Review of partially coherent diffraction imaging

Xu Wen-Hui; Ning Shou-Cong; Zhang Fu-Cai

doi:10.7498/aps.70.20211020

Abstract

Coherent diffraction imaging (CDI), a type of lensless imaging method, relies on the use of light source with high-degree coherence to compute highly resolved complex-valued objects. The coherence of light source consists of temporal coherence and spatial coherence. In practice, it is difficult to obtain a fully coherent source. Spatial decoherence can be generated in the following three scenarios: no synchronization mechanism for the whole radiation source, a finite (non-zero) point spread function of the detector, and the sample variation within exposure time. Partial temporal coherence means that the beam is not quasi-monochromatic, behaving as the energy spread of the illumination. The consequence of reduced degree of temporal and/or spatial coherence in CDI is the decrease of visibility in the measured diffraction intensity. A fundamental assumption of CDI is the full temporal and spatial coherence, and even a relatively small deviation from full coherence can prevent the phase retrieval algorithm from converging accurately. It is necessary to break the barrier of limited coherence by improving the experimental setups directly or optimizing the phase retrieval algorithms to mitigate decoherence. Based on the Wolf’s model of coherence-mode of light and the framework of CDI using partially coherent light proposed by Nugent et al., various methods have been proposed to solve the problems induced by low coherence. Those methods generally experience a similar development process, that is, from the requirement for measuring the spatial (coherent length or complex coherent factor) or temporal (spectrum distribution) coherence properties to without the need for such priori knowledge. Here in this work, the principles of partial coherent CDI, and the major progress of CDI with partial spatial- and temporal-coherent light are reviewed.

Keywords:

Authors and contacts

1.
Harbin Institute of Technology, Harbin 150001, China
2.
Department of Electrical and Electronic Engineering, Southern University of Science and Technology, Shenzhen 518055, China
3.
Department of Materials Science and Engineering, National University of Singapore, Singapore 117575, Singapore

Corresponding author: Zhang Fu-Cai, zhangfc@sustech.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11775105, 12074167), the Shenzhen Science and Technology Program, China (Grant No. KQTD20170810110313773), and the Centers for Mechanical Engineering Research and Education at MIT and SUSTech (MechERE Centers at MIT and SUSTech) (Grant No. 6941806).

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Cited By

图 1 用于研究部分相干CDI的照明方式和成像系统^[61]　(a)平面波和球面波照明示意图, 透镜会引入额外的相位曲率, 假设透镜是理想的并且在物平面的相干长度是相同的; (b)实验系统模型示意图

Figure 1. Schematics of the two illumination conditions and experimental system for investigating partially coherent CDI^[61]: (a) Illumination by planar and curved component waves, respectively. Assuming an ideal perfect lens, both have identical intensities and coherence lengths at the sample plane. However, the illumination with a lens has an additional phase curvature by the focal length of the lens; (b) the experimental system being modeled and the coordinate system used.

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图 2 低空间相干光照明时, 采用文献[63]多模算法(a)和单模算法(b)分别重构的物波强度信息; 在高空间相干光照明但已知相干长度信息不准确的情况下, 采用文献[63]多模算法(c)和单模算法(d)分别重构的物波强度信息^[63]

Figure 2. Reconstruction of the magnitude of the wave leaving the sample. At the low-spatial coherent illumination, the reconstruction using the known coherence information (a) and assuming perfect spatial coherence (b). In the high-spatial coherent condition, reconstruction utilized the known coherence information (c) and full-spatial coherence assumption (d)^[63]

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图 3 散射实验中的退相干现象^[83]　(a)由照明探针的横向相干性和有限的带宽引起的模态混合; (b)样品中的混合模态, 包括量子混合态和快速平稳随机过程, 如振动、开关或稳定流; (c)检测器的点扩散造成的模态混合; (d)对于远场衍射, 退相干会造成散射强度分布的可见度降低. 图(d)右半边是模拟样品振动给衍射图样带来的影响

Figure 3. Decoherence in scattering experiments^[83]: (a) Mixed states within the probing radiation may include all sources of mixing that manifest themselves as transverse partial coherence or finite bandwidth; (b) mixed states in the object of interest may occur from quantum mixtures and fast stationary stochastic processes such as vibrations, switching or steady flows; (c) mixed states caused by the detector point spread; (d) the signature of decoherence in far-field diffraction is a decrease in visibility of the scattered intensity distribution. Here the right half of the simulated diffraction pattern shows the effect of sample vibration.

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图 4 伊辛模型成像模拟^[83]　(a) 在样本的布局中, 灰色方块代表每个自旋, 在照明光上产生±π/2的相移. 蓝色线条表示铁磁耦合, J = –1, 红色线条表示反铁磁耦合, J = +1; (b) 减去平均值后的4个主要物体模态. 相邻自旋正交化的结果, 在(b)中用蓝色和红色框勾勒出铁磁键(c)和反铁磁键(d), 相对模态的振幅与期望值非常吻合

Figure 4. Imaging a simulated Ising model^[83]. (a) In the sample layout, grey squares mark individual spins, which incur a ±π/2 phase shift on the illumination. Blue lines represent ferromagnetic couplings, J = –1, red lines antiferromagnetic ones, J = +1. (b) The four dominant modes after subtraction of the mean. (c) and (d) The results of orthogonalization of neighbouring spins, outlined by blue and red frames in panel (b), for a ferromagnetic bond (c) and an antiferromagnetic bond (d). The relative mode amplitudes are in good agreement with expected values.

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图 5 (a)传统ptychography收集数据的模式, 即每移动距离r_j到1个位置静止后, 检测器才开始曝光进行数据的采集^[86]; (b) 样品以速度v进行连续运动时的数据采集模式, 每个r_j位置的衍射图样都是由在检测器曝光时间T内的$ \left| v \right|T $个位置的物波光子的叠加而得^[86]; (c) 样品以速度v进行连续运动时的数据采集模式可以被离散化, 即$ \left| v \right|T $可以离散化为n个小距离$ v\Delta t $^[86]; (d)—(f)样品相位的重构结果(左边1 mode是假设完全相干的情况, 右边5 modes是假设部分相干而将照明探针分解为5个模态的情况): (d)由原始采集数据直接重构的结果; (e)将相邻4个位置(总横跨距离是400 nm)采集的衍射图样叠加后重构的结果, 这里用相邻扫描位置对应衍射图的叠加来近似样品的连续运动^[86]; (f)将相邻7个位置(总横跨距离是700 nm)采集的衍射图样叠加后重构的结果^[86]

Figure 5. (a) Conventional ptychography uses data collected with discrete translations r_j ^[86]; (b) for a continuously scanned sample with constant velocity v, the diffraction data will be made up of the positions that occupy a length $ \left| v \right|T $ around the scan positions r_j due to an integration time of T ^[86]; (c) the continuous scan can be discretized with steps separated by a small distance $ v\Delta t $^[86]; (d)−(f) reconstructed object phase (1 mode assuming full coherence, left and 5 modes assuming partial coherence, right): (d) Reconstruction for the original data^[86]; (e) reconstruction using new positions that consist of summing four adjacent positions (over 400 nm horizontally), approximating a continuously scanned sample^[86]; (f) reconstruction for a larger (seven positions, 700 nm) level of summing, which would be equivalent to an increased scanning speed^[86].

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图 6 (a)—(h)单层WS₂样品在不同电子剂量条件(58000和3300 e·Å^–2)下ptychography重构的结果; (i)—(p)单层WS₂样品在不同扫描步长(0.85和5.08 Å)下ptychography重构的结果; (a), (b)和(i), (j)分别分解为3个模态和2个模态重构的结果; (e), (f)和(m), (n)为不使用模态分解方法重构的结果; (c)和(d), (g)和(h), (k)和(l)以及(o)和(p)分别为对应于(a)和(b), (e)和(f), (i)和(j)以及(m)和(n)的衍射花样^[95]

Figure 6. Ptychographic reconstructions of a monolayer WS₂ sample in different illumination dose (58000 and 3300 e·Å^–2, (a)−(h)) and different scan step size (0.85 and 5.08 Å, (i)−(p)) conditions^[95]: Mixed reconstructions from datasets using three ((a), (b)) and two ((i), (j)) probe modes, respectively; ((e), (f)) and ((m), (n)) single mode reconstructions; ((c), (d)), ((g), (h)), ((k), (l)) and ((o), (p)) corresponding diffractograms of ((a), (b)), ((e), (f)), ((i), (j)) and ((m), (n)).

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图 7 (a) 准单色光(右上)和宽带光(左下)照明所对应的部分衍射强度数据^[64]; (b)—(e) 重构的待测样品幅值信息(空间相干长度已知): (b) 准单色数据重构结果, 数据采集曝光时间为500 × 3 s, 重构时假设照明光完全相干^[64]; (c) 复色数据重构结果, 数据采集曝光时间为500 × 50 ms, 重构时假设照明光完全相干^[64]; (d) 复色数据重构结果, 重构时假设照明光完全相干, 但是采用从图(b)获得的物体精确尺寸作为初始支撑^[64]; (e) 复色数据重构结果, 采用和图(b)和(c)相同的重构过程, 但是重构算法为复色光衍射算法(polychromatic diffraction algorithm, PolyCDI)^[64]

Figure 7. (a) Comparison of the quasi-monochromatic (upper right) and broadband (lower left) data^[64]. (b)−(e) Reconstructed amplitude of the sample (all reconstructions presented here use known spatial coherence information): (b) Reconstruction from quasi-chromatic data, collected with 500 × 5 s exposures, assuming full temporal coherence^[64]; (c) reconstruction from polychromatic data, collected with 500 × 50 ms exposures, assuming full temporal coherence^[64]; (d) reconstruction from polychromatic data assuming full temporal coherence but using the exact dimensions of the object (obtained from panel (b)) as the initial support^[64]; (e) reconstruction from polychromatic data using an identical procedure to panel (b) and (c) but using the polychromatic diffraction algorithm (PolyCDI)^[64].

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图 8 衍射图样数值单色化的原理图^[105]. 传统的CDI假设照明光源是单色的. 对于阿秒宽带光源, 衍射图样是所有波长所对应的单色衍射图样的加权叠加. 这些单色的衍射图样可以认为只有几何缩放的差别. 这种缩放是数值可逆的, 由矩阵C来表示该缩放矩阵. 确定了C之后, 就可以通过求解矩阵向量问题b = Cm从宽带光衍射图样b中提取出单色化的衍射图样m (图中右下部分所示为对应于一维衍射图样的简化情形). 然后就能直接将m代入到传统的相位恢复算法中重构出样品的复振幅信息. ℓ是复色光所包含的离散化光波的波长个数

Figure 8. Principle of the numerical monochromatization^[105]. Conventional CDI assumes a monochromatic source. In the case of an attosecond broadband source, the diffraction pattern is the incoherent, spectrally weighted sum of the monochromatic diffraction patterns corresponding to all wavelengths presents in the sources. These monochromatic patterns are identical except for a geometric scaling. In the presented method this scaling is numerically inverted, and the scaling matrix is denoted by matrix C. The monochromatization method consists in the inversion of the matrix-vector problem b = Cm in order to retrieve the monochromatic diffraction pattern m from the broadband measurement b (The lower right part of the figure shows the simplified case for 1D diffraction pattern). Then m can be directly put into the conventional phase retrieval algorithm. ℓ is the index that corresponds to the spectral discretization.

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表 1 不同部分时间相干CDI方法的比较

Table 1. Comparison of different methods of partially temporal-coherent CDI.

方法	是否需要光谱信息	可适用的波的类型	最大带宽	实验结果
方法	是否需要光谱信息	可适用的波的类型	最大带宽	中心波长/nm	空间分辨率
Multiwavelength CDI^[100]	Yes	Discrete	Several harmonics	(35 ± 1)	(165 ± 5) nm
Two-step CDI^[101]	Yes	Discrete	Several harmonics	810	1 μm
PolyCDI^[64]	Yes	Continuous	11%	0.888	(150 ± 5) nm
Numerical monochromatization^[105]	Yes	Continuous	>10%	710	2.5 μm
Two-pulse imaging^[113]	No	Continuous	No limitation	47	6.7 μm

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