The stochastic parallel gradient descent (SPGD) algorithm is one of the most commonly used control algorithms for wavefront sensorless adaptive optics (AO) systems. This method usually uses the driving voltages of deformable mirrors as control parameters, and the number of actuators is equal to the number of dimensions of the control parameters. It is simple and suitable for AO systems that have a small number of actuators and do not have any requirement for the convergence speed. With the increasing applications of AO, the number of actuators required has gradually increased.

In wavefront sensorless AO systems, when taking the driving voltages of the deformable mirrors as control parameters, an increase in the number of actuators leads to a greater number of dimensions of the control parameters and a larger optimization space of the algorithm, which will lead to slower convergence of the algorithm. Various modal coefficients are often used as control parameters to reduce the dimensions of control parameters. When the modal coefficients are modeled as control parameters, the optimization space of the algorithm can be reduced, and the convergence speed can be improved.

The Zernike polynomial, which was introduced by Zernike to represent the diffraction effects on concave mirrors, is often used to describe optical wavefront aberrations. When using the Zernike mode to calculate the covariance matrix of the amplitude in Kolmogorov turbulence, namely, the Noll matrix, there are nonzero elements outside the diagonal. This inherent modal crosstalk indicates the statistical dependence between modes, which limits the correction ability of AO systems based on these modes. In this study, the Karhunen–Loève (K-L) modal coefficients derived from the Zernike mode are used as the control parameter of a wavefront sensorless AO system. First, the rationality of the K-L mode is analyzed. The aberration-fitting ability of the deformable mirror (DM) to the K-L and Zernike modes is then discussed. Finally, the convergence speed and correction effect of the AO system are compared when the driving voltages of the actuators, K-L modes, and Zernike modes are used as control parameters.

Generally, the order of the mode needs to be determined based on the fitting ability of the deformable mirror, so that the dimension of control parameters can be relatively small while ensuring the correction ability of the deformable mirror. The Zernike modes and K-L modes are fitted with several deformable mirrors with 32, 61, 97, and 127 actuators, respectively. The results show that the fitting ability is relatively stable for K-L modes while fluctuations appear for Zernike modes (Fig.3). We use the error rate (η) as the evaluation standard. The fitting is effective if η<1. The lower the error rate, the better the fitting ability. Notably, 32-element, 61-element, 97-element, and 127-element deformable mirrors can fit the first 22-order, 55-order, 79-order, and 91-order K-L modes, respectively, while they can fit the first 20-order, 36-order, 54-order, and 68-order Zernike modes (Fig.4), respectively. It can be seen from the above data that the ability to fit K-L modes for the deformable mirror is greater than that to fit Zernike modes. A greater number of modes indicates better correction ability, which implies that the correction capability and convergence speed of AO systems can be improved when K-L modal coefficients are used as the control parameters.

The two modal methods only need 20 modes as control parameters when the atmospheric turbulence strength (D/r₀) is 5, and the convergence of the conventional SPGD is used as the reference index. When the Strehl ratio (SR) is up to 0.8, the K-L modal method, Zernike modal method, and conventional SPGD require 122, 139, and 180 iterations, respectively. The convergence speed of the K-L modal method and Zernike modal method is 47.5% and 29.4% greater than that of the conventional SPGD control algorithm, respectively (Fig.5). The correction results also show that when the D/r₀ is 10 (Fig. 6), 15 (Fig.7), and 20 (Fig.8), the correction performance and convergence speed obtained using K-L modal coefficients are better than those obtained using Zernike modal coefficients as control parameters (Table 1).

The SPGD control algorithm, based on optimizing the actuator voltages, is widely used as a control algorithm for wavefront sensorless AO systems. The number of actuators in the deformable mirror determines the dimensions of the control parameters. Generally, the greater the number of actuators, the better the correction effect. Moreover, the more actuators tend to reduce the convergence speed of the AO system. The SPGD algorithm, which is based on optimizing the modal coefficients, can effectively resolve this contradiction. When the control parameters are modal coefficients, the optimization space of the algorithm can be reduced, and the convergence speed can be improved.

The fitting capability of the DM to the aberrations of K-L modes and Zernike modes is compared and analyzed. The convergence speed and correction performance of the AO system are investigated when the voltages of the actuators, K-L modes, and Zernike modes are used as control parameters under various turbulence strengths. The convergence speed of the K-L modal method and Zernike modal method is 47.5% and 29.4% greater than that of the conventional SPGD control algorithm, respectively. The results under several turbulence strengths also show that the correction performance and convergence speed of the K-L modal method are better than those of the Zernike modal method. The results of the study can provide a reference for the practical application of the SPGD control algorithm based on K-L modes.