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 information IN of 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:
在介绍基于平行平晶的三步衍射成像系统之前, 先简要讨论采用精密移动平台完成SBMIR技术方案。典型的SBMIR光学成像系统如图 1(a)所示。图像传感器CCD固定在精密的机械移动平台上, 通过电机转动使平移台沿着轨道方向移动, 平移台每移动一段距离Δz, CCD就记录一次物体的衍射强度信息IN。另一种方案则是通过移动样品位置来实现。设CCD采集记录的强度信息与物体傅里叶变换的幅度成平方关系为:
Where F denotes the Fourier transform operator, IN is the object plane and RZ+ΔZ is 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表示傅里叶变换算子, IN为物平面, RZ+ΔZ为衍射距离。通常情况下, 典型的SBMIR系统需要采集记录的衍射强度图不少于3幅, 一般为10~20幅, 并以此完成的样品的波前重建工作。
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 distance z. 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 information I1 of 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 information I2 of 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 information I3 of the third diffraction plane. After completing these three steps, the diffractive surface intensity information of I1, I2, I3 are 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统如图 1(b)所示, 物体与图像传感器CDD的相对位置是固定不变的, P1和P2表示两块平行平晶, 采用相干光照明, 系统中的单色平面波, 经针孔滤波器与透镜组合成的准直扩束系统后, 垂直照射到物体平面, 经过一段衍射距离z后到达图像传感器CDD记录面。系统经过3个步骤完成成像过程:第一步, 搭建与固定好系统器件后, 直接用CCD采集记录得到第一衍射面的强度信息I1; 第二步, 保持CCD与物体的位置不变, 在它们之间的任意位置插入一块平行平晶P1, CCD采集记录后得到第二衍射面的强度信息I2; 第三步, 在第二步系统结构位置不变的基础上, 在物体与CCD之间任意位置插入另一块平行平晶P2, CCD再次记录下第三幅衍射图的强度信息I3。最终, 一共得到3个不同衍射面的强度信息I1, I2, I3。整个过程中, 物体和CCD的位置是不需要移动及改变的, 完成整个实验约需30 s。该系统不使用精度平移台, 没有系统抖动、精度问题, 也无过采样的复杂实验过程。
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 distance z0 to meet a Fresnel plane, which is the first step of the system. The image then travels the distance z1 to a second Fresnel plane, which is the second step of the system. Finally, it then continues to travel the distance z2 to 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统的原理分析如图 1(c)所示, 用单色相干平面波照射物体, 物体经过距离z0的衍射后得到第一衍射面, 即前文所述的第一步; 继续传播距离z1后得到第二衍射面, 即前文所述的第二步; 再继续传播距离z2后得到第三衍射面, 即前文所述的第三步。以上过程不是通过精密平移台移动CCD或物体来实现的, 而是采用在CCD与物体之间插入平行平晶得到单光束多强度的信息重建。当然, 需要指出的是此系统上述步骤可以反向实施, 即可先将所有的平行平晶插入, 再每次抽取一块完成实验研究过程。
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统的关键算法是以G-S算法为基础, 由原来的一幅衍射图样增加为3幅不同衍射距离的衍射图样, 以实现对样品的恢复重建, 从而提高迭代算法的计算精确程度, 及收敛速度和恢复重建效果。同时, 三步衍射成像系统方法的一个最大优势在于可以对复振幅型的物体进行恢复重建。
The algorithm key steps are shown in Fig. 2, assuming
关键算法步骤如图 2所示, 设
for the complex amplitude distribution on the object plane after k iterations(the initial object plane light field distribution g(0)(x0, y0) can be substituted for any non-zero complex amplitude distribution as the initial amplitude and phase of the calculation process mentioned in the study are given randomly), G1(k)(x1, y1), G2(k)(x2, y2) and G3(k)(x3, y3) are the complex amplitude distributions after k iterations on the first, second, and third diffraction planes and |F1(x1, y1)|2, |F2(x2, y2|2 and |F3(x3, y3)|2 are the actual measured light intensity distributions on the first, second, and third diffraction planes, respectively. FrT represents the Fresnel diffraction positive transformation and IFrT represents the Fresnel diffraction inverse transformation. The specific formula and operation steps are as follows:
为第k次迭代后物平面上的复振幅分布(初始物平面光场分布g(0)(x0, y0)可取任意非零的复振幅分布, 研究计算过程中初始的振幅和相位均采用随机给定的方式), G1(k)(x1, y1), G2(k)(x2, y2), G3(k)(x3, y3)依次为第1、第2、第3衍射平面上第k次迭代后的复振幅分布, |F1(x1, y1)|2, |F2(x2, y2)|2, |F3(x3, y3)|2分别为在第1、第2、第3衍射平面上实际测得的光强分布, FrT表示菲涅耳衍射正变换, IFrT表示菲涅耳衍射逆变换。具体的公式及运算步骤如下:
(1) From the object plane positive to the first diffractive surface:
(2) From the first diffractive surface positive to the second diffractive surface:
(3) From the second diffractive surface positive to the third diffractive surface:
(4) Reverse from the third diffractive surface to the second diffractive surface:
(5) Reverse from the second diffractive surface to the first diffractive surface:
(6) Reverse from the first diffractive surface to the object plane:
When the sample is a pure amplitude type object, there is:
When the sample is a complex amplitude type object, there is:
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 coefficient Co is used to judge the effect of restoration and reconstruction, and the range is generally [0, 1]. The closer the Co value 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 the Co value of the amplitude can quickly reach 1.
为了进一步论证三步衍射成像系统方法的可行性, 首先进行了计算机数值模拟分析, 结果如图 3所示。分别计算了单步衍射成像(图 3(a)), 两步衍射成像(图 3(b))、三步衍射成像(图 3(c))。为方便比较, 特将算法的迭代次数都设置为200次, 采用常用图像评价函数相关系数Co来判断恢复重建效果, 其取值范围一般为[0, 1]。Co值越接近1说明恢复重建的物体越接近真实的物体。其值越小说明恢复质量越差, 越偏离真实物体, 成像效果越差, 并以此来判断和选择算法的迭代停止条件。对于纯振幅型的物体而言, 由于没有相位, 所以较为简单。就不在给出其详细的数值模拟结果, 但需要指出的是其收敛速度非常的快, 且振幅的Co值能快速达到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 6.45×10-6 m/pixel, the laser′s wavelength is 632.8×10-9 m. The sample is of the complex amplitude type, its phase distribution range is set to [-π, π], and the diffraction distances are set to z0=100 mm, z1=10 mm, z2=10 mm. The reconstruction of the complex amplitude of the sample is completed, and the correlation Co coefficient′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 of Co at a desired point in Fig. 3 (a), the cursor included with the Matlab software package is used.
在进行计算机数值模拟分析过程中, 使用的样品图样为灰度图, 尺寸大小为256 pixel×256 pixel, 图像传感器CCD的像素尺寸为6.45×10-6 m/pixel, 激光波长为632.8×10-9 m, 且样品为复振幅型, 其相位分布范围设为[-π, π], 将衍射距离设为z0=100 mm, z1=10 mm, z2=10 mm, 完成对样品复振幅的恢复重建。对振幅分布与相位分布分别用相关系数Co值表示。在数值模拟结果中, 其中实线为振幅部分相关系数变化值, 蓝色点线为相位部分相关系数变化值。为了标注Co在某点的数值大小, 在图 3(a)中, 使用了Matlab软件中自带的游标。
From the simulation results shown in Fig. 3(a), under the same conditions, the Co value 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 of z0+z1, the recovery and reconstruction effect resulting from two-step diffraction imaging is significantly improved, the Co values 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 convergence Co value 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 the Co value reaches the optimal ideal value.
图 3(a)数值模拟结果表明, 在相同的条件下, 单步衍射成像恢复重建的振幅与相位部分的Co数值均低于0.5。说明仅有单幅衍射强度图是无法对复振幅型的物体进行恢复重建的, 单步相干衍射成像方法不适用于复振幅型物体。然而, 在增加一幅距离为z0+z1的衍射图后, 两步衍射成像的恢复重建效果得到了明显提升, 振幅及相位部分的Co值均高于0.5, 且有收敛的趋势, 但算法迭代到后面则出现了轻微的降低趋势。所以综合评价没有达到理想结果, 结果如图 3(b)。相比之下, 三步相干衍射成像在算法迭代到60次时开始收敛且到70次左右完全收敛, 后续没有任何下降的趋势。并且, 无论是振幅部分还是相位部分, 最终收敛的Co值均达到1。结果说明由一幅衍射图增加至三幅衍射的过程中, 系统的成像质量不断的提升, 且到第三幅时算法完全收敛, Co值也达到最佳的理想值。
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). The x-value indicates that the system gradually increases the noise from 0, and the step size increases by 2%. When the noise increases to 20%, the Co value 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.
为了分析三步相干衍射成像系统方案的鲁棒性, 对增加噪声系统进行了的数值模拟计算, 结果如图 3(d)所示。其中横坐标表示加入的噪声从0逐渐增加, 增加步长为2%, 当噪声增加至20%时, 振幅与相位的Co值继续高于0.94, 且变化幅度非常小。数值模拟结果表明, 三步衍射成像对抗噪声能力良好。
Traditional SBMIR Technology and System Implementation
Three-step Coherent Diffraction Imaging System Based on Parallel Plates