Single-frame far-field wavefront retrieval method based on Walsh function modulation
Kong Qingfeng1,2,3,4, Wang Shuai1,3, Yang Ping1,3, Lin Haiqi1,3,4, Liu Yong2, Xu Bing1,3     
1. Key Laboratory of Adaptive Optics, Chinese Academy of Sciences, Chengdu, Sichuan 610209, China;
2. School of Optoelectronic Science and Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, China;
3. Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu, Sichuan 610209, China;
4. University of Chinese Academy of Sciences, Beijing 100049, China

Overview: Phase retrieval (PR) is an iterative process of wavefront recovering from known intensity distribution. Owing to high detection accuracy and lower environment requirement, PR has become an attractive candidate to the wavefront sensor (WFS) in adaptive optics. With the development of computer speed, PR will have greater potentiality in active optic systems.

Gerchberg and Saxton proposed firstly the GS algorithm to recover the wavefront aberration. This algorithm can achieve typical convergence, but its iterative process easily falls into stagnation. For instance, the true pupil field f(x, y) and its twin f*(-x, -y) have the same Fourier modulus, so the algorithm tries to recover both together and goes nowhere. For overcoming the two-fold ambiguity of GS algorithm, phase diversity (PD) algorithm was proposed by Gonsalves in 1979. However, this method introduces a defocused plane to increase the constrained intensity information to reconstruct the wavefront, which will sacrifice the simplicity. Löfdahl proposed PD sensor with a beam splitter to make the focus and defocused images be captured by one CCD. For the same purpose a PD sensor with a distorted diffraction grating was presented by Blanchard in 2000. Nevertheless, more complex structure, interaction of high frequency information and dynamic range of CCD limit the practical application of these methods in the PD.

In 2008, Min Li and Xinyang Li proposed a method based on the linear phase retrieval (LPR) to reconstruct small aberrations from a single far field image. In 2010, the linearized focal plane technique (LIFT) was presented by Serge Meimon, which is only effective for lower order aberration. Bing Dong et al demonstrated a hybrid phase retrieval algorithm using a combination of LPR and GS in 2015. The estimation result of LPR is used as a prior knowledge to speed convergence of GS. In this way, the higher order aberrations are basically recovered, but the problem of multiple solutions remains unsolved.

In order to solve the problems mentioned above in traditional single-frame phase retrieval algorithm, a feasible method is proposed to break the rotational symmetry of near-field wavefront phase. If continuous aberration is used for wavefront modulation, it is equivalent to adding a wavefront on the measured wavefront. This cannot solve the problem of multiple solutions.

A wavefront reconstruction method based on Walsh function two-dimensional discrete phase modulation is proposed in this paper. This method can effectively break the symmetry of near-field wavefront and overcome problem of PR multiple solutions. The basic principle and limitation of this method are introduced. The wavefront retrieval accuracy of the proposed algorithm is compared to the traditional GS algorithm through simulations and experiments. The results show that the wavefront retrieval algorithm based on phase modulation of Walsh function can accurately reconstruct the wavefront aberration under the condition of single far-field image.

Supported by National Natural Science Foundation of China (61805251) and Chinese Academy of Sciences Youth Innovation Promotion Agency (2017429)