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摘要: 从斜率复原波前是夏克-哈特曼波前传感器这一类斜率采样探测器的核心流程。传统的复原算法中，区域法对局部波前的复原效果好，但易受斜率噪声的影响，同时空间分辨率较低；模式法抗噪能力强，但没有精确复原局部波前的能力。本文提出了基于B样条函数的快速复原算法，将波前展开为B样条曲面的线性组合，并将复原问题从斜率最小二乘问题转化为泊松方程，利用斜率的Taylor展开式估计散度，再通过超松驰迭代法进行快速求解。该方法将B样条函数的理论散度积分和实际散度估计分离，可以方便地扩展到不同阶次和不同节点数量的B样条基复原算法中。另外，通过改变散度估计的计算区域，可以灵活控制并平衡算法的局部复原能力和抗噪能力。对变形镜驱动器响应函数的测量实验表明，该方法具有较好的局部复原能力、抗噪能力和任意精度的空间分辨率。
Abstract: Traditional schemes for Shack-Hartmann wavefront reconstruction can be classified into zonal and modal methods. The zonal methods are good at reconstructing the local details of the wavefront, but are sensitive to the noise in the slope data. The modal methods are much more robust to the noise, but they have limited capability of recovering the local details of the wavefront. In this paper, a B-spline based fast wavefront reconstruction algorithm in which the wavefront is expanded to the linear combination of bi-variable B-spline curved surfaces is proposed. Then, a method based on successive over relaxation (SOR) algorithm is proposed to fast reconstruct the wavefront. Experimental results show that the proposed algorithm can recover the local details of the wavefront as good as the zonal methods, while is much more robust to the slope noise.-
Key words:
- B-spline function /
- wavefront reconstruction /
- Hartmann wavefront sensor
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图 1 B样条基曲面。(a) 1阶B样条基曲面；(b) 2阶B样条基曲面
Figure 1. Surfaces of B-spline basis. (a) First-order B-spline surface; (b) Second-order B-spline surface
图 2 方形排布的B样条基位置与子孔径之间的关系。(a) Fried；(b) Southwell
Figure 2. Positional relation between B-spline basis and subaperture with a square layout. (a) Fried model; (b) Southwell model
图 5 4号驱动器的干涉仪测量数据
Figure 5. Measurement data of No.4 actuator obtained with ZYGO interferometer
图 6 不同复原方法复原结果。(a) 本文算法重建波前；(b) 本文算法的残余波前；(c) 基于Zernike多项式的模式法重建波前；(d) 模式法的残余波前；(e) Fried区域法重建波前；(f) 区域法的残余波前
Figure 6. Wavefronts restored by different methods. (a) Wavefront restored by our method; (b) Residual wavefront error of (a); (c) Wavefront restored by the modal method; (d) Residual wavefront error of (c); (e) Wavefront restored by the zonal method; (f) Residual wavefront error of (e)
图 7 不同驱动器复原残差
Figure 7. PV and RMS results of residual wavefront reconstructed by different methods
表 1 不同复原方法结果比较(4#驱动器)
Table 1. Comparison results of different wavefront reconstruction methods
数据类型 波前 残差 相关值/% PV/μm RMS/μm PV/μm RMS/μm 原始 1.579 0.159 - - - 区域法 1.493 0.184 0.277 0.026 94.8 Zernike模式法(35项) 1.163 0.259 1.185 0.232 85.1 本文方法 1.612 0.167 0.162 0.028 97.2 
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表 2 不同驱动器复原残差及相关值
Table 2. Residual reconstruction error and correlation values of different actuators
驱动器序号 区域法 Zernike模式法(35项) 本文方法 PV/μm RMS/μm 相关值/% PV/μm RMS/μm 相关值/% PV/μm RMS/μm 相关值/% 1 0.305 0.031 99.4 1.772 0.248 84.3 0.219 0.030 99.8 2 0.355 0.038 99.1 1.715 0.218 82.5 0.299 0.035 99.7 3 0.343 0.033 99.7 1.767 0.237 81.2 0.241 0.031 99.6 4 0.277 0.026 94.8 1.185 0.232 85.1 0.162 0.028 97.2 5 0.258 0.031 98.7 1.825 0.258 85.6 0.196 0.023 99.8 6 0.323 0.028 99.6 1.754 0.243 78.3 0.172 0.027 99.7 7 0.332 0.034 99.5 2.004 0.261 82.9 0.217 0.033 99.6
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