孔慶峰，王 帥，楊 平，林海奇，劉 永，許 冰

基于Walsh函數(shù)調(diào)制的單幀遠(yuǎn)場(chǎng)波前反演方法

孔慶峰1,2,3,4，王 帥1,3*，楊 平1,3*，林海奇1,3,4，劉 永2，許 冰1,3

1中國(guó)科學(xué)院自適應(yīng)光學(xué)重點(diǎn)實(shí)驗(yàn)室，四川 成都 610209；2電子科技大學(xué)光電科學(xué)與工程學(xué)院，四川 成都 610054；3中國(guó)科學(xué)院光電技術(shù)研究所，四川 成都 610209；4中國(guó)科學(xué)院大學(xué)，北京 100049

單幀遠(yuǎn)場(chǎng)圖像重構(gòu)波前像差的方法在結(jié)構(gòu)簡(jiǎn)便性方面有獨(dú)特的優(yōu)勢(shì)。然而，傳統(tǒng)的基于單幀遠(yuǎn)場(chǎng)圖像的波前重構(gòu)算法存在多解問(wèn)題，算法收斂過(guò)程容易陷入停滯。本文在分析單幀相位反演方法的多解問(wèn)題成因的基礎(chǔ)上，提出了一種基于Walsh函數(shù)的二維離散相位調(diào)制的波前重構(gòu)方法。此種方法可以有效地打破近場(chǎng)波前的對(duì)稱性，解決多解問(wèn)題。仿真結(jié)果表明，該方法只需一個(gè)遠(yuǎn)場(chǎng)圖像就可以精確地重建波前像差。

相位反演；像差；遠(yuǎn)場(chǎng)；二維離散調(diào)制

1 引 言

相位反演技術(shù)(Phase retrieval，PR)是一種通過(guò)已知的二維光強(qiáng)分布來(lái)恢復(fù)波前像差的方法。由于其檢測(cè)精度高、環(huán)境要求低，它被廣泛應(yīng)用于激光質(zhì)量評(píng)價(jià)[1]、自適應(yīng)光學(xué)[2-3]、X射線相位對(duì)比成像[4-5]、天文觀測(cè)[6-8]等領(lǐng)域。

Gerchberg-Saxton(GS)算法是經(jīng)典的利用遠(yuǎn)場(chǎng)圖像重構(gòu)波前的復(fù)原算法，其結(jié)構(gòu)簡(jiǎn)單，易于實(shí)現(xiàn)[9]。GS算法利用遠(yuǎn)場(chǎng)和近場(chǎng)強(qiáng)度間的傅里葉變換關(guān)系來(lái)計(jì)算波前像差，但其算法在迭代過(guò)程容易陷入停滯。這種迭代停滯是由于實(shí)際的近場(chǎng)復(fù)振幅和它的旋轉(zhuǎn)180°共軛復(fù)振幅在遠(yuǎn)場(chǎng)上具有相同的光強(qiáng)分布而產(chǎn)生的[10]。因此，GS算法很容易收斂到局部最小偽解或兩個(gè)全局最小模糊解之一[11]。

Gonsalves提出了相位差(phase difference，PD)算法來(lái)克服GS算法的多解問(wèn)題[2]，該算法通過(guò)添加一副離焦面光強(qiáng)分布信息來(lái)提高波前恢復(fù)的精度。然而，此方法需要使用兩個(gè)CCD相機(jī)，犧牲了光學(xué)結(jié)構(gòu)的簡(jiǎn)便性。

2008年，李敏、李新陽(yáng)提出了一種基于線性相位反演的方法來(lái)解決單幀相位反演的問(wèn)題，此方法對(duì)較小的像差有效[12-13]。2010年，Meimon提出了一種基于單個(gè)圖像重建波前的線性焦面反演算法，但該算法僅對(duì)低階像差復(fù)原有效[14]。Greenbaum在2016年提出了一種引入非冗余掩模(non-redundant mask，NRM)打破相位的符號(hào)不確定性的方法來(lái)重建單幀焦面圖像的近場(chǎng)波前[11]。然而，NRM進(jìn)出光路的精度和探測(cè)裝置的復(fù)雜性限制了該算法在實(shí)際波前探測(cè)中的應(yīng)用。

為解決傳統(tǒng)單幀反演算法的上述問(wèn)題，本文提出了一種基于Walsh函數(shù)的二維離散相位調(diào)制的反演方法，打破了近場(chǎng)像差的旋轉(zhuǎn)翻轉(zhuǎn)對(duì)稱性，通過(guò)調(diào)制后的特殊光強(qiáng)分布，可以快速恢復(fù)出原始波前的準(zhǔn)確相位信息。

本文的結(jié)構(gòu)如下：第二節(jié)著重分析了傳統(tǒng)單幀相位反演方法的像差多解問(wèn)題，并介紹了基于Walsh函數(shù)的二維離散相位片調(diào)制的波前復(fù)原算法的基本原理和限定條件；在第三節(jié)中，通過(guò)仿真實(shí)驗(yàn)，比較了基于不同Walsh函數(shù)相位調(diào)制的復(fù)原結(jié)果；并在第四節(jié)對(duì)本文工作進(jìn)行了總結(jié)。

2 基本原理

2.1 傳統(tǒng)單幀相位反演方法的像差多解問(wèn)題

當(dāng)兩組波前像差具有互為旋轉(zhuǎn)180°復(fù)共軛關(guān)系時(shí)(類似于奇函數(shù))，它們將具有相同的遠(yuǎn)場(chǎng)光斑分布，即兩個(gè)不同近場(chǎng)對(duì)應(yīng)同一個(gè)遠(yuǎn)場(chǎng)光強(qiáng)分布。下面用數(shù)學(xué)表達(dá)式來(lái)分析此問(wèn)題。

均勻光波入射下，通過(guò)振幅歸一化，入射平面波復(fù)振幅可以表示為

遠(yuǎn)場(chǎng)復(fù)振幅可以通過(guò)近場(chǎng)復(fù)振幅的傅里葉變換獲得，在忽略系數(shù)的情況下，可以表示如下：

比較式(2)和式(4)，根據(jù)積分對(duì)稱性，兩者實(shí)部相等、虛部符號(hào)相反，光強(qiáng)值為振幅的平方。因此，兩組具有旋轉(zhuǎn)180°復(fù)共軛關(guān)系的波前具有相同的遠(yuǎn)場(chǎng)光斑分布，即：

因此，利用單個(gè)遠(yuǎn)場(chǎng)光斑反演近場(chǎng)波前像差，算法就可能由于多解問(wèn)題陷入停滯，得不到準(zhǔn)確的波前相位信息。

2.2 二維離散像差調(diào)制

為了解決基于單幀焦面圖像的相位反演算法的多解問(wèn)題，一種可行的方法是打破近場(chǎng)波前相位的旋轉(zhuǎn)對(duì)稱性。如果采用連續(xù)像差進(jìn)行波前調(diào)制，則相當(dāng)于在待測(cè)波前的基礎(chǔ)上疊加了一個(gè)波面，雖然可能對(duì)原來(lái)的兩組旋轉(zhuǎn)對(duì)稱像差的遠(yuǎn)場(chǎng)光強(qiáng)一致性起到了破壞，但并不能解決多解問(wèn)題。

Wang于2009年首次提出了利用Walsh函數(shù)來(lái)表征離散二元像差的模式，如圖1所示[15-16]，此組函數(shù)的黑白區(qū)間總面積相等。這些圖案與傳統(tǒng)連續(xù)像差具有明顯不同的分布方式。

如果以這些離散圖案制作成相位片，即黑色區(qū)域相對(duì)白色區(qū)域具有相位臺(tái)階，將其置于光路中，則有可能打破原有近場(chǎng)相位的旋轉(zhuǎn)對(duì)稱性。由此本文提出了一種基于Walsh函數(shù)調(diào)制的波前復(fù)原方法，其光路系統(tǒng)如圖2所示。在近場(chǎng)插入一片帶有相位臺(tái)階的相位片，其具有類似圖1的空間分布方式。通過(guò)此相位片的空間調(diào)制后，波前經(jīng)過(guò)透鏡聚焦到相機(jī)靶面。

比較式(6)和式(7)，兩者具有明顯差別，一般情況下：

將式(9)代入式(7)，可得：

圖1 Walsh函數(shù)分布

圖2 光學(xué)系統(tǒng)模型

比較式(6)和式(10)，兩者實(shí)部相等，虛部符號(hào)相反，即同樣可以得到：

分別用正、負(fù)離焦像差和一對(duì)隨機(jī)旋轉(zhuǎn)復(fù)共軛隨機(jī)像差為例，來(lái)驗(yàn)證以上結(jié)論。其中正、負(fù)離焦像差可以看作為特殊的旋轉(zhuǎn)復(fù)共軛對(duì)。仿真模擬中，透鏡焦距為500 mm，波長(zhǎng)625 nm，光束半徑為2 mm，CCD像素尺寸為10mm，相位片的相位臺(tái)階為p/2。

2.3 基于Walsh函數(shù)二維離散調(diào)制的相位反演算法

由式(1)可知，遠(yuǎn)場(chǎng)復(fù)振幅可以近似看成近場(chǎng)復(fù)振幅的傅里葉變換：

對(duì)上式進(jìn)行逆傅里葉變換，可以得到：

具體的基于Walsh函數(shù)二維離散調(diào)制的相位反演算法流程如圖4所示。首先設(shè)定待測(cè)波前初始值為0，之后通過(guò)二維調(diào)制，計(jì)算得到遠(yuǎn)場(chǎng)光強(qiáng)，比較計(jì)算所得光強(qiáng)與實(shí)際測(cè)得光強(qiáng)是否足夠接近，可用誤差平方和(sum of squares due to error，SSE，用SSE表示)來(lái)評(píng)價(jià)，其公式表達(dá)式如下：

3 仿真實(shí)驗(yàn)

在本節(jié)中，對(duì)基于Walsh函數(shù)二維調(diào)制波前復(fù)原方法的有效性和測(cè)量精度進(jìn)行了數(shù)值仿真。與傳統(tǒng)GS算法之間的對(duì)比仿真實(shí)驗(yàn)也被闡述。

仿真參數(shù)如下：透鏡焦距為500 mm，波長(zhǎng)625 nm，光束半徑為2 mm，CCD像素尺寸為10mm，相位片的相位臺(tái)階為p/2。評(píng)估標(biāo)準(zhǔn)SSE被設(shè)置為小于10-6。如果300次迭代后不能達(dá)到判定標(biāo)準(zhǔn)，程序循環(huán)將終止。

首先，利用前65階Zernike多項(xiàng)式生成了PV值為2.5218 rad和RMS值為0.3719 rad的隨機(jī)波前像差，如圖5所示。

使用傳統(tǒng)GS算法、基于Walsh函數(shù)W3的調(diào)制反演算法、基于Walsh函數(shù)W5的調(diào)制反演算法分別重構(gòu)上述隨機(jī)入射波前。

圖4 基于Walsh函數(shù)調(diào)制的相位反演算法流程圖

圖5 待測(cè)波前。(a) 波形；(b) Zernike系數(shù)

為了驗(yàn)證不同相位臺(tái)階對(duì)二維離散調(diào)制方法的影響，以四象限結(jié)構(gòu)W3為例，進(jìn)行仿真，評(píng)價(jià)指標(biāo)為10E-7，相位臺(tái)階深度分別是p/5、p/2和7p/10。對(duì)應(yīng)的波前復(fù)原迭代曲線如圖9所示，可以看出在p/2的相位臺(tái)階下，整體復(fù)原效果最好，p/5效果稍好，而綠色曲線7p/10效果稍差一些，但都能準(zhǔn)確復(fù)原波前。相位臺(tái)階過(guò)小時(shí)，調(diào)制效果不夠明顯；過(guò)大時(shí)，臺(tái)階效應(yīng)較大，不利于復(fù)原波面。

圖7 傳統(tǒng)GS算法和Walsh函數(shù)調(diào)制算法的迭代曲線

圖8 不同相位反演方法的殘余波前RMS值比較圖

圖9 不同相位臺(tái)階下評(píng)價(jià)指標(biāo)ESSE的收斂曲線

4 結(jié) 論

本文詳細(xì)分析了傳統(tǒng)單幀反演算法的多解問(wèn)題，提出了一種基于Walsh函數(shù)調(diào)制的波前重構(gòu)方法。介紹了該方法的基本原理和限定條件，即調(diào)制相位本身應(yīng)不具備180°旋轉(zhuǎn)翻轉(zhuǎn)對(duì)稱性。通過(guò)仿真實(shí)驗(yàn)，比較了提出算法和傳統(tǒng)GS算法的波前復(fù)原精度。結(jié)果表明，基于Walsh函數(shù)相位調(diào)制的波前反演算法在單幀遠(yuǎn)場(chǎng)圖像條件下可以精確地復(fù)原出待測(cè)波前像差。當(dāng)然，從相位調(diào)制片制作工藝難度和對(duì)波前尺寸的普適性角度來(lái)講，實(shí)際應(yīng)用中基于W3的四象限調(diào)制方式將更有優(yōu)勢(shì)。

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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

1Key Laboratory of Adaptive Optics, Chinese Academy of Sciences, Chengdu, Sichuan 610209, China;2School of Optoelectronic Science and Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, China;3Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu, Sichuan 610209, China;4University of Chinese Academy of Sciences, Beijing 100049, China

Distribution of Walsh functions

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(,) and its twin*(-, -) 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. L?fdahl 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.

Citation: Kong Q F, Wang S, Yang P,Single-frame far-field wavefront retrieval method based on Walsh function modulation[J]., 2020, 47(6): 190323

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

1Key Laboratory of Adaptive Optics, Chinese Academy of Sciences, Chengdu, Sichuan 610209, China;2School of Optoelectronic Science and Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, China;3Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu, Sichuan 610209, China;4University of Chinese Academy of Sciences, Beijing 100049, China

The reconstruction of wavefront from single far-field image data has unique advantages in simplicity of structure. However, the traditional wavefront reconstruction algorithm has multiple solutions based on single far-field image, its iterative process easily falls into stagnation. In this paper, based on the analysis of the multi-solution problem of single-frame phase retrieval method, a wavefront reconstruction method based on Walsh function two-dimensional discrete phase modulation is proposed. This method can effectively break the symmetry of near-field wavefront and overcome problem of multiple solutions. The simulation results show that the method can accurately reconstruct wavefront aberration with only one far-field image.

phase retrieval; aberration; far-field; two-dimensional discrete modulation

10.12086/oee.2020.190323

TN247

A

: Kong Q F, Wang S, Yang P,. Single-frame far-field wavefront retrieval method based on Walsh function modulation[J]., 2020,47(6): 190323

孔慶峰，王帥，楊平，等. 基于Walsh函數(shù)調(diào)制的單幀遠(yuǎn)場(chǎng)波前反演方法[J]. 光電工程，2020，47(6): 190323

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

* E-mail: wangshuai@ioe.ac.cn; pingyang2516@163.com

2019-06-12；

2019-09-16

國(guó)家自然科學(xué)基金資助項(xiàng)目(61805251)；中國(guó)科學(xué)院青年創(chuàng)新促進(jìn)會(huì)資助項(xiàng)目(2017429)

孔慶峰(1983-)，男，博士研究生，主要從事波前復(fù)原技術(shù)的研究。E-mail：kqfengxx@sina.com

王帥(1988-)，男，博士，副研究員，主要從事新型波前探測(cè)技術(shù)的研究。E-mail：wangshuai@ioe.ac.cn楊平(1980-)，男，博士，研究員，主要從事自適應(yīng)光學(xué)系統(tǒng)的研究。E-mail：pingyang2516@163.com