^{1}Japan Synchrotron Radiation Research Institute (JASRI), Sayo, Hyogo 679-5198, Japan^{2}Research and Services Division of Materials Data and Integrated System, National Institute for Materials Science, Tsukuba, Ibaraki 305-0047, Japan^{3}Graduate School of Frontier Sciences, The University of Tokyo, Kashiwa, Chiba 277-8561, Japan
Received October 11, 2021; Accepted January 5, 2022; Published February 1, 2022
Elucidating magnetic interactions, such as ferromagnetic interaction and anisotropy, is important for improving the performance of magnetic materials and devices. In addition to those intrinsic factors, extrinsic ones such as grain size and grain boundaries also play a significant role in permanent magnets and soft magnetic materials. For capturing such information, a powerful approach is to monitor in detail the temporal evolution of the ferromagnetic domain structure during the magnetization or demagnetization process and then to estimate the determinants of physical properties. For measuring magnetic patterns with high spatio-temporal resolution, single-shot X-ray diffractive imaging is an effective measurement technique. However, it is usually difficult to extract valid physical property information because the measurement data contains errors and missing information. Here, we propose a phase retrieval algorithm based on total variation and L_{2} regularizations for reconstructing ferromagnetic domain images from the diffraction pattern. The algorithm has been validated on emulated diffraction data, and is expected to be a powerful analysis method for extracting physical property information from the single-shot diffraction data.
This article is published by the Physical Society of Japan under the terms of the Creative Commons Attribution 4.0 License. Any further distribution of this work must maintain attribution to the author(s) and the title of the article, journal citation, and DOI.
Dynamic physical properties of magnetic textures, such as variation of magnetic domains pattern and motion of magnetic skyrmions, attracts much attention from both basic science and applied research viewpoints. Since the topological magnetic textures interact with various external fields, such as irradiation of light or electric fields, clarifying the temporal variation of magnetic patterns under the applied external field is important to understand the determinants of physical properties and to improve the performance in magnetic materials and magnetic devices.^{1}^{–}^{3}^{)} For example, in permanent magnets, it has been suggested that their performance is strongly correlated with the ferromagnetic domain pattern.^{4}^{)} The determinants of physical properties, such as a coercive magnetic field and a remanence, are expected to be estimated by studying the change in magnetic domain pattern during the magnetization or demagnetization process in detail. Changes in the domain pattern during the magnetization process have been usually observed by microscopy using magneto-optical effects, i.e., the Kerr effect and the Faraday effect.^{5}^{)} Although sub-µm resolution is the limit of the microscopy of visible light, there is a possibility that the physical properties are attributed to a finer magnetic texture, which requires microscopy methods with higher spatial resolution for observation.
The use of soft X-rays with nanometric wavelength enables microscopic observations with higher spatial resolution and element-selective observations by tuning the soft X-ray energy to absorption edges of the target magnetic element. Their properties are very important for understanding magnetic properties, as they allow us to observe magnetic domain patterns for individual element.^{6}^{,}^{7}^{)} Focusing imaging microscopy, scanning transmission X-ray microscopy and coherent X-ray diffractive imaging (CDI) are known as microscopic techniques to observe the magnetic textures using X-rays.^{8}^{–}^{11}^{)} Here, we focus on the CDI method, which is relatively easy to achieve high resolution and suitable for observing magnetic dynamics. Diffraction measurement in the reciprocal space would provide a more precise information of quasi-periodic magnetic pattern and then is suitable for observing irreversible processes by single-shot measurement. Despite such characteristics, it is necessary to reconstruct real-space images by analyzing diffraction pattern which correspond to the Fourier transformation of real-space images. Since the observed diffraction pattern lacks the phase information in the Fourier space, phase retrieval algorithms and/or holographic techniques are mandatory to obtain the phase information, and then to reconstruct the real-space images by the inverse Fourier transformation.^{9}^{–}^{11}^{)} Comparing the phase retrieval algorithms with the holography techniques, the phase retrieval algorithms provide higher spatial resolution because their resolutions are not restricted by reference holes used in the holography.
Reconstruction of a magnetic domain image from the magnetic diffraction pattern has been reported using the phase retrieval algorithm, especially so-called the hybrid-input-output (HIO) algorithm.^{9}^{)} Conventional algorithms are highly susceptible to overfitting to noises and errors, which may cause difficulties in the single-shot measurements where intensities of diffraction pattern with poor signal-to-noise (S/N) ratio are assumed. The HIO algorithm needs special techniques to reconstruct the real-space image when a diffraction pattern lacks information, which increases the analysis time.^{9}^{)} In the present study, to suppress overfitting and the impact of lack of information, we consider introducing some regularizations in the iterative Fourier transformation of the phase retrieval algorithms. For example, utilizing the sparseness of isolated magnetic skyrmions, i.e., few skyrmions in real space, the sparse phase retrieval algorithm based on the \(L_{1}\) regularization (\(L_{1}\) algorithm) can reconstruct the real-space images from the diffraction patterns with a low S/N ratio and lack of information as reported in a previous study.^{12}^{)} Here, we propose a sparse phase retrieval algorithm based on total variation (\(\mathit{TV}\)) regularization for reconstructing magnetic domain patterns possessing sparse differential images.
2. Method
Sparse phase retrieval algorithm based on total variation and \(L_{2}\) regularizations
The phase retrieval algorithm reconstructs a spatial distribution of transmittance \(f(\boldsymbol{r})\) depending on the magnetic moment parallel to the X-rays by minimizing the residual between the absolute value of the Fourier transform \(\mathcal{F}[f(\boldsymbol{r})]\) and the square root of the measured value \(F_{\textit{obs}}\) (\(=\sqrt{I_{\textit{obs}}}\)), and can be formulated as, \begin{equation} f(\boldsymbol{r}) = \mathop{\mathrm{arg\,min}}_{f(\boldsymbol{r})}\left\{\frac{1}{2N^{2}}\|{F}_{\textit{obs}}(\boldsymbol{q}) - | \mathcal{F}[f(\boldsymbol{r})] |\|_{l_{2}}^{2} \right\} \end{equation} (1) with the total number of pixels \(N^{2}\). Although \(f(\boldsymbol{r})\) is generally complex valued, here we consider the real value because it can be tuned by selecting the X-ray energy at around the peak of absorption spectra.^{13}^{)} We consider adding a regularization term as prior information that the magnetic domain structure has many smooth regions and the domain wall region where the magnetic moment reverses is sparse. Such sparseness can be introduced as the \(\mathit{TV}\) regularization, which is known to be a powerful method in the compressed sensing and the denoising of images. Since spatial distribution of X-ray transmittance in the nonmagnetic state can be measured beforehand, this information can be also incorporated as the \(L_{2}\) regularization. In the \(\mathit{TV}\)-\(L_{2}\) algorithm, a phase retrieval problem is expressed as \begin{align} f(\boldsymbol{r}) &= \mathop{\mathrm{arg\,min}}_{f(\boldsymbol{r})}\biggl\{\frac{1}{2N^{2}}\|{F}_{\textit{obs}}(\boldsymbol{q}) - | \mathcal{F}[f(\boldsymbol{r})] |\|_{l_{2}}^{2} + \lambda_{l_{2}}\|f(\boldsymbol{r}) \notag\\ &\quad- f_{\textit{base}}\|_{l_{2}}^{2 \in S} + \lambda_{\textit{TV}}\|\nabla f(\boldsymbol{r})\|_{\textit{TV}}^{1 \in S} + \lambda_{l_{1}}'\|{f}(\boldsymbol{r})\|_{l_{1}}^{1 \notin S} \biggr\}, \end{align} (2) where \(f_{\textit{base}}\) represents the spatial distribution of transmittance. S indicates the area where the specimen exists,^{14}^{)}\(\|\cdot \|_{l_{2}}\) and \(\|\cdot \|_{l_{1}}\) are \(\ell_{2}\) and \(\ell_{1}\) norms, respectively, \(\|\cdot \|_{\textit{TV}}\) denotes the \(\mathit{TV}\) norm defined by \(\|\nabla f(\boldsymbol{r})\|_{\textit{TV}}^{1} =\sum_{i,j}\sqrt{\{f(r_{i + 1,j}) - f(r_{i,j})\}^{{2}} + \{f(r_{i,j + 1}) - f(r_{i,j})\}^{{2}}}\) for \(i,j = 1,2,\ldots,N\). \(\lambda_{l_{2}}\), \(\lambda_{\textit{TV}}\), and \(\lambda_{l_{1}}'\) are regularization parameters. Phase retrieval algorithms start from a random phase pattern \(\theta_{1}(\boldsymbol{q})\) and calculate forward and inverse Fourier transforms iteratively. Using Parseval's theorem \(\frac{1}{N^{2}}\sum_{i}| F(q_{i}) |^{2} =\sum_{j} | f(r_{j}) |^{2}\) with \(F(\boldsymbol{q}) =\mathcal{F}[f(\boldsymbol{r})]\), the first term in Eq. (2) can be transformed into the absolute error in real space. Then, the real-space constraint of the \(\mathit{TV}\)-\(L_{2}\) algorithm is expressed as \begin{equation} f_{n + 1}(\boldsymbol{r}) = \mathop{\mathrm{arg\,min}}_{f(\boldsymbol{r})} \begin{cases} \dfrac{1}{2}\|{f}'_{n}(\boldsymbol{r}) - f(\boldsymbol{r})\|_{l_{2}}^{2} + \lambda_{l_{2}}\|{f}(\boldsymbol{r}) - f_{\textit{base}}\|_{l_{2}}^{2} + \lambda_{\textit{TV}}\|\nabla f(\boldsymbol{r})\|_{\textit{TV}}^{1} &\text{$\boldsymbol{r}\in S$}\\ \dfrac{1}{2}\|{f}'_{n}(\boldsymbol{r}) - f(\boldsymbol{r})\|_{l_{2}}^{2} + \lambda_{l_{1}}'\|{f}(\boldsymbol{r})\|_{l_{1}}^{1} &\text{otherwise}\end{cases}, \end{equation} (3) here \(f'_{n}(\boldsymbol{r})\) is calculated using the Fourier transform of \(F_{n}'(\boldsymbol{q}) = | F_{\textit{obs}}(\boldsymbol{q}) |\exp \{i\theta_{n}(\boldsymbol{q}) \}\). Equation (3) inside S can be rewritten by expanding the first two terms and completing the square for \(f(\boldsymbol{r})\) as \begin{align} f_{n + 1}(\boldsymbol{r}) = \mathop{\mathrm{arg\,min}}_{f(\boldsymbol{r})}\left\{\frac{1}{2}\|{f}_{n}^{*}(\boldsymbol{r}) - f(\boldsymbol{r})\|_{l_{2}}^{2} + \lambda_{\textit{TV}}^{*}\|\nabla f(\boldsymbol{r})\|_{\textit{TV}}^{1} \right\}&\quad \notag\\ \boldsymbol{r}\in S,& \end{align} (4) where \(f_{n}^{*}(\boldsymbol{r})\) and \(\lambda_{\textit{TV}}^{*}\) are defined as \(f_{n}^{*}(\boldsymbol{r})\equiv \frac{f'_{n}(\boldsymbol{r}) + 2\lambda_{l_{2}}f_{\textit{base}}}{2\lambda_{l_{2}} + 1}\) and \(\lambda_{\textit{TV}}^{*}\equiv \frac{\lambda_{\textit{TV}}}{2\lambda_{l_{2}} + 1}\).
Equation (4) is solved by the primal-dual splitting method which does not require to calculate the inverse matrices that takes time to compute.^{15}^{)} By introducing \(\boldsymbol{p}= (p^{1}(\boldsymbol{r}),p^{2}(\boldsymbol{r}))\) as the dual variable of \(f(\boldsymbol{r})\), the update equation for the k-th iteration is expressed as \begin{equation} (\boldsymbol{p}_{k + 1})_{i,j} = \frac{(\boldsymbol{p}_{k})_{i,j} + \tau (\nabla (\mathop{\text{div}}\nolimits\boldsymbol{p}_{k}) - f_{n}^{*}/\lambda_{\textit{TV}}^{*})_{i,j}}{1 + \tau | \nabla (\mathop{\text{div}}\nolimits\boldsymbol{p}_{k}) - f_{n}^{*}/\lambda_{\textit{TV}}^{*}|}, \end{equation} (5) where τ denotes the hyper-parameter for updating the dual variable and \(\mathop{\text{div}}\nolimits\boldsymbol{p}_{k}\) is defined by \(\mathop{\text{div}}\nolimits\boldsymbol{p}_{k}\equiv \nabla\cdot \boldsymbol{p}_{k}\). The primal-dual splitting method starts from the initial value \(\boldsymbol{p}_{0} = (O_{N,N},O_{N,N})\) and updates \(\boldsymbol{p}\). After performing enough iterations to converge, the solution of Eq. (4) is obtained using the dual variable at the \(k'\)-th iteration as \begin{equation} f_{n + 1}(\boldsymbol{r}) = f_{n}^{*}(\boldsymbol{r}) - \lambda_{\textit{TV}}^{*}\mathop{\text{div}}\nolimits\boldsymbol{p}_{k'}. \end{equation} (6) Hence, Eq. (3) can be solved as \begin{equation} f_{n + 1}(\boldsymbol{r}) = \begin{cases} f_{n}^{*}(\boldsymbol{r}) - \lambda_{\textit{TV}}^{*}\mathop{\text{div}}\nolimits\boldsymbol{p}_{k'} &\text{$\boldsymbol{r}\in S$}\\ S_{\lambda_{l_{1}}'}(f'_{n}(\boldsymbol{r})) &\text{otherwise}\end{cases}, \end{equation} (7) where \(S_{\lambda}(x)\) corresponds to the soft threshold function defined by \begin{equation} S_{\lambda}(x) = \begin{cases} x - \lambda &\text{($x > \lambda$)}\\ 0 &\text{($- \lambda \leq x \leq \lambda$)}\\ x + \lambda &\text{($x < - \lambda$)}\end{cases}. \end{equation} (8)
Tuning the parameters in the \(\mathit{TV}\)-\(L_{2}\) algorithm
Here we set \(\lambda_{l_{1}}' = 1\) based on the prior information that there are very weak transmitted X-rays outside S. We selected the \(\lambda_{l_{2}}\) value from values which can reproduce the base transmittance \(f_{\textit{base}}\). \(\lambda_{\textit{TV}}\) was determined using one-standard-error rule of four-fold cross-validation error.^{16}^{)}τ was set as \(\tau = 1/8\) because the convergence of the algorithm was proved in the range of \(\tau\leq 1/8\).^{15}^{)}
Generation of domain patterns using time-dependent Ginzburg–Landau equation
Magnetic domain patterns were generated using time-dependent Ginzburg–Landau (TDGL) equation.^{17}^{)} The TDGL equation is expressed as \begin{align} \frac{\partial \phi (\boldsymbol{r})}{\partial t} &= \alpha \lambda (\boldsymbol{r})[\phi (\boldsymbol{r}) - \phi (\boldsymbol{r})^{3}] + \beta \nabla^{2}\phi (\boldsymbol{r}) \notag\\ &\quad- \gamma \int \phi (\boldsymbol{r}')G(\boldsymbol{r},\boldsymbol{r}')\,d\boldsymbol{r}' + h(t), \end{align} (9) where the first term corresponds to the anisotropy energy with the magnitude α. \(\lambda (\boldsymbol{r})\) is expressed as \(\lambda (\boldsymbol{r}) = 1 +\mu (\boldsymbol{r})/4\); here, \(\mu (\boldsymbol{r})\) denotes zero-mean normal random numbers with variance \(\sigma^{2}\). The second term corresponds to the exchange interaction energy with the magnitude β. The third term denotes the dipole interaction energy with the magnitude γ. \(G(\boldsymbol{r},\boldsymbol{r}')\) is defined by \(G(\boldsymbol{r},\boldsymbol{r}')\equiv \frac{1}{|\boldsymbol{r}-\boldsymbol{r}'|^{3}}\). The fourth term is the external magnetic field at a time point t expressed as \(h(t) = h_{0} - vt\) with the initial magnetic field \(h_{0}\) and changing velocity v. We generated three different domain patterns, i.e., the maze-, intermediate-, and island-type domain patterns, using the parameter values presented in Table I. The generated domain patterns are shown in Fig. 1.
Table I. Parameter values of TDGL equation for generating three different magnetic domain patterns. The value of v plays a dominant role for generating different types of patterns.
Emulated diffraction patterns of the single-shot imaging experiment
We generated the emulated diffraction patterns of the single-shot imaging experiment using X-ray free electron laser (XFEL). Figure 2(a) shows the model image of the maze-type domain pattern with the asymmetric real-space support S. The intensity profile along the horizontal line of the model image is shown in Fig. 2(b). In the emulation, the X-ray magnetic circular dichroism signal is set as \((f_{\uparrow} - f_{\downarrow})/(f_{\uparrow} + f_{\downarrow}) = 0.05\), which is comparable to that of a typical magnetic element such as Fe or Co at the \(L_{3}\) absorption edge.^{18}^{)} The base transmittance of the specimen is defined by \((f_{\uparrow} + f_{\downarrow})/2\) and set as \(f_{\textit{base}} = 1\). To emulate the observed diffraction pattern, we calculated the squared Fourier transform of the model image \(|\mathcal{F}[f] |^{2}\) and multiplied it by a constant value based on the photon counts that reach a detector. Assuming that the incident photon number of XFEL was above \(10^{11}\) photons/pulse at around 700 eV,^{19}^{)} the order of the photon counts that reach a detector after transmitting a specimen was set at 10^{8}. Considering a detector as a charge coupled device (CCD) camera, a counting error occurs according to the diffraction intensity and follows a Poisson distribution. Additional statistical error of the background of the image, such as thermal noise and read-out error in the CCD itself, is also assumed to follow a Poisson distribution. In the present study, the averaged background counts were set to 600 from a rule of thumb. These errors and the background were added to the image. To protect the CCD from the intense direct beam, a direct beam stopper was placed at the center of the diffraction pattern so as not to exceed 50000 counts, which results in the lack of information. Figure 2(c) shows the diffraction pattern of the maze-type magnetic domain pattern generated by emulating the single-shot imaging experiment. The structural form factor pattern \(F_{\textit{obs}}(\boldsymbol{q})\), which was obtained by the square root of the normalized diffraction pattern to the model image after subtracting the averaged background, is shown in Fig. 2(d).
Figure 2. (Color online) (a) Model image of a maze-type magnetic domain. (b) Intensity profile along the dotted line in the model image. (c) Diffraction pattern generated by emulating the single-shot imaging experiment of the maze-type magnetic domain. (d) Diffraction pattern obtained from the square root of the normalized diffraction pattern to the model image after subtracting the averaged background.
3. Results and Discussion
First, we performed the phase retrieval from the perfect diffraction pattern without noise or lack of information, comparing the conventional algorithms [error-reduction (ER) and HIO^{20}^{)}] with the regularization algorithms (\(L_{1}\),^{12}^{)}\(L_{2}\), and \(\mathit{TV}\)-\(L_{2}\)). Next, we demonstrated the phase retrieval from the emulated diffraction pattern of the single-shot imaging experiment with noise and lack of information. Then, the domain pattern dependence of the phase retrieval using the \(\mathit{TV}\)-\(L_{2}\) algorithm is shown. Finally, we quantitatively evaluated the real-space images reconstructed by the phase retrieval algorithms. In this study, the iteration number of iterative Fourier transformation and the inner loop number for the \(\mathit{TV}\) regularization are set as \(n = 400\) and \(k = 20\), respectively. While the parameters in the \(\mathit{TV}\)-\(L_{2}\) algorithm were determined as described in Sect. 2.3, \(\lambda_{l_{1}}\) and \(\lambda_{l_{2}}\) in the \(L_{1}\) and \(L_{2}\) algorithms are optimized using the error in real space.
Phase retrieval from the perfect diffraction pattern without noise or lack of information
The reconstructed real-space images from the perfect diffraction pattern using the conventional algorithms (ER and HIO) and the regularization algorithms (\(L_{1}\), \(L_{2}\), and \(\mathit{TV}\)-\(L_{2}\)) are shown in Fig. 3. In the ER algorithm, the reconstructed image [Fig. 3(b)] is similar to the model one [Fig. 2(a)]. However, the reconstructed image has the stripe pattern that is different from the model one, indicating that the ER algorithm cannot reach the true solution once the algorithm is trapped by a local minimum. On the other hand, the HIO algorithm can escape from the local minima and reach the true solution. Thus, the reconstructed image [Fig. 3(c)] is in good agreement with the model one. The regularization algorithms also reconstructed the real-space images, which are almost the same as the model one as shown in Figs. 3(d)–3(f).
Figure 3. (Color online) (a) Perfect diffraction pattern without noise or lack of information. Reconstructed real-space images from the perfect diffraction pattern using the (b) ER, (c) HIO, (d) \(L_{1}\), (e) \(L_{2}\), and (f) \(\mathit{TV}\)-\(L_{2}\) algorithms. The reconstructed images are enlarged for clear visualization.
Phase retrieval from the emulated diffraction pattern of the single-shot imaging experiment
The reconstructed real-space images from the emulated diffraction pattern of the single-shot imaging experiment using the conventional and regularization algorithms are shown in Fig. 4. The reconstructed images using the conventional algorithms are quite different from the model one as shown in Figs. 4(b) and 4(c). It is considered that the reconstruction became worse due to the overfitting to the noise and the lack of information in the low-q region. On the other hand, the regularization algorithms can suppress the overfitting to the noise and become robust against the lack of information by using the prior information of base transmittance. The domain patterns reconstructed by the \(L_{1}\) and \(L_{2}\) algorithms are lower contrast than that in the original image, indicating that the prior information on the base transmittance is not sufficient for reconstructing the domain pattern from the noisy data obtained by single-shot imaging. The \(\mathit{TV}\)-\(L_{2}\) algorithm — the combined method of the \(\mathit{TV}\) and \(L_{2}\) regularizations — reconstructed the domain pattern with higher contrast than the \(L_{1}\) and \(L_{2}\) algorithms. The reconstruction becomes possible by using the prior knowledge concerning the sparseness of the differential image.
Figure 4. (Color online) (a) Emulated diffraction pattern of the single shot imaging experiment. [Also shown in Fig. 2(d).] Reconstructed real-space images from the emulated diffraction pattern using the (b) ER, (c) HIO, (d) \(L_{1}\), (e) \(L_{2}\), and (f) \(\mathit{TV}\)-\(L_{2}\) algorithms. The reconstructed images are enlarged for clear visualization.
Domain pattern dependence of the phase retrieval using the \(\mathit{TV}\)-\(L_{2}\) algorithm
To figure out the effectiveness of the \(\mathit{TV}\)-\(L_{2}\) algorithm, we also demonstrated the phase retrieval of the intermediate- and island-type domain patterns shown in Figs. 1(b) and 1(c). The enlarged model images of the intermediate- and island-type domain patterns are shown in Figs. 5(a) and 5(e), respectively. The emulated diffraction patterns of the single-shot imaging experiment are shown in Figs. 5(b) and 5(f). After subtracting the background, the diffraction patterns are obtained as the square root of the normalized diffraction patterns to the model images as shown in Figs. 5(c) and 5(g). The reconstructed real-space images from the diffraction patterns are shown in Figs. 5(d) and 5(h). Although the domain patterns are different from the maze-type, the \(\mathit{TV}\)-\(L_{2}\) algorithm can extract the information on the domain patterns and reconstruct the real-space images equivalent to the original ones.
Figure 5. (Color online) (a) Enlarged model image of the intermediate-type domain pattern. (b, c) Diffraction patterns obtained by emulating the single-shot imaging experiment. (d) Reconstructed real-space image from the diffraction pattern using the \(\mathit{TV}\)-\(L_{2}\) algorithm. (e) Enlarged model image of the island-type domain pattern. (f, g) Diffraction patterns obtained by emulating the single-shot imaging experiment. (h) Reconstructed real-space image from the diffraction pattern using the \(\mathit{TV}\)-\(L_{2}\) algorithm.
Quantitative evaluation of the phase retrieval algorithms in the various magnetic domain patterns
We quantitatively evaluated the phase retrieval algorithms using the mean squared error (MSE) defined by \(\mathit{MSE}=\frac{1}{2N^{2}}\|{f}_{\textit{model}}(\boldsymbol{r}) - f_{\textit{reconst.}}(\boldsymbol{r})\|_{l_{2}}^{2}\) with \(\boldsymbol{r}\in S\), where \(f_{\textit{model}}(\boldsymbol{r})\) and \(f_{\textit{reconst.}}(\boldsymbol{r})\) correspond to the original and reconstructed images in real space, respectively. Figure 6 shows the comparison of MSE between the conventional and regularization algorithms. In the case of the phase retrieval from the perfect diffraction patterns without the noise or lack of information, the HIO algorithm provided the lowest MSE. The MSEs in the regularization algorithms are lower than that in the ER algorithm, indicating that the prior information based on sparse modeling works properly. The \(L_{2}\) algorithm works better than the \(L_{1}\) algorithm in all the domain patterns, which indicates that the prior information on the base transmittance of the specimen should be introduced by the \(L_{2}\) regularization. The \(\mathit{TV}\)-\(L_{2}\) algorithm works better than the \(L_{2}\) algorithm. Therefore, the prior information on the sparseness of the differential image is considered to play an important role in reconstructing the domain patterns. On the other hand, in the case of the phase retrieval from the emulated diffraction patterns of the single-shot imaging experiment, the phase retrieval using the HIO algorithm became worse due to the noise and lack of information. In the ER algorithm, the MSE became worse by approximately triple digits compared with that in the phase retrieval from the perfect diffraction patterns. However, the regularization algorithms were not so affected by the noise and lack of information and provided better reconstructions than those in the conventional algorithms by more than double digits. The regularization algorithms may become more powerful techniques in the phase retrieval from noisy data with lack of information. As with the phase retrieval from the perfect diffraction patterns, the \(\mathit{TV}\)-\(L_{2}\) algorithm works best among the regularization algorithms. The \(\mathit{TV}\)-\(L_{2}\) algorithm realized the MSE as small as the order of \(10^{-5}\) in all the domain patterns, indicating that the algorithm can reproduce the real-space images and work properly even if the domain pattern is changed. Therefore, it is expected that the \(\mathit{TV}\)-\(L_{2}\) algorithm can extract the information on time-dependent variation of magnetic domain patterns in actual single-shot imaging measurements.
Figure 6. (Color online) Comparison of mean squared errors between the conventional and regularization algorithms. Domain pattern dependence is also shown. (a) and (b) correspond to the phase retrieval from the perfect diffraction patterns and the emulated diffraction patterns of the single-shot imaging experiment, respectively.
4. Conclusion
We developed a phase retrieval algorithm based on the \(\mathit{TV}\) and \(L_{2}\) regularizations and demonstrated the effectiveness of the \(\mathit{TV}\)-\(L_{2}\) algorithm in the single-shot imaging. In the ER, HIO, \(L_{1}\), and \(L_{2}\) algorithms, it is difficult to reconstruct the real-space image from the emulated diffraction patterns of the single-shot imaging experiment. However, the \(\mathit{TV}\)-\(L_{2}\) algorithm can extract the information on the domain patterns using the prior information on the sparseness of the differential image. We also clarified that the \(\mathit{TV}\)-\(L_{2}\) algorithm works well even if the domain patterns were changed from the maze-type to island-type. When applying the \(\mathit{TV}\)-\(L_{2}\) algorithm to experimental data, there are some difficulties especially in the \(\mathit{TV}\)-\(L_{2}\) algorithm such as experimentally obtaining the base transmittance and tuning the regularization parameters as well as those in the conventional algorithms such as determining the real-space support and subtracting the background. After overcoming these difficulties, the \(\mathit{TV}\)-\(L_{2}\) algorithm is expected to realize the phase retrieval in actual single-shot imaging experiments and advance the research on magnetic dynamics.
Acknowledgement
This work was supported by JSPS KAKENHI (19K20603 and 19H04399), JST CREST (JPMJCR1761 and JPMJCR1861), and JST PRESTO (JPMJPR177A).
References
1 A. Hubert and R. Schäfer, Magnetic Domains (Springer, Berlin, 1998). Google Scholar
2 X. Z. Yu, N. Kanazawa, W. Z. Zhang, T. Nagai, T. Hara, K. Kimoto, Y. Matsui, Y. Onose, and Y. Tokura, Nat. Commun. 3, 988 (2012). 10.1038/ncomms1990 Crossref, Google Scholar
3 A. Rosch, Nat. Nanotechnol. 8, 160 (2013). 10.1038/nnano.2013.21 Crossref, Google Scholar
4 M. Seul and D. Andelman, Science 267, 476 (1995). 10.1126/science.267.5197.476 Crossref, Google Scholar
5 T. H. Johansen and D. V. Ahantsev, Magneto-Optical Imaging (Kluwer Academic, Boston, MA, 2003). Google Scholar
6 I. Radu, K. Vahaplar, C. Stamm, T. Kachel, N. Pontius, H. A. Dürr, T. A. Ostler, J. Barker, R. F. L. Evans, R. W. Chantrell, A. Tsukamoto, A. Itoh, A. Kirilyuk, Th. Rasing, and A. V. Kimel, Nature 472, 205 (2011). 10.1038/nature09901 Crossref, Google Scholar
7 K. Yamamoto, S. E. Moussaoui, Y. Hirata, S. Yamamoto, Y. Kubota, S. Owada, M. Yabashi, T. Seki, K. Takanashi, I. Matsuda, and H. Wadati, Appl. Phys. Lett. 116, 172406 (2020). 10.1063/5.0005393 Crossref, Google Scholar
8 S. Woo, K. Litzius, B. Krüger, M. Im, L. Caretta, K. Richter, M. Mann, A. Krone, R. M. Reeve, M. Weigand, P. Agrawal, I. Lemesh, M. Mawass, P. Fischer, M. Kläui, and G. S. D. Beach, Nat. Mater. 15, 501 (2016). 10.1038/nmat4593 Crossref, Google Scholar
9 S. Flewett, S. Schaffert, J. Mohanty, E. Guehrs, J. Geilhufe, C. M. Günther, B. Pfau, and S. Eisebitt, Phys. Rev. Lett. 108, 223902 (2012). 10.1103/PhysRevLett.108.223902 Crossref, Google Scholar
10 V. Ukleev, Y. Yamasaki, D. Morikawa, N. Kanazawa, Y. Okamura, H. Nakao, Y. Tokura, and T. Arima, Quantum Beam Sci. 2, 3 (2018). 10.3390/qubs2010003 Crossref, Google Scholar
11 S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, Nature 432, 885 (2004). 10.1038/nature03139 Crossref, Google Scholar
12 Y. Yokoyama, T. Arima, M. Okada, and Y. Yamasaki, J. Phys. Soc. Jpn. 88, 024009 (2019). 10.7566/JPSJ.88.024009 Link, Google Scholar
13 A. Scherz, W. F. Schlotter, K. Chen, R. Rick, J. Stöhr, J. Lüning, I. McNulty, Ch. Günther, F. Radu, W. Eberhardt, O. Hellwig, and S. Eisebitt, Phys. Rev. B 76, 214410 (2007). 10.1103/PhysRevB.76.214410 Crossref, Google Scholar
14 R. W. Gerchberg and W. O. Saxton, Optik 35, 237 (1972). Google Scholar
15 A. Chambolle, J. Math. Imaging Vision 20, 89 (2004). 10.1023/B:JMIV.0000011325.36760.1e Crossref, Google Scholar
16 K. P. Murphy, Machine Learning: A Probabilistic Perspective (MIT Press, Cambridge, MA, 2012). Google Scholar
17 E. A. Jagla, Phys. Rev. E 70, 046204 (2004). 10.1103/PhysRevE.70.046204 Crossref, Google Scholar
18 C. T. Chen, Y. U. Idzerda, H.-J. Lin, N. V. Smith, G. Meigs, E. Chaban, G. H. Ho, E. Pellegrin, and F. Sette, Phys. Rev. Lett. 75, 152 (1995). 10.1103/PhysRevLett.75.152 Crossref, Google Scholar