental results showed that the root mean square of the DM-corrected residual after 55 eigenmodes' correction was 0.09 \(\upmu\)m. After decoupling correction, the resolution of the 700-1700 nm wide band improved from 1.1 to 1 times the diffraction limit, which is 1.2 times better than that obtained using the traditional Zernike mode correction. a +
Footnote †: journal: Applied Physics Letters A  
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amplitude, while the PV values of high-order aberrations are relatively small, so we call this small amplitude. To simultaneously compensate for the low-order large amplitude and high-order small amplitude aberrations, some researchers have proposed a woofer-tweeter dual DM AO system. The woofer was used to correct large amplitude, low spatial frequency aberrations and the tweeter was employed to correct small amplitude, high spatial frequency aberrations [8]. Currently, the DM with the most units worldwide is installed on the PALM-3000 system on the 5 m Haier Telescope in the United States. For this DM, the number of actuators is 3888, and the driving stroke is only 2 \(\upmu\)m. It must be used in conjunction with a 241-actuator DM to obtain the image of Jupiter's Ganymede in the visible waveband. Due to its complicated coupling and control problems, no follow-up report on imaging in the visible waveband is available, and since 2010, the number of DM actuators has not increased [9; 10]. Liquid crystal (LC) wavefront correctors present the advantages of high density, large amplitude, and low cost, which can compensate for the shortcomings of DMs. The principles and advantages of LC devices can be found in detail from reference [11]. However, due to the slow response of LCs in the long infrared waveband and severe LC dispersion in the broadened waveband, the working waveband of LCs is limited to 700-950 nm for a fast response without dispersion, which is narrower than that of DMs [12]. In 2016, researchers from the Changchun Institute of Optics, Fine Mechanics and Physics proposed a combination of a DM and LC as a wavefront corrector to completely exploit their advantages, overcome their shortcomings, and use the nondispersion feature of DMs to correct large amplitude low-order aberrations in the infrared and visible wavebands. LC is used to correct small amplitude high-order aberrations in the visible waveband, to meet the requirements of wide-band aberration correction [13; 14]. Irrespective of whether a device is a dual DM system or an LC-DM system, the aberrations to be corrected are measured by a wavefront sensor (WFS). The distribution and decoupling methods of aberrations must be studied to prevent the correction loss caused by the opposite aberrations of each corrector, and the AO correction can be rendered unstable [15].