薄膜磁性多层[PT（2.7 nm）/CO60FE20B20（0.8 nm）/MGO（1.5 nm）] X15 X15在TA（2.3 nm）/pt（2.3 nm）/pt（3.7 nm）上的pt（2.7 nm）上限层上置于硅nitridembrane上。TA，PT和CO60FE20B20由D.C.种植Magnetron的溅射和MGO是通过射频溅射在AR压力为3.5 mtorr（用于PT）和3个MTORR的所有其他材料的AR压力上生长的。如前所述，该材料的质地温度为650 K23。该材料的零温度，零场磁接地状态是平面磁性顺序参数MZ的平行条带形状域的状态，但是这些条带域的任何弯曲构型在能量上几乎相等，实际上所有这些配置都是能量性的，均能变性23。
用[Cr（5 nm）/Au（55 nm）] X20涂层硝酸硅膜的背面，以阻断入射的软X射线41。通过聚焦离子束光刻磨碎了三个圆形孔，将直径定义为720 nm，直径为720 nm，直径为60 nm和40 nm的两个参考孔，可提供全息图的参考光束。这些孔直径代表了一个几何形状，该几何形状是针对空面磁性域41,42,43的时间分辨X射线全息图开发的。它们代表了对小变化的敏感性（用较小的对象孔揭示）和上下文信息（首选较大的视野是首选）的良好折衷。
该实验是在布鲁克黑文国家实验室的国家同步加速器光源II的连贯的软X射线散射光束线（CSX）上进行的。部分相干，圆形极化的X射线聚焦在透射几何形状中的样品上。使用50 µm上游针孔来提高相干性并减少整体通量以限制梁引起的加热。通过将X射线能量调整为CO L3吸收边缘（778 eV），可以实现磁对比度，其中X射线磁性圆形二色性（XMCD）对比度为最大44。Sydor快速CCD摄像头，在样品下游33厘米内安装在Vacuum中，记录了相干散射和相位编码的X射线作为全息图。直接光束被梁挡堵塞。为了映像时间序列中的动力学，连续记录全息图（帧），并与各自的时间戳单独存储。为了将磁性与地形对比度分开（见下文），每100帧将螺旋性反转。每个单帧的暴露时间设置为0.3 s，略低于单个检测器像素的饱和度限制。有效的读数时间为0.087 s。在120秒内记录了两种螺旋脉的完整堆栈。这包括更改解散器的时间。框架的时间戳是通过梁线控制和采集系统记录的，与曝光和读数时间相比，计算机精确度并因此可以忽略不计。也就是说，我们实验的时间分辨率为0.387 s。使用液态氦流低温恒温器用∆T控制样品温度< 0.01 K precision.
Topography holograms Isum were recorded as a sum of positive and negative helicity holograms. only holograms without dynamics during the acquisition were used for topography data. That is, we only considered sets where the magnetic reconstruction from the difference of positive and negative helicity holograms showed full contrast across the entire field of view. The topography holograms were manually updated along the temporal frame sequence as needed. In total we used 26 topography holograms, each averaged over at least 100 frames per helicity. To determine if the topography image still represented the current state of the sample, we determined if the topographically induced airy pattern from the circular aperture was successfully suppressed in the difference holograms. Moreover, we checked that our topography holograms were free of any magnetic contrast via the pair correlation maps. To this end, we note that spurious magnetic contribution would produce an offset in the magnetic pair correlation map. The sign of this offset would be inverted at every helicity change, leading to a characteristic checkerboard pattern in the correlation maps. We confirmed that such patterns are absent in our data.
Subsequently, we subtracted Isum from all magnetic holograms I± recorded with either positive or negative helicity X-rays to obtain the difference hologram Idiff without mask scattering. To compensate for intensity drift, we used a dynamic factor α, that is, Idiff = ±(I± − αIsum). The dynamic factor was automatically determined for each image from the scalar product of the full topographic and the magnetic holograms as α = I±, Isum/Isum, Isum. This equation is based on the approximation that topography and magnetic signal are uncorrelated and hence orthogonal. importantly, this approach leaves only one time-dependent signal in the reconstruction process (I±) and, moreover, produces the same output Idiff for the same state regardless of whether it was recorded with positive or negative helicity light. Thus, after this step, the temporal resolution of CCI equals the temporal resolution (here the acquisition time) of the original single frame I±.
Images in Figs. 1, 3 were reconstructed by Fourier-transform holography (FTH)42,45,46, which is a linear operation and therefore best suited to illustrate the effect of limited signal-to-noise ratio (SNR) and blind averaging. The final, high-SNR images of the magnetic states and modes (Figs. 2, 4, Extended Data Figs. 4, 6, 7, 9 and supplementary videos) were reconstructed by holographically aided iterative phase retrieval47,48, which offers increased resolution and contrast of the resulting images as long as the underlying scattering data are of low noise.
Whereas FTH images were processed based on difference holograms, our phase retrieval algorithm relies on the positive values in the scattering plane. Therefore, we started by calculating the single helicity holograms for every mode discerned by our CCI algorithm using the following equation:
where I±,k is a frame belonging to the set of frames ϕi assigned to a particular mode i, Isum,k is the topography image associated with that specific frame and αk, Idiff,k are its associated dynamic factor and difference image, respectively (see Methods section ‘Algorithm to subtract static topography holograms’ for the definition of the dynamic factor, α). Such a procedure allowed us to obtain single helicity images for each frame with the photon statistics of both helicities.
The resulting holograms were then centred, and any present (readout) offset was subtracted. A support mask for the phase retrieval algorithm was manually generated by selectively thresholding the absolute value of the FTH reconstruction of the average hologram. The same mask was used to reconstruct all images. The holograms were then fed to an iterative phase retrieval algorithm designed for dichroic imaging48. First, the positive helicity exit wave of the sample was reconstructed using a combination of two different phase retrieval algorithms. The starting guess used was derived from the support mask multiplied by a constant, adjusted according to the intensity of the input hologram. We first applied a modified version of the iterative relaxed averaged alternating reflections (RAAR) algorithm49 (700 iteration steps with a relaxation parameter β smoothly going from 1 to 0.5), where the nonnegativity reflector constraint was removed, owing to its inadequacy to deal with complex holograms. The positive helicity reconstruction was completed by the error reduction algorithm50 (50 iteration steps).
Then, the negative helicity exit wave was reconstructed using the error reduction algorithm (50 iteration steps) using the phase of the reconstructed positive helicity complex hologram as a starting guess. This approach enables us to avoid any shift between the two different helicity images. Finally, the magnetic reconstruction was obtained by computing the ratio between the difference and the sum of the two reconstructed images to extract the pure XMCD contrast.
The phase retrieval algorithm yields reconstructions with higher contrast and better resolution compared to the FTH reconstruction. We achieve diffraction-limited resolution of 18 nm for internal modes with 250 frames or more contributing to the reconstruction. The resolution was determined via Fourier ring correlation51 using the half-bit threshold criterion52. Displaying the ratio between the difference and sum, instead of the bare difference, allows to neglect topographic features and widens the field of view of the technique to 810 nm, as the borders of the object hole partially covered in gold present the same contrast as the centre, despite less light being transmitted.
In our X-ray holography geometry, we can determine the magnetic pair correlation directly in Fourier space, as illustrated in Extended Data Fig. 1. We start by subtracting the topography hologram from the recorded single-helicity hologram to obtain the difference hologram hdiff (Extended Data Fig. 1a). Fourier-transforming the difference hologram yields the so-called Patterson map (Extended Data Fig. 1b). The Patterson map includes four reference-reconstructed real-space images of the out-of-plane magnetization mz(x, t) in the off-centre region, and the interference between object hole charge scattering and magnetic scattering in the centre.
We next eliminate the reference-induced modulations as these are mostly adding noise, owing to their sensitivity to fluctuations of the incident beam. We therefore first crop the central part of the Patterson map excluding the reconstructions and, second, transform back to Fourier space. The resulting filtered difference hologram , depicted in Extended Data Fig. 1c, exhibits high intensity at small reciprocal vectors q and needs to be flattened before performing the correlation analysis. Mathematically, , where is the Airy-disk scattering of the approximately binary transmission Tmask of the circular mask defining the field of view. This identity is derived in ref. 53. We therefore extract the pure magnetic scattering by dividing the filtered difference hologram by the square root of the topographic sum hologram . Then, the Fourier-space pair correlation matrix (Extended Data Fig. 1d), calculated using
is essentially equivalent to the pair correlation matrix of the underlying real-space textures mz(x, t), see ref. 53. Note that ., . is the scalar product of the unravelled pixel arrays and . is the norm according to this scalar product.
Our method to classify frames corresponding to the same physical state is illustrated in Fig. 3. It is based on two ingredients: (i) a low-noise metric dij that quantifies the pair-wise similarity (‘distance’) of two camera frames ϕi and ϕj recorded at times ti and tj and (ii) an iterative, hierarchical clustering algorithm3,4 that groups frames into same-state sets.
We start with the frame-wise pair correlation map of magnetic textures, cij, see previous section. As shown in Fig. 3a and Extended Data Fig. 2, already cij resolves clear grid-shaped fingerprints of state changes at a single-frame level, with approximately ten times better signal-to-noise ratio than a similar analysis based on holographic real-space reconstructions (see Extended Data Fig. 3). We further reduce the noise of the metric by the following consideration: If Sα and Sβ represent the same state (A) then the correlation of Sα with any arbitrary frame should be the same as the correlation of Sβ with that frame. This follows from the transitivity of the equivalence relation. Hence, each pair correlation cαγ for any γ is a fingerprint of state A and the entire vector c(Sα) ≡ (cα1, cα2, …, cαn) has considerably better photon statistics than the single scattering pattern Sα alone. We therefore define our low-noise fourth-order Pearson distance metric dij as
where is the mean of the vector c(Si). That is, the matrix is the normalized Pearson correlation matrix of cij. The resulting distance matrix is shown in Fig. 3b.
On the basis of the measure of distance, we identify distinctive clusters of highly similar frames through an iterative variant (Fig. 3a–c) of a concept know from gene identification3 as UPGMA (unweighted pair group method with arithmetic mean) to construct a hierarchical cluster tree from dij. In this bottom-up process, the nearest two clusters are combined into a higher-level cluster, starting with single frames on the lowest level3,4 and ending with a single final cluster. The UPGMA clustering was performed using the MATLAB function ‘linkage’ specified with the ‘correlation’ metric to compute the proximity of two scattering patterns Si and Sj by autocorrelation of the input matrix dij. This additional application of the correlation function results in a nonlinear similarity metric enhancing the numerical separation of high correlation values (similar states) and low correlation values (dissimilar states). The resulting so-called dendrogram is presented in Fig. 3c. The vertical level, or height, of each link in the dendrogram corresponds to the distance (abstract, nonlinear in our case) between the linked clusters. If two linked clusters represent the same physical state of the sample, the step size, or height difference, between the old and the new hierarchy level is small. Conversely, a large step, also known as an inconsistency, indicates a natural division in the data set, that is, the merging of different physical states. The largest step size is naturally observed at the topmost node in the tree. We aim to make the most robust classification and therefore split the tree only at the top node into two subclusters. Instead of dividing the dendrogram into a complete set of clusters in a single step, we iterate through the whole clustering procedure, separating the dendrogram in each iteration at most into two subgroups (at the topmost link). The iterations enable us to extract domain configurations beyond the conventional UPGMA, for example, to separate configurations A1 and A2 in Fig. 3c.
The next iteration step is to decide if our two identified subclusters represent pure or mixed states, that is, if they comprise frames from a single state or multiple states. To this end, we quantify the distinctness of the topmost link in the dendrogram by the inconsistency coefficient ξ, where ξ is given as the ratio of the average absolute deviation of the step height to the standard deviation of all lower-level links included in the calculation. Both the step height and the standard deviation are indicated in Fig. 3c. On this basis, we identify all clusters that undercut an inconsistency threshold as pure-state clusters whereas, complementarily, mixed states are present in clusters exceeding this threshold.
The iteration steps are repeated for mixed-state clusters until—within our temporal resolution and available signal-to-noise ratio—complete separation into clusters containing only pure states is achieved. The frames of each pure-state cluster are then averaged, and a real-space image of the corresponding domain configuration is reconstructed (Fig. 3d). The knowledge of which frames entered each cluster, and the time stamps of when these frames were recorded, yields the temporal reconstruction, as shown in Fig. 3e (see Methods section ‘Estimation of the temporal discrimination threshold and reconstruction of the 32 states’ for more details about the temporal reconstruction).
Applied to our data, we find that the modified clustering algorithm performs well for thresholds of the inconsistency coefficient ξ in the range 1.7 to 2.2. The upper limit is determined by the minimal sensitivity needed to separate the most evident mixed-states (for example, identified by motion-blurred contrast, as in Fig. 1b). The lower limit is set by the total number of frames (here 99%) that can be assigned to clusters that yield acceptable real-space reconstructions, which we assume as 15 frames per cluster. Specifically, we chose ξ = 1.85 because it produces the lowest frame misclassification probability while assigning 99.5% of our 28,800 frames to clusters with >15帧（有关框架错误分类的评估的更多详细信息，请参见方法“对32个状态的时间歧视阈值和重建的估计”部分。
我们注意到，分层聚类的主要缺点是所谓的链式现象和对异常值的敏感性，合并不一致的群集或创建少量元素的次要群集。尽管前者通过选择足够低的不一致阈值来有效地最大程度地减少了迭代过程，但后者导致纯状态分解为几个不可分割的簇。在我们的情况下，该算法找到119个磁性构型，其中许多磁性构型明显无法区分。为了消除此组中的重复项，我们通过平均所有属于这些群集的框架ϕA和ϕB组合的对cab cab cab来计算所有已识别的簇A和B之间的相似性。然后，使用常规（非读图）凝集的UPGMA算法，我们处理了所得的119×119单元格群集相关矩阵，将实际代表相同域配置的组簇变为组簇。我们通过应用与上述相同的迭代方案来验证我们的最终集群是纯状态。
结果是一组72个域配置，为进一步分离为离散的内部模式（仅是空间分辨的域配置的离散版本）和状态（时空分辨出的配置）的基础。
为了从有时通过相检索算法重建的有时嘈杂或不均匀照明的真实空间图像中提取域壁的位置，我们将群集分析中确定的72个域配置离散化（参见扩展数据图4）。为此，我们从低通滤波器开始：首先用圆形二进制掩码裁剪了簇平均全息图，然后再重建真实空间图像以减少图像中的高频噪声。对于每个内部模式，单独确定傅立叶空间中滤波器的半径为最大Q，超过该模式的所有帧的总磁散射信号。尽管此滤镜减少了空间分辨率，但域壁的位置仍然不受影响。域分割基于复杂值图像的重建相φ（r）的符号，而不是其实际部分，因为该相显示了向上和下磁化的最明显的分离。重建相位的集中在零附近，以实现通过域壁位置的相变±π观察到的最大对比度。我们认为小于50像素的域是由噪声引起的人工制品，并将其与周围的域合并。手动调整受伪像的强烈影响的图像。
72个内部模式的一些原始真实空间重建显示了混合态叠加的证据，请参见图4b。如果域状态在采集单个帧中发生变化，则这种混合状态不可避免地会出现（即，如果动态比我们的最小时间分辨率更快）。因此，仅将域图像二进制为扩展数据中所示的图像将构成信息丢失。但是，我们发现整个二进制内部模式构成了代表性的基础，基于该模式可以分解原始的灰度图像，如扩展数据中所示。实际上，我们通过真实空间相关性评估了域图像和所有二元内部模式之间的相似性53。下一步，我们应用了多线性回归，将每个单独的灰度图像分解为与灰度图像具有> 88％相似性的二进制内部模式。我们发现，内部模式的加权叠加准确地表示所有域配置，而状态32除了手动创建了一个附加的二进制域配置73（我们将其归因于状态32是我们时间序列中的最后一个状态，并且不足数据可自动分解IT）。在扩展数据中显示了所有72个内部域模式的离散表示图6。
聚类分析是将数据分类为具有相似特征的自然划分的一种方法，例如，在这项工作中，同时磁性域状态。应用于嘈杂数据的任何此类算法最终都会产生分配错误。这些错误在时间重建中具有至关重要的作用，在时间重建中，每个错误分类的框架都是错误检测或被忽视的过渡。但是，对于非常相似的状态，发现真正的分配变得越来越困难（请参见图3D中的域配置A1和A2）。在聚类算法确定的72个内部模式中，有些表现出高达99％的真实空间相似性。鉴于这种非常接近的相似性，问题出现了单个框架对特定磁状态的分配程度的准确性。我们通过估计框架错误分类概率来解决这个问题，这是我们从两个簇之间分配帧的鲁棒性来得出的。
在我们的分析中使用的自下而上的聚类方法中，在较低级别创建的集群之间的链接强烈影响更高级别的聚类。整个链接树（即树状图）对低速高噪声数据以最低级别创建的链接特别敏感，在该链接中，初始距离的小变化可能导致最终分配偏离。相反，针对初始帧集的变化的鲁棒性是衡量聚类分配的质量的度量。基于此基本原理，我们开发了以下分析，以验证聚类分析的准确性和灵敏度。
在我们的分析中，我们验证了每对簇A和B之间帧的聚类分配。我们假设我们的完整群集分析已经分别输出了成员ϕA和ϕB的代表性估计，分别属于A和B。为了进行错误分析，我们现在比较聚类以不同的初始帧集开始 - 一方面是原始帧集ϕA+ϕB，另一方面是十个新组装的子集。每个子集并分别包含一个随机选择的父帧集ϕA和ϕB的一半。我们使用相同的迭代集聚集的UPGMA算法将组装的帧集分为两个簇，如前所述。通过将原始帧与群集分配与新分配进行比较，即如果在A或B中分组的框架分别分别分组或分别分组或分类，则可以评估聚类的鲁棒性。否则，将帧视为错误分类，我们将错误分类的概率定义为错误分类帧的平均分数。
对于大多数模式对（87％），我们发现聚类与随机子集的完美巧合。我们发现，如果至少一个簇包含非常少的帧数，则框架分类的鲁棒性会大大下降。我们从进一步的分析中排除了少于100帧的群集（单个全息图堆栈的标准fth图像的帧数），因为记录更多的帧并不是一个基本的挑战。
在扩展数据图5中，我们显示了所有剩余模式对的框架错误分类概率作为其相似性的函数，以其对相关性表示。仅针对非常相似的域配置发生了明显的错误分类率。具体而言，我们发现框架错误分类概率低于1％的相似性，高达93.8％。我们将其定义为分析中的灵敏度阈值。将具有较高相似性的内部域模式分组，使其时间歧视具有可接受的错误分类率，如图3E所示。这些模式被称为空间和时间分辨的磁性“状态”。在实践中，我们连续合并了最近的模式，直到两个不同状态的任何对模式之间的距离超过了100％ - 93.8％= 6.2％的相似性距离，该距离由灵敏度阈值定义。图6中显示了对已识别状态及其相关内部域模式的概述。在这种情况下，我们注意到，在视野的不同部分，域墙跳跃之间的相关性不是集群算法的人工制品，而是从材料的材料范围内构成的远处，而远离材料的愿望，而是从远处构成的范围，而是从远处进行构成，从而构成了范围的范围，从而构成了势力的竞争，从而构成了势力的范围。场能。
状态和内部模式的数量可能会随着模型参数的特定选择而变化，但是文本中得出的结论在上面讨论的广泛参数中保持稳健。我们进一步指出，这种类似配置的“套在一起”的方法使CCI甚至可以广泛地适用于表现出非经常动力学的系统，尤其是与固定和有限的视野结合在一起时。
过渡网络（图4C）在其原始表示中已经在其原始表示状态下将视觉划分为状态的聚集体。为了量化这一观察结果，我们定义了一个新的距离度量，该指标考虑了相似性距离以及状态之间的过渡频率。为此，我们使用最简单的方法，在其中，我们将两个距离度量添加为
如果DAB是状态A和B之间的距离，则根据该新的度量，CSTATE是状态相似性距离，是A和B之间的过渡频率θ的对数，T是实验的持续时间，表示在所有状态下观察到的最高。然后，我们应用了UPGMA层次聚类算法以根据此距离度量来从状态创建树状图（扩展数据图10）。在此树状图中，手动将定义聚集物的分离阈值手动设置在0.38的大台阶上。
我们确定有吸引力的固定位点位置的方法是基于域壁职业概率和平均域壁曲率。首先，我们提取了在扩展数据中显示的所有内部模式中至少有20％在图7C中观察到的域壁位置。在这些热点中，有一些点状区域和线形区域，其职业概率增加，表明固定位置。我们使用最大职业概率的点来直接确定点状热点的固定位点。
其次，我们使用了在扩展数据中说明的平均域壁曲率图7D来定位线形热点的固定位点。也就是说，我们提取具有大于1.5 µm -1的平均曲率的扩展区域。在这些区域中，我们将最高曲率的点定义为线形热点中固定位点的位置。为了确保足够的空间分离，只考虑了至少分开8像素的固定位点。
在扩展数据表1中，我们将CCI与现有的时间分辨相干成像方法进行了比较。这些数字基于以下考虑：第1行：这项工作中常规相干成像的结果。当平均约250帧以上时，可以实现18 nm的衍射限值分辨率（请参见“实时空间图像重建”的方法，该部分对应于〜100 s。当然，要实现最佳分辨率，系统必须在整合时间内保持静态。第2行：通过使用CCI实现这项工作实现的时空分辨率。为了达到报告的衍射有限的空间分辨率，必须平均与常规成像相同的帧数，但由于CCI而没有时间分辨率损失。在指定的分辨率中未包含状态分配给框架的统计错误以及将几种内部模式分组为状态分组的不确定性，并且必须额外考虑。第3行：传统的相干X射线成像的预期潜力，分辨率低于10 nm（通过非合并X射线成像技术难以实现的制度）。随着目标空间分辨率的增加，积分时间大大增加，因为散射信号通常在高Q处强烈衰减。第4行：同步辐射源的CCI。此处报告的帧速率对应于Moench detector54的检测器帧速率，这是软X射线状态中特别快的检测器。第5行：在高重复率X射线自由电子激光器时进行CCI。这里给出的观点基于欧洲XFEL（EUXFEL）55的规格。时间分辨率受到最大束频率（当前在Euxfel处的1/（220 ns））的限制，并且最大持续时间受束列车的长度（目前在Euxfel时为600 µs）的限制。第6行：XFEL的泵 - 探针方案中的CCI， 如果时间分辨率受脉冲持续时间和持续时间的限制，则最大泵 - 探针延迟可用。第7行：基于参考。43。第8行：基于参考。56,57。第9行：基于参考文献的组合。58，其中展示了无损的单发全息磁成像，并参考。59，其中显示出在两个不同时间的同一标本的图像可以用这种单发全息图编码。