LUO Yong, LI Tuo, LI Gui-lin, SHI Yi-shi*

(1.School of Optoelectronics,University of Chinese Academy of Sciences,Beijing 100049,China； 2.Academy of Opto-Electronics,Chinese Academy of Sciences,Beijing 100094,China； 3.School of Science,Xijing University,Xi′an 710123,China)

Abstract: In a traditional single beam multiple-intensity reconstruction(SBMIR) system, error is accumulated by multiple translational image sensors, which reduces the imaging effect and the effective resolution of the photoelectric imaging system. In this paper, a three-step coherent diffraction imaging system based on parallel plates is proposed. Three different diffraction planes are obtained by inserting or extracting two parallel plates and imaging and restoration reconstruction of complex amplitude objects are achieved. The numerical simulation and experiments show that the system overcomes the error accumulation problem of several translations in the SBMIR system, and one only needs to record three diffraction surfaces to avoid oversampling. The proposed optical system is easy to implement and has high repeatability.

Key words: the single-beam multiple-intensity;coherent diffraction imaging;parallel plates;complex amplitude

1 Introduction

引 言

Coherent Diffraction Imaging(CDI) is a lensless diffraction imaging technique[1-3]that has been rapidly developing with applications in adaptive X-ray imaging and related fields[4-8]. The CDI methods implemented in nowadays involve holography[9-10], wavefront detection reconstruction[11]and Gerchberg-Saxton′s(GS) algorithm for multiple diffraction information surfaces[12-13]. In general, single-step diffraction imaging is not suitable for complex amplitude recovery reconstruction because only one diffraction pattern is recorded. Therefore, an improved GS algorithm was produced[14-15], along with random binary pure phase modulation[16], rotational phase modulation[17], single-beam multi-intensity wavefront reconstruction(SBMIR) and other technical solutions[18-21]. However, among the many schemes, most of them are limited to the use of computers for numerical simulation. Also, specific optical imaging experiments for the schemes have not been implemented and they have no proposed specific experimental procedure. For this reason, further verification is needed to give plausibility to the methods and reproducibility of their experiments. Traditional SBMIR technology has been studied using numerical simulation analysis and specific experimentation. The phase recovery problem has also been solved. SBMIR methodology generally involves fixing a CCD camera on a precision stage and adopts a mechanical stepping mode. This easily causes the experimental image to have problems that result from shaking equipment, thereby reducing image resolution and quality. Furthermore, this technique usually requires that 10-20 diffractive faces be collected and recorded, which introduces the issue of oversampling defects. The above problems lead to difficulty repeating experiments and cause the technology to be less useful in real-world applications.

相干衍射成像(Coherent Diffraction Imaging，CDI)，是一種無透鏡衍射成像技術(shù)[1-3]，從提出至今快速發(fā)展并應(yīng)用于自適應(yīng)成像、X射線成像等相關(guān)領(lǐng)域[4-8]。CDI的實(shí)現(xiàn)方法有基于全息術(shù)[9-10]、波前檢測(cè)重建[11]和多個(gè)衍射信息面的Gerchberg-Saxton(GS)算法等[12-13]。通常，單步衍射成像因?yàn)橹挥涗浺环苌鋱D像，無法適用于復(fù)振幅的恢復(fù)重建，所以基于此類相位恢復(fù)重建的問題，通常采用改進(jìn)型的GS算法[14-15]、隨機(jī)二元純相位調(diào)制[16]、旋轉(zhuǎn)相位調(diào)制[17]、單光束多強(qiáng)度波前重建(SBMIR)等技術(shù)方案[18-21]。然而，大部分方法都只局限于利用計(jì)算機(jī)進(jìn)行數(shù)值模擬，具體的光學(xué)成像實(shí)驗(yàn)并未實(shí)現(xiàn)，也沒有提出具體的實(shí)驗(yàn)方案。所以方法的實(shí)用性及實(shí)驗(yàn)的可重復(fù)性需要進(jìn)一步驗(yàn)證。傳統(tǒng)的SBMIR技術(shù)對(duì)數(shù)值模擬分析和具體實(shí)驗(yàn)都進(jìn)行了研究，相位恢復(fù)問題得到了解決。但是，傳統(tǒng)的SBMIR技術(shù)大多是將圖像傳感器CCD固定于精密平移臺(tái)上，采用機(jī)械移動(dòng)的步進(jìn)方式，導(dǎo)致實(shí)驗(yàn)圖像有抖動(dòng)問題。從而降低了成像分辨率和質(zhì)量，而且此技術(shù)通常需采集記錄10～20個(gè)衍射面，有過采樣的缺陷，上述問題及缺陷導(dǎo)致實(shí)驗(yàn)重復(fù)性較差，不利于技術(shù)方案的實(shí)際應(yīng)用。

In order to solve and avoid the above issues, a three-step coherent diffraction imaging system based on parallel plates is proposed. The position of the CCD camera and the sample is fixed and two parallel plates are inserted in or extracted from the system. By doing so, three intensity information diffraction planes are quickly obtained and the sample pattern is eventually reconstructed by the recovery algorithm. The results of computer numerical simulation and actual optical experiments show that the system effectively avoids and solves the problem with shaking and the oversampling defects that exist in the traditional technical solutions. The proposed method is simple, quick to perform and repeatable while also producing images that are of significantly higher quality

為了解決和避免上述問題及缺陷，本文提出一種基于平行平晶的三步相干衍射成像系統(tǒng)，圖像傳感器CDD和樣品的位置固定不變，采用依次在系統(tǒng)中插入或抽出2塊平行平晶的方法，快速獲得3幅強(qiáng)度信息衍射圖，通過恢復(fù)算法最終重建樣品圖像。計(jì)算機(jī)數(shù)值模擬和實(shí)際的光學(xué)實(shí)驗(yàn)結(jié)果表明：該系統(tǒng)有效解決了傳統(tǒng)技術(shù)方案的抖動(dòng)問題與過采樣的缺陷，最重要的是系統(tǒng)的成像效果顯著提升，且具有實(shí)驗(yàn)可重復(fù)性高，操作簡(jiǎn)單快捷的特點(diǎn)。

2 Imaging System Structure and Method Principle Analysis

成像系統(tǒng)結(jié)構(gòu)及方法原理分析

2.1 Traditional SBMIR Technology and System Implementation

傳統(tǒng)SBMIR技術(shù)及系統(tǒng)實(shí)現(xiàn)

Before introducing the three-step coherent diffraction imaging system based on parallel plates, a brief discussion on the single-beam multi-strength reconstruction(SBMIR) technology scheme using a precision mobile platform will be presented. A typical SBMIR optical imaging system is shown in Fig.1(a). The CCD camera is fixed on a motorized precision stage where the rotation of a motor causes the precision stage to move along its track. The CCD camera records the diffraction intensity informationINof the object every time the precision stage moves by a distance of Δz. Anther scheme will be done by moving the position of the sample. Let the square of the intensity of the CCD acquisition recorded and the amplitude of the Fourier transform of the object be:

在介紹基于平行平晶的三步衍射成像系統(tǒng)之前，先簡(jiǎn)要討論采用精密移動(dòng)平臺(tái)完成SBMIR技術(shù)方案。典型的SBMIR光學(xué)成像系統(tǒng)如圖1(a)所示。圖像傳感器CCD固定在精密的機(jī)械移動(dòng)平臺(tái)上，通過電機(jī)轉(zhuǎn)動(dòng)使平移臺(tái)沿著軌道方向移動(dòng)，平移臺(tái)每移動(dòng)一段距離Δz，CCD就記錄一次物體的衍射強(qiáng)度信息IN。另一種方案則是通過移動(dòng)樣品位置來實(shí)現(xiàn)。設(shè)CCD采集記錄的強(qiáng)度信息與物體傅里葉變換的幅度成平方關(guān)系為：

IN=[F(ON)RZ+ΔZ]2, (1)

WhereFdenotes the Fourier transform operator,INis the object plane andRZ+ΔZis the diffraction distance. Typically, an SBMIR system requires that at least 3 diffraction intensity maps be recorded for wavefront reconstruction of the completed sample but often as many as 10-20 are used.

式中，F(xiàn)表示傅里葉變換算子，IN為物平面，RZ+ΔZ為衍射距離。通常情況下，典型的SBMIR系統(tǒng)需要采集記錄的衍射強(qiáng)度圖不少于3幅，一般為10～20幅，并以此完成的樣品的波前重建工作。

2.2 Three-step Coherent Diffraction Imaging System Based on Parallel Plates

基于平行平晶的三步衍射成像系統(tǒng)

The proposed three-step coherent diffraction imaging system based on parallel plates is shown in Fig.1(b). The relative position of the object and the CCD camera is fixed. P1 and P2 represent two parallel flat crystal plates, which are illuminated by coherent light. The monochromatic plane wave in the system is vertically irradiated to the object plane after being collimated by a pinhole filter and a lens. It then reaches the CCD camera which records the surface after a diffraction of distancez. The system can complete the imaging processing in three steps:For the first step, after constructing and fixing the system device, the CCD camera is directly used to record the intensity informationI1of the first diffractive surface; In the second step, with the relative position of the CCD camera and the object unchanged, a parallel crystal plane P1 is inserted at any position between them. The CCD camera then collects and records the intensity informationI2of the second diffractive surface; In the third step, without disturbing the setup from the second step, another parallel flat crystal P2 is inserted in an arbitrary position between the object and the CCD camera, then the CCD camera is once again used to record the intensity informationI3of the third diffraction plane. After completing these three steps, the diffractive surface intensity information ofI1,I2,I3are each known. The positions of the object and CCD camera do not need to be moved or changed throughout the entire process and the completion time of the entire experiment is about 30 s. It does not involve the use of a precision stage, has no shaking in its system, no accuracy problems and has a simple experimental procedure that does not suffer from issues caused by oversampling.

研究提出的基于平行平晶的三步衍射成像系統(tǒng)如圖1(b)所示,物體與圖像傳感器CDD的相對(duì)位置是固定不變的，P1和P2表示兩塊平行平晶，采用相干光照明，系統(tǒng)中的單色平面波，經(jīng)針孔濾波器與透鏡組合成的準(zhǔn)直擴(kuò)束系統(tǒng)后，垂直照射到物體平面，經(jīng)過一段衍射距離z后到達(dá)圖像傳感器CDD記錄面。系統(tǒng)經(jīng)過3個(gè)步驟完成成像過程：第一步，搭建與固定好系統(tǒng)器件后，直接用CCD采集記錄得到第一衍射面的強(qiáng)度信息I1；第二步，保持CCD與物體的位置不變，在它們之間的任意位置插入一塊平行平晶P1，CCD采集記錄后得到第二衍射面的強(qiáng)度信息I2；第三步，在第二步系統(tǒng)結(jié)構(gòu)位置不變的基礎(chǔ)上，在物體與CCD之間任意位置插入另一塊平行平晶P2，CCD再次記錄下第三幅衍射圖的強(qiáng)度信息I3。最終，一共得到3個(gè)不同衍射面的強(qiáng)度信息I1,I2,I3。整個(gè)過程中，物體和CCD的位置是不需要移動(dòng)及改變的，完成整個(gè)實(shí)驗(yàn)約需30 s。該系統(tǒng)不使用精度平移臺(tái)，沒有系統(tǒng)抖動(dòng)、精度問題，也無過采樣的復(fù)雜實(shí)驗(yàn)過程。

Fig.1 Structural comparison and principle analysis of proposed system 圖1 系統(tǒng)結(jié)構(gòu)比較及原理分析

The principle of the three-step coherent diffraction imaging system based on parallel plates is shown in Fig.1(c). The object is illuminated by a monochromatic coherent wave, and the object plane is diffracted by distancez0to meet a Fresnel plane, which is the first step of the system. The image then travels the distancez1to a second Fresnel plane, which is the second step of the system. Finally, it then continues to travel the distancez2to meet a third Fresnel plane, being the third step of the system. The above process is not completed by moving the CCD camera or the object through the precision stage, but instead by inserting parallel flat crystals between the CCD camera and the object, allowing the information to be reconstructed using the multiple intensities of the single beam. Of course, it should be pointed out that the above steps can be implemented in reverse, meaning that all the parallel flat crystals can be inserted first and then sequentially removed.

基于平行平晶的三步衍射成像系統(tǒng)的原理分析如圖1(c)所示，用單色相干平面波照射物體，物體經(jīng)過距離z0的衍射后得到第一衍射面，即前文所述的第一步；繼續(xù)傳播距離z1后得到第二衍射面，即前文所述的第二步；再繼續(xù)傳播距離z2后得到第三衍射面，即前文所述的第三步。以上過程不是通過精密平移臺(tái)移動(dòng)CCD或物體來實(shí)現(xiàn)的，而是采用在CCD與物體之間插入平行平晶得到單光束多強(qiáng)度的信息重建。當(dāng)然，需要指出的是此系統(tǒng)上述步驟可以反向?qū)嵤纯上葘⑺械钠叫衅骄Р迦?，再每次抽取一塊完成實(shí)驗(yàn)研究過程。

3 Key Algorithms and Numerical Simulation Analysis

關(guān)鍵算法及數(shù)值模擬分析

3.1 Key Algorithm Methodology

關(guān)鍵算法步驟

The algorithm of the three-step coherent diffraction imaging system based on parallel plates is based on the GS algorithm. The original diffraction plane is increased to three diffraction planes with different diffraction distances to recover and reconstruct the sample. It is through this method that the accuracy of the iterative algorithm and the convergence speed and recovery reconstruction effects are improved. An added benefit of the three-step coherent diffraction imaging system is that it has the ability to recover and reconstruct complex amplitude objects.

基于平行平晶的三步衍射成像系統(tǒng)的關(guān)鍵算法是以G-S算法為基礎(chǔ)，由原來的一幅衍射圖樣增加為3幅不同衍射距離的衍射圖樣，以實(shí)現(xiàn)對(duì)樣品的恢復(fù)重建，從而提高迭代算法的計(jì)算精確程度，及收斂速度和恢復(fù)重建效果。同時(shí)，三步衍射成像系統(tǒng)方法的一個(gè)最大優(yōu)勢(shì)在于可以對(duì)復(fù)振幅型的物體進(jìn)行恢復(fù)重建。

The algorithm key steps are shown in Fig.2, assuming

Fig.2 Block diagram of the key algorithm 圖2 關(guān)鍵算法框圖

關(guān)鍵算法步驟如圖2所示，設(shè)

g(k)(x0，y0)=|g(k)(x0,y0)|·

exp[kφ0(x0,y0)] , (2)

(1)From the object plane positive to the first diffractive surface:

從物平面正向到第一衍射面：

(3)

(2)From the first diffractive surface positive to the second diffractive surface:

第一衍射面正向到第二衍射面：

(4)

(3)From the second diffractive surface positive to the third diffractive surface:

第二衍射面正向到第三衍射面：

(5)

(4)Reverse from the third diffractive surface to the second diffractive surface:

第三衍射面逆向到第二衍射面：

(6)

(5)Reverse from the second diffractive surface to the first diffractive surface:

第二衍射面逆向到第一衍射面：

(7)

(6)Reverse from the first diffractive surface to the object plane:

第一衍射面逆向到物平面：

(8)

When the sample is a pure amplitude type object, there is:

當(dāng)樣品為純振幅型物體時(shí)，則有

g(k+1)(x0,y0)=|g(k)′(x0,y0)| ， (9)

When the sample is a complex amplitude type object, there is:

樣品為復(fù)振幅型物體時(shí)，則有

g(k+1)(x0,y0)=g(k)′(x0,y0) . (10)

3.2 Numerical Simulation Analysis

數(shù)值模擬分析

In order to further demonstrate the feasibility of the method using the three-step coherent diffraction imaging system, a numerical simulation analysis of the computer was first carried out, with the results shown in Fig.3. The single-step coherent diffraction image is calculated and analyzed is shown in Fig.3(a), the two-step diffraction imaging is shown in Fig.3(b) and the three-step diffraction imaging is shown in Fig.3(c). For convenience of comparison, the number of algorithm iterations is set to 200 times, the commonly used image evaluation function correlation coefficientCois used to judge the effect of restoration and reconstruction, and the range is generally [0,1]. The closer theCovalue is to 1, the closer the reconstruction is to the real object. If the value is smaller, the recovery quality is worse. Furthermore, the higher the deviation from the real object, the worse the imaging effect, affecting the iterative break and selection algorithm's number of iterations. For a pure amplitude type object, since there is no phase, the calculation is relatively simple and its detailed numerical simulation results are omitted. However, it should be pointed out that the convergence speed is very fast and theCovalue of the amplitude can quickly reach 1.

為了進(jìn)一步論證三步衍射成像系統(tǒng)方法的可行性，首先進(jìn)行了計(jì)算機(jī)數(shù)值模擬分析，結(jié)果如圖3所示。分別計(jì)算了單步衍射成像(圖3(a))，兩步衍射成像(圖3(b))、三步衍射成像(圖3(c))。為方便比較，特將算法的迭代次數(shù)都設(shè)置為200次，采用常用圖像評(píng)價(jià)函數(shù)相關(guān)系數(shù)Co來判斷恢復(fù)重建效果，其取值范圍一般為[0，1]。Co值越接近1說明恢復(fù)重建的物體越接近真實(shí)的物體。其值越小說明恢復(fù)質(zhì)量越差，越偏離真實(shí)物體，成像效果越差，并以此來判斷和選擇算法的迭代停止條件。對(duì)于純振幅型的物體而言，由于沒有相位，所以較為簡(jiǎn)單。就不在給出其詳細(xì)的數(shù)值模擬結(jié)果，但需要指出的是其收斂速度非常的快，且振幅的Co值能快速達(dá)到1。

In the process of computationally calculated numerical simulation, the sample pattern used is a grayscale image with a size of 256 pixel×256 pixel, the CCD camera′s pixel size is 6.45×10-6m/pixel, the laser′s wavelength is 632.8×10-9m. The sample is of the complex amplitude type, its phase distribution range is set to [-π,π], and the diffraction distances are set toz0=100 mm,z1=10 mm,z2=10 mm. The reconstruction of the complex amplitude of the sample is completed, and the correlationCocoefficient′s value is represented for the amplitude distribution and the phase distribution, respectively. In the simulated results, the black solid line is the value of the amplitude part correlation coefficient change, and the blue dotted line is the value of the phase part correlation coefficient change. In order to mark the value ofCoat a desired point in Fig.3 (a), the cursor included with the Matlab software package is used.

在進(jìn)行計(jì)算機(jī)數(shù)值模擬分析過程中，使用的樣品圖樣為灰度圖，尺寸大小為256 pixel×256 pixel，圖像傳感器CCD的像素尺寸為6.45×10-6m/pixel，激光波長(zhǎng)為632.8×10-9m，且樣品為復(fù)振幅型，其相位分布范圍設(shè)為[-π,π]，將衍射距離設(shè)為z0=100 mm,z1=10 mm,z2=10 mm，完成對(duì)樣品復(fù)振幅的恢復(fù)重建。對(duì)振幅分布與相位分布分別用相關(guān)系數(shù)Co值表示。在數(shù)值模擬結(jié)果中，其中實(shí)線為振幅部分相關(guān)系數(shù)變化值，藍(lán)色點(diǎn)線為相位部分相關(guān)系數(shù)變化值。為了標(biāo)注Co在某點(diǎn)的數(shù)值大小，在圖3(a)中，使用了Matlab軟件中自帶的游標(biāo)。

From the simulation results shown in Fig.3(a), under the same conditions, theCovalue range of the amplitude and phase fractions of the single-step diffraction imaging recovery reconstruction is less than 0.5. It is shown that only obtaining a single coherent diffraction intensity map is impossible to recover and reconstruct a complex amplitude object, which is why the method of single-step coherent diffraction imaging is not applicable to such objects. However, with the addition of a diffraction plane that has a distance ofz0+z1, the recovery and reconstruction effect resulting from two-step diffraction imaging is significantly improved, theCovalues of the amplitude and phase are higher than 0.5 and there is a tendency to converge. Nevertheless, late in the algorithm iteration, there is a slight decrease in morphology so a comprehensive evaluation shows that it cannot achieve the desired result. These results are shown in Fig.3(b). In contrast, the three-step coherent diffraction imaging process starts to converge when the algorithm passes 60 iterations and completely converges after about 70 iterations without any subsequent regression. Moreover, final convergenceCovalue of the recovery results, either the amplitude portion or the phase portion, reaches 1. These results show that the imaging quality of the system is continuously improved from one diffraction plane to three diffractions, that the algorithm completely converges to the third image, and that theCovalue reaches the optimal ideal value.

Fig.3 Numerical simulation analysis and comparison 圖3 數(shù)值模擬分析及比較

圖3(a)數(shù)值模擬結(jié)果表明，在相同的條件下，單步衍射成像恢復(fù)重建的振幅與相位部分的Co數(shù)值均低于0.5。說明僅有單幅衍射強(qiáng)度圖是無法對(duì)復(fù)振幅型的物體進(jìn)行恢復(fù)重建的，單步相干衍射成像方法不適用于復(fù)振幅型物體。然而，在增加一幅距離為z0+z1的衍射圖后，兩步衍射成像的恢復(fù)重建效果得到了明顯提升，振幅及相位部分的Co值均高于0.5，且有收斂的趨勢(shì)，但算法迭代到后面則出現(xiàn)了輕微的降低趨勢(shì)。所以綜合評(píng)價(jià)沒有達(dá)到理想結(jié)果，結(jié)果如圖3(b)。相比之下，三步相干衍射成像在算法迭代到60次時(shí)開始收斂且到70次左右完全收斂，后續(xù)沒有任何下降的趨勢(shì)。并且，無論是振幅部分還是相位部分，最終收斂的Co值均達(dá)到1。結(jié)果說明由一幅衍射圖增加至三幅衍射的過程中，系統(tǒng)的成像質(zhì)量不斷的提升，且到第三幅時(shí)算法完全收斂，Co值也達(dá)到最佳的理想值。

In order to illustrate the robustness of the three-step coherent diffraction imaging system, a numerical simulation of the system′s ability to combat noise is added, as shown in Fig.3(d). Thex-value indicates that the system gradually increases the noise from 0, and the step size increases by 2%. When the noise increases to 20%, theCovalue of the amplitude and phase continues to exceed 0.94 with only minor fluctuations. Numerical simulation results show that the three-step diffraction imaging can effectively combat noise.

為了分析三步相干衍射成像系統(tǒng)方案的魯棒性，對(duì)增加噪聲系統(tǒng)進(jìn)行了的數(shù)值模擬計(jì)算，結(jié)果如圖3(d)所示。其中橫坐標(biāo)表示加入的噪聲從0逐漸增加，增加步長(zhǎng)為2%，當(dāng)噪聲增加至20%時(shí)，振幅與相位的Co值繼續(xù)高于0.94，且變化幅度非常小。數(shù)值模擬結(jié)果表明，三步衍射成像對(duì)抗噪聲能力良好。

4 Experimental results and analysis

實(shí)驗(yàn)結(jié)果及分析

4.1 Optical experiment results

光學(xué)實(shí)驗(yàn)結(jié)果

In order to demonstrate the feasibility of the three-step coherent diffraction imaging system based on parallel plates, the actual optical system imaging and recovery reconstruction work was carried out, and the corresponding SBMIR experiment based on precision translation stage was completed. These system structures are shown in Fig.1(a) and Fig.1(b). The experiment uses the following parameters: the laser is a coherent light source of a single plane with a wavelength ofλ=632.8 nm. The CCD camera′s pixel size is 6.45 μm/pixel, the number of intensities recorded by the experiment is 3 and the image size is 800 pixel×800 pixel. The above parameters are identical for both experiments. The difference between these experiments is that the traditional SBMIR system uses a precision translation stage where the moving CCD camera obtains diffractive surfaces at different distances. The accuracy of the translation stage is 0.01 μm and the maximum movement range is 15 mm. The three-step coherent diffraction imaging system based on parallel plates uses three parallel flat crystals, which are sequentially inserted into the system to obtain different diffractive surfaces. To an extent, the size of the parallel flat crystals does not affect the experimental operation so there is no outlined requirement, which is an advantage of this method. For more diffractive surfaces, the number of parallel flat crystals can be increased and the system will not need to be modified. Furthermore, there is no issue with shaking equipment or error caused by movement.

為了論證基于平行平晶的三步衍射成像系統(tǒng)的可行性，進(jìn)行了實(shí)際的光學(xué)系統(tǒng)成像及恢復(fù)重建工作，并完成相對(duì)應(yīng)的基于精密平移臺(tái)的SBMIR實(shí)驗(yàn)，系統(tǒng)結(jié)構(gòu)如圖1(a)和圖1(b)所示。實(shí)驗(yàn)參數(shù)設(shè)置如下：激光器為單色平面波的相干光源，波長(zhǎng)λ=632.8 nm,圖像傳感器CCD像素尺寸為6.45 μm/pixel,實(shí)驗(yàn)所采集記錄的強(qiáng)度信息圖數(shù)量為3幅，采集圖像尺寸為800 pixel×800 pixel,兩個(gè)系統(tǒng)的實(shí)驗(yàn)參數(shù)完全相同。不同的是，傳統(tǒng)的SBMIR采用的是精密平移臺(tái)，移動(dòng)CCD得到不同距離的衍射面，平移臺(tái)的精度為0.01 μm,最大移動(dòng)量程為15 mm。而基于平行平晶的三步衍射成像系統(tǒng)，使用的是三塊平行平晶，依次插入系統(tǒng)中，以此得到不同的衍射面。在不影響實(shí)驗(yàn)操作的情況下，平行平晶的規(guī)格尺寸并沒有嚴(yán)格要求，這也是此系統(tǒng)的一個(gè)優(yōu)勢(shì)，具有一定的自由度。如果需要更多衍射面，則增加平行平晶數(shù)量即可，不需要改動(dòng)系統(tǒng)，故沒有移動(dòng)器件導(dǎo)致的系統(tǒng)抖動(dòng)問題。

The experimental samples used were USAF 1951 resolution targets. The experimental results of recovery and reconstruction are shown in Fig.4. For the traditional SBMIR system, using a precision mobile platform requires that the CCD camera and sample be affixed to the platform, then, using movements of the platform, the CCD camera records the diffraction surface intensity information at different positions. In the experimental results given, in order to facilitate the comparison, the position of the sample to be tested and the CCD camera are fixed on the platform and the test was performed using moving steps of 3 mm. The translation stage is moved twice to create three steps and the CCD was allowed to record three experimental results. Finally, reconstruction was performed with the results shown in Fig.4(a).

實(shí)驗(yàn)樣品均為USAF 1951分辨率板，恢復(fù)重建結(jié)果如圖4所示。傳統(tǒng)SBMIR系統(tǒng)使用了精密移動(dòng)平臺(tái)，其將CCD或待測(cè)樣品固定于平臺(tái)上，然后移動(dòng)平臺(tái)，CCD記錄不同位置處的衍射面強(qiáng)度信息圖。在給出的實(shí)驗(yàn)結(jié)果中，為了便于比較，待測(cè)樣品位置固定不變，把CCD固定于平臺(tái)上移動(dòng)，移動(dòng)步長(zhǎng)為Δz=3 mm，分3個(gè)步驟移動(dòng)2次平移臺(tái)，CCD記錄3次實(shí)驗(yàn)結(jié)果。最終恢復(fù)重建結(jié)果如圖4(a)所示。

Fig.4 Comparison of phase distribution after restoration and reconstruction 圖4 恢復(fù)重建后相位分布比較

In order to perform the three-step diffraction imaging system based on parallel plane, parallel flat crystals were inserted in three steps. The CCD camera recorded three experimental results which were eventually recovered and reconstructed. These results are shown in Fig.4(b).

基于平行平晶的三步衍射成像系統(tǒng)，使用平行平晶，分3個(gè)步驟兩次插入平行平晶，CCD記錄3次實(shí)驗(yàn)結(jié)果，最終進(jìn)行恢復(fù)重建，結(jié)果如圖4(b)所示。

Some areas in Figs.4(a) and (b) are highlighted in dotlines and then magnified. It can be clearly seen that the quality of (b) is better than that of (a), showing higher quality in experimental results given by the three-step coherent diffraction imaging system, when compared to the traditional SBMIR system method using a mobile platform.

圖4(a)和(b)中,將虛線圈標(biāo)出的部分放大相同倍數(shù)進(jìn)行觀察比較。通過對(duì)比可以明顯看出圖4(b)比圖4(a)的質(zhì)量好。實(shí)驗(yàn)結(jié)果表明：平行平晶的三步衍射成像系統(tǒng)比傳統(tǒng)的使用移動(dòng)平臺(tái)的SBMIR系統(tǒng)實(shí)際成像效果好。

The experimental results also show that the phase distributions of some parts of the sample are reversed. The reason for this may be that the sample is tilted during the experiment, so that the parallel beam is inconsistent when it is irradiated onto the surface of the sample, causing errors in the reconstructed results.

同時(shí)實(shí)驗(yàn)結(jié)果顯示，樣品某些部位的相位分布存在翻轉(zhuǎn)情況，原因可能是在實(shí)驗(yàn)過程中，樣品有傾斜，使得平行光束照射到樣品表面時(shí)并不一致，所以記錄后重構(gòu)結(jié)果有誤差。

In order to further illustrate the three-step coherent diffraction imaging system based on parallel flat crystals and how a complex amplitude type object can be imaged and restored, the experiment of using a biological slice as a sample was performed with results shown in Fig.5. The experimental parameters used were the same as the preceding experiment with only the test sample being different.

為了進(jìn)一步的說明基于平行平晶的三步衍射成像系統(tǒng)，能夠?qū)?fù)振幅型物體成像并恢復(fù)重建，完成了以生物切片為樣品的實(shí)驗(yàn)，如圖5所示。所使用的實(shí)驗(yàn)參數(shù)與上述相同，只更換待測(cè)樣品。

Fig.5 Experimental results of complex amplitude type samples 圖5 復(fù)振幅型樣品的實(shí)驗(yàn)結(jié)果

Fig.5(a) is a microscopic photograph of an original bio-slice sample. Fig.5(b) shows the reconstructed phase distribution under ordinary coherent diffraction imaging and Fig.5(c) is the amplitude distribution of the sample resulting from the parallel crystal plate system. The amplitude distribution after recovery is restored under a three-step diffraction imaging system. Fig.5(d) is the distribution of the phase portion. Fig.5(b) shows the recovery reconstruction results under ordinary(single-step) diffraction imaging technology. Since there is only one diffractive surface, the information that can be obtained is extremely limited. There is no effective constraint on the phase recovery of the sample so it is impossible to effectively restore and reconstruct the complex amplitude type object. From the results of Fig.5(c) and (d), it is clear that the three-step coherent diffraction imaging system based on parallel flat crystals can effectively image and recover reconstruction of complex amplitude objects.

在圖5中，5(a)為原始生物切片樣品的顯微圖，5(b)為普通相干衍射成像恢復(fù)重建后的相位分布，5(c)為基于平行平晶的三步衍射成像系統(tǒng)下恢復(fù)重建后的振幅分布，5(d)為相位部分的分布。圖5(b)為普通(單步)衍射成像技術(shù)下的恢復(fù)重構(gòu)結(jié)果。由于只有一個(gè)衍射面，所能夠獲得的信息極為有限，對(duì)樣品的相位恢復(fù)沒有有效的約束條件，所以無法對(duì)復(fù)振幅型的物體進(jìn)行有效的恢復(fù)重建。圖5(c)、5(d)的結(jié)果說明，基于平行平晶的三步衍射成像系統(tǒng)能夠?qū)?fù)振幅型的物體進(jìn)行有效的成像及恢復(fù)重建。

The experimental results show that the three-step coherent diffraction imaging system based on parallel plates in which flat crystals are simply inserted or extracted can decrease error resulting from system shaking and movement of a traditional SBMIR precision mobile platform. It can also effectively capture an image of amplitude and complex amplitude objects and restore them without oversampling. Furthermore, all of this is possible with high repeatability.

實(shí)驗(yàn)結(jié)果表明，基于平行平晶的三步衍射成像系統(tǒng)，采用插入或抽取平行平晶的方法，使系統(tǒng)避免了因機(jī)械移動(dòng)而產(chǎn)生的系統(tǒng)抖動(dòng)。能有效對(duì)振幅型，復(fù)振幅型的物體進(jìn)行成像并恢復(fù)重建，不需要過采樣，并且系統(tǒng)的重復(fù)性高。

A further benefit of the 3-step process is that it combats the problem where imaging systems often encounter field-of-view problems. Because the system's used sample size is generally much smaller than the size of the parallel flat crystals, and both are in the near-field range, the field of view of the system is unaffected by the flat crystals.

成像系統(tǒng)通常會(huì)涉及到視場(chǎng)問題，系統(tǒng)中由于使用的樣品尺寸遠(yuǎn)小于平行平晶的尺寸，且都是在近場(chǎng)范圍成像，因此系統(tǒng)的視場(chǎng)不會(huì)受到平晶的影響。

4.2 Error Discussion

誤差討論

The experimental results show that the imaging system has a certain amount of error and that the reconstructed sample has some ripples. In response to these problems, error analysis is performed from the following few aspects.

實(shí)驗(yàn)結(jié)果表明，成像系統(tǒng)有一定的誤差，恢復(fù)重構(gòu)的樣品有一些波紋。針對(duì)這些問題，從以下幾個(gè)方面進(jìn)行誤差分析。

Influence of optical component uniformity:The parallel crystal three-step coherent diffraction imaging system is influenced by the uniformity of the optical components of which it is comprised. In the process of manufacturing such optical components, the level and quality are not always ideal. One of the most important devices within this system is the parallel flat crystal. The light in the experiment was deviated after passing through the flat crystal, causing reconstruction results to be inaccurate and corrugated. Presumably, the lens of the system may also have this problem.

光學(xué)元件均勻性的影響，基于平行平晶的三步相干衍射成像系統(tǒng)，其中最重要的器件之一平行平晶，此類光學(xué)元件在加工制造過程中，工藝水平達(dá)不到理想要求，所以導(dǎo)致實(shí)驗(yàn)中衍射光束通過平晶后出現(xiàn)偏差，最終恢復(fù)重建結(jié)果有誤差及波紋，當(dāng)然系統(tǒng)中的透鏡也可能有此問題。

The influence of the tilt of the optics in the imaging system:The need to insert or extract flat crystals in sequence in this operation cannot completely guarantee perfect alignment. Even the sample may be tilted. These tilts will cause the distance from the diffraction plane to the CCD camera to be inconsistent, which may cause ripples and errors in the experimental results.

光學(xué)器件的傾斜的影響，在成像系統(tǒng)中，需要依次插入或抽取平晶，在進(jìn)行此操作時(shí)，無法完全保證絕對(duì)的平直而沒有傾斜，同時(shí)樣品也有可能出現(xiàn)傾斜，這些傾斜則會(huì)導(dǎo)致樣品衍射到CCD記錄面的距離不一致，因此可能會(huì)導(dǎo)致實(shí)驗(yàn)結(jié)果出現(xiàn)了波紋和誤差。

External force vibration and air disturbance effects: In these optical experiments, all instruments are exposed to the environment, so external forces may cause the optical platform to vibrate. Such air disturbances in the system will affect the CCD recorded results which also has an impact on restoration and reconstruction.

外力振動(dòng)及空氣的擾動(dòng)影響，進(jìn)行光學(xué)實(shí)驗(yàn)，基所有的儀器都裸露在外，因此外力因素可能導(dǎo)致光學(xué)平臺(tái)振動(dòng)，系統(tǒng)周圍的空氣擾動(dòng)，都會(huì)影響到CCD記錄的結(jié)果。這同樣對(duì)恢復(fù)重建的結(jié)果造成了一定的影響。

5 Conclusion

結(jié) 論

Based on the GS algorithm and the single-step diffraction lensless imaging method, a three-step coherent diffraction imaging system based on parallel flat crystals is proposed and compared with traditional SBMIR technology using a precision mobile platform. Using computerized numerical simulation and actual optical experimentation, it was demonstrated that a three-step coherent diffraction imaging system based on parallel flat crystals: is unlike the traditional SBMIR technology in that the required diffraction patterns are reduced from between 10 and 20 to merely 3; has no mechanical movement, or evidence of shaking in resulting imagery; does not suffer from reduced imaging resolution in photoelectric imaging systems resulting from movement; and is easily repeatable. As the system′s diffractive surfaces are added simply added to the system, the system succeeds in imaging and recovery reconstruction complex amplitude type objects and overcomes the hurdles that the single-step diffraction imaging cannot. At the same time, the proposed imaging system, being a lensless coherent diffraction imaging system, has no lens aberration problem. However, the parallel crystal plates used in the system, the required level of precision, as well as the tilt problem during plate insertion can have a negative impact on results. The three-step coherent diffraction imaging system based on parallel flat crystals proposed in this paper has a wide range of applications and a high value in the fields of diffraction imaging measurement, multi-wavelength imaging, biological microscopy, and optical information security.

在G-S算法和單步衍射無透鏡成像方法的基礎(chǔ)上，提出了基于平行平晶的三步衍射成像系統(tǒng)，并與使用精密移動(dòng)平臺(tái)的傳統(tǒng)SBMIR技術(shù)進(jìn)行比較，從計(jì)算機(jī)數(shù)值模擬和實(shí)際的光學(xué)實(shí)驗(yàn)進(jìn)行論證。與傳統(tǒng)SBMIR技術(shù)不同，基于平行平晶的三步衍射成像系統(tǒng)所需記錄的衍射圖由10～20幅，降低為3幅，更值得注意的是系統(tǒng)沒有機(jī)械移動(dòng)，沒有圖像抖動(dòng)導(dǎo)致光電成像系統(tǒng)的成像分辨率降低的問題，也不需考慮移動(dòng)的精度問題，且實(shí)驗(yàn)的重復(fù)性好；由單步衍射成像1個(gè)衍射面提升至3個(gè)衍射面，系統(tǒng)實(shí)現(xiàn)了對(duì)復(fù)振幅型物體的成像及恢復(fù)重建，有效克服了單步衍射成像對(duì)復(fù)振幅型物體無法恢復(fù)的缺陷。提出的成像系統(tǒng)，雖然是無透鏡的相干衍射成像系統(tǒng)，沒有透鏡的像差問題，但系統(tǒng)中使用的平行平晶光學(xué)器件，其工藝水平及精度，還有插入過程中的傾斜問題對(duì)實(shí)驗(yàn)結(jié)果造成一定的影響。研究所提出的基于平行平晶的三步衍射成像系統(tǒng)，在衍射成像的測(cè)量，多波長(zhǎng)成像，生物顯微，光學(xué)信息安全等領(lǐng)域具有廣泛的應(yīng)用價(jià)值。