† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11174274, 11174279, 61205021, 11204299, 61475152, and 61405194) and State Key Laboratory of Applied Optics, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences.
Adaptive optics (AO) systems are widespread and considered as an essential part of any large aperture telescope for obtaining a high resolution imaging at present. To enlarge the imaging field of view (FOV), multi-laser guide stars (LGSs) are currently being investigated and used for the large aperture optical telescopes. LGS measurement is necessary and pivotal to obtain the cumulative phase distortion along a target in the multi-LGSs AO system. We propose a high precision phase reconstruction algorithm to estimate the phase for a target with an uncertain turbulence profile based on the interpolation. By comparing with the conventional average method, the proposed method reduces the root mean square (RMS) error from 130 nm to 85 nm with a 30% reduction for narrow FOV. We confirm that such phase reconstruction algorithm is validated for both narrow field AO and wide field AO.
Adaptive optics (AO) technology is widely used to correct the distorted wavefront induced by atmospheric turbulence in real time, and restore the imaging resolution close to the diffraction limit of a telescope.[1] As an efficient wavefront corrector, phase-only liquid crystal on silicon (LCOS) possesses a prominent advantage of high pixel density, which makes it possible to be used in the AO system for large aperture telescopes.[2–7] To obtain images reaching the diffraction limit, a bright source such as a guide star must be available within the isoplanatic patch for aberration measurement. However, the brightness requirement is quite rigorous for arbitrary sky/space objects, which results in a low probability of finding a sufficient number of bright nature guide stars (NGSs) for AO correction. A solution to avoid the scarcity of bright NGSs is to produce artificial laser guide stars (LGS) from the ground by using the back-scattered (or back-radiated) light.[8] Thus, only one NGS is required to sense overall tilt since very faint NGS can still be used for image motion sensing. Several LGSs are used for high order aberration sensing in LGS AO systems.
Unfortunately, a light cone rather than a desired cylinder is produced from the LGS to the telescope, thereby resulting in an error called “focus anisoplanatism” (FA) or the cone effect. This is a serious impediment to the general application of LGS in large aperture telescope AO systems. Foy and Labeyrie in 1985 suggested to use an array of laser beacons for astronomical AO to reduce the FA error.[9,10] In their concept, the cumulative phase distortion along a target can be obtained solely from the LGS measurements. There are some works for this problem,[11–14] but all of these approaches have one aspect in common in that they require a priori knowledge of a real-time-varied profile. Hart proposed an approach for ground layer AO (GLAO) that the signals from the all beacons are sensed separately and averaged to obtain the mean wavefront as the estimation of the aberration.[15–17] In this approach, the weight factors of all LGSs are the same. However, as the FA error depends on the propagation path, it should be different for every LGS so that the weight factors should also be different. To resolve this problem and decrease the wavefront tomographic error, we propose a novel algorithm to calculate these weight coefficients based on the interpolation, and thereby estimate the phase for target without the measurement of the atmospheric turbulence refractive index structure constant
In this paper, the phase reconstruction algorithm is described in detail theoretically, and is then validated in simulation. The algorithm of phase reconstruction is presented in Section 2. We compare the simulation results and performances with the previous reported ones and analyze them in Section 3. Finally, the conclusions are given in Section 4.
Assume that there are n different LGSs located at height H.
![]() | Fig. 1. Schematic diagram of LGSs’ distribution and objective star with elevation angle θ and azimuth angle ψ. Light from LGS: dashed line; light from objective star: dot-dashed line. |
The direction vector of a light from an objective star can be described as (cosθ cosψ, cosθ sinψ, sinθ). Define a vector in the xy plane as
According to the Kolmogorov theory, the turbulence field is a locally uniform and isotropic field. Its statistical properties are independent of the space position within the limited space. Therefore, the statistical aberration moments c and their corresponding square values
We estimate the unknown aberration from the target star along the direction (θ, ψ) by the weighted summation of the detected data. The phase delay at any point on the telescope pupil can be estimated by the interpolation of n phase values measured from LGSs at the same point
To find an array of optimized weight coefficients, we have to analyze the estimation error J, expanded as the following polynomial, at first:
![](cpb_25_9_094214/cpb161108ieqn3.gif)
![](cpb_25_9_094214/cpb161108ieqn1.gif)
LGSs are generally arranged in regular polygon. We consider 6 LGSs that are arranged at the vertices of a hexagon as shown in Fig.
![]() | Fig. 2. Arrangement of six LGSs and their geometric relationship, L = θfovH: (a) plan view, (b) side view. |
The results for Jm, which is a function of ρ, when L = 0 are calculated and shown in Fig.
For the same LGS, the weight coefficient changes with the location of the point in the aperture. Calculate the weight coefficient of each point over the telescope for each LGS. All the weight coefficients when ρ/D = 0.37 are shown in Fig.
![]() | Fig. 4. The weighted coefficients comparison between the proposed method (top) and the average method (bottom) for ρ = 0.37D. |
The optimal ρ for different FOVs are different. Figure
For this work, we have chosen to use the well-known Hufnagel–Valley 5/7 mode. Such profile Troxel determines that four layers can be used to accurately model the turbulence,[18] and the altitudes and strengths of the layers are listed in Table
![]() | Table 1. Altitudes and relative layer strengths of four-layer turbulence model. . |
In our simulation, a 6 m telescope is adopted with six LGSs at height 20 km. The positions of the LGSs have been shown in Fig.
We first consider a narrow FOV with θfov = 0′ so that the target is at an infinite distance and zenith direction. As aforementioned optimization, L = 0, and the optimal ρ is 0.37D = 2.22 m. Therefore, the zenith angle of LGS is 18.3″. The RMS after compensating is shown at Fig.
This method is also applicable in the case of large FOV. Consider that the diameter of FOV is 1 arcmin. Thus, L = 1.2D, and the optimal ρ is 0.55D, i.e., 3.3 m, from Fig.
In this paper, a phase reconstruction algorithm is proposed to estimate the phase for a target with an uncertain turbulence profile based on the interpolation. This algorithm is independent of the complicated measurement of turbulence
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