Sodium laser guide star (LGS) adaptive optics (AO) mitigates the aberrations induced by atmospheric turbulence to achieve a near diffraction limited image, and acts as an essential setup for large ground-based optical telescopes. Unfortunately, the classical AO system with a single sodium LGS achieves good resolution within a field of view (FOV) smaller than several arcseconds in the visible wavelength range, which severely limits the application of AO system for observations of more objects. In order to increase the FOV, multi-conjugated adaptive optics (MCAO) was proposed. In MCAO, the atmosphere turbulence is sensed by sodium LGSs in different sights with independent wavefront sensors respectively, then a three-dimensional map of the atmosphere is reconstructed by tomography, and finally, the turbulence at different altitudes are compensated by a series of deformable mirrors. To enhance the performance of the MCAO, it is usually necessary to optimize the conjugated heights and number of deformable mirrors (DMs), enhance the photon return of the LGSs, and so on. However, the number and configuration of the sodium LGSs are the main factors which can affect the tomography performance of the MCAO system. In our study, we report a method to optimize the number and configuration of the LGSs to maximize the tomography performance and uniformity within the FOV. We hope this work will be of guiding significance to the engineers of AO systems.

In order to study the influence of sodium LGS constellation on the atmosphere turbulence tomography performance of the MCAO, for a 10 m diameter telescope with a FOV of 1′, an atmosphere model for Lijiang Gaomeigu [median value of Fried parameter (r₀) is 8 cm at a wavelength of 550 nm] with a two-layer MCAO system is established, with the two layers located at 0 km and 8 km. The projection matrix for three sodium LGSs equispaced at the edge of the FOV can be calculated. Then, the LGSs are seen by three 25×25 Shack-Hartmann wavefront sensors independently to compute the MCAO high-order correction. At last, tomography is performed for the two turbulent layers. The research is numerically studied by using the OOMAO simulation platform. The Strehl ratio (SR) map at H band in the FOV is obtained, as shown in Fig. 4. The mean value of SR, RSˉ, and the normalized standard deviation (STD) of the SR map are used to evaluate the tomography performance and uniformity of the MCAO.

For the three sodium LGSs equispaced at the edge of the FOV, the simulation results show that the reconstructed wavefront at the ground layer is more accurate than that at the high layer. The reason is that the footprint from each LGS illuminates only a portion of the meta-pupil at the high layer, while the footprint and the meta-pupil are equal at the ground layer. Meanwhile, the maximum SR along the sight of LGS reaches 0.69, and the minimum SR 0.45 appears at the intersection of two LGS sights. At the same time, the performance evolution of 500 frame images for the MCAO system tomography with three sodium LGSs during 1 s is simulated. The mean values of SR and STD are 0.42 and 0.226, respectively.

Then, the turbulence tomography is performed for 3-6 sodium LGSs located at the vertices of regular polygons or the vertices combined with the center of FOV (Fig. 7), and the influence of the zenith angle of the regular polygons is also analyzed. Under the condition that the zenith angle is 41.459″, the maximum value of RSˉ is 0.53 for the configurations of 3+1 and 5+1 LGSs. For all configurations of the constellations, the values of STD in the FOV range from 0.22 to 0.23. For the zenith angles ranging from 6″ to 41.459″, the performance of three LGSs located at the vertices of a triangle reaches the optimum when the zenith angle is half of the FOV. For other constellations, the SR is improved from 0.4 to 0.55 with the increase of the zenith angle (Fig. 9) and the STD is decreased when the zenith angle is increased (Fig. 10).

In our study, for a MCAO with several LGSs, the SR reaches the maximum along the sight of LGS while the minimum appears at the intersection of two LGS sights. For the FOV of 1′, the turbulence can be detected comprehensively by the constellation of 3+1 LGSs with a large zenith angle of the regular polygons. For the zenith angle of the regular polygons less than half of the FOV, the SR reaches the maximum with the constellation pattern of 4 LGSs, while for the zenith angle larger than half of the FOV, the optimal performance is reached with the constellation configuration of 5+1 LGSs. For all the configurations of constellations, the uniformity is improved with the increase of the zenith angle, and for the configuration of constellation of 4 LGSs, a preferable uniformity can be obtained.